Modulation Transfer Function - mtf modulation transfer function

 2024-08-29 04:33:40

Thanks Yosuke for such an interesting and clear post. The model I am currently working on includes a Gaussian beam focused by a high NA objective lens. Clearly, this is too tightly focused for the paraxial approximation to hold, and I encountered the problems you have described above. However, searching around the web I wasn’t able to find out so far anyone coming up with a workaround to these limitations. More in general, is there a way to simulate in COMSOL the point spread function of a high NA lens? You can imagine I am now really looking forward to the follow-up post you promised describing the solutions! I am wondering, are you planning to publish this any soon? Would you be able meanwhile to point to me some useful information on this matter? With kind regards, Attilio

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高斯光束

If spherical and/or coma is present in a system suffering from astigmatism, then shifting the stop can help minimize the blur from astigmatism. Lens bending can also help minimize astigmatism. Since astigmatism is proportional to the square of the field angle and increases linearly with aperture, choosing a smaller field angle or pupil size can be an option if astigmatism is too great. In the case that stop shift, bending, field angle and pupil size are insufficient, adding additional lenses to the system can contribute the opposite sign astigmatism to cancel it out.

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.

We can think of distortion as varying transverse magnification with field as seen in figure 1.18. Magnification is equal to the ratio of the image height to the object height. In a distortion-less system, this ratio would be the same across the field, but when distortion is present this ratio is variable. In figure 1.18, the transverse magnification is greater for the larger field position (blue line) and is an example of pincushion distortion. Barrel distortion would result in the larger field position having a smaller magnification.

Using an achromatic doublet, as in figure 1.3, significantly improves image quality by reducing the blur. Bringing all three wavelengths to a common focus can further reduce this blur. This type of lens, called an apochromat, is generally made of 3 elements.

One of the COMSOL modes named “Nanorods” with application library path: Wave_Optics_Module/Optical_Scattering/ nanorods. In the “Model Definition” section at page 1 of this model, the author determined that the rods have dimensions less than wavelength, as my case, and as I understand he overcame the problem of Gaussian beam is an approximation solution by the following sentence and I will write it as it was reminded ” For tightly focused beams you also need to include an electric field component in the propagation direction”.

Spherical aberration (SA) is the only monochromatic aberration that is present on the optical axis. It is similar to axial chromatic in this regard as well as the fact that it is the same everywhere in the field. Spherical aberration occurs when light rays at or near the edge (or margin) of the lens focus at a different location than those that enter the lens at or near the center as seen in figure 1.7. Like measuring ACA, SA is sometimes measured as a longitudinal or transverse aberration. We refer to longitudinal spherical aberration as LSA and transverse spherical as TSA, as shown below.

Dear Yosuke Mizuyama： Thanks Yosuke for such an interesting and clear post.My current work is a single crystal fiber laser, and I encountered the problem you described above while simulating the propagation of light in the pump!I am here to ask you what method can I use to simulate the propagation of a Gaussian beam (W0 =0.147mm) in a rod with a diameter of 1mm and display the light intensity distribution!I used the ray tracing module, but the results are too poor. Can you tell me how to implement my simulation?Thank you very much!

You can imagine I am now really looking forward to the follow-up post you promised describing the solutions! I am wondering, are you planning to publish this any soon? Would you be able meanwhile to point to me some useful information on this matter?

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.

As seen in figure 1.13, there are two focal positions. At one focal position the image of an off-axis point appears as a line segment oriented sagittally (on a line passing through the optical axis), while a small distance away it appears as a line tangent to a circle centered on the optical axis. The two surfaces where the sagittal and tangential line segments appear are the sagittal and tangential foci. There is a position between the sagittal and tangential focal planes, called the medial focus, where the image is a more circular blur that is smaller in diameter than either of the line segments.

Thanks Yosuke, Could you please guide me how I can write an expression for a gaussian beam (in 2D) propagating in x-direction while the polarization in y-direction? And, also how I can define a coordinate transfer in expression for an incident angle of the beam?

We can also use symmetry to correct for LCA. Designing a lens system so that it is close to symmetric about the stop will decrease LCA.

Dear Simon, Thank you for your interest in my blog. The above formula is written for beams in vacua or air for simplicity. But the formula still holds if you read k as the wave number in a material, that is, if you use n*k instead of k, where n is the refractive index of the material. In COMSOL, the Gaussian beam settings in the background field feature in the Wave Optics module are set for the vacuum by default, i.e., the wave number is set to be “ewfd.k0”. But you can change it to “ewfd.k” for more general cases. COMSOL will automatically take care of the local “k” depending on where you have different materials in your domain. There is a tricky thing you have to keep in mind in this situation: You have to know the waist position wherever it is positioned. If a Gaussian beam is incident from air to glass and makes a focus in the glass, the waist position will be different from the case where the material doesn’t exist (See Applied Optics, Vol. 27, No. 9, p.1834-1839 (1988) ). You have to calculate the focus position first, and then enter the focus position in COMSOL. Best regards, Yosuke

{ Error in user-defined function. – Function: dE_dE__z__internalArgument Failed to evaluate variable. – Variable: comp1.emw.Ebx – Defined as: exp(i*phase)*(!(comp1.isScalingSystemDomain)*(comp1.es.Ex+((j*d((unit_V_cf*E(x/unit_m_cf,y/unit_m_cf,z/unit_m_cf))/unit_m_cf,z))/comp1.emw.k0))) Failed to evaluate expression. – Expression: comp1.emw.Ebx Failed to evaluate operator. – Operator: mean – Geometry: geom1 }

Dear Yasmien, Can you please go through our technical support, support@comsol.com? We need to see your model to solve your problem. Thank you. Yosuke

Hi, I am rather new to Comsol. Thanks for this good explanation of Gaussian beam. I have a question: What should I change/add to incident a Gaussian beam at interface with some degree of angle if the scattered field formulation is chosen (as you have shown in window above)? Regards, Jana

Using a combination of several thin lenses to adjust the Petzval curvature can also be an option. Generally, there are no limits on reducing the size of the curvature if there are equal amounts of negative and positive powered lenses in the system.

If the lens needs additional correction, making the lens aspherical can further reduce SA. Lens designers can use combinations of these techniques and more to achieve the level of correction needed.

Dear Jana, Thank you for reading this blog. There are some limitations for the built-in Gaussian beam feature. 1. You can only propagate it along the x or y or z axis. Due to this limitation, you will have to rotate your material in order to simulate a beam at an angle. 2. The focus position needs to be known a priori. This is a little bit tricky to explain but you need to know the focus position inside your material and enter the position in “Focal plane along the axis” section because COMSOL won’t automatically calculate the focus position shift if you only know the field outside your material. Because of the convergence of a Gaussian beam, there will be a refraction at a material interface, which causes the focus shift. For more details about the Gaussian beam focus shift at interfaces, please refer to this paper: Shojiro Nemoto, Applied Optics, Vol. 27, No. 9 (1988). If you would like a more flexible way, you can define a paraxial Gaussian beam in Definition and also define a coordinate transfer. Please send support@comsol.com a question on this method since it’s a little bit difficult to explain here. Best regards, Yosuke

Dear Attilio, Thank you for reading my blog post and for your comment. We will publish a follow-up blog post with rigorous solutions in a few months. In the meantime, you may want to check out this reference: P. Varga et al., “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation”, Optics Communications, 152 (1998) 108-118. In this paper, the authors give an exact formula for a nonparaxial Gaussian wave. Best regards, Yosuke

The result of LCA in telescopes is that off-axis images of stars appear to be little line segments that are blue at one end and red at the other as seen in figure 1.6. The line segments are in a sagittal orientation, which means that they lie on lines that pass through the optical axis.

Dear Daniel, Thank you for reading this blog. When we assumed time-harmonic waves to derive the Helmholtz equation from the time-dependent wave equation, we factored out exp(i*omega*t). Remembering this process, we get a time-dependent wave by putting the factor back, i.e., by replacing exp(-ik*x) with exp(i*(omega*t -k*x)) in the formula in this blog. This is implemented in second_harmonic_generation.mph in our Application Libraries under Wave Optics Module > Nonlinear Optics. I hope this helps! Best regards, Yosuke

Dear Yosuke Mizuyama Very interested topic. I have one question, please. Why lambda is equal to 500nm and used in COMSOL as the default value for the calculation of frequency (f=c_const/500[nm])?

“Stopping down” the lens is another way to reduce ACA. This just means allowing a smaller diameter of light through the systems. The term “stopping down” comes from the size of the stop in the lens, which is just the limiting aperture in the optical system. For the achromat in figure 1.3, the stop size is just the lens diameter and is also equal to the entrance pupil diameter used in calculating the f/# (as described in the Basic Optics Terms section.)

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.

If ACA is present in a system, LCA is linear with respect to stop position. Thus it’s helpful to choose a stop position where the LCA is zero. Once LCA is corrected, ACA can be corrected by achromatizing (changing elements to achromats) elements that are not close to the stop. Once ACA is corrected, the stop can be moved to its original position since LCA depends on the presence of ACA in a lens.

laguerre-gaussianbeam

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., E_z(x,y) = A(x,y)e^{-ikx}, where the propagation axis is in x and A(x,y) is the slowly varying function. This will yield an identity

An image suffering from purely coma will have good image quality in the center, but as you move farther off-axis the image will degrade linearly with field position. Looking at figure 1.11 we can see the difference between a un-aberrated image (a) and an image from a lens suffering from coma (b). If the object was something other than a random point field such as stars, like an image of a sample in a microscope, we would see a sharp focus in the center and a linearly increasing blur as we get closer to the edge of the image.

We can see examples of both pin-cushion and barrel distortion in figure 1.19, using a grid of lines as the object. Pin-cushion distortion is characterized by increased magnification with increased field (distance from the optical axis). This is shown by the magnification of the red arrow being approximately equal to the un-aberrated image magnification and the blue arrow having a larger magnification than the un-aberrated image. Barrel distortion is just the opposite, characterized by decreased magnification with increased field, as shown.

Dear Yosuke, Thanks for your clarification and I got the idea in using mesh. The explanation of the reason of existence an electric field component in the propagation direction is still unclear to me, I am sorry I did not understand it well. Also, why do we represent this component by differentiating the gaussian beam field according to the polarization direction? Could you recommend a source for reading?

If we image stars through a lens with astigmatism, as shown in figure 1.14, the image shape will depend on which plane of focus we selected as the image plane. If we position the image plane on the tangential focus, the off-axis image of the stars would resemble lines tangent to an imaginary circle centered on the optical axis. When we position the image plane at the medial focus, the off-axis image will simply look out of focus when compared to the un-aberrated image. When positioning the image plane at the sagittal focus, the image of the stars will resemble short line segments that rest on lines that go through the optical axis.

Thanks for your kind reply, it is very helpful, and yes I want to focus the beam to the size of the nano-particle with 6 nm radius, but I have 2 questions if you kindly allow:

Since SA varies with the cube of the entrance pupil diameter, stopping down the lens will greatly reduce SA. Changing the glass type to one with a higher index can help reduce the curvature needed to bend the light and thereby reduce SA.

Dear Yasmien, Thank you very much for reading my blog and for your interest. Do you have to focus your beam to the size of the nano-particle? The beam waist size is determined depending on how much you have to focus. For that wavelength range, the least possible waist radii are as large as 127 nm to 159 nm, though. And for these numbers, the paraxial formula will not give you an accurate result. If a slow (gently focusing) beam works for your characterization, the waist radius of 4 um or larger would work and our Gaussian beam background feature gives you a correct result.

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., |\partial^2 A/ \partial x^2| \ll |2k \partial A/\partial x|. 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, {\rm abs} \left ( {\rm real} \left ( (\partial^2 A/ \partial x^2) / (2ik \partial A/\partial x) \right ) \right ).

The blur due to TSA varies as the cube of the f/#. The f/#, as stated on our Basic Optics Terms page, is the focal length of the lens divided by the entrance pupil diameter, or in the case of a single lens, the diameter of the lens (f/# = f/d). So if the focal length is held constant and the entrance pupil diameter is increased the blur from spherical aberration will also increase (as the f/# decreases). Considering this, the cube of the f/# is therefore inversely proportional to TSA. For example, this means that the blur is 8X as large for a lens at f/2 as for the same lens stopped down to f/4. The characteristic shape of spherical aberration is a circular blur and it creates a haziness across the entire image as seen in figure 1.8.

where w(x), R(x), and \eta(x) are the beam radius as a function of x, the radius of curvature of the wavefront, and the Gouy phase, respectively. The following definitions apply: w(x) = w_0\sqrt{1+\left ( \frac{x}{x_R} \right )^2 }, R(x) = x +\frac{x_R^2}{x}, \eta(x) = \frac 12 {\rm atan} \left ( \frac{x}{x_R} \right ), and x_R = \frac{\pi w_0^2}{\lambda}.

Dear Yasmien, That means there is no purely linearly polarized beam for non-paraxial Gaussian beams. There is a reference in the pdf document for the nanorods model, M. Lax, W.H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics”, Physical Review A, vol. 11, no. 4, pp. 1365-1370 (1975). You don’t want to simulate what really doesn’t exist, do you? If your beam is really a tightly focused beam, it has a propagation component inevitably. So you have to add it no matter how it’s a different component than your preferred plane to which you want to believe it’s polarized.

It wasn’t until 1757 that John Dolland discovered combining two different types of glass could reduce this problem. This technique, called an achromat (from the Greek ‘a‘ (without) and ‘chromos‘ color), brings the blue and red foci to the same location and leaves the green light at a different focal length. See figure 1.3 for an example.

ACA, along with other aberrations, can be measured two ways. The first as a distance along the optical axis (longitudinal) and the second as a distance perpendicular from the optical axis (transverse). These are shown in figure 1.1b. It’s helpful, when measuring the transverse and longitudinal ray aberrations, to designate abbreviations for each aberration. For transverse axial chromatic aberrations we will use the abbreviation “T-ACA” and “L-ACA” for the longitudinal axial chromatic aberration. If you would like to learn more about how to measure aberrations, follow this link.

The Gaussian beam is recognized as one of the most useful light sources. To describe the Gaussian beam, there is a mathematical formula called the paraxial Gaussian beam formula. Today, we’ll learn about this formula, including its limitations, by using the Electromagnetic Waves, Frequency Domain interface in the COMSOL Multiphysics® software. We’ll also provide further detail into a potential cause of error when utilizing this formula. In a later blog post, we’ll provide solutions to the limitations discussed here.

Dear Yosuke, Thank you so much for this reliable blog. I have a question about one of limitations of paraxial gaussian beam. I am trying to study the optical characteristics of gold nano-particle (radius = 6nm) on a wavelength spectrum extended from 400 nm to 500 nm, and I do not know how I can determine the beam radius waist (w0) value. I think it will be less than wavelength and this will not match with the paraxial approximation for Maxwell equation that used in the suggested gaussian beam in your blog. Could you tell me the proper choice for the value of w0 and how can I use the gaussian beam formula as a background source in my case. Thanks in advance

And I wrote the component of electric field in propagation direction as following: (j*d(E(x,y,z),z)/emw.k0) where (z) is the polarization direction and I used it to overcome the paraxial approximation problem if you rememer, but I get this error:

COMSOL Gaussianbeam

Note that figure 1.1 functions as an aid in the understanding of aberrations and is not to scale. The actual distances between foci (focal lengths) for different wavelengths would be much smaller.

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.

Petzval curvature or field curvature differs from the previous aberrations; it does not blur the image at all. Rather, it causes the image to lie on a surface that is not a plane. As a first approximation, the surface is spherical. Lenses are typically used to inspect things that lie on a plane, and most detectors, whether CCD, CMOS or film, are also planar. This means that even though Petzval curvature doesn’t blur the image on the Petzval image surface, it does result in a blur on a plane image surface (as shown in figure 1.15), and the blur increases with the square of the image height. The radius of the Petzval surface is completely insensitive to f/#, but the blur on a flat image plane caused by the Petzval curvature increases linearly with entrance pupil diameter.

Dear Jana, That was a typo. Correct: y2 = x*sin(theta)+y*cos(theta) Other than that, if you have more questions on this particular one, please send your question to support@comsol.com with your model. That’d be more effective. Best regards, Yosuke

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!

Dear Yasmien, The solution is one of the valid methods for both 3D and 2D. Mesh refinement works for increasing the accuracy of finite element solutions. If you use a loosely focused Gaussian beam, yes, your paraxial Gaussian beam in your finite element model will become closer to the closed-form paraxial Gaussian beam. But if you use a very small waist size in the paraxial Gaussian beam formula, mesh refinement will not work to improve the error coming from the paraxial approximation. It doesn’t change the scalar paraxial approximation nature. The technique used in the model you referred to is actually a remedy to the fact that the Gaussian beam starts to show its vectorial nature when it’s tightly focused, which is negligibly small when the focusing is not tight where the scalar paraxial Gaussian beam formula is valid. Best regards, Yosuke

Dear Yasmien, Can you please go through our technical support, support@comsol.com? We need to see your model to solve your problem. Thank you. Yosuke

When I write an expession for x2, as you mentioned above, it shows Syntax error in expression – Expression: x*cos(theta) – y*sin(theta) – Subexpression: – y*sin(the – Position: 14 Error in automatic sequence generation.

Thank you for this clear and informative demonstration of the paraxial beam functionality in COMSOL! At the moment I am working on in bulk laser material processing of sapphire where I need to define an Gaussian beam entering the material and focusing in the bulk. Since this process is time dependent ( we want to study the behavior ), I am looking for a time dependent description of the Gaussian beam to take into account the varying pulse energy. I noticed that the Gaussian beam is only available in the frequency domain, what do I need to know and model to study this in a time dependent study?

Dear Yosuke Mizuyama Very interested topic. I have one question, please. Why lambda is equal to 500nm and used in COMSOL as the default value for the calculation of frequency (f=c_const/500[nm])?

The last of the five Seidel aberrations is distortion. Like Petzval curvature, it does not blur the image, but unlike Petzval curvature it does not curve the image. Instead, it either compresses the edges of an image, resulting in barrel distortion, or expands them, resulting in pincushion distortion.

Dear Yosuke, Thanks for your kind reply, it is very helpful, and yes I want to focus the beam to the size of the nano-particle with 6 nm radius, but I have 2 questions if you kindly allow: 1) I realized that you determine the waist radii depending on the wavelength only, Do you divide it by (pi)?, ignoring the particle radius. The second part of my question is should I depend on one factor only in determining w0 that is wavelength only? 2) When you write that 4 um is the proper w0, Do you mean that I can use this value for the whole previous wavelength spectrum? I mean is it constant? Best regards

Gaussianbeamcalculator

For a rotated one at an angle theta, please replace x and y in the above expression with x2 and y2 and define x2 = x*cos(theta) – y*sin(theta) y2 = x*sin(theta) + x*cos(theta)

As you know the gaussian beam source that I asked you about I used it in 3D structure and was represented in my model by analytic functon with the next formula: E(x,y,z)= E0*w0/w(x)*exp(-(y^2+z^2)/w(x)^2)*exp(-i*(k*x-eta(x)+k*(y^2+z^2)/(2*R(x))))

I am trying to study the optical characteristics of gold nano-particle (radius = 6nm) on a wavelength spectrum extended from 400 nm to 500 nm, and I do not know how I can determine the beam radius waist (w0) value. I think it will be less than wavelength and this will not match with the paraxial approximation for Maxwell equation that used in the suggested gaussian beam in your blog.

A custom lens designer can use a few techniques to reduce spherical aberration. One technique called “lens bending” requires adjusting the lens curvatures while keeping the lens power the same. This effectively adjusts the shape of the lens to minimize SA. For systems used with infinite conjugates (object or image at infinity), it is best to bend the lens so the greatest curvature is toward infinity. Looking at figure 1.7, this is not the case. The greatest curve faces away from the incident light. This causes a large SA as discussed above. Referring now to figure 1.9, the upper lens layout shows the same lens flipped around so the greatest curve of the lens faces the incident light from infinity. This greatly reduces the SA.

Dear Yasmien, 1) You can not focus a beam to an infinitely small size. Yes, the number I gave you is lambda/pi. I have no proof for this but it is what I know as the smallest possible spot size for a wavelength no matter what your particle size is. The wavelength is not the determining factor of w0. The minimum beam waist radius is determined by how the laser beam has originally been generated inside a laser cavity. You can’t change it. You can focus the beam by a focusing lens but you can only worsen it or at most you can keep it as it is depending on the lens quality. So when you simulate a focusing laser beam, you should have the specification of the laser beam. 2) I gave w0 = 10 lambda. If your wavelength is 400 nm, 10×400 nm = 4 um is the waist radius for which the paraxial Gaussian beam is a good approximation. For 500 nm, it’d be 5 um. Best regards, Yosuke

Besselbeam

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

2) When you write that 4 um is the proper w0, Do you mean that I can use this value for the whole previous wavelength spectrum? I mean is it constant?

Dear Yosuke, Is the background method applicable to the case of an interface? Suppose the beam is incident from air to glass, is this formular still valid? If not, how to implement the correct one? Thank you Best regards Simon

Dear Yosuke, I read your kind answer carefully and understood it. I am really thankful to this discussion with you because I do learn from it, so excuse me in this extra question; One of the COMSOL modes named “Nanorods” with application library path: Wave_Optics_Module/Optical_Scattering/ nanorods. In the “Model Definition” section at page 1 of this model, the author determined that the rods have dimensions less than wavelength, as my case, and as I understand he overcame the problem of Gaussian beam is an approximation solution by the following sentence and I will write it as it was reminded ” For tightly focused beams you also need to include an electric field component in the propagation direction”. My question: is this solution appropriate in 3D or in 2D structures only? Away from the previous question, do you think that decreasing the mesh size would increase the accuracy of gaussian beams in small structures? I will wait your kind answer and really thank you in advance.

If the object and image are interchanged, the sign of distortion will flip. We can use this principle to create a distortion-less system. If two identical lenses are used and a stop is placed midway between them, distortion will be zero as seen in figure 1.20. If it is not possible to utilize exact symmetry, another option is to add more elements to the system in an attempt to “balance” the distortion seen such that the sum of all element contributions to distortion = zero.

Although we list these aberrations individually, they normally occur in combinations. Most lenses have all of the above aberrations, as well as chromatic variation of the monochromatic aberration (e.g. the spherical aberration in red is different from the spherical aberration in blue). Things also become more complex as the f/# is reduced and the field angle increased. The Seidel aberrations are also known as third order aberrations, and there are fifth and higher order aberrations in addition. If you want to learn more about this topic, I’d suggest reading Welford’s Aberrations of Optical Systems. You can also visit our How to Measure Aberrations page to get an introduction to some different methods.

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.

Dear Yosuke, As you know the gaussian beam source that I asked you about I used it in 3D structure and was represented in my model by analytic functon with the next formula: E(x,y,z)= E0*w0/w(x)*exp(-(y^2+z^2)/w(x)^2)*exp(-i*(k*x-eta(x)+k*(y^2+z^2)/(2*R(x)))) And I wrote the component of electric field in propagation direction as following: (j*d(E(x,y,z),z)/emw.k0) where (z) is the polarization direction and I used it to overcome the paraxial approximation problem if you rememer, but I get this error: { Error in user-defined function. – Function: dE_dE__z__internalArgument Failed to evaluate variable. – Variable: comp1.emw.Ebx – Defined as: exp(i*phase)*(!(comp1.isScalingSystemDomain)*(comp1.es.Ex+((j*d((unit_V_cf*E(x/unit_m_cf,y/unit_m_cf,z/unit_m_cf))/unit_m_cf,z))/comp1.emw.k0))) Failed to evaluate expression. – Expression: comp1.emw.Ebx Failed to evaluate operator. – Operator: mean – Geometry: geom1 } Could you tell me the problem here? Thanks

Ernst Abbe, a man who is responsible for many advances in microscopy, came up with a formula called the Abbe sine condition. The sine condition has two forms as shown in figure 1.12. The first is for the case of finite object and image distances. In this case, if the ratio of the sine of the object angle to the sine of the image angle is constant for all rays, the condition is met. In the case of an infinite object, if the ratio of object ray height to the sine of the image angle is constant for all rays, the condition is met. If a lens corrects spherical aberration and coma, it meets the sine condition. We call this an aplanat.

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.

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.

Gaussianbeamq parameter

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 \left ( \int_\Omega |E_{\rm sc}|^2dxdy / \int_\Omega |E_{\rm bg}|^2dxdy \right )^{0.5}, where \Omega 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.

I read your kind answer carefully and understood it. I am really thankful to this discussion with you because I do learn from it, so excuse me in this extra question;

Early telescope users, like Galileo, were troubled by ACA because there was always a colored blur around everything they observed. Human eyes are most sensitive to green light, so that’s where they focused their telescopes, leaving red and blue out of focus. This resulted in a magenta (red + blue) blur as shown in figure 1.2. It’s easy to see how this image results in a blur. First place an image plane at the green focus in figure 1.1b. Then consider how the T-ACA from the blue and red light rays combine, forming a blurred magenta ring around the central white dot.

Another technique called “lens splitting” is when a single lens is split into multiple lenses in close proximity that have a total power of the original single lens. This is effective because SA is highly dependent on angle of incidence, therefore if the lens can be split into multiple lenses, we can decrease the angles of incidence while keeping the same power. Figure 1.9 shows an example of lens bending and splitting.

A common ploy is to attempt to introduce Petzval curvature into a lens to flatten out the astigmatic focal surfaces, giving a smaller blur on the image plane. This, known as an artificially flattened field, results in acceptable but not exceptional image quality.

If our familiar example object of stars were imaged through a system with Petzval field curvature the image would look in focus in the center of the image, with the focus falling off near the edges as shown in figure 1.16. This is very similar to the image at the medial focus of a system suffering from astigmatism.

Lateral chromatic aberration (LCA) increases linearly with the distance from the optical axis. This means that it is zero on axis because the distance from the axis is zero. If you’re looking for another way to think about lateral color, consider this: lateral color is a chromatic difference in magnification – red objects appear bigger than blue or vice versa. As seen figure 1.4, the red image height is larger than the green and blue image heights. The difference between the extreme image heights is equivalent to the amount of LCA is in a system.

Since the magnification difference from distortion increases with the cube of field, one option to minimize it is to decrease the field. Although decreasing the field of view can help correct distortion, it is not usually acceptable, as most lenses have a specific field angle that is desirable for their application.

LCA is the result of dispersion of the chief ray. Thus, when a lens exhibits LCA, light is affected very similarly to light traveling through a dispersive prism (figure 1.5). Imagining the prism in figure 1.5 is the tip of a lens shows us how the phenomenon occurs in a lens like one from figure 1.4. White light that is incident on the first surface of the prism (or lens) bends according to Snell’s law. Since the angle of refraction is dependent on the wavelength of light, the optical paths for different wavelengths diverge and we get dispersion – (separation of white light into all wavelengths across the visible spectrum). Since the index for blue light (nBLUE) is higher, it bends more strongly than green and red light.

The special solution to this paraxial Helmholtz equation gives the paraxial Gaussian beam formula. For a given waist radius w_0 at the focus point, the slowly varying function is given by

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 w_0 or the far-field divergence angle \theta 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.

Thanks for your clarification and I got the idea in using mesh. The explanation of the reason of existence an electric field component in the propagation direction is still unclear to me, I am sorry I did not understand it well. Also, why do we represent this component by differentiating the gaussian beam field according to the polarization direction?

In the scattered field formulation, the total field E_{\rm total} is linearly decomposed into the background field E_{\rm bg} and the scattered field E_{\rm sc} as E_{\rm total} = E_{\rm bg} + E_{\rm sc}. Since the total field must satisfy the Helmholtz equation, it follows that (\nabla^2 + k^2 )E_{\rm total} = 0, where \nabla^2 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

Petzval curvature will be zero for a meniscus lens with equal radii. The power of the lens is proportional to the thickness, so if one desires to change the power without adding to the Petzval curvature, using a thick meniscus can work.

There are five monochromatic aberrations: spherical aberration, coma, astigmatism, Petzval field curvature, and distortion. Each can be present even if the optical system is being used with monochromatic light, e.g. a laser. They differ in appearance and their dependence on f/# and field height (distance from the optical axis). Discovered over several decades in the late nineteenth century and codified by L. Seidel, these monochromatic aberrations are thus known as the Seidel aberrations.

Once a lens is corrected for spherical aberration, coma and astigmatism we call it anastigmatic. The name is derived from the Greek ‘ana’, meaning ‘up from’, and ‘stigma’, meaning ‘point’.

Dear Jana, Here’s the expression: Ex = 0 Ey = sqrt(w0/w(x))*exp(-y^2/w(x)^2)*exp(-i*k*x-i*k*y^2/(2*R(x))-eta(x)) Ez = 0 w0 = given waist radius, k = 2*pi/lambda w(x) = w0*sqrt(1+(x/xR)^2) xR = pi*w0^2/lambda R(x) = x+xR^2/x eta(x) = atan(x/xR)/2 For a rotated one at an angle theta, please replace x and y in the above expression with x2 and y2 and define x2 = x*cos(theta) – y*sin(theta) y2 = x*sin(theta) + x*cos(theta) Best regards, Yosuke

Axial chromatic aberration (ACA) occurs everywhere in the field of view, and is pretty much the same everywhere. The fact that the refractive index (nλ) of glass is different for different wavelengths causes this. The result is that a simple lens (e.g. a magnifying glass) has a different focal length for each wavelength of light as shown in figure 1.1a below.

The two chromatic aberrations are axial chromatic and lateral chromatic. The distinction is that axial chromatic aberration is present on the optical axis, as well as everywhere else. Lateral chromatic aberrations only occurs off-axis.

Choosing the correct shape of the lens can be very helpful in reducing coma. Using a convex-planar lens with the convex side facing infinity will help to minimize coma. This is close to the bending needed to minimize spherical aberration. Also, since the amount of blur is proportional to the square of the entrance pupil diameter, stopping down the lens will help minimize coma.

This factorization is reasonable for a wave in a laser cavity propagating along the optical axis. The next assumption is that |\partial^2 A/ \partial x^2| \ll |2k \partial A/\partial x|, which means that the envelope of the propagating wave is slow along the optical axis, and |\partial^2 A/ \partial x^2| \ll |\partial^2 A/ \partial y^2|, 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.,

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:

laguerre-gaussian modes

It is important to note that stopping down the lens does not change L-ACA but instead can greatly decrease T-ACA. This is because stopping down the lens does not change the focal length difference between colors, it only decreases the angle between the light rays and the optical axis. Therefore, it decreases the distance from the optical axis that the out of focus light rays cross the image plane (T-ACA).

Gaussianbeampropagation simulation

Since the blur from Petzval curvature on a planar image surface increases with the square of field, decreasing the field is one option to minimize it. Although this can help correct the blur, it is not usually acceptable as most lenses have a specific field angle that is desirable for their application.

Dear Yosuke, Thank you for this clear and informative demonstration of the paraxial beam functionality in COMSOL! At the moment I am working on in bulk laser material processing of sapphire where I need to define an Gaussian beam entering the material and focusing in the bulk. Since this process is time dependent ( we want to study the behavior ), I am looking for a time dependent description of the Gaussian beam to take into account the varying pulse energy. I noticed that the Gaussian beam is only available in the frequency domain, what do I need to know and model to study this in a time dependent study? Daniel

Here, x_R 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 1/e^2 intensity position diverges at an approximate divergence angle of \theta = \lambda/(\pi w_0).

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.

1) I realized that you determine the waist radii depending on the wavelength only, Do you divide it by (pi)?, ignoring the particle radius. The second part of my question is should I depend on one factor only in determining w0 that is wavelength only?

Astigmatism is closely related to field curvature and when both are present in a system the result is two image surfaces as shown in figure 1.17. The tangential and sagittal image surfaces will converge on the Petzval surface if astigmatism is corrected for. This can be done by lens bending, stop shift, moving elements or changing optical glasses in a system.

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.

Could you tell me the proper choice for the value of w0 and how can I use the gaussian beam formula as a background source in my case.

Shifting the stop is also helpful in correcting coma if spherical is present in the system. This is similar to how stop shift corrects for LCA if ACA is present. Also like LCA, we can reduce coma by using a close to symmetric system.

Thanks Yosuke, When I write an expession for x2, as you mentioned above, it shows Syntax error in expression – Expression: x*cos(theta) – y*sin(theta) – Subexpression: – y*sin(the – Position: 14 Error in automatic sequence generation. Is last expression for y2 right, because in both parts there is x? Can I define x and y are equal to 1? Regards Jana

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.

Away from the previous question, do you think that decreasing the mesh size would increase the accuracy of gaussian beams in small structures?

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.

Another option is to add a negative field-flattener lens to the system placed close to the image surface. This lens’ design allows it to have the opposite field curvature of the system to cancel it out. Due to its position near the focal plane it will not affect other aberrations greatly.