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. For that wavelength range, the least possible waist radii are as large as 127 nm to 159 nm, though. That is, I am looking for monochromatic solutions of the Maxwell equations which look ~ What are good non-paraxial gaussian . 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. You can only propagate it along the x or y or z axis. \begin{equation} \omega(z=0) = \omega_0 \end{equation}. Dear Yasmien, Screenshot of the settings for the Gaussian beam background field. 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)? I have one question, please. Dear Daniel, Two sources of radiation in the plane, given mathematically by a function f, which is zero in the blue region The real part of the resulting field A, A is the solution to the inhomogeneous Helmholtz equation ( 2 k 2) A = f.. where 2 is the Laplace operator (or "Laplacian"), k 2 is the eigenvalue, and f is the (eigen)function. Yosuke, Dear Yosuke, These qualities are why lasers are such attractive light sources. Since the paraxial equation is just the Helmholtz equation with simplifying assumptions, we can use our basic solution to the Helmholtz, the angular spectrum, multiplied by a transfer function to find the field at an arbitrary distance z: \begin{equation} E(x,y,z) = \iint^{\infty}_{-\infty}A(k_x,k_y;0)H(k_x,k_y;z)e^{-i(k_xx+k_yy)}dk_xdk_y \end{equation}. The above formula is written for beams in vacua or air for simplicity. The paraxial Helmholtz equation 2 TA2jk A z =0 Plug the derivatives and cancel the common factor, A0 z2 exp jkx2+y2 2z k k x2 z +j k k y2 z +j 2jk jk x2 +y2 2z 1 =0 k x2 z k y2 z 2j jk x2 +y2 2z =0 k x2 z k y2 z +k x2 +y2 z =0 0=0 The paraboloidal wave is indeed a . Best regards, Thank you for reading my blog post and for your comment. The well known paraxial approximation to equation ( 1) is, followed by, Discretization of the first equation using the Crank-Nicholson scheme results in a tridiagonal set of equations to be solved in order to propagate the wavefield from a level z to . Since the solution must be periodic in from the definition of . The paraxial approximation of the Helmholtz equation is: where is the transverse part of the Laplacian. ( ) . Green's Function for the Helmholtz Equation If we fourier transform the wave equation, or alternatively attempt to find solutions with a specified harmonic behavior in time , we convert it into the following spatial form: (11.41) (for example, from the wave equation above, where , , and by assumption). Write expressions for the beam. The paraxial Gaussian beam formula is an approximation to the Helmholtz equation derived from Maxwells equations. We can recognize the propagation factor exp {-ikz} as well as the transverse variation of the amplitude : From a mathematical point of view, the spherical wave is a solution of the propagation equation. 2) I gave w0 = 10 lambda. The limitation appears when you are trying to describe a Gaussian beam with a spot size near its wavelength. Read more about this topic: Helmholtz Equation. At z = $z_R$, the beam waist is $\sqrt{2}\omega_0$ and the beam diameter is $2\sqrt{2}\omega_0$. 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. . To circumvent this drawback, families of so-called finite-energy Airy-type beams have been proposed in the literature . To describe the Gaussian beam, there is a mathematical formula called the paraxial Gaussian beam formula. 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. Todays blog post has covered the fundamentals related to the paraxial Gaussian beam formula. That means there is no purely linearly polarized beam for non-paraxial Gaussian beams. Do you have to focus your beam to the size of the nano-particle? A schematic illustrating the converging, focusing, and diverging of a Gaussian beam. 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 its tightly focused, which is negligibly small when the focusing is not tight where the scalar paraxial Gaussian beam formula is valid. The results shown above clearly indicate that the paraxial Gaussian beam formula starts failing to be consistent with the Helmholtz equation as its focused more tightly. 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 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. Since we want to incorporate spatial variations in velocity, this limitation is ultimately to be avoided, so after getting the paraxial equation in the Fourier domain, ik z is replaced by ,and ik x is replaced . Yosuke. In the above plot, we saw the relationship between the waist size and the accuracy of the paraxial approximation. Expansion and cancellation yields the following: Because of the paraxial inequalities stated above, the 2A/z2 factor is neglected in comparison with the A/z factor. Syntax error in expression A paraxial ray is a ray which makes a small angle () to the optical axis of the system, and lies close to the axis throughout the system. Please send support@comsol.com a question on this method since its a little bit difficult to explain here. The angular spread of the Gaussian beam is then defined as: \begin{equation} \theta = \frac{\lambda}{\pi\omega_0} \end{equation}. COMSOL will automatically take care of the local k depending on where you have different materials in your domain. Failed to evaluate operator. Physics: I am playing around with some optics manipulations and I am looking for beams of light which are roughly gaussian in nature but which go beyond the paraxial regime and which include non-paraxial vector-optics effects like longitudinal polarizations and the like. In the last section, we started with a general solution (angular spectrum) to the Helmholtz equation: \begin{equation} (\nabla^2+k^2)E(x,y,z) = 0\end{equation}. We will then find solutions for this equation (in next part of this page, in fact!). For 500 nm, itd be 5 um. [2] Inhomogeneous Helmholtz equation [ edit] The inhomogeneous Helmholtz equation is the equation 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 I define x and y are equal to 1? In the meantime, you may want to check out this reference: P. Varga et al., The Gaussian wave solution of Maxwells equations and the validity of scalar wave approximation, Optics Communications, 152 (1998) 108-118. You signed in with another tab or window. Airy beams are solutions to the paraxial Helmholtz equation known for exhibiting shape invariance along their self-accelerated propagation in free space. (We have derived this equation during previous lecture Check out YOul lecture notes) Verify that Eq: (3.1-3) and (3.14) are solutions of the paraxial Helmholtz equation Verify that Eq: (3.1-S)is solution of the paraxial Helmholtz equation Using the complex number inversion procedures shown the class derive and define equations (3.1-6) -(..-H . There are some limitations for the built-in Gaussian beam feature. Contents 1 Motivation and uses 2 Solving the Helmholtz equation using separation of variables 2.1 Vibrating membrane 2.2 Three-dimensional solutions The interface features a formulation option for solving electromagnetic scattering problems, which are the Full field and the Scattered field formulations. The phase will asymptotically approach $\pi/2$ as z $\rightarrow \infty$. Equation (1) retains the full spatial symmetry of the NLH model, and is a more convenient framework for comparing new results with those obtained from paraxial calculations. Ez = 0 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}. Best regards, To second order, the approximations above for sine and tangent do not change (the next term in their Taylor series expansion is zero), while for cosine the second order approximation is. Because of the convergence of a Gaussian beam, there will be a refraction at a material interface, which causes the focus shift. Failed to evaluate variable. The parabolodal wave is an exact solution to the paraxial wave equation as we found above, so we can thus use this to find the Gaussian solution to the paraxial wave equation. In the paraxial approximationof the Helmholtz equation, the complex amplitudeAis expressed as A(r)=u(r)eikz{\displaystyle A(\mathbf {r} )=u(\mathbf {r} )e^{ikz}} where urepresents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Thus, we can shift to the position: 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. A different approach for seeing the same trend is shown in our Suggested Reading section. 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. 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. Paraxial-Helmholtz-equation Paraxial Helmholtz equation y= ( (i) /2*k )2/x2, which looks like the unsteady heat equation except that y is % the time-like variable and the coefficient of the second derivative is imaginary. This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. 4, pp. Correct: y2 = x*sin(theta)+y*cos(theta) Since $\zeta$ is just a number, it can also be imaginary, so we can just try substituting $\zeta = -iz_R$ and find the envelope as: \begin{equation} \varepsilon(x,y,z) = \frac{E_0}{q(z)}e^{-ik(x^2+y^2)/2q(z)} \end{equation}. Thank you On the other hand, this formulation can be rewritten in the form of an inhomogeneous Helmholtz equation as. The minimum beam waist radius is determined by how the laser beam has originally been generated inside a laser cavity. We will publish a follow-up blog post with rigorous solutions in a few months. Paraxial Helmholtz equation y= ((i) /2*k )2/x2, which looks like the unsteady heat equation except that y is % the time-like variable and the coefficient of the second derivative is imaginary. has flat wavefronts) before the divergence due to diffraction becomes significant. The NLS equation can be recovered from Eq. In the paraxial approximation of the Helmholtz equation, [1] the complex amplitude A is expressed as where u represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. By providing your email address, you consent to receive emails from COMSOL AB and its affiliates about the COMSOL Blog, and agree that COMSOL may process your information according to its Privacy Policy. If the background field doesnt satisfy the Helmholtz equation, the right-hand side may leave some nonzero value, in which case the scattered field may be nonzero. So you have to add it no matter how its a different component than your preferred plane to which you want to believe its polarized. Hi, Thank you very much for reading my blog and for your interest. (Helmholtz equation) 2 . Show that the wave with complex envelope A (r) = [A_1/q (z)] exp [-jk (x^2 + y^2)/2q (z)], where q (z) = z +jz_0 and z_0 is a constant, also satisfies the paraxial Helmholtz equation. Today, well learn about this formula, including its limitations, by using the Electromagnetic Waves, Frequency Domain interface in the COMSOL Multiphysics software. When we assumed time-harmonic waves to derive the Helmholtz equation from the time-dependent wave equation, we factored out exp(i*omega*t). Sketch the intensity of the Gaussian beam in the plane z=0. Thank you so much for this reliable blog. Lets check this condition on the x-axis. Dear 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. OSTI.GOV Journal Article: Stability of a modified Peaceman-Rachford method for the paraxial Helmholtz equation on adaptive grids Journal Article: Stability of a modified Peaceman-Rachford method for the paraxial Helmholtz equation on adaptive grids Thank you. Best regards, Simon, Dear Simon, equation and the paraxial wave equation. Now we can check the assumptions that were discussed earlier. Attempt Separation of Variables by writing. Heres the expression: This field can be regarded as an error of the background field. . The paraxial approximation places certain upper limits on the variation of the amplitude function A with respect to longitudinal distance z. That was a typo. Thanks Yosuke, To specify a paraxial Gaussian beam, either the waist radius w_0 or the far-field divergence angle \theta must be given. Why lambda is equal to 500nm and used in COMSOL as the default value for the calculation of frequency (f=c_const/500[nm])? Dear Attilio, Paraxial Helmholtz equation y= ((i) /2*k )2/x2, which looks like the unsteady heat equation except that y is % the time-like variable and the coefficient of the second derivative is imagina. Now, if we were to substitute this equation into the Helmholtz equation, we would first have via Chains Rule (hehe): \begin{equation} \nabla^2E(x,y,z) = (\nabla^2\varepsilon 2ik\hat{z}\cdot\nabla{\varepsilon}-k^2\varepsilon)e^{-ikz} \end{equation}, \begin{equation} \nabla^2\epsilon 2ik\frac{\partial\varepsilon}{\partial{z}} = 0 \end{equation}, If we then consider that the envelope varies slowly, such that, \begin{equation} |\frac{\partial^2\varepsilon}{\partial{z}^2}| << 2k|\frac{\partial\epsilon}{\partial{z}}| \end{equation}, \begin{equation} \frac{\partial^2\varepsilon}{\partial{x^2}}+\frac{\partial^2\varepsilon}{\partial{y^2}} 2ik\frac{\partial\epsilon}{\partial{z}} = 0 \end{equation}. If you would like a more flexible way, you can define a paraxial Gaussian beam in Definition and also define a coordinate transfer. This then gives us the physical intuition into what the Rayleigh range means: its a measure of how far the beam is approximately collimated (i.e. Yosuke. The Helmholtz differential equation can be solved by the separation of variables in only 11 coordinate systems. In the paraxial approximation, the complex magnitude of the electric field E becomes. Substituting u(r) = A(r) eikz then gives the paraxial equation for the original complex amplitude A : The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation. Dear Yosuke Mizuyama But you can change it to ewfd.k for more general cases. Would you be able meanwhile to point to me some useful information on this matter? where as before we had the Rayleigh range defined as: \begin{equation} z_R = \frac{\pi\omega_0^2}{\lambda} \end{equation}. The paraxial wave equation, in homogeneous or in random media, is a model used for many applications, for instance in communication and imaging [19]. As the beam propagates further into the far field, ie many Rayleigh ranges away, the beam expands essentially linearly with distance such that the shape approaches a cone with a divergence angle $\theta$. It doesnt change the scalar paraxial approximation nature. The focus position needs to be known a priori. In this paper, the authors give an exact formula for a nonparaxial Gaussian wave. Best regards Assuming polarization in the x direction and propagation in the +z direction, the electric field in phasor (complex) notation is given by: Rigurously speaking, nonparaxial beams are solutions of the wave equation without the paraxial approximation, in other words, they are solutions of the Helmholtz wave equation 2E +k2E = 0, with k the wave number. 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. 2.4.1 Paraxial Wave Equation We start from the Helmholtz Equation (2.18) +k2 0 e E(x,y,z,)=0, (2.214) withthefreespacewavenumberk0 = /c0. In these approximations, the transfer function of free space is: \begin{equation} H(k_x,k_y;z) = e^{-ikz}e^{i(k_x^2+k_y^2)z/2k} \end{equation}. Mesh refinement works for increasing the accuracy of finite element solutions. and the Paraxial Helmholtz Equation, which describes collimated beams: $$ \nabla^2 \psi (x,y,z)= -2 i k \frac{\partial \psi (x,y,z)}{\partial z} $$ The above equation describes a beam propagating through the "z" direction. The paraxial approximation is accurate within 0.5% for angles under about 10 but its inaccuracy grows significantly for larger angles. (1) then the Helmholtz Differential Equation becomes. 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}. I will wait your kind answer and really thank you in advance. Another approach is to find a differential equation that approximates paraxial field propagation. More in general, is there a way to simulate in COMSOL the point spread function of a high NA lens? Clearly, this is too tightly focused for the paraxial approximation to hold, and I encountered the problems you have described above. P. Vaveliuk, Limits of the paraxial approximation in laser beams. We will discuss this topic in a future blog post. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. A statement of the approximation involves the optical axis, which is a line that passes through the center of each lens and is oriented in a direction normal to the surface of the lens (at the center).The paraxial approximation approximation is valid for rays that make a small angle to the optical axis of the . Similar (scalar) equations must be obeyed by each component of e and b. HELMHOLTZ EQUATION If the field is monochromatic at frequency , e and b are represented by the phasors A and B: e = Re {Aexp(-j t)} b = Re{ Bexp(-j t)} Maxwell's equations for free space then become E = j B (6.9) . However, there is a limitation attributed to using this formula. Expansion and cancellation yields the following: Because of the paraxial inequalities stated above, the 2A/z2 factor is neglected in comparison with the A/z factor. 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? [ ] . The following plot is the result of the calculation as a function of x normalized by the wavelength. Note that the variable name for the scattered field is ewfd.relEz. Specifically: These conditions are equivalent to saying that the angle between the wave vector k and the optical axis z must be small enough so that. Instead, you simply need to specify the waist radius, focus position, polarization, and the wave number. If not, how to implement the correct one? The equations in polar coordinates can be similarly transformed, the paraxial approximation . There is the laplacian, amplitude and wave number associated with the equation. As such, it would be reasonable to want to simulate a Gaussian beam with the smallest spot size. This wave, called the Gaussian beam, is the subject of Chapter 3. In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). Due to this limitation, you will have to rotate your material in order to simulate a beam at an angle. We will prove this boundary condition in Section 3, but for now we shall simply use the result of applying this boundary condition with the Helmholtz equation to find that E ( r) = E x x ^ + E y y ^ + E z z ^ satisfies the Helmholtz with constituents given as: (7) E x = E x, 0 c o s ( m x x L) s i n ( m y y L) s i n ( m z z L) Stay tuned! Can you tell me how to implement my simulation?Thank you very much! This consent may be withdrawn. 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 wont automatically calculate the focus position shift if you only know the field outside your material. 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)))). As the paraxial Helmholtz equation is a complex equation, let's take a look at the real part of this quantity, . Show that the wave whose complex envelope is given by A(r) [A1/q(2)] exp[- jk(z? 4.2 Paraxial Wave Equation For optical wave propagation, we can further reduce the Helmholtz equation (3) to what is called the paraxial wave equation. We can see that the paraxiality condition breaks down as the waist size gets close to the wavelength. The exact monochromatic wave equation is the Helmholtz equation (1) where is the angular frequency and v ( x, z) is the wave velocity at the point ( x, z ). It has the form of an evolution equation that describes waves prop-agating along a privileged axis and it can be obtained by neglecting backscatter- where A represents the complex-valued amplitude of the electric field, which modulates the sinusoidal plane wave represented by the exponential factor. And I wrote the component of electric field in propagation direction as following: You can imagine I am now really looking forward to the follow-up post you promised describing the solutions! 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. It is, however, not a cut-off number, as the approximation assumption is continuous. Its up to you to decide when you need to be cautious in your use of this approximate formula. Standard integral transform methods are used to obtain general . Eqs (6.6) and (6.7) are vector wave equations. In this equation, is a complex variable representing the phase and amplitude of the wave and k is the wave number equal to 2/, where is the wavelength. 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. The nonlinear paraxial equation has exact soliton solutions (Huser et al., 1992) that correspond to a balance between nonlinearity and dispersion in the case of temporal solitons or between nonlinearity and diffraction in the case of spatial solitons. Most lasers emit beams that take this form. When we use the term Gaussian beam here, it always means a focusing or propagating Gaussian beam, which includes the amplitude and the phase. Away from the previous question, do you think that decreasing the mesh size would increase the accuracy of gaussian beams in small structures? Show that the wave with complex envelope A (r)= [A_1 / q (z)]exp [-jk (x^2+y^2)/2q (z)], where q (z)=z+jz_0 and z_0 is constant, also satisfies the paraxial Helmholtz equation. I am really grateful to this discussion with you. Note: The term Gaussian beam can sometimes be used to describe a beam with a Gaussian profile or Gaussian distribution. To obtain the tightest possible focus, most commercial lasers are designed to operate in the lowest transverse mode, called the Gaussian beam. This is then the most general solution to the paraxial wave equation. Thanks for this good explanation of Gaussian beam. The U.S. Department of Energy's Office of Scientific and Technical Information

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