Classical Thermal Radiation Theory

Candlelight was the first man-made light source. Much later, gas lights emerged. Then, in the 19^th century, the first electric incandescent lamp was invented. 140 years later, the incandescent lamp is being replaced by the LED.

Candles, gas lights, and incandescent lamps all use thermal radiation, the only heat transfer phenomenon that works without the presence of heat-conductive media. Without this special property, we wouldn’t be able to live on Earth: Solar energy would pour in by this mechanism through a vacuum.

An incandescent light bulb.

One of the most interesting stories in physics is when Max Planck discovered the formula of the spectral distribution, now called the Planck distribution, for black body radiation. The black body is an idealized physical body that absorbs all incident electromagnetic radiation; equivalently, a body that emits maximized thermal radiation.

The discovery of the black body theory opened the door to quantum mechanics. With Planck’s work, we now know how much power we get from a black body object at temperature T in an equilibrium state, which is called the Stefan-Boltzmann law:

where is the Stefan-Boltzmann constant and n is the refractive index of the media.

We now consider more practical objects, called gray bodies. A gray body is an imperfect black body; i.e., a physical object that partially absorbs incident electromagnetic radiation. The ratio of a gray body’s thermal radiation to a black body’s thermal radiation at the same temperature is called the emissivity of the gray body.

The emissivity of a black body is unity, while that of a gray body is larger than 0 and smaller than 1. The emissivity is a function of the geometry of the radiative surface, its physical properties, and the wavelength. So, how is emissivity determined from certain structures?

For the sake of simplicity, we will consider the diffuse gray surface, which doesn’t account for the spectral dependency of the radiation, throughout this section. The following theory is based on Gouffé’s paper (Ref. 1), which outlines the classical gray body radiation theory. The math is something that you might have learned in middle or high school, but the physical concept is a little more complicated. Gouffé’s paper was cited by many researchers at the time it was published (Ref. 7). In this blog post, we will try to include what is missing from the original paper in order to provide a complete explanation.

First of all, let’s define our terms and conventions. When we mention reflectivity and absorptivity in this blog post, we must differentiate between the “material” and “apparent” quantities. This is very important in order to avoid confusion when learning about thermal radiation. Because gray bodies typically have certain (tiny) structures on their surfaces, the “apparent” quantity viewed from a far field is different from the raw material quantity. For example, the apparent reflectivity of a rough surface is always lower than its material reflectivity.

Our goal is to calculate the apparent emissivity from a given material reflectivity and the geometry of the structure. Here, we will differentiate “material” quantities from “apparent” quantities by appending “_0″ in the notation. Otherwise, it is understood that we are talking about a quantity in general. The nomenclature of the terms is as follows:

• Material reflectivity,
• Material absorptivity,
• Material emissivity,
• Apparent reflectivity,
• Apparent absorptivity,
• Apparent emissivity,
• Cavity’s internal area,
• Opening area,
• View angle,
• View factor (normalized solid angle),
• View factor (area ratio),

From the conservation law of energy, we have the following relation for opaque materials between the reflectivity and the absorptivity :

Kirchhoff’s law states that the material emissivity at thermodynamic equilibrium is equal to the absorptivity ; i.e.,

From the two relations above, we get the emissivity from the reflectivity; i.e.,

We consider a single surface structure, as shown in the following figure. The structure can be anything, but in this blog post, we use a spherical cavity with an opening on top with area (circle of radius ) at distance from the bottom of the cavity. We assume that the material reflectivity is uniform over the internal surface of the cavity and the reflection takes place according to Lambert’s law; i.e., the intensity of the reflection is , where is the viewing angle, as depicted in the figure. We want to calculate how much apparent reflection we get out of an incident light of the energy of unity.

Surface structure for calculating the apparent reflectivity.

As the first-order approximation, the reflection from the bottom of the cavity is:

where is called the view factor.

Note that in Gouffé’s paper, the view factor is assumed to be uniform over the cavity surface. In the COMSOL® software, it is instead a function of the position, which is always true. (So, we need integration to compute the emissivity.)

In this case, the view factor is the normalized solid angle to the opening from the first reflection point. The solid angle is calculated as:

Note that the total solid angle for the hemisphere is , not . This is due to the Lambertian factor, .

As a result, the view factor is

To this approximation, the apparent reflectivity is

from which the apparent emissivity is derived as

Roughly speaking, the smaller the opening, the more the cavity becomes a black body due to the view factor.

Next, let’s improve the approximation. After the first reflection, which we already calculated, the rest is absorbed by the cavity material or contributes to further reflections. The material absorption is from the conservation of energy. The energy left for the subsequent reflections is:

Now, assuming again that further reflections take place in a uniform way, the reflection that gets out of the cavity at the second reflection should be the above quantity multiplied by the material reflectivity one more time and another view factor defined by the area ratio, , where is the area of the cavity, including the opening area . The apparent reflectivity for the second reflection is

Similarly to the first-order approximation, the apparent emissivity for the second-order approximation is:

To this approximation, the apparent emissivity should become more accurate with the additional term.

Finally, we can take all of the reflections into account by calculating the following converging infinite series:

In the case of a sphere, it can be shown that , which reduces this result to

Now, let’s rewrite and explicitly in terms of the geometry parameters and . From the geometry, it’s easy to prove that , from which can be rewritten as

Next, the opening area can be calculated by

where is the radius of the sphere and is the angle between the plane intersecting the sphere center, which is parallel to the opening perimeter.

From the geometry, there are relations that , which yields that and . By using these relations, the opening area can be rewritten as

Since the total sphere area is , the view factor is given as

which proves that .

Note that in the COMSOL® software, we don’t have the view factor , only the view factor , which is the purely geometrical quantity.

COMSOL Multiphysics® Simulation Versus Theory

So far, we’ve learned the classical gray body theories. Now, let’s compute our final goal, the apparent emissivity for a gray body consisting of a spherical shell with an opening, by using the Heat Transfer with Surface-to-Surface Radiation interface. Then, we can compare the computed dependency of the opening size on the emissivity with the approximate formulas.

Before moving to the main results, let’s check one thing. In COMSOL Multiphysics, we can calculate the view factor values (corresponding to in the above theory, but as a function of position) using the radopu and radopd operators. The quantity that is obtained by using these operands is purely geometric, so there is no need to run a study to use this operator. If you change the geometry, you can just update the solution before using the operator.

In this model, there is only one radiating surface because the model is an open cavity. So, COMSOL Multiphysics can calculate only the view factor for the cavity surface itself. We’ll call this the self view factor. The view factor , which we have discussed so far as the factor that views outside of the cavity from a point on the cavity surface, can be called the ambient view factor. We can calculate the view factor by subtracting the self view factor from unity; i.e., 1-intop1(comp1.ht.radopu(1,0))/intop1(1).

As mentioned earlier, the view factor is position dependent in general, and so is the view factor in the COMSOL® software. To calculate the contribution from all points on the cavity surface, we need to integrate the view factor as a function of position. The operator intop1 is an integration operator defined on the cavity surface. The following figure compares the calculated results with the classical view factor theory, showing very good agreement.

Comparison between the classical theory and the radopu operator calculation in COMSOL Multiphysics for the ambient view factor.

Now, let’s move on to the settings. To calculate surface-to-surface radiation, a Surface-to-Surface Radiation physics interface and a Diffuse Surface node for the internal surface of the sphere need to be added, and a Heat Transfer with Surface-to-Surface Radiation multiphysics coupling node needs to be added (see below). COMSOL Multiphysics will compute the view factor by performing surface integration for each mesh node except when the geometry is axisymmetric where a closed-form expression is used. For the sake of simplicity, we don’t include ambient radiation; i.e., . A Temperature boundary condition is set to the outer boundary of the spherical shell with = 2500 deg K. The material emissivity is set to 0.5 in order to see the difference more easily.

Settings for the Diffuse Surface node.

The following plots show the computed ambient view factor, radiosity, and temperature for various opening radii. Following Gouffé’s paper, the geometry aspect ratio defined by , where is the opening’s radius and is the cavity height, is used as a sweep parameter.

Computational results for the ambient view factor, radiosity, and temperature versus .

These results qualitatively suggest that the smaller the size of the opening, the more the cavity emits thermal radiation and the higher the surface temperature.

To calculate the apparent emissivity for the computed radiosity result, we can now use the Stefan-Boltzmann law. The radiosity of the black body at temperature is . Dividing the computed radiosity by this number, we get the apparent emissivity of the gray body spherical cavity, which is intop1(ht.J)/intop1(1)/(sigma_const*T0^4). Let’s now compare the results with the various theories we have discussed so far.

Comparison between COMSOL Multiphysics results and various theoretical approximations for the apparent emissivity for a spherical cavity.

The cyan line shows the first-order approximation, which underestimates the amount of reflection causing the higher emissivity, which is inaccurate. The second order obviously improves the accuracy, but it’s not enough. The plot for the infinite series approximation (orange line) gives a much better result, but it’s still not very close to the COMSOL Multiphysics results.

The reason for this discrepancy is mentioned in the last part of Gouffé’s paper. As some readers might have noticed in the previous figure, the surface temperature is different for each geometry, regardless of the same outer boundary temperature of 2500 deg K. This is called radiation cooling. We have yet to include this effect, which takes place for any real material with a finite thermal conductivity. The effect changes the temperature of the inner surface of the shell temperature, depending on the opening area. Therefore, the temperature needs to be compensated in order to calculate the emissivity at the same temperature for all opening areas.

The correction factor is, owing to the Stefan-Boltzmann law, the fourth power of the temperature ratio; i.e., (maxop1(T))^4/T0^4, where maxop1 is an operator defined on the cavity surface that finds the maximum value on the surface. Finally, with this correction, the red curve is the most accurate theoretical prediction, which agrees very well with the COMSOL Multiphysics results (blue line).

Designing an Efficient Incandescent Lamp with the COMSOL® Software

The light source in an incandescent lamp is created by a twisted tungsten filament. The material emissivity of tungsten is 0.462 at 2500 deg K for a 0.467-um wavelength (Ref. 2). In the past, researchers proposed that we can design more efficient incandescent lamps if we fabricate some microstructures on the surface of the tungsten filament. This is true. As we just learned, the apparent emissivity can be close to 1 if we have a cavity with a very tiny opening, which we can call a black body filament. In addition, research (Ref. 3–4, 8–9) also proposed that if the maximum cavity size is about a half of 0.78 um, then any infrared light with a wavelength larger than 0.78 um may be suppressed due to the waveguide’s cutoff effect. Then, the efficiency for the visible light may be significantly improved.

This effect may be more than the emissivity enhancement, because thermal radiation below tungsten’s melting temperature consists of mostly infrared light that is wasted to heat. It would be fantastic if we could really cut the infrared off.

Our “dream” incandescent lamp with an infrared suppressing black body filament. Background image in the public domain, via Wikimedia Commons.

Unfortunately, this “dream lamp” can’t be a reality for various reasons.

First, it isn’t possible for the surface to be made up of “all” holes. To make holes with a low view factor, meaning a smaller opening area than the hole size, the surface can’t be densely filled with the openings; i.e., the surface needs a flatter area between the openings, which radiate infrared. Alternatively, we can make deep holes to decrease the ambient view factor and then make each hole as close to the adjacent holes as possible. However, this makes it more difficult to fabricate deeper holes.

Also, the surface energy of the surface with holes seems to be higher than that of a flat surface. So, the surface with holes tends to melt down at a lower temperature than the bulk melting temperature.

Third, it is very difficult to make holes on a twisted wire filament because it’s a 3D structure. It is easier to make holes on a flat ribbon filament because it’s 2D, but flat filaments are not electrically convenient because they need more current to achieve the same temperature than the twisted wire filament (higher voltage and lower current is convenient for our current infrastructure).

Concluding Remarks

Although gray body radiation theory was developed a long time ago, the theory is still well organized. There are many sources for closed-form view factor formulas (Ref. 5; note that it’s also called “configuration factor”). This theory has been proven by experiments and is still applied to many real-world applications, but it can seem too difficult to understand. In this blog post, we learned the theory for the simplest example and saw a very successful benchmark result with COMSOL Multiphysics. The Heat Transfer with Surface-to-Surface Radiation interface is a reliable tool that makes complicated radiation computations easier.

We can come up with a more interesting incandescent lamp by making a special view factor. If we make tiny periodic grooves on a flat ribbon filament, we will get a “directional” incandescent lamp. Usually, a filament emits light to all directions, but this special filament illuminates only a certain direction (Ref. 6). We can explore many more ideas and many more applications of gray body radiation theory with simulation.

Next Step

Try the Gray Body model yourself by clicking the following button, which will take you to the Application Gallery. There, you can download the related MPH-file if you have a valid software license and COMSOL Access account.

References

1. A. Gouffé, “Corrections d’ouverture des corps-noirs artificiels compte tenu des diffusions multiples internes (Corrections of emissivity for the artificial black-body considering multiple internal diffusions)” (in French), Revue d’Optique, t. 24, no. 1–3, 1945.
2. C.J. Smithells, Tungsten, Chapman & Hall, Ltd., 1926.
3. J.F. Waymouth, Proceedings of LS-5, 1989.
4. J.F. Waymouth, J. Illum. Engng. Inst. Jpn., 74, 12, pp. 800–805, 1990.
5. J.R. Howell, “A Catalog of Radiation Heat Transfer Configuration Factors“, University of Texas at Austin.
6. W.Z. Black, “Radiative heat transfer characteristics of specially prepared V-groove cavities”, PhD Thesis to Purdue University, 1968.
7. Precision Measurement and Calibration, Radiometry and Photometry (Vol. 7), U.S. Department of Commerce, National Bureau of Standards.
8. M. Sugimoto, T. Fujioka, T. Inoue, H. Fukushima, Y. Mizuyama, S. Ukegawa, T. Matsushima, and M. Toho, “The Infrared Suppression in the Incandescent Light from a Surface with Submicron Holes”, Journal of Light & Visual Environment, Vol. 18, No. 2 (1994).
9. T. Kondo, S. Hasegawa, T. Yanagishita, N. Kimura, T. Toyonaga, and H. Masuda, “Control of thermal radiation in metal hole array structures formed by anisotropic anodic etching of Al”, Optics Express Vol. 26 No. 21 (2018).

Paraxiality of Gaussian Beams

The well-known Gaussian beam formula is only valid for paraxial Gaussian beams. Paraxial means that the beam mainly propagates along the optical axis. There are several papers that talk about paraxiality in a quantitative sense (see Ref. 1).

Roughly speaking, if the beam waist size is near the wavelength, the beam propagates at a higher angle to a focus. Therefore, the paraxiality assumption breaks down and the formulation is no longer accurate. To alleviate this problem and to provide you with a more general and accurate formulation for general Gaussian beams, we introduced a nonpariaxial Gaussian beam formulation. In the user interface this is referred to as Plane wave expansion.

The method is based on the angular spectrum of plane waves (Ref. 2) and is sometimes referred to as the angular spectrum method (Ref. 3).

Angular Spectrum of Plane Waves

Let’s briefly review the paraxial Gaussian beam formula in 2D (for the sake of better visuals and understanding).

We start from Maxwell’s equations assuming time-harmonic fields, from which we get the following Helmholtz’s equation for the out-of-plane electric field with the wavelength for our choice of polarization:

where .

The angular spectrum of plane waves is based on the following simple fact: an arbitrary field that satisfies the above Helmholtz equation can be expressed as the following plane wave expansion:

where is an arbitrary function.

The integration path is a circle of radius for real and . (For complex and , the integration domain extends to a complex plane.) The function is called the angular spectrum function. One can prove that this satisfies Helmholtz’s equation by direct substitution.

Now that we know that this formulation always gives exact solutions to Helmholtz’s equation, let’s try to understand it visually. From the constraint, , we can set and and rewrite the above equation as:

The meaning of the above formula is that it constructs a wave as a sum, or integral, consisting of many waves propagating in various directions, all with the same wave number . This is shown in the following figure.

Visualization of the angular spectrum of plane waves.

When actually solving a problem using this formula, all you have to do is find the angular spectrum function that satisfies the boundary conditions. By assuming that the profile of the transverse field (perpendicular to the propagating direction, i.e., optical axis) is also a Gaussian shape (see Ref. 4), one can derive that , where is the spectrum width.

By some more mathematical manipulations, we get a relationship between the spectrum width and the beam waist radius . For example, for a slow Gaussian beam, the angular spectrum is narrow. A plane wave, on the other hand, is the extreme case where the angular spectrum function is a delta function. For a fast Gaussian beam, the angular spectrum is wider, and vice versa.

This was a quick summary of the underlying theory for nonparaxial Gaussian beams. To recap what we have shown so far, let’s rewrite the formula once more by using polar coordinates, :

This is the formulation that Born and Wolf (Ref. 2) use in their book.

The 3D formula is more complicated and looks different due to polarization, but the basic idea is the same as seen in the references mentioned above. It can also look different depending on whether or not you consider evanescent waves. The Plane Wave Expansion method used in the Wave Optics Module and the RF Module, although based on the angular spectrum theory, is adapted for numerical computations.

Plane Wave Expansion: Settings and Results

Let’s compare the new feature, Plane wave expansion, with the previously available feature, Paraxial approximation. The Settings window covering both methods is shown below.

The Plane Wave Expansion feature settings.

With the new feature, you have two options if the Automatic setting doesn’t give you a satisfactory approximation:

1. Wave vector count
2. Maximum transverse wave number

The first option determines the number of discretization levels, depending on how fine you want to represent the Gaussian beam. The more plane waves, the finer it gets. The second option is related to the integral bound in the previous equation; i.e., . This integral bound can be the maximum for the smallest possible spot size and can be more shallow for slower beams, depending on how fast the Gaussian beam is. You need more angled plane waves with a larger transverse wave number to represent faster (more focused) beams.

The following results compare the two formulas for the case where the spot radius is , which is considerably nonparaxial. As in the previous blog post, the simulation is done with the Scattered Field formulation and the domain is surrounded by a perfectly matched layer (PML). This way, the scattered field represents the error from the exact Helmholtz solution.

The left images below show the new feature, while the images on the right show the paraxial approximation. The top images show the norm of the computed Gaussian beam background field, ewfd.Ebz, while the bottom images show the scattered field norm, ewfd.relEz, which represents the error from the exact Helmholtz solution. Obviously, the error from the Helmholtz solution is greatly reduced in the nonparaxial method.

Comparison between the angular spectrum of plane waves and the paraxial formula.

Concluding Remarks

We have discussed the theory and results for an approximation method for nonparaxial Gaussian beams using the new plane wave expansion option. Remember that this formulation is extremely accurate, but is still an approximation under assumptions. First, we have made an assumption for the field shape in the focal plane. Second, we assume that the evanescent field is zero. If you are interested in the field coupling to some nanostructure near the focal region in a fast Gaussian beam, you may need to calculate the evanescent field.

Next Step

Learn more about the formulations and features available for modeling optically large problems in the COMSOL® software by clicking the button below:

Note: This functionality can also be found in the RF Module.

References

1. P. Vaveliuk, “Limits of the paraxial approximation in laser beams”, Optics Letters, vol. 32, no. 8, 2007.
2. M. Born and E. Wolf, Principles of Optics, ed. 7, Cambridge University Press, 1999.
3. J. W. Goodman, Fourier Optics.
4. G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation”, Phys. Rev. a. 27, pp. 1693–1695, 1983.

Comparing Methods for Solving Large Wave Optics Models

In electromagnetic simulations, the wavelength always needs be resolved by the mesh in order to find an accurate solution of Maxwell’s equations. This requirement makes it difficult to simulate models that are large compared to the wavelength. There are several methods for stationary wave optics problems that can handle large models. These methods include the so-called diffraction formulas, such as the Fraunhofer, Fresnel-Kirchhoff, and Rayleigh-Sommerfeld diffraction formula and the beam propagation method (BPM), such as paraxial BPM and the angular spectrum method (Ref. 1).

Most of these methods use certain approximations to the Helmholtz equation. These methods can handle large models because they are based on the propagation method that solves for the field in a plane from a known field in another plane. So you don’t have to mesh the entire domain, you just need a 2D mesh for the desired plane.

Compared to these methods, the Electromagnetic Waves, Beam Envelopes interface in COMSOL Multiphysics (which we will refer to as the Beam Envelopes interface for the rest of the blog post) solves for the exact solution of the Helmholtz equation in a domain. It can handle large models; i.e., the meshing requirement can be significantly relaxed if a certain restriction is satisfied.

A beam envelopes simulation for a lens with a millimeter-range focal length for a 1-um wavelength beam.

We discuss the Beam Envelopes interface in more detail below.

Theory Behind the Beam Envelopes Interface

Let’s take a look at the math that the Beam Envelopes interface computes “under the hood”. If you add this interface to a model and click the Physics Interface node and change Type of phase specification to User defined, you’ll see the following in the Equation section:

Here, is the dependent variable that the interface solves for, called the envelope function.

In the phasor representation of a field, corresponds to the amplitude and to the phase, i.e.,

The first equation, the governing equation for the Beam Envelopes interface, can be derived by substituting the second definition of the electric field into the Helmholtz equation. If we know , the only unknown is and we can solve for it. The phase, , needs to be given a priori in order to solve the problem.

With the second equation, we assume a form such that the fast oscillation part, the phase, can be factored out from the field. If that’s true, the envelope is “slowly varying”, so we don’t need to resolve the wavelength. Instead, we only need to resolve the slow wave of the envelope. Because of this process, simulating large-scale wave optics problems is possible on personal computers.

A common question is: “When do you want the envelope rather than the field itself?” Lens simulation is one example. Sometimes you may need the intensity rather than the complex electric field. Actually, the square of the norm of the envelope gives the intensity. In such cases, it suffices to get the envelope function.

What Happens If the Phase Function Is Not Accurately Known?

The math behind the beam envelope method introduces more questions:

• What if the phase is not accurately known?
• Can we use the Beam Envelopes interface in such cases?
• Are the results correct?

To answer these questions, we need to do a little more math.

1D Example

Let’s take the simplest test case: a plane wave, , where for wavelength = 1 um, it propagates in a rectangular domain of 20 um length. (We intentionally use a short domain for illustrative purposes.)

The out-of-plane wave enters from the left boundary and transmits the right boundary without reflection. This can be simulated in the Beam Envelopes interface by adding a Matched boundary condition with excitation on the left and without excitation on the right, while adding a Perfect Magnetic Conductor boundary condition on the top and bottom (meaning we don’t care about the y direction).

The correct setting for the phase specification is shown in the figure below.

We have the answer , knowing that the correct phase function is or the wave vector is a priori. Substituting the phase function in the second equation, we inversely get , the constant function.

How many mesh elements do we need to resolve a constant function? Only one! (See this previous blog post on high-frequency modeling.)

The following results show the envelope function and the norm of , ewbe.normE, which is equal to . Here, we can see that we get the correct envelope function if we give the exact phase function, constant one, for any number of meshes, as expected. For confirmation purposes, the phase of , arg(E1z), is also plotted. It is zero, also as expected.

The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the correct phase function k₀x, computed for different mesh sizes.

Now, let’s see what happens if our guess for the phase function is a little bit off — say, instead of the exact . What kind of solutions do we get? Let’s take a look:

The envelope function (red), the electric field norm (blue), and the phase of the envelope function (green) for the wrong phase function, 0.95 k₀x, computed for different mesh sizes.

What we see here for the envelope function is the so-called beating. It’s obvious that everything depends on the mesh size. To understand what’s going on, we need a pencil, paper, and patience.

We knew the answer was , but we had “intentionally” given an incorrect estimate in the COMSOL® software. Substituting the wrong phase function in the second equation, we get . This results in , which is no longer constant one. This is a wave with a wavelength of = 20 um, which is called the beat wavelength.

Let’s take a look at the plot above for six mesh elements. We get exactly what is expected (red line), i.e., . The plot automatically takes the real part, showing . The plots for the lower resolutions still show an approximate solution of the envelope function. This is as expected for finite element simulations: coarser mesh gives more approximate results.

This shows that if we make a wrong guess for the phase function, we get a wrong (beat-convoluted) envelope function. Because of the wrong guess, the envelope function is added a phase of the beating (green line), which is .

What about the norm of ? Look at the blue line in the plots above. It looks like the COMSOL Multiphysics software generated a correct solution for ewbe.normE, which is constant one. Let’s calculate: Substituting both the wrong (analytical) phase function and the wrong (beat-convoluted) envelope function in the second equation, we get , which is the correct fast field!

If we take a norm of , we get a correct solution, constant one. This is what we wanted. Note that we can’t display itself because the domain can be too large, but we can find analytically and display the norm of with a coarse mesh.

This is not a trick. Instead, we see that if the phase function is off, the envelope function will also be off, since it becomes beat-convoluted. However, the norm of the electric field can still be correct. Therefore, it is important that the beat-convoluted envelope function be correctly computed in order to get the correct electric field. The above plots clearly show that. The six-element mesh case gives the completely correct electric field norm because it fully resolves the beat-convoluted envelope function. The other meshes give an approximate solution to the beat-convoluted envelope function depending on the mesh size. They also do so for the field norm. This is a general consequence that holds true for arbitrary cases.

No matter what phase function we use in COMSOL Multiphysics, we are okay as long as we correctly solve the first equation for and as long as the phase function is continuous over the domain. When there are multiple materials in a domain, the continuity of the phase function is also critical to the solution accuracy. We may discuss this in a future blog post, but it is also mentioned in this previous blog post on high-frequency modeling.

2D Example

So far, we have discussed a scalar wave number. More generally, the phase function is specified by the wave vector. When the wave vector is not guessed correctly, it will have vector-valued consequences. Suppose we have the same plane wave from the first example, but we make a wrong guess for the phase, i.e., instead of . In this case, the wave number is correct but the wave vector is off. This time, the beating takes place in 2D.

Let’s start by performing the same calculations as the 1D example. We have and the envelope function is now calculated to be , which is a tilted wave propagating to direction , with the beat wave number and the beat wavelength .

The following plots are the results for θ = 15° for a domain of 3.8637 um x 29.348 um for different max mesh sizes. The same boundary conditions are given as the previous 1D example case. The only difference is that the incident wave on the left boundary is . (Note that we have to give the corresponding wrong boundary condition because our phase guess is wrong.)

In the result for the finest mesh (rightmost), we can confirm that is computed just like we analyzed in the above calculation and the norm of is computed to be constant one. These results are consistent with the 1D example case.

The electric field norm (top) and the envelope function (bottom) for the wrong phase function , computed for different mesh sizes. The color range represents the values from -1 to 1.

Simulating a Lens Using the Beam Envelopes Interface

The ultimate goal here is to simulate an electromagnetic beam through optical lenses in a millimeter-scale domain with the Beam Envelopes interface. How can we achieve this? We already discussed how to compute the right solution. The following example is a simulation for a hard-apertured flat top incident beam on a plano-convex lens with a radius of curvature of 500 um and a refractive index of 1.5 (approximately 1 mm focal length).

Here, we use , which is not accurate at all. In the region before the lens, there is a reflection, which creates an interference. In the lens, there are multiple reflections. After the lens, the phase is spherical so that the beam focuses into a spot. So this phase function is far different from what is happening around the lens. Still, we have a clue. If we plot , we see the beating.

Plot of . The inset shows the finest beat wavelength inside the lens.

As can be seen in the plot, a prominent beating occurs in the lens (see the inset). Actually, the finest beat wavelength is in front of the lens. To prove this, we can perform the same calculations as in the previous examples. The finest beat wavelength is due to the interference between the incident beam and reflected beam, but we can ignore this because it doesn’t contribute to the forward propagation. We can see that the mesh doesn’t resolve the beating before the lens, but let’s ignore this for now.

The beat wavelength in the lens is for the backward beam and for the forward beam for n = 1.5, which we can also prove in the same way as the previous examples. Again, we ignore the backward beam. In the plot, what’s visible is the beating for the forward beam. The backward beam is only a fraction (approximately 4% for n = 1.5 of the incident beam, so it’s not visible). The following figure shows the mesh resolving the beat inside the lens with 10 mesh elements.

The beat wavelength inside the lens. The mesh resolves the beat with 10 mesh elements.

Other than the beating for the propagating beam in the lens, the beating in the subsequent air domain is pretty large, so we can use a coarse mesh here. This may not hold for faster lenses, which have a more rapid quadratic phase and can have a very short beat wavelength. In this example, we must use a finer mesh only in the lens domain to resolve the fastest beating.

The computed field norm is shown at the top of this blog post. To verify the result, we can compute the field at the lens exit surface by using the Frequency Domain interface, and then using the Fresnel diffraction formula to calculate the field at the focus. The result for the field norm agrees very well.

Comparison between the Beam Envelopes interface and Fresnel diffraction formula. The mesh resolves the beat inside the lens with 10 mesh elements.

The following comparison shows the mesh size dependence. We get a pretty good result with our standard recommendation, , which is equal to . This makes it easier to mesh the lens domain.

Mesh size dependence on the field norm at the focus.

As of version 5.3a of the COMSOL® software, the Fresnel Lens tutorial model includes a computation with the Beam Envelopes interface. Fresnel lenses are typically extremely thin (wavelength order). Even if there is diffraction in and around the lens surface discontinuities, the fine mesh around the lens part does not significantly impact the total number of mesh elements.

Concluding Remarks

In this blog post, we discuss what the Beam Envelopes interface does “under the hood” and how we can get accurate solutions for wave optics problems. Even if we get beating, the beat wavelength can be much longer than the wavelength, which makes it possible to simulate large optical systems.

Although it seems tedious to check the mesh size to resolve beating, this is not extra work that is only required for the Beam Envelopes interface. When you use the finite element method, you always need to check the mesh size dependence for accurately computed solutions.

Next Steps

Try it yourself: Download the file for the millimeter-range focal length lens by clicking the button below.

References

1. J. Goodman, Fourier Optics, Roberts and Company Publishers, 2005.

Introduction to Laser Simulation

Albert Einstein discovered stimulated emission in 1916, but the laser was invented by one of three American scientists, C.H. Townes, A.L. Schawlow, or G. Gould, presumably in 1957. (There was a patent war over who first theoretically invented the laser and there is still controversy about who came up with the idea.) The first laser was built by Theodore Maiman in 1960.

Lasers are very useful in many applications, such as cutting, perforation, melting, ablation, telecommunication, measurement, and spectroscopy among others. In the 1960s, A.G. Fox, T. Li, and H. Kogelnik made further progress in the research of lasers by coming up with ways to calculate the field distribution, formulate Gaussian beams, and predict the laser stability based on the paraxial theory of optics. These theories constituted the fundamentals for handling and designing lasers, even now, 60 years later.

Among the theories, Kogelnik’s laser stability theory is particularly useful for laser engineering. The theory consists of two parts:

1. The paraxial wave optics theory predicts what Gaussian beam is generated in a laser cavity
2. The geometrical optics theory predicts when the laser beam stays stable in the cavity

A laser may stop working after it is turned on or pumped harder. This is due to the thermal lensing effect. The laser crystal is pumped by another light source or sources to cause the inversion population. This process generates undesired heat by the residual energy of the pump light that is not used to induce the stimulated emission. The undesired heat introduces an undesired change in the refractive index and causes bulging of the crystal surface.

Kogelnik’s theory covers laser cavities including a rod lens, which approximates the thermal lensing effect by parabola fitting the temperature distribution in the crystal.

A titanium-doped sapphire femtosecond laser. The green laser from an Ar laser is used to pump the Ti:sapphire crystal emitting orange glare for the population inversion. The laser crystal is a circular rod, but has an angled cut called the Brewster cut. The pump beam enters the crystal surface at the Brewster angle to avoid undesired Fresnel reflections. Image by Hankwang — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.

Methods for Analyzing Laser Stability

The idea of the stability part of Kogelnik’s theory is quite simple: The laser beam is stable if a ray bouncing back and forth between the laser cavity mirrors does not escape from the cavity after a certain number of round trips. This theory was developed in a more concise and organized way by using the ABCD matrix theory based on paraxial optics rather than real ray tracing.

In this theory, thermal lensing is modeled as a thin lens that has a certain focal length. The focal length is calculated by fitting a parabola to the temperature distribution in the laser crystal. Thus, the ABCD matrix theory and the parabola fitting method are simple to perform and create beautiful results. You just need the distances of the optics, the focal length of any lenses involved, and the radius of curvature of the mirrors.

All folding mirrors are replaced by lenses having an equivalent focal length and an approximate focal length is used to represent the thermal lensing effect instead of giving the pump power in watts. Optics can’t be tilted and the optical layout is flattened to 1D in a straight line.

This method is too simple to handle more complex and more practical laser cavities. What if you want to see the results in 3D, or add a prism in your stability analysis? What if you want to fold your laser beam by folding mirrors, or tilt one of the optics? What about giving a pump power in watts, not an approximate focal length? Also, why not perform ray tracing and thermal analyses without any restrictions or approximations?

Performing a laser stability analysis without taking the thermal lensing effect into account is almost useless, because every laser gets pumped. So, we need a thermal analysis as a counterpart of the stability analysis, which can’t be separated from ray tracing. By coupling the Heat Transfer and Geometrical Optics interfaces in COMSOL Multiphysics, we have all the tools we need.

Analyzing a Laser with a Confocal Cavity in COMSOL Multiphysics®

Let’s take a look at a simple example model of a laser cavity and briefly outline how the ABCD matrix method works and how it agrees with COMSOL Multiphysics results. This section complements part of one of Kogelnik’s own research papers (Ref. 1). In the ABCD matrix method, while following Hecht’s notation (Ref. 2), a ray is characterized by the ray angle θ and the ray position y in a 2×1 column vector as

All optical processes including propagation, refraction, reflection, focusing, and defocusing are denoted by a 2×2 matrix called the ray transfer matrix.

For example, propagation through a distance L is denoted by a 2×2 matrix

and reflection in a mirror with the radius of curvature R in the air by

Our example laser cavity has a pair of identical mirrors that have the curvature of radius R = 0.1 m and cavity length L = R/2 = 0.05 m. This cavity looks like the below figure and is called the “confocal cavity”.

Ray tracing in COMSOL Multiphysics for a confocal cavity after a couple of bounces.

Let’s set θ₀ = 0.1 rad and y₀ = 0.01 m as the initial ray angle and position. (For the ABCD matrix method, the x-position really doesn’t matter, but let’s set the initial ray on x = 0 m; i.e., on the center of the left mirror.) The ABCD matrix method tells us that the next angle and the position of the ray after propagating the cavity length is calculated as

This is the ray angle and position on the right mirror (before it gets reflected). After the reflection, the next ray angle and position are still on the right mirror, but toward the left. They are calculated to be

Then, another travel of the cavity length lets the ray return to the original left mirror as

Finally, the end of one round-trip is the reflection by the left mirror; i.e.,

which can be confirmed in the above figure.

The beauty of this method is that you can only perform matrix operations to know the behavior of rays. We can calculate a matrix M that represents the sequence of the ray propagation and reflection as

which gives the same result as the previous result:

This transfer matrix M has the property that M³ = I; i.e., it becomes the identical matrix after three round-trips. Actually, this can be proved analytically by means of Sylvester’s theorem without calculating a matrix product:

where .

Substituting n = 3 in the above formula for our M matrix, we get M³ = I. We can confirm this with ray tracing in the next figure, in which the ray paths are colored by the number of round-trips.

Ray tracing results for the same confocal cavity as the previous plot, showing that the ray is returning to the initial position after 3 round-trips.

In this case, it is said that this cavity is stable, which means that the ray inside this cavity will not escape from it. The above formula is particularly useful for deriving a general consequence that the sequence represented by this matrix is stable when it holds for the trace of the matrix that . In our case, the trace is calculated to . So, it’s stable.

The results can be also viewed by plotting the y-component of the unit wave vector as in the below figure. The plot shows that the ray direction periodically alters but never grows and returns to the same value after 3 round-trips. Roughly speaking, ±0.15 radians is the upper and lower bound for the ray to be inside the cavity and the plot below indeed proves that this is the case here.

The y-component of the unit wave vector. The direction of the ray is bounded by a finite minimum and maximum, which means that the ray will not diverge beyond the cavity mirrors.

Changing the Laser’s Cavity Length

Let’s discuss another example. We’ll choose the cavity length to be m. This cavity is a little bit off from the confocal cavity. This intentionally tweaked cavity length guarantees that and in the formula. Therefore, the transfer matrix becomes the identical matrix after 20 round-trips and the initial ray angle and position are obtained. The transfer matrix in this case is

and the product of this transfer matrix for 20 times will be M²⁰ = I. Let’s confirm this with ray tracing in the following figure. The red arrow is indeed back to the original mirror center and pointing to the original direction.

Ray tracing results for a laser cavity with the same mirrors and a little longer cavity length after 20 round-trips.

The y-component of the unit wave vector.

In general, the transfer matrix for the right mirror with the radius of curvature R₁, the left mirror with R₂, and the cavity length L for one round-trip is written as

After some trivial arithmetic, the stability condition derived from the trace of the above matrix is expressed as

This stability formula is further extended to the case where a lens is involved in the cavity. Introducing a lens inside a cavity makes the transfer matrix complicated and it is no longer simple to write it out here, but the stability criterion is still simple. Adding a lens with focal length f modifies this to

where L₀ = L₁ + L₂ – L₁L₂/f and L₁ and L₂ are the distance from the lens to the left and right mirrors, respectively. Let’s add a focusing lens with focal length f = 50 mm in the previous cavity, for which the stability criterion number is 0.10978, predicting that the cavity is stable.

Ray tracing results for the previous cavity with a thin lens added to it.

The y-component of the unit wave vector.

So far, we have studied some basic laser cavity stability analyses with the ABCD method side by side with the ray tracing capabilities in COMSOL Multiphysics. The stability results from both approaches are in good agreement.

Simulating Lasers with Varying Focal Lengths

Now, let’s move on to the last example, in which the focal length in the previous example is parameterized and varied, and the variation of the cavity stability is plotted in the famous stability chart. In Kogelnik’s theory, the previous stability condition formula is rewritten as

by using a set of new variables defined by g₁ = 1 – L₂/f – L₀/R₁ and g₂ = 1 – L₁/f – L₀/R₂.

The stability regions are plotted as the hatched area in the following famous stability chart with g₁ and g₂ as the horizontal axis and the vertical axis, respectively. If g₁ and g₂, which are derived by the original cavity parameters R₁, R₂, and L₀, are within the hatched area, the laser cavity is stable according to Kogelnik’s theory. For the previous example, we used the same radius of curvature for the two mirrors. So R₁ = R₂ and therefore L₁ = L₂ = L₀/2 from symmetry, which results in g₁ = g₂. This is a 45-degree line. When the cavity parameters change, the values of g₁ and g₂ change along the line through the first and third quadrants.

Let’s take a look at how this is visualized. We fix the mirror curvature to 0.1 m and the cavity length to 0.084357 m while changing the focal length of the lens from 15 mm to 40 mm with 5-mm increments. The following figure shows Kogelnik’s theoretical results for this particular laser cavity with a lens. The laser is stable for f = 25, 30, 35, and 40 mm, giving g₁g₂ = 0.67091, 0.43100, 0.29200, and 0.20545, respectively.

These values are plotted in the hatched area, whereas the focal lengths of 15 and 20 mm make this laser unstable as their g₁g₂ values exceed 1.0; i.e., 1.1299 and 2.1593, respectively. The theoretical prediction for this laser cavity is plotted in the figure below. The ray tracing result for each focal length is depicted in the sides of the figure showing a perfect match with the theory. In the COMSOL computation, the built-in stop condition terminates the computation once the ray gets out of the cavity. The ratio of the actual computation time T to the preset computation time T₀ (50 ns in this case), T/T₀ represents the cavity stability if T₀ is appropriately chosen. Any value of this stability index other than 1.0 may not mean much, as we know that a real laser turns off really quickly when it gets unstable; in other words, the change of laser stability is almost digital, 1 (on) or 0 (off).

Comparison between Kogelnik’s stability theory and the COMSOL Multiphysics ray tracing results for an off-confocal cavity with a thin lens.

This last example shows that ray tracing can perform a laser cavity stability analysis with the thermal lensing effect taken into account, since this example is Kogelnik’s model for a laser cavity with thermal lensing. In Kogelnik’s theory, a laser crystal heated by a pump laser is replaced by a thin lens with a certain focal length. Once the temperature distribution inside the crystal is known, the refractive index distribution is calculated by n = n₀ + (T – T₀)dn/dT to the first-order approximation, where n, n₀ are the refractive index distribution with and without pumping; T, T₀ are the temperature distribution with and without pumping; and dn/dT is the coefficient of index variation per temperature change.

The refractive index distribution of a pumped laser crystal is approximately a parabolic shape around the center if the crystal is a rod; i.e., n(r) = n₀(1 – 2r²/a²), where r is the radial distance in the polar coordinate system with its z-axis along the optical axis of the pumping. The calculated refractive index distribution is fitted to this parabola to get the fitting parameter a. Once this parameter is calculated, the focal length that represents the thermal lensing effect can be approximated by f ~ a²/(4n₀L), where L is the crystal length (Ref. 3). Thermal lensing also includes surface bulging, which causes refraction angle changes. This can be analyzed in a similar way.

Concluding Remarks

With COMSOL Multiphysics, we don’t have to do the fitting. We just do heat and mechanical simulation with the Heat Transfer and Solid Mechanics interfaces to know the new refractive index distribution and surface bulging and then simply carry out ray tracing with the ray optics interfaces to know the stability of your laser cavity under the thermal lensing effect. We will further discuss the ray tracing capabilities of COMSOL Multiphysics in an upcoming blog post. Here, we see a full-fledged model analyzing the stability of a Ti:sapphire laser cavity with a double-pumped Brewster-cut crystal and a pair of dispersion compensation prisms. Stay tuned!

A stability analysis model for a Ti:sapphire laser cavity solving for temperature, mechanical displacement, and ray position and angle.

References

1. H. Kogelnik and T. Li, Applied Optics, Vol. 5, No. 10 (1966)
2. E. Hecht, Optics, Addison Wesley
3. W. Koechner, Solid-State Laser Engineering, Springer

Additional Resources

• Browse these laser cavity tutorial models:
  • Laser Cavity with a Thin Lens
  • Two-Mirror Laser Cavity
  • Bow-Tie Laser Cavity
• Read more about laser simulation on the COMSOL Blog:
  • Understanding the Paraxial Gaussian Beam Formula
  • Modeling Laser-Material Interaction in COMSOL Multiphysics

Sequential, Nonsequential, and Exact Ray Tracing Algorithms

Ray tracing algorithms can be categorized as sequential and nonsequential methods. The ray tracing method used in the Ray Optics Module can be categorized as a nonsequential wavefront method.

Ray tracing through an optical system largely takes place in two alternating steps, regardless of the details of the algorithm being used:

1. Given an initial position and ray direction, either at a point on an object or on a surface in the optical system, the ray tracing algorithm determines where the ray will hit the next surface and trace the ray to that point.
2. The algorithm then continues by adjusting the ray direction by means of applying a boundary condition, such as reflection or refraction, where the ray hits a surface. This changes the direction of the reflected or refracted ray and prepares it to be traced through subsequent media.

Even in complex systems with a large number of surfaces, the entire ray tracing process can usually be broken down into successive iterations of these two steps, as shown below.

Several sequential ray tracing algorithms are described in some of the standard texts on geometrical optics (see, for example, Refs. 1, 2, and 3). By sequential ray tracing, it is meant that the order of the rays or mirrors to be encountered by the ray is specified as an input to the ray tracing algorithm; in other words, you know that the ray will first pass through lens 1, then lens 2, and so on, but you don’t know exactly where the ray will pass through each lens surface or what the angles of the reflected and refracted rays will be.

In a nonsequential ray tracing method, the order in which the rays encounter lenses and mirrors is not specified beforehand. Instead, you specify the light source and the method will automatically determine the interaction between the rays and the optical components in the system.

Some important sequential ray tracing algorithms make use of paraxial approximations and assume that the angles between rays and the optic axis are small. In contrast, certain sequential and nonsequential ray tracing methods can be classified as exact methods (1, 2, 3) in that the shape of the lens is taken into account when predicting the intersection point of the ray with the lens surface, and the paraxial approximation is not used. Thus, rays intersect the actual surface of a curved lens, not the vertex plane.

In exact analytical ray tracing methods, the intersection point of the ray with the surface of a lens or mirror is computed by solving an algebraic equation. For example, if the surface of a lens is spherical, then given the initial position and direction of a ray, its intersection point with the surface of the lens can be computed exactly by solving a quadratic equation. For more elaborate, nonspherical surfaces, the equation for the intersection point of the ray with the surface might not have a closed-form analytic solution and may require a numerical approach.

In the following discussion, we’ll compare the approach used in the COMSOL® software to the exact ray tracing method described above. We’ll ignore the simpler methods of paraxial optics, because ray tracing in COMSOL Multiphysics always uses the full implementation of Snell’s law and never uses the small-angle approximation.

Homogeneous vs. Graded-Index Materials

Many traditional ray tracing methods require that each material is homogeneous; that is, the refractive index has zero gradient in each medium and only changes discontinuously at boundaries. An approach based on algebraically solving for the intersection point with the next lens, as described above, only works for nongraded media. This is because the rays in graded-index media can be curved, so the ray’s initial position and direction are not sufficient to determine where it will intersect a surface.

Real-world optical systems may have components with graded-index, or heterogeneous, materials either by design or due to multiphysics effects. Examples of graded refractive index distributions by design include the Luneburg lens, Maxwell’s fish-eye lens, and GRIN lenses fabricated by electrophoresis. A lens subject to an environment with nonuniform temperature distributions will inadvertently have a graded refractive index due to the material properties being dependent on temperature and strain. These are some of the most important cases for which the wavefront method excels.

An example of a ray path in a heterogeneous medium is shown in the following plot. In the image, the thermal color table shows the temperature in the modeling domain, with lighter colors corresponding to higher temperatures. The change in the refractive index is proportional to the change in the temperature, so the ray path becomes more noticeably curved where the temperature gradient is largest. The color along the ray path shows the instantaneous group velocity magnitude of the ray, with the greatest magnitude shown in red and the lowest magnitudes in blue and violet. This simulation, exaggerated for demonstration purposes, conceptually depicts what happens in pumped laser crystals in laser systems.

A single ray follows a curved path (exaggerated) through a heated material.

The Wavefront Ray Tracing Method

The ray tracing method in COMSOL Multiphysics numerically solves a set of coupled first-order ordinary differential equations (ODEs) for the components of the instantaneous ray position q and wave vector k. These coupled equations are analogous to the Hamiltonian formulation in classical mechanics,

where the angular frequency ω takes the place usually occupied by the Hamiltonian H (Ref. 4).

When the refractive index is homogeneous, the Hamiltonian formulations above are reduced to the expressions that account for a constant speed and ray direction of light, as follows:

When there is a discontinuity of the refractive index at an interface, COMSOL Multiphysics numerically computes the direction of the refracted ray using Snell’s law. This formulation makes it possible for the COMSOL® software to compute not only special cases with homogeneous refractive indices, but also more general cases such as thermal lensing in laser engineering (both straight rays and curved rays). Note that this is a time-stepping method and the results show the rays at different instances of time during propagation.

Discretization of Surfaces and Geometry Shape Order

As an example of a geometric shape, consider a rotationally symmetric aspheric lens surface sag described by the following analytical expression:

(1)

Here, are the lens surface sag, curvature, conic constant, and the coefficients for the even polynomials, respectively; and , where x and y are the two directions transverse to the optic axis.

In COMSOL Multiphysics, lenses and mirrors can have their shapes be input as analytic functions or as full-fledged 3D CAD models. Analytically given shapes are entered as parametric curves or surfaces and are approximated by high-order nonuniform rational spline (NURBS). In the next step, the surfaces (or curves) are approximated with a finite element mesh, which further approximates the shape using polynomials.

In the user interface, the maximum order of these polynomials is determined by the Geometry shape order in the Settings window for the model component. The accuracy of these approximations can be increased by refining the mesh and/or increasing the geometry shape order. The rays then interact with the individual mesh elements for the purposes of predicting intersection points and applying Snell’s law.

The default option Automatic for the Geometry shape order in most cases approximates surfaces using quadratic polynomials, although linear shape order is sometimes used to avoid creating inverted mesh elements. For most applications, the order can be increased by the user to 7^th-order (Septic) polynomials.

Settings window for geometry shape order.

Let’s now consider an aspheric lens, the shape of which was described earlier. To make it easier to see the discretization, the lens is here meshed extremely coarsely. In the figures below, the linearized mesh elements are indicated by gray lines, while the numeric representation of the curved surface is shown as a black outline.

The left figure is a ray tracing result for the default (Automatic, i.e., mostly quadratic) setting and the right figure is for the case of Linear geometry shape order. The color of the rays represents the y-component (vertical axis) of the wave vector. You can see where rays get refracted by looking for the position where the color changes. In any choice of geometry shape order, the surface shape can no longer be written by the above aspheric surface formula; i.e., it’s a collection of piecewise continuous polynomials whose order is governed by the Geometry shape order setting.

An aspheric lens surface and its approximate surface shape using quadratic (left) and linear (right) geometry shape order.

Rays and the Finite Element Mesh

There are several reasons why the ray tracing algorithm in COMSOL Multiphysics uses the finite element mesh to reflect and refract rays at surfaces. As previously mentioned, this emphasizes the COMSOL® software’s strength as a multiphysics simulation tool by making it possible to trace rays through graded-index media where the refractive index distribution is a function of a field computed by another physics interface. Another situation where an analytic expression for the lens surface may not be available is when the surface is deformed as a result of thermal expansion or due to other external forces on the lens. In conclusion, almost any physics coupled to the optical system is modeled using the finite element method, so the coupling takes place through a finite element mesh.

Because the ray tracing algorithm uses the finite element mesh, a minimum requirement for ray tracing in COMSOL Multiphysics is that all boundaries that reflect or refract the rays must have a boundary mesh. The boundary mesh elements are used to compute surface properties such as the normal direction and the principal radii of curvature that are needed when the rays are reflected or refracted. The boundary mesh also provides information to a mesh search algorithm that predicts the instant when rays will reach the boundary, allowing the quantities that are computed along ray paths, such as intensity and phase, to be computed more accurately.

In some cases, the domains that the rays propagate through should also have a domain-level finite element mesh. For example, domains containing a graded-index medium must always be meshed. If the refractive index is temperature dependent, then in order to compute the refractive index at the ray’s instantaneous position in the medium, it is necessary to query a value from the temperature field. The temperature can be interpolated at the ray’s exact location within a mesh element using the finite element basis functions defined in that element. Then, this temperature value can be substituted into a temperature-dependent expression for the refractive index.

A domain-level mesh is also required when the rays propagate in an absorbing medium. The deposited ray power in the medium is discretized using piecewise discontinuous functions that are defined on the mesh elements, so the mesh is necessary to confine the contribution of each ray to the total heat source to a region of finite extent.

How Mesh Resolution Affects the Solution

So far, we’ve learned that ray tracing in the COMSOL® software is an approximate solution that treats each surface as a piecewise continuous polynomial based on the finite element mesh. Therefore, the accuracy of the solution depends on the resolution of the mesh. Higher geometry shape order and finer meshes reduce the discretization errors. In the right plot of the previous figure, some rays pass through the same boundary element, which spuriously results in the same wave vectors (lines with the same colors). This is one type of approximation error that we can visibly see when the geometry shape order is low and the mesh is very coarse.

The following plots show the y-component of the unit wave vector of the refracted rays after the first refraction by the same lens in the previous example. The horizontal axis represents the initial ray position in the y axis. The error introduced by an underresolved mesh is most significant when the geometry shape order is linear.

Plots of the y-component of the unit wave vector for quadratic geometry shape order (left) and linear shape order (right) with different mesh element sizes (shown by different colors).

The Effect of Mesh Refinement on Spot Diagrams

We’ve seen that the level of mesh refinement affects the accuracy of the ray trajectories, so it should come as no surprise that it also affects the accuracy of spot diagrams. Results from exact ray tracing methods usually show near-perfect symmetry and continuity if the distributions of initial ray positions and directions are symmetric and regular. The spot diagram for an aspheric lens is always symmetric around the optical axis and the ray aberrations are always smooth and continuous. For the finite-element-based wavefront method, the results will only be symmetric if the mesh inherits the symmetry of the original lens geometry.

Even for symmetrical optical systems with symmetrical initial distributions of ray position and direction, ray tracing results in COMSOL Multiphysics can give you slightly unsymmetric results if the mesh is unsymmetric. But these deviations from symmetry will gradually be reduced as the mesh is refined, since the smaller mesh elements on either side of the symmetry plane converge toward the same representation of the surface. This is just another manifestation of the mesh discretization error. The symmetry of the solution can be improved by using a structured mesh. However, it should be noted that a symmetric solution doesn’t necessarily mean that the solution is more accurate.

The next figure shows the effect of the maximum mesh element size on the spot diagram and a plot of the RMS spot radius for a plano-convex lens. This focusing lens is designed to have a diffraction-limited spot. You can see how the RMS spot radius converges to a certain value as the mesh size drops as expected. Let’s take a closer look. The spot for the coarsest mesh is larger than the diffraction limit. It is also very asymmetric and the intersection points are distributed over a very large area. This mesh size doesn’t look particularly bad for other physics like heat and mechanical problems. However, for ray tracing with the default Automatic geometry shape order, this mesh is too coarse to produce an accurate spot diagram. Only the last two mesh sizes give a concentrated and symmetric-looking spot.

Comparison of spot diagram at the focus between quadratic and cubic geometry shape orders depending on the max mesh size. The red circles show the diffraction limit size for the lens at the wavelength of 0.66 um.

Log-log plot of the RMS spot radii as a function of the max mesh size for quadratic and cubic geometry shape order compared to the diffraction limit size.

Interpreting the Solution Data Generated by the Ray Tracing Method in COMSOL Multiphysics®

Let’s now have a look at how the format of the solution data may differ from that of the traditional (stationary) analytical ray tracing algorithms described above.

The following is an example that compares the time-dependent solution obtained in COMSOL Multiphysics to the typical output from the standard sequential ray tracing plane-to-plane method for the same focusing lens in the previous example. A plano-convex lens focuses rays into a spot. In the familiar results from the conventional plane-to-plane method, all of the rays are drawn to the end plane. On the other hand, in COMSOL Multiphysics, each ray is drawn to be at the position where the ray is supposed to be at the specified time step. So, in general, the rays are not laid in the same plane.

This is why ray trajectory results in COMSOL Multiphysics almost always show some curvature. This curvature is nothing but the wavefront in terms of the geometrical optics regime. In the example, the wavefront is curved by the quadratic phase function that the lens introduces to the rays. To draw all rays to the end plane, use a later solution time so that all rays have had sufficient time to reach the plane and stop propagating.

Simulation results for time-dependent ray tracing in COMSOL Multiphysics® (left) and the standard sequential plane-to-plane ray tracing (right).

While it’s convenient to see raw wavefronts, we need to make sure that all of the rays we simulate have passed the plane in which we want to evaluate a quantity. The following sequential spot diagrams illustrate a spot diagram for the above example at different solution times. In COMSOL Multiphysics, spot diagrams are generated using the Poincaré Map plot type. Because the wavefront is converging, outer rays arrive at the plane first and inner rays come after. Thus, in-plane evaluations may vary depending on the time we choose.

Spot diagrams generated at different solution times.

To make sure that we evaluate all rays in the plane we are interested in, COMSOL Multiphysics is equipped with operators called attimemin and attimemax. For example, the following expression gives the RMS spot radius in the yz-plane at x = 0.1 m:

sqrt( gop.gopaveop1( attimemin(0,0.4[ns],(qx-0.1[m])^2, qy^2+qz^2) ) )

The operator gop.gopaveop1() takes the average of the expression in parentheses over all rays. The argument to this operator,

attimemin(0,0.4[ns],(qx-0.1[m])^2, qy^2+qz^2)

calls the attimemin operator, which takes four arguments.

The first two arguments of attimemin define the beginning and end of the time period in which we want to compute the minimum value of an expression. We have to make sure that the time when the ray crosses the plane is in this range. The third argument is the expression to minimize. At the time when the minimum value of this third argument is detected for the ray, the value of the fourth argument is returned. The attimemin operator uses interpolation between the stored solution times, so it is possible to accurately get the intersection point of each ray with the plane even if this intersection point doesn’t coincide with a time step stored in the solution.

In words, the previous expression might read:

“Compute the square root of the average over all rays of the value of the radial coordinate at the time between 0 and 0.4 ns for which the ray’s distance from the plane x = 0.1 reaches a minimum.”

The next example shows how the storage of the solution at discrete time steps can affect the trajectory plots when rays are reflected or refracted at boundaries. We usually don’t know in advance the exact time or optical path length at which each ray will interact with each surface, so these intersection times will usually not coincide exactly with the stored solution times. As a result, the COMSOL Multiphysics ray tracing results show some small incompleteness in the trajectories near an interface or a reflective wall, as shown in the figure below where rays seem to change direction a short distance away from the boundary instead of exactly on it. However, this is expected for this type of time-dependent ray tracing method and does not affect the accuracy. Even if a ray doesn’t look like it’s hitting an interface or wall in the results, the COMSOL® software computes an accurate interaction time for each ray, typically using a second-order Taylor method to extrapolate the ray position from the previous stored solution time.

Reflection of rays at a reflective wall. Not all of the rays are snapped to the wall in the plot, but all ray trajectories are accurately computed.

Concluding Remarks

In this blog post, we have discussed the unique time-dependent and mesh-based method used in COMSOL Multiphysics for ray optics simulations. We also went over what types of anomalies you might encounter in the ray tracing results, especially when the time step is large and the mesh is coarse. As in other physics interfaces, refining the mesh or improving the resolution in time can clear up these unusual-looking behaviors and yield an accurate, converged solution.

Further Resources

Learn more about the ray optics and ray tracing capabilities of COMSOL Multiphysics on the COMSOL Blog:

• Using the New Ray Tracing Algorithm in COMSOL Multiphysics 5.2a
• Modeling Thermally Induced Focal Shift in High-Powered Laser Systems
• How to Model Solar Concentrators with the Ray Optics Module

References

1. R.R. Shannon, The Art and Science of Optical Design, Cambridge, 1997.
2. D.C. O’Shea, Elements of Modern Optical Design, Wiley, 1985.
3. R. Ditteon, Modern Geometrical Optics, Wiley, 1998.
4. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed., Butterworth-Heinemann, Oxford, 1975.

Fourier Transformation with Fourier Optics

Implementing the Fourier transformation in a simulation can be useful in Fourier optics, signal processing (for use in frequency pattern extraction), and noise reduction and filtering via image processing. In Fourier optics, the Fresnel approximation is one of the approximation methods used for calculating the field near the diffracting aperture. Suppose a diffracting aperture is located in the plane at . The diffracted electric field in the plane at the distance from the diffracting aperture is calculated as

where, is the wavelength and account for the electric field at the plane and the plane, respectively. (See Ref. 1 for more details.)

In this approximation formula, the diffracted field is calculated by Fourier transforming the incident field multiplied by the quadratic phase function .

The sign convention of the phase function must follow the sign convention of the time dependence of the fields. In COMSOL Multiphysics, the time dependence of the electromagnetic fields is of the form . So, the sign of the quadratic phase function is negative.

Fresnel Lenses

Now, let’s take a look at an example of a Fresnel lens. A Fresnel lens is a regular plano-convex lens except for its curved surface, which is folded toward the flat side at every multiple of along the lens height, where m is an integer and n is the refractive index of the lens material. This is called an m^th-order Fresnel lens.

The shift of the surface by this particular height along the light propagation direction only changes the phase of the light by (roughly speaking and under the paraxial approximation). Because of this, the folded lens fundamentally reproduces the same wavefront in the far field and behaves like the original unfolded lens. The main difference is the diffraction effect. Regular lenses basically don’t show any diffraction (if there is no vignetting by a hard aperture), while Fresnel lenses always show small diffraction patterns around the main spot due to the surface discontinuities and internal reflections.

When a Fresnel lens is designed digitally, the lens surface is made up of discrete layers, giving it a staircase-like appearance. This is called a multilevel Fresnel lens. Due to the flat part of the steps, the diffraction pattern of a multilevel Fresnel lens typically includes a zeroth-order background in addition to the higher-order diffraction.

A Fresnel lens in a lighthouse in Boston. Image by Manfred Schmidt — Own work. Licensed under CC BY-SA 4.0, via Wikimedia Commons.

Why are we using a Fresnel lens as our example? The reason is similar to why lighthouses use Fresnel lenses in their operations. A Fresnel lens is folded into in height. It can be extremely thin and therefore of less weight and volume, which is beneficial for the optics of lighthouses compared to a large, heavy, and thick lens of the conventional refractive type. Likewise, for our purposes, Fresnel lenses can be easier to simulate in COMSOL Multiphysics and the add-on Wave Optics Module because the number of elements are manageable.

Modeling a Focusing Fresnel Lens in COMSOL Multiphysics®

The figure below depicts the optics layout that we are trying to simulate to demonstrate how we can implement the Fourier transformation, applied to a computed solution solved for by the Wave Optics, Frequency Domain interface.

Focusing 16-level Fresnel lens model.

This is a first-order Fresnel lens with surfaces that are digitized in 16 levels. A plane wave is incident on the incidence plane. At the exit plane at , the field is diffracted by the Fresnel lens to be . This process can be easily modeled and simulated by the Wave Optics, Frequency Domain interface. Then, we calculate the field at the focal plane at by applying the Fourier transformation in the Fresnel approximation, as described above.

The figures below are the result of our computation, with the electric field component in the domains (top) and on the boundary corresponding to the exit plane (bottom). Note that the geometry is not drawn to scale in the vertical axis. We can clearly see the positively curved wavefront from the center and from every air gap between the saw teeth. Note that the reflection from the lens surfaces leads to some small interferences in the domain field result and ripples in the boundary field result. This is because there is no antireflective coating modeled here.

The computed electric field component in the Fresnel lens and surrounding air domains (vertical axis is not to scale).

The computed electric field component at the exit plane.

Implementing the Fourier Transformation from a Computed Solution

Let’s move on to the Fourier transformation. In the previous example of an analytical function, we prepared two data sets: one for the source space and one for the Fourier space. The parameter names that were defined in the Settings window of the data set were the spatial coordinates in the source plane and the spatial coordinates in the image plane.

In today’s example, the source space is already created in the computed data set, Study 1/Solution 1 (sol1){dset1}, with the computed solutions. All we need to do is create a one-dimensional data set, Grid1D {grid1}, with parameters for the Fourier space; i.e., the spatial coordinate in the focal plane. We then relate it to the source data set, as seen in the figure below. Then, we define an integration operator intop1 on the exit plane.

Settings for the data set for the transformation.

The intop1 operator defined on the exit plane (vertical axis is not to scale).

Finally, we define the Fourier transformation in a 1D plot, shown below. It’s important to specify the data set we previously created for the transformation and to let COMSOL Multiphysics know that is the destination independent variable by using the dest operator.

Settings for the Fourier transformation in a 1D plot.

The end result is shown in the following plot. This is a typical image of the focused beam through a multilevel Fresnel lens in the focal plane (see Ref. 2). There is the main spot by the first-order diffraction in the center and a weaker background caused by the zeroth-order (nondiffracted) and higher-order diffractions.

Electric field norm plot of the focused beam through a 16-level Fresnel lens.

Concluding Remarks

In this blog post, we learned how to implement the Fourier transformation for computed solutions. This functionality is useful for long-distance propagation calculation in COMSOL Multiphysics and extends electromagnetic simulation to Fourier optics.

Next Steps

Download the model files for the Fresnel lens example by clicking the button below.

Read More About Simulating Wave Optics

• Simulating Holographic Data Storage in COMSOL Multiphysics
• How to Simulate a Holographic Page Data Storage System
• How to Implement the Fourier Transformation in COMSOL Multiphysics

References

1. J.W. Goodman, Introduction to Fourier Optics, The McGraw-Hill Company, Inc.
2. D. C. O’Shea, Diffractive Optics, SPIE Press.

Gaussian Beam: The Most Useful Light Source and Its Formula

Because they can be focused to the smallest spot size of all electromagnetic beams, Gaussian beams can deliver the highest resolution for imaging, as well as the highest power density for a fixed incident power, which can be important in fields such as material processing. These qualities are why lasers are such attractive light sources. To obtain the tightest possible focus, most commercial lasers are designed to operate in the lowest transverse mode, called the Gaussian beam.

As such, it would be reasonable to want to simulate a Gaussian beam with the smallest spot size. There is a formula that predicts real Gaussian beams in experiments very well and is convenient to apply in simulation studies. However, there is a limitation attributed to using this formula. The limitation appears when you are trying to describe a Gaussian beam with a spot size near its wavelength. In other words, the formula becomes less accurate when trying to observe the most beneficial feature of the Gaussian beam in simulation. In a future blog post, we will discuss ways to simulate Gaussian beams more accurately; for the remainder of this post, we will focus exclusively on the paraxial Gaussian beam.

A schematic illustrating the converging, focusing, and diverging of a Gaussian beam.

Note: The term “Gaussian beam” can sometimes be used to describe a beam with a “Gaussian profile” or “Gaussian distribution”. When we use the term “Gaussian beam” here, it always means a “focusing” or “propagating” Gaussian beam, which includes the amplitude and the phase.

Deriving the Paraxial Gaussian Beam Formula

The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwell’s equations. This is the first important element to note, while the other portions of our discussion will focus on how the formula is derived and what types of assumptions are made from it.

Because the laser beam is an electromagnetic beam, it satisfies the Maxwell equations. The time-harmonic assumption (the wave oscillates at a single frequency in time) changes the Maxwell equations to the frequency domain from the time domain, resulting in the monochromatic (single wavelength) Helmholtz equation. Assuming a certain polarization, it further reduces to a scalar Helmholtz equation, which is written in 2D for the out-of-plane electric field for simplicity:

where for wavelength in vacuum.

The original idea of the paraxial Gaussian beam starts with approximating the scalar Helmholtz equation by factoring out the propagating factor and leaving the slowly varying function, i.e., , where the propagation axis is in and is the slowly varying function. This will yield an identity

This factorization is reasonable for a wave in a laser cavity propagating along the optical axis. The next assumption is that , which means that the envelope of the propagating wave is slow along the optical axis, and , which means that the variation of the wave in the optical axis is slower than that in the transverse axis. These assumptions derive an approximation to the Helmholtz equation, which is called the paraxial Helmholtz equation, i.e.,

The special solution to this paraxial Helmholtz equation gives the paraxial Gaussian beam formula. For a given waist radius at the focus point, the slowly varying function is given by

where , , and are the beam radius as a function of , the radius of curvature of the wavefront, and the Gouy phase, respectively. The following definitions apply: , , , and .

Here, is referred to as the Rayleigh range. Outside of the Rayleigh range, the Gaussian beam size becomes proportional to the distance from the focal point and the intensity position diverges at an approximate divergence angle of .

Definition of the paraxial Gaussian beam.

Note: It is important to be clear about which quantities are given and which ones are being calculated. To specify a paraxial Gaussian beam, either the waist radius or the far-field divergence angle must be given. These two quantities are dependent on each other through the approximate divergence angle equation. All other quantities and functions are derived from and defined by these quantities.

Simulating Paraxial Gaussian Beams in COMSOL Multiphysics®

In COMSOL Multiphysics, the paraxial Gaussian beam formula is included as a built-in background field in the Electromagnetic Waves, Frequency Domain interface in the RF and Wave Optics modules. The interface features a formulation option for solving electromagnetic scattering problems, which are the Full field and the Scattered field formulations.

The paraxial Gaussian beam option will be available if the scattered field formulation is chosen, as illustrated in the screenshot below. By using this feature, you can use the paraxial Gaussian beam formula in COMSOL Multiphysics without having to type out the relatively complicated formula. Instead, you simply need to specify the waist radius, focus position, polarization, and the wave number.

Screenshot of the settings for the Gaussian beam background field.

Plots showing the electric field norm of paraxial Gaussian beams with different waist radii. Note that the variable name for the background field is ewfd.Ebz.

Looking into the Limitation of the Paraxial Gaussian Beam Formula

In the scattered field formulation, the total field is linearly decomposed into the background field and the scattered field as . Since the total field must satisfy the Helmholtz equation, it follows that , where is the Laplace operator. This is the full field formulation, where COMSOL Multiphysics solves for the total field. On the other hand, this formulation can be rewritten in the form of an inhomogeneous Helmholtz equation as

The above equation is the scattered field formulation, where COMSOL Multiphysics solves for the scattered field. This formulation can be viewed as a scattering problem with a scattering potential, which appears in the right-hand side. It is easy to understand that the scattered field will be zero if the background field satisfies the Helmholtz equation (under an approximate Sommerfeld radiation condition, such as an absorbing boundary condition) because the right-hand side is zero, aside from the numerical errors. If the background field doesn’t satisfy the Helmholtz equation, the right-hand side may leave some nonzero value, in which case the scattered field may be nonzero. This field can be regarded as an error of the background field. In other words, under certain conditions, you can qualify and quantify exactly how and by how much your background field satisfies the Helmholtz equation. Let’s now take a look at the scattered field for the example shown in the previous simulations.

Plots showing the electric field norm of the scattered field. Note that the variable name for the scattered field is ewfd.relEz. Also note that the numerical error is contained in this error field as well as the formula’s error.

The results shown above clearly indicate that the paraxial Gaussian beam formula starts failing to be consistent with the Helmholtz equation as it’s focused more tightly. Quantitatively, the plot below may illustrate the trend more clearly. Here, the relative L2 error is defined by , where stands for the computational domain, which is compared to the mesh size. As this plot suggests, we can’t expect that the paraxial Gaussian beam formula for spot sizes near or smaller than the wavelength is representative of what really happens in experiments or the behavior of real electromagnetic Gaussian beams. In the settings of the paraxial Gaussian beam formula in COMSOL Multiphysics, the default waist radius is ten times the wavelength, which is safe enough to be consistent with the Helmholtz equation. It is, however, not a “cut-off” number, as the approximation assumption is continuous. It’s up to you to decide when you need to be cautious in your use of this approximate formula.

Semi-log plot comparing the relative L2 error of the scattered field with the waist size in the units of wavelength.

Checking the Validity of the Paraxial Approximation

In the above plot, we saw the relationship between the waist size and the accuracy of the paraxial approximation. Now we can check the assumptions that were discussed earlier. One of the assumptions to derive the paraxial Helmholtz equation is that the envelope function varies relatively slowly in the propagation axis, i.e., . Let’s check this condition on the x-axis. To that end, we can calculate a quantity representing the paraxiality. As the paraxial Helmholtz equation is a complex equation, let’s take a look at the real part of this quantity, .

The following plot is the result of the calculation as a function of x normalized by the wavelength. (You can type it in the plot settings by using the derivative operand like d(d(A,x),x) and d(A,x), and so on.) We can see that the paraxiality condition breaks down as the waist size gets close to the wavelength. This plot indicates that the beam envelope is no longer a slowly varying one around the focus as the beam becomes fast. A different approach for seeing the same trend is shown in our Suggested Reading section.

Real part of the paraxiality along the x-axis for paraxial Gaussian beams with different waist sizes.

Concluding Remarks on the Paraxial Gaussian Beam Formula

Today’s blog post has covered the fundamentals related to the paraxial Gaussian beam formula. Understanding how to effectively utilize this useful formulation requires knowledge of its limitation as well as how to determine its accuracy, both of which are elements that we have highlighted here.

There are additional approaches available for simulating the Gaussian beam in a more rigorous manner, allowing you to push through the limit of the smallest spot size. We will discuss this topic in a future blog post. Stay tuned!

Editor’s note, 7/2/18: The follow-up blog post, “The Nonparaxial Gaussian Beam Formula for Simulating Wave Optics“, is now live.

Suggested Reading

• P. Vaveliuk, “Limits of the paraxial approximation in laser beams”, Optics Letters, Vol. 32, No. 8 (2007)
• Browse related topics here on the COMSOL Blog:
  • Simulating Holographic Data Storage in COMSOL Multiphysics
  • How to Simulate a Holographic Page Data Storage System

Understanding Fourier Transformation with a Fraunhofer Diffraction Example

The ability to implement the Fourier transformation in a simulation is a useful functionality for a variety of applications. Besides Fourier optics, we use Fourier transformation in Fraunhofer diffraction theory, signal processing for frequency pattern extraction, and image processing for noise reduction and filtering.

In this example, we calculate an image of the light from a traffic light passing through a mesh curtain, shown below. To simplify the model, we assume the electric field of the lights is a plane wave of uniform intensity; for instance, 1 V/m. Let the mesh geometry be measured by the local coordinates and in a plane perpendicular to the direction of the light propagation, and let the image pattern be measured by the local coordinates and near the eye in a plane parallel to the mesh plane.

A Fraunhofer diffraction pattern as a Fourier transform of a square aperture in a mesh curtain.

According to the Fraunhofer diffraction theory, then, we can calculate the image above simply by Fourier transforming the light transmission function, which is a periodic rectangular function if the mesh is square. Let’s consider a simplified case of a single mesh whose transmission function is a single rectangular function. We will discuss the case of a periodic transmission function later on.

We are interested in the light hitting one square of the mesh and getting diffracted by the sharp edges of the fabric while transmitting in the center of the mesh. In this case, the light transmission function is described by a 2D rectangular function. By implementing a Fourier transformation into a COMSOL Multiphysics simulation, we can more fully understand this process.

Utilizing Data Sets in COMSOL Multiphysics

In order to learn how to implement Fourier transformation, let’s first discuss the concept of data sets, or multidimensional matrices that store numbers. There are two possible types of data sets in COMSOL Multiphysics: Solution and Grid. For any computation, the COMSOL software creates a data set, which is placed under the Results > Data Sets node.

The Solution data set consists of an unstructured grid and is used to store solution data. To make use of this data set, we specify the data to which each column and row corresponds. If we specify Solution 1 (sol1), the matrix dimension corresponds to that of the model in Study 1. If it is a time-dependent problem, for example, the data set has a three-dimensional array, which may be written as with . Here, is the number of stored time steps, is the number of nodes, and is the number of the space dimension. Similarly, the data set for a time-dependent parametric study consists of a 4D array. Again, note that the spatial data (other than the time and parameter data) links with the nodal position on the mesh, not necessarily on the regular grid.

On the other hand, the Grid data set is equipped with a regular grid and is provided for functions and all other general purpose uses. All numbers stored in the Grid data set link to the grid defined in the Settings window. This data set is automatically created when a function is defined in the Definition node and by clicking on Create Plot. This creates a 1D Grid data set in the Data Sets node.

You also need to specify the range and the resolution of your independent variables. By default, the resolution for a 1D Grid data set is set to 1000. If the independent variable (i.e., x) ranges from 0 to 1, the Grid data set prepares data series of 0, 0.001, 0.002, …, 0.999, and 1. The default resolution is 100 for 2D and 30 for 3D. For Fourier transformation, we use the Grid data set. We can also use this data set as an independent tool for our calculation, as it does not point to a solution.

Implementing the Fourier Transformation

To begin our simulation, let’s define the built-in 1D rectangular function, as shown in the image below.

Defining the built-in 1D rectangular function.

Then, we click on the Create Plot button in the Settings window to create a separate 1D plot group in the Results node.

A plot of the built-in 1D rectangular function.

Let’s look at the Settings window of the plot. We expand the 1D Plot Group 1 node and click on Line Graph 1 to see the data set pointing to Grid 1D. In the Grid 1D node settings, we see that the data set is associated with a function rect1.

Settings for the built-in 1D rectangular function.

Settings for the 1D Grid data set.

We can create a 2D rectangular function by defining an analytic function in the Definitions node as rect1(x)*rect1(y). For learning purposes, we will create and define a 2D Grid data set and plot it manually instead of automatically. The results are shown in the following series of images.

In the Grid 2D settings, we choose All for Function because the 2D rectangular function uses another function, rect1. We also assign and as independent variables, which we previously defined as the curtain’s local coordinates, and set the resolution to 64 for quicker testing. To plot our results, we choose the 2D grid data, renamed to Grid 2D (source space), for the data set in the Plot Group settings window.

Defining the function in the Grid 2D settings.

Creating and defining a 2D data set.

Setting the 2D plot group for the 2D rectangular function.

A 2D plot of the 2D rectangular function.

Now, let’s implement a Fourier transform of this function by calculating:

Here, and represent the destination space (Fourier/frequency space) independent variables, as we previously discussed.

Since we already created a 2D data set for and , now we can create a Grid 2D data set, renamed to Grid 2D (Destination space), for and (shown below). We choose Function from Source and All from Function because the rect function calls the rect1 function as well. We can change the resolution to 64 here, as we did for the 2D data set, for quicker calculation.

Settings for the Grid 2D data set for the Fourier space.

Now, we are at the stage in our simulation where we can type in the equations by using the integrate operator.

Entering the equation for the Fourier transform of the 2D rectangular function.

We finally obtain the resulting Fourier transform, as shown in the figure below. Compare this (more accurately, the square of this) to each twinkling colored light in the photograph of the mesh curtain. In practice, this image hasn’t been truly seen yet. To calculate the image on its final destination, the retina of the eye, we would need to implement the Fourier transformation one more time.

The Fourier transform of the 2D rectangular function.

Concluding Remarks on Fourier Transformation

In COMSOL Multiphysics, you can use the data set feature and integrate operator as a convenient standalone calculation tool and a preprocessing and postprocessing tool before or after your main computation. Note that the Fourier transformation discussed here is not the discrete Fourier transformation (FFT). We still use discrete math, but we carry out the integration numerically by using Simpson’s rule. This function is used in the integration operator in COMSOL Multiphysics, while the discrete Fourier transform is formed by the operation of number sequences. As a result, we don’t need to be concerned with the aliasing problem, Fourier space resolution issue, or Fourier space shift issue.

There is more to discuss on this subject, but let’s comment on the two cases that we simplified earlier. We calculated for a single mesh. In practice, the mesh curtain is made of a finite number of periodic square openings. It sounds like we have to redo our calculation for the periodic case, but fortunately, the end result differs only by an envelope function of the periodicity. For details, Hecht’s Optics outlines this topic very well.

The second simplification was that we assumed a sharp rectangular function for the mesh transmission function. In COMSOL Multiphysics, all functions other than the user-defined functions are smoothed to some extent for numerical stability and accuracy reasons. You may have noticed that our rectangular function had small slopes. This may be a complication rather than a simplification because the simplest case is a rectangular function with no slopes and we used a smoothed rectangular function instead of a sharp one.

The Fourier transforms of the two extreme cases are known; i.e., a rectangular function with no slopes is transformed to a sinc function (sin(x)/x) and a Gaussian function to another Gaussian function. A sinc function has ripples around the center representing a diffraction effect, while a Gaussian function decays without any ripples. Our smoothed rectangular function is somewhere between these two extremes, so its Fourier transform is also somewhere between a sinc function and a Gaussian function. As we previously mentioned, the curtain fabric can’t have sharp edges, so our results may be more accurate for this example case anyway.

Further Reading

• Check out these blog posts about simulating holographic data storage systems:
  • Simulating Holographic Data Storage in COMSOL Multiphysics
  • How to Simulate a Holographic Page Data Storage System
• Find more information in these introductory books on optics:
  • J.W. Goodman, Introduction to Fourier Optics, W. H. Freeman, 2004.
  • E. Hecht, Optics, Pearson Education Limited, 2014.

Designing a Simulation for a Holographic Page Data Storage System

In a previous blog post discussing bit-by-bit hologram simulation, we introduced holographic data storage, its applications in consumer electronics, and how to simulate a bit-by-bit hologram. Now, we’ll discuss the other form of holographic data storage: page data storage. A page is a block of data represented by a spatial light modulator (SLM) that is either transmissive or reflective by using microelectromechanical systems (MEMS) or liquid crystal on silicon (LCoS).

As mentioned in the previous blog post, simulation for holographic data storage has traditionally been performed by the beam propagation method, which can handle very large computational domains, but cannot correctly handle a large focusing angle. COMSOL Multiphysics, on the other hand, uses a full-wave method, which can handle any kind of beam, but uses relatively more memory. With COMSOL Multiphysics, we can simulate a page (multibyte) data storage system in a small domain. To demonstrate, let’s consider a rectangular domain similar to that used in the previous study. This time, we will cipher one-byte (or eight bits) of data.

A typical optical layout of page-type holographic data storage (the character code of my name is encoded in binary data in the SLM).

For this simulation, we will use the binary data converted from the character code of a part of my own name in its native language. 01001101, which means “water”, can be seen in the fifth row in the SLM in the image above. To be more realistic, we’ll use a set of Fourier lenses to focus the object beam into the holographic material to record, expand, and visualize the retrieved object beam onto the detector in the retrieval process. Of course, we won’t model a lens, but instead make a focused beam by Fourier transforming the electric field amplitude after the SLM and providing it as the incident field in the scattering boundary condition on the incident boundary.

To image the retrieved object beam on the detector, we again Fourier transform the retrieved electric field amplitude and square the norm to get an intensity that a charge-coupled device (CCD) or complementary metal-oxide semiconductor (CMOS) sensor detects as a signal. More signal processing takes place afterward to create a cleaner signal and lessen the bit error rate to a significantly smaller level, but we will not go into this process here.

A holographic page data storage system, carrying one-byte data.

Defining the Reference Beam

In our previous discussion, we used a slightly diverging super Gaussian beam. For this simulation, the domain size will be inevitably wider along the direction of the reference beam propagation, which we will discuss later. So, if we use a diverging beam, the beam will eventually touch the boundaries, which needs to be avoided. Instead of launching a 10 um beam with a flat phase on the left boundary, we will add the following quadratic phase function so the beam slightly focuses in the middle of the domain, assuming the out-of-plane electric field solved for

where is the waist radius of the reference beam, is the refractive index of the holographic material, is the wave number in the vacuum, and is the wavefront curvature at distance from the beam waist (focal plane) position defined by

Here, is the Rayleigh range in which the beam is almost straight.

For = 10 um, = 1 um, and = 1.35, it gives = 424 um. We will see later that this number is far larger than our domain size, which means that the beam is almost collimated in the computational domain. To define the wavefront curvature, we have borrowed the paraxial Gaussian beam formula. We ignored a constant phase and the Gouy phase, which are not necessary here. The image below shows how to enter the incident field with a right curvature at the left boundary ().

Defining the reference beam with a wavefront curvature.

Defining the Object Beam

As we are using a 10 um beam radius, the vertical domain size, , of 30 um is large enough. The biggest obstacle here is how to determine the horizontal domain size, , for the object beam entrance. Now, the aperture through which the object beam transmits is a 1 x 8 SLM with 8 pixels. The SLM behaves like a diffraction grating with a period of . When the object beam transmits through the SLM and is focused, the zeroth-order beam is focused into a circle of the so-called Airy ring radius and the diffracted beams of higher orders will spread out at angles corresponding to the diffraction orders.

To get sufficient information from the SLM and store correct data in the holographic material, we want to capture up to at least the first-order beams (0^th and ±1^st). Otherwise, we may get some retrieved signal, but the signal might not fully restore the original data. Another reason why we only take up to the first orders is because all other higher orders will be too weak in intensity to be recorded in the holographic material.

The first requirement is that the zeroth-order beam radius, , must be 10 um, which determines the numerical aperture (NA) of the lens system. The Airy ring radius, , is given by the Airy ring radius formula

where is the wavelength in the air.

We want the Airy ring radius to be 10 um. From this requirement, we get the NA for a given and as

On the other hand, the NA is originally defined as

where is the focusing angle, is the number of SLM pixels, is the half size of the SML pixel, and is the focal length of the Fourier lens.

From this equation, a ratio, , is derived as

We apply the grating equation for the first order

where is the diffraction angle of the first-order beams.

We get the deviation of the beam position of the first-order beams from the zeroth-order beam at a distance as

Inserting the known numbers, = 8 and = 10 um, we get = 65.6 um. Adding some margin to capture the “whole” first-order beams, half of may be 80 um; that is, = 160 um. It’s worth mentioning that this particular figure is one of the key elements of holographic technology.

Other than this number, , , and are undetermined. Now that we know all of the domain sizes, we can estimate the number of meshes needed from the maximum mesh size, , where is the refractive index of the holographic material and is the intersecting angle between the object and reference beams. With the RAM capacity of my own computer, = 1 um seems to be the shortest wavelength. Then, we get = 131.1, of which the numbers and are dependent. For now, let be 40 um, followed by = 5.2 mm. We now have all of the simulation parameters.

To prepare the 1 x 8 pixel data, we can define the primitive built-in rectangular function to represent a single pixel. To make pixel data, the rectangular function is shifted and added up. 01001101 is defined as an analytic function, as shown in the figure below. The open subapertures stand for “1″.

An SLM aperture opacity function, representing the eight-bit data of 01001101.

Implementing a Fourier Transformation in COMSOL Multiphysics

Next, we focus the object beam. In Fourier optics, the image of the input electric field that is focused by a Fourier lens in the focal plane is the Fourier transform of the input field. The complex electric field amplitude in the image plane focused by a Fourier lens with the focal length is calculated by

where is the spatial coordinate in the Fourier/image space and represents the spatial frequency.

Do we need to use additional software to implement the Fourier transformation? No. By using COMSOL Multiphysics, all of the required capabilities are included in one package. You can also use COMSOL Multiphysics as a convenient scientific computational software in the GUI of the same platform as other finite element computations.

The Settings window is shown in the figure below, followed by the result of the Fourier transformation of the page data 01001101, calculated by the COMSOL software.

The settings for the incident object beam, which is the Fourier transform of the electric field amplitude after the SLM.

The computed incident object beam as the Fourier transform of the binary data 01001101.

The center beam is the zeroth-order beam and the two side beams with the opposite phase are the first-order beams. This is a typical Fraunhofer diffraction pattern of a grating. As we calculated before, our computational domain fits these three beams exactly. This electric field amplitude is given as the Electric Field boundary condition for the object beam. The following figures are the result of the page data recording.

The electric field amplitude (top) and intensity (bottom) for the page data recording.

Our hologram simulation is starting to look more interesting thanks to our encoding and ciphering work. The data for my name has been encoded by an industrial standard and then converted to a binary code. Then, it was Fourier transformed by a Fourier lens, which can be thought of as another ciphering process. Finally, the code was ciphered in a hologram. Of course, you can’t crack the code by simply looking at any of the images above.

Retrieving the Holographic Data

Next, we move on to the data retrieval step. To retrieve the data, as was described in the previous blog post, we can use the same COMSOL Multiphysics feature to turn the functionalities on and off. We do this by adding the Wave Equation, Electric 2 node with a user-defined refractive index, which specifies the modulated index.

The Settings window for the modulated refractive index.

The modulated refractive index. The modulation amplitude corresponds to the position where the electric field intensity exceeds the threshold.

By turning the object beam off and keeping the reference beam on, as well as having the modulated index, we get the result of the retrieval simulation.

The electric field amplitude (top) and intensity (bottom) for the page data retrieval.

The electric field amplitude at the bottom edge during page data retrieval (cross section).

Now, we want to image this retrieved data onto the CCD surface by using the other Fourier lens. To do so, we will Fourier transform the retrieved electric field amplitude again and take the square of this amount. The following figure is the final result. The CCD detects the 1 positions in the original code, 01001101. We finally see the code again!

The retrieved data on the CCD surface. The dashed line represents the position of 1 in the original code.

Concluding Remarks on Holographic Page Data Storage

We have implemented a holographic page data storage model using the wave optics capabilities of COMSOL Multiphysics. Though the rigorous Maxwell solver persuades us to pay more attention to some specific restrictions, we were able to catch a glimpse of the holography created by the design calculation we performed prior to the simulation. We also went over some helpful and convenient uses of COMSOL Multiphysics as a scientific calculator. As we learned, the COMSOL software can perform all of these tasks in one environment, with sequential finite element computations and other scientific calculations performed simultaneously.

Further Reading

• Check out the COMSOL Blog:
  • Take a look at the first part of this series for an introduction to holographic data storage
  • Read about the built-in integration operators for the Fourier transformation
• Learn more about Fourier optics with the book Introduction to Fourier Optics by J.W. Goodman

The Rise of Holography in Consumer Electronics

About a decade ago, a surprising number of researchers and engineers in the U.S., Japan, and other countries worked to discover the next generation of optical storage devices to succeed the Blu-ray drive. Holography was strongly believed to be the only solution. Researchers expected that consumer demand for digital data storage would increase infinitely and in turn developed various types of holograms for a quick time-to-market. Although holographic storage was not very commercially competitive against solid-state memory, it is still a technology that any optoelectronic engineer should understand fully.

Over the last few years, as computational hardware has improved, simulation software has flourished. Software simulations let the engineer address device sensitivity, determine how much can be overwritten in one fraction of volume, and reduce the signal-to-noise ratio. Traditionally, simulation in this area has been performed by the so-called beam propagation method (BPM). The advantage of this method is that it can handle problems that involve interference, diffraction, and scattering in a domain that is 1000 times that of the wavelength. Also, the computational cost is cheap. However, the disadvantage is that it cannot correctly compute lights with a large focusing angle.

COMSOL Multiphysics has two different approaches for solving Maxwell’s equations for such holographic storage problems. One approach, the full-wave approach, can model interference and scattering, but only for modeling domain sizes that are comparable to the wavelength. The other approach, called the beam envelope method, can compute interference for a large scale, but cannot compute arbitrary scattering. In this blog series, we will look at using the full-wave approach to simulate a small-volume hologram to study how the hologram deciphers the code by the reference wave — one of the most exciting factors of holography.

Simulating Holographic Data Storage in COMSOL Multiphysics

As mentioned in a previous blog post, in general holography, the object beam is a beam scattered from an arbitrary object. In holographic data storage, the object beam is a single beam carrying one-bit data or a beam passing through a spatial light modulator (SLM) carrying multibit data. The former system is called bit-by-bit holographic data storage, while the latter is referred to as holographic page data storage.

In these processes, the object beam transmits through the aperture and comes across the reference beam to generate a complex interference fringe pattern in a holographic material. The interference fringe is the cipher that carries your information. This process is called recording. The light sources for the object and reference beams need to be coherent to each other and the coherence length needs to be appropriately long. To satisfy these conditions, the light source for holography is typically chosen from solid-state lasers such as a YAG laser; gas lasers such as a He-Ne laser; and nonmodulated semiconductor lasers, such as GaN and GaAs laser diodes with direct current operation.

To have a mutual coherence, the light source is originally a single laser that is split into two beams by a beam splitter. When the optical path difference between the two beams is controlled to be much less than the coherence length of the laser beam, the two beams generate an interference pattern, which is a standing wave of the laser beam at the intersectional volume in a holographic material.

Typical commercial holographic materials are made of certain photopolymers. This nonmoving stationary intensity modulation of the electric field initiates polymerization, which slightly changes the local refractive index from the original raw index. The refractive index change is , which is typically less than 1%. The value is nonlinearly proportional to the electric field intensity.

After the refractive index modulation has been set in, only the reference beam is shone on the holographic material. Then, the reference beam is scattered by the interference fringe and the scattered beam creates the objective beam as if the objective beam is present. This process is called retrieval. The retrieved object beam is detected by any single-pixel photodetector, such as GaP, Si, InGaAs, or Ge photodiodes for the bit-by-bit data storage, as well as by CMOS or CCD image sensors.

A typical optical layout of page-type holographic data storage (the character code of my name is encoded in binary data in the SLM in this figure).

How to Design a Single-Bit Hologram Simulation

Now, let’s simulate a bit-by-bit holographic data storage example. There is a single open aperture for the object beam instead of an SLM, so the object beam carries one-bit data, which can mean “1 or 0″ or “exist or not exist”. Our computational domain is a square and the layout of the beams is such that the object beam enters from the top side, while the reference beam comes from the left. Note that this 90-degree configuration is a simplified example to demonstrate the simulation setup and is not a very realistic scenario.

A schematic of bit-by-bit holographic data storage. The objective is to compute the electromagnetic fields within a small region of the holographic material.

Let’s go through each of the steps of the simulation process for holographic data storage, including preparation, recording, retrieval, and an overview of the appropriate settings in COMSOL Multiphysics.

Our first task is to appropriately set up the model of the laser beam. This process looks very simple, but it requires knowledge of electromagnetics and computer simulation beyond just the usage of COMSOL Multiphysics. The following points must be considered when setting up a model of a laser beam.

Beam Collimation

First of all, we want to have straight beams that uniformly propagate through the material and a wide spatial overlap between the two beams. To achieve this, the beam width has to be chosen carefully. The lower bound on the beam width is controlled by the uncertainty principle. If we try to specify a beam width that is very narrow compared to the wavelength, this means that we are trying to specify the position within a very small region. When the position is well specified, the light’s momentum becomes more uncertain, which equivalently leads to more spreading out of the beam and the beam diverges.

How much the light diverges for a given beam size is quantitatively well described by the paraxial Gaussian beam theory, which defines the beam divergence via the spread angle . This spread angle is related to the paraxial Gaussian beam waist radius as , where is the wavelength. It is obvious from this formula that the light diverges if we make the beam waist radius small compared to the wavelength. In the figure shown below and to the left, we can see a case where the waist radius equals the wavelength. You can see that the small beam waist leads to a quickly diverging beam.

If you instead specify a waist radius ten times the wavelength, then the divergence angle is , which is approximately 32 mrad. This angle is good enough for our purposes. A slightly diverging but almost collimated Gaussian beam is depicted in the figure below on the right. Super Gaussian or Lorentzian beam shapes can also be used to describe such a collimated beam.

A beam with a narrow waist (left) diverges, while a beam with a wide waist (right) diverges negligibly. The electric field magnitude is plotted, along with arrows showing the Poynting vector.

Domain Size

Our modeling domain must be large enough to capture all of the relevant phenomena that we want to capture, but not too large. This can be visualized from the image above of the two crossed beams. The modeling domain need only be large enough to enclose the region where the beams are intersecting. It doesn’t need to be too large, since we aren’t interested in the fields far away from the beam, which we know will be small. The domain also doesn’t need to be too small because we would lose information.

Boundary Conditions

The boundaries of our modeling domain must achieve two purposes. First, we must launch the incoming beams, and second, the beam must be able to propagate freely out of the modeling domain. Within COMSOL Multiphysics, both of these conditions can be realized with the Second-Order Scattering boundary condition, which mimics an open boundary and also allows an incoming field representing a source from outside of the modeling domain to be specified.

It is also important that the scattering boundary conditions are placed far enough away from the beam centerline, such that the beams are only normally incident upon the boundaries. The beam should not have any significant component in parallel incidence upon the boundary, since this will lead to spurious reflections, as described in our earlier blog post on boundary conditions for electromagnetic wave problems.

We can use the information about the beam waist to choose a domain size that is sufficiently wider than the beam, such that the electric field intensity falls off by six orders of magnitude at the boundary, as shown in the figure below.

If the domain width relative to the beam width is sufficiently large, there will be no spurious reflections (left). If the scattering boundary conditions are placed too close (right) to the beam centerline, there are observable spurious reflections.

Meshing Requirements

This problem solves for beams propagating in different directions and computes scattering and interference patterns in a material with a known refractive index. Since we know the wavelength and the refractive index, we can use this information to choose the element size. The element size must be small enough to resolve the variations in the propagating electromagnetic waves. We know from the Nyquist criterion that we need at least two sample points per wavelength, but this would give us very low accuracy. A good rule of thumb is to start with an element size of , or eight elements per wavelength in a material with peak refractive index .

Of course, you will always want to perform a mesh refinement study. For this type of problem, an element size of will typically be sufficient. Also be aware that the smaller you make the elements (the higher the accuracy), the more time and computational resources your model will take. For a detailed discussion about how to predict the size of the model, please see our blog post on the memory needed to solve a model.

Considering all of these factors, we will simulate a laser beam with a vacuum wavelength of 1 um and a beam waist profile of , a sixth-order super Gaussian beam. We will solve for the out-of-plane electric field, which means that we solve a scalar Helmholtz equation.

Simulating the Recording Step

Now that we have appropriate settings for the beam and the domain, we are ready for the recording simulation. The figure below shows the results of the recording process. The object beam and reference beam make an interference fringe pattern at a slant angle of 45 degrees and with a periodicity of 0.524 um. This 45-degree fringe is the cipher for a single of 1 recorded in the holographic material.

The computed electric field and intensity for the one-bit data recording.

Next, the holographic material modulates its refractive index in the portion where the electric field intensity is above a certain threshold value. In the case of photopolymers, polymerization starts in this high-intensity region. Now, let the distribution of this high-intensity portion be denoted , as it adds up modulation on the raw index . This means that the global refractive index can be written as . is the modulation depth, which is dependent on the material’s photochemical properties.

The function shape of the modulation also depends on the material and process. The new index takes the shape of a biased and periodic rectangular function swinging around the raw index. The next figure plots the new refractive index and its cross section after recording. In this simulation example, we have used and . The modulation function can be expressed by a logical expression, ( (ewfd.normE)/maxop1(ewfd.normE) )>threshold, where the maxop operator calculates the maximum value inside the domain, normalizing the electric field norm. threshold is a given threshold value for polymerization.

A contour map of the electric field intensity for the binary recording that is cut off at a threshold and binarized.

A cross-sectional plot of the modulated index.

Simulating the Retrieval Step

Next, we simulate the retrieval process, which includes:

• Turning off the object beam
• Shining the reference beam only

After these settings change, we get the final results, as shown in the next two plots. The reference beam is diffracted/scattered by the interference fringes and creates a new beam, which restores the amplitude and the phase information overlooking a multiplicative constant. Note that the retrieved object beam is not symmetric because the reference beam slightly diverges.

The computed electric field and intensity for the retrieval of the object beam carrying one-bit data.

Automated COMSOL Multiphysics Settings

So far, we have gone through the simulation procedure in a step-by-step manner, but it is possible to perform this sequential simulation all at once. In COMSOL Multiphysics, there is a helpful feature in the Solver settings that we can use to perform this two-step sequence, the recording and retrieval processes, in one click of the Compute button. To do this, we select the Modify physics tree and variables for study step check box in each study step.

For recording, we apply the scattering boundary condition with the incident field of the super Gaussian beam (Reference SBC) on the left edge, the scattering boundary condition with the incident field of the super Gaussian beam (Object SBC) on the top edge, and the scattering boundary condition with no incidence for the rest of the boundaries (Open SBC).

Settings for Study 1 and Step 1 of the recording process.

Adding the Wave Equation, Electric 2 node for index modulation.

To set up a modulated refractive index, we add one more Wave Equation, Electric node, in which the previous result specifies a new user-defined refractive index. Here, we have used the withsol() operator, which lets users apply the previous solution to evaluate an arbitrary expression. In this example, the new refractive index is given by n1+dn*withsol('sol2',((ewfd.normE/maxop1(ewfd.normE))^2>threshold)-0.5), where 'sol2' is the solution for Step 1 (the recording process) and the threshold is 0.4.

Settings for Study 1 and Step 2 for the retrieval process.

In the retrieval process, we turn off the object beam by disabling the Object SBC. To switch to the modulated refractive index, the original Wave Equation, Electric 1 node is disabled and the Wave Equation, Electric 2 node is turned on. Finally, Open SBC is replaced by a new scattering boundary condition with no incidence for the top, bottom, and right boundaries (Open SBC 2).

Concluding Remarks on Simulating Holographic Data Storage

Today, we discussed how to determine electromagnetic beam settings, which can be a very complex problem. Then, we demonstrated a simple holographic data storage simulation, called a bit-by-bit hologram. We also learned how to implement several steps in COMSOL Multiphysics to run a series of simulation steps at one time. Stay tuned for the next part of this holography series, in which we will simulate a more interesting, complicated, and realistic system of multibit holograms called holographic page data storage.

Further Reading

• Read the blog post Shaping Future Holography for the history, principles, applications, and implications of holograms
• Watch this archived webinar for a full demonstration on how to simulate wave optics problems in COMSOL Multiphysics
• Have any questions? Contact us for support and guidance on modeling your own holography problems in COMSOL Multiphysics

The Mathematical Conventions of Piezoelectric Theory

Before we start discussing piezoelectric physics, let’s first review a couple of mathematical conventions. Piezoelectric materials have properties that allow strain to produce electric polarization and vice versa. The former is known as the direct piezoelectric effect, while the latter is referred to as the inverse piezoelectric effect. In mathematics, two forms of a set of matrix equations are conventionally used to describe these effects: the strain-charge form (also known as the d-form) and the stress-charge form (or the e-form). The two forms can be derived from each other by a transformation.

Note: The two forms are convertible, thus you can choose whichever option is preferred. In COMSOL Multiphysics, the strain-charge form is converted to the stress-charge form internally.

Here, we’ll choose the d-form, in which the relations are written as

where the field quantities , , , and are the strain, electric displacement, stress, and electric field, respectively.

The material parameters , , and are the compliance, coupling matrix, and relative permittivity, respectively. The superscript stands for transpose. The first equation expresses the relation that the stress field and electric field produce strain. The second equation, meanwhile, shows the electric polarization generated by the stress field and electric field. The second term of the first equation accounts for the electric contribution to the strain.

In solid mechanics, we typically consider an imaginary, small differential volume to understand the force that is exerted on a volume. Let’s consider a volume where the surfaces are aligned to each axis of the global Cartesian coordinate system (xyz-axes).

There are six distinct quantities that describe the forces per unit area on the surface of the imaginary volume. Three of them are normal to each surface and are usually denoted by , , and . The other three are parallel to each surface and are denoted by , , and . (, , and are omitted because of symmetry.)

The first index represents the normal vector of the surface under consideration. The second index indicates the direction of the force. For example, is a force along the x-axis that is exerted on a yz-plane (a plane perpendicular to the x-axis); is a force along the z-axis on an xz-plane; and so on.

The force components in a small differential cubic volume.

These six quantities are often expressed as a column vector. In COMSOL Multiphysics, the order of the quantities has two conventions: one is the standard notation and the other is the Voigt notation. In COMSOL Multiphysics, it is important to make sure that the Voigt notation is used in the Piezoelectric Devices interface. (The standard notation is used by default in the Solid Mechanics interface.)

In the Voigt notation, the stress is written as a 6 x 1 column vector (special attention should be paid to the ordering of yz-, xz-, and xy- components):

Similarly, the strain can be written as

In COMSOL Multiphysics, the coupling matrix is defined by

Here, the notation reads as the coupling coefficient for the strain component caused by the electric field component . Now, in the Voigt notation, all of the subscripts are numeric with the rule:

As such, the coupling coefficient matrix is

Thus, the electric contribution can be rewritten as

In the view of the material property settings in COMSOL Multiphysics, the matrix is flattened into a 1 x 18 row vector to be concise. It then looks like the following:

We always have the option to view the expression in the matrix form. We can do so by clicking on the Edit button in the “Output properties” section, as shown below:

A screenshot of the COMSOL Desktop shows the d coefficients in a matrix form.

Basic Piezoelectric Simulations

Now that we’ve reviewed some of the basics of piezoelectric theory, let’s turn our attention to performing simulations. Here, we’ll enter a nonzero number only in . This means that the piezoelectric material makes a normal force along the z-axis by the z-component of the electric field . We also assume that the bottom surface is mechanically fixed.

The results shown below are obtained, with the surface color representing the total displacement and the arrows indicating the electric field. The contribution of the electric field to the strain is given by . The black outline indicates the initially undeformed shape. As indicated by the plot, the volume is stretched in the z-axis direction.

An example of the d coefficients with the only nonzero in . Under an electric field along the z-axis, the volume stretches along the z-axis.

Keeping the same electric field, we’ll now enter a negative coefficient only in . With the yz-plane mechanically fixed, the volume shrinks in the x-axis direction.

An example of the d coefficients with the only nonzero negative in . Under an electric field along the z-axis, the volume shrinks in the x-axis.

The last preliminary example shows a shear force, with a nonzero number entered only in . The xz-plane is mechanically fixed and an electric field is applied along the y-axis.

An example of the d coefficients with the only nonzero in . Under an electric field along the y-axis, the volume experiences a shear along the yz-plane.

Building on these basic elements, we’ll now introduce you to a helpful trick for designing piezoelectric actuators — a piezoelec”trick”, if you will.

Improving the Range of Motion in Piezoelectric Actuators: A Novel Design

Simulating a Piezoelectric Cantilever, a Simple MEMS Device

On its own, a piece of piezoelectric material is not a MEMS device. To become such a device, it must be attached to other elastic materials. Cantilevers are one of the simplest MEMS examples, and there are two different piezoelectric types. The first is the unimorph type, which is typically a sheet made by attaching a single layer of piezoelectric material to an elastic material. The other is the bimorph type, which is composed of two piezoelectric layers and other elastic materials. The materials used for attachment can be almost anything, as long as it is possible to put an electrode layer on their surface. Here, for simplicity, we will consider a unimorph type of MEMS device.

It is worth mentioning that all of the piezoelectric material properties provided by COMSOL Multiphysics are assumed to be poled in the z-axis of the local coordinate system. If the material is poled along another direction, you need to define a coordinate system so that its third direction is aligned with the poling direction.

COMSOL Multiphysics provides convenient functionalities to set up local coordinate systems. To learn more, you can refer to page 106 in the Structural Mechanics Module Users Guide. You can also consult the Piezoelectric Shear-Actuated Beam and Thickness Shear Mode Quartz Oscillator models, found in the Application Libraries under the Piezoelectric Devices interface in the MEMS Module and also on our website.

Under the assumption referenced above, we model a unimorph piezoelectric cantilever that is fixed at one end. From the Material Libraries, we select barium titanate (BaTiO3) as a piezoelectric material and silica (SiO2) as the substrate.

The principle of the MEMS device is simple. An electric field is applied in the z-direction, causing shrinkage in the x-direction due to (=-7.8e-11 [C/N]). Note that the elastic properties of the electrode are not modeled here. To achieve the z-directional electric fields, an electric potential is applied on one side of the barium titanate, while the other side is grounded. With the left end mechanically fixed, the MEMS cantilever moves about 9 um at the free end. As the top material (BaTiO3) shrinks, it pulls the bottom material (SiO2), causing the entire device to bend in the upward direction.

A unimorph cantilever with a fixed end shows the total displacement.

But what if we have to fix both ends of the device for some reason — perhaps mechanical ruggedness? The result is obvious: There is almost no displacement, as the top material cannot shrink along the x-axis at all. More accurately, the only displacement is due to the effect, which is not intended.

A unimorph cantilever with both ends fixed shows almost no displacement.

It is, however, possible to overturn such a situation. We can do so by modifying the piezoelectric device with a little “trick”. As the resulting plot below shows, the displacement is now back to the same order of magnitude. If you look at the electric field directions, what we have done to obtain this result is apparent. The electric field is reversed in the center part of the beam. You can perform the trick by patterning the electrode into three parts: a center and two sides, grounding them alternately and applying a voltage alternately in the opposite order as in the first of the next two figures.

Material and voltage configurations in a unimorph cantilever with both ends fixed under alternate electric fields.

A unimorph cantilever with both ends fixed. Applying alternate electric fields improves the total displacement.

Let’s look at another example. It may seem a bit mysterious at first sight, as the entire electric field is aligned in the same direction. Still, we have the same performance as the previous result without alternating the electric field. We instead alternate the piezoelectric material, splitting it into three parts (again, a center and two sides) and flipping the center part.

With regards to simulation, this can be done either by using different local coordinate systems or, in a simple case like this, by changing the sign of the coupling coefficient, . In practice, we can really split the piezoelectric material up and put the flipped part in the center. An easier and more practical approach is to pattern the electrode in the same manner as the previous example, but pole the center part in the opposite direction. There is no need to split the material in such a situation.

For the unimorph cantilever with both ends fixed, alternate material and voltage configurations.

For the unimorph cantilever with both ends fixed, alternating the poling direction also improves the total displacement.

Simulating a Piezoelectric Membrane MEMS Device

Our last example is particularly interesting. Without the use of a trick, we would really have no displacement, since the entire perimeter is fixed and, as such, there is no room for any component to play a role. Applying the same trick from the previous examples, the entire-perimeter-fixed membrane can have a tremendous displacement. The simulation shown here can be used as a MEMS device that compensates for, to name an example, optical aberration (particularly spherical and astigmatism).

Alternate material and voltage configurations in a membrane MEMS device with the entire perimeter fixed.

A membrane MEMS device with the entire perimeter fixed. Applying an alternate electric field or poling direction improves the total displacement.

From the Basics to a Helpful Design Trick

We have reviewed some of the fundamental mathematics and conventions that are necessary when simulating piezoelectric materials. Further, we have walked you through the steps of performing basic simulations, then diving into more interesting applications using our piezoelec”trick”.

When it comes to the trick, the secret to significantly improving the displacement is the “inflection zone”, which is created by alternating the electric field or the material property. (Refer to US Patent 7,369,482 for more details and other examples.) The local coordinate system functionalities of COMSOL Multiphysics make it very easy to set up a simulation for such systems with alternating material orientations.

If you enjoyed today’s blog post, be sure to check out some other related blog posts that might spark your interest:

• Piezoelectric Materials: Crystal Orientation and Poling Direction
• Piezoelectric Materials: Understanding the Standards
• Piezoelectric Materials: Applying the Standards