Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation

Abstract

We study multi-parameter solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation which include oscillating laser beams in a parabolic waveguide, spiral light beams, and other important families of propagation-invariant laser modes in weakly varying media. A “smart” lens design and a similar effect of superfocusing of particle beams in a thin monocrystal film are also discussed. In the supplementary electronic material, we provide a computer algebra verification of the results presented here, and of some related mathematical tools that were stated without proofs in the literature. We also demonstrate how computer algebra can be used to derive some of the presented formulas automatically, which is highly desirable as the corresponding hand calculations are very tedious. In numerical simulations, some of the new solutions reveal quite exotic properties which deserve further investigation including an experimental observation.

Appendices

Appendix 1: From Maxwell to paraxial wave

We follow [9] with somewhat different details. In dielectrics (no free current, no free charge, isotropic, homogeneous, material linear), the Maxwell equations for the complex electric ${\varvec{E}}$ and magnetic ${\varvec{H}}$ fields for a monochromatic wave varying as ${\text{e}}^{-i\omega t}$ are given by

$$\begin{aligned} {\text {curl}}{\varvec{E}}=i\frac{\omega }{c}\mu {\varvec{H}} ,\qquad \ \ {\text {div}}\left( \mu {\varvec{H}}\right) =0, \end{aligned}$$

(5.1)

$$\begin{aligned} {\text {curl}}{\varvec{H}}=-i\frac{\omega }{c}\varepsilon {\varvec{E}} ,\qquad {\text {div}}\left( \varepsilon {\varvec{E}}\right) =0, \end{aligned}$$

(5.2)

where $\varepsilon$ is the permittivity and $\mu$ is the permeability of the material (see, for example, [12, 91, 143, 144]). Let us consider a “polarized” wave of the form,

$$\begin{aligned} {\varvec{E}}=f(x,y,z){\text{e}}^{ikz}{\varvec{e}}_{x}+g(x,y,z){\text{e}}^{ikz} {\varvec{e}}_{z},\qquad k^{2}=\varepsilon \mu \frac{\omega ^{2}}{c^{2}}, \end{aligned}$$

(5.3)

where $\left\{ {\varvec{e}}_{x},{\varvec{e}}_{y},{\varvec{e}} _{z}\right\}$ are orthonormal vectors in $\left. {\mathbb {R}} \right. ^{3}.$ From the first Equation (5.1) one gets:

$$\begin{aligned} i\frac{\omega }{c}\mu {\varvec{H}}&={\text {curl}}{\varvec{E}} \\&=\frac{\partial g}{\partial y}{\text{e}}^{ikz}{\varvec{e}}_{x}+\left( \frac{\partial f}{\partial z}+ikf-\frac{\partial g}{\partial x}\right) {\text{e}}^{ikz}{\varvec{e}}_{y}-\frac{\partial f}{\partial y}{\text{e}}^{ikz}{\varvec{e}} _{z} \end{aligned}$$

(5.4)

and the second Equation (5.1) is automatically satisfied. In addition, from the second Equation (5.2):

$$\begin{aligned} \frac{\partial f}{\partial x}+\frac{\partial g}{\partial z}+ikg=0. \end{aligned}$$

(5.5)

If $g\equiv 0,$ then $f_{x}=0$ and the only transversal solution is a plane wave, ${\varvec{E}}={\text{e}}^{ikz}{\varvec{e}}_{x},$ up to a constant multiple (cf. [91]).

In a similar fashion, from the first Equation (5.2) and (5.4) we obtain:

$$\begin{aligned} k^{2}{\varvec{E}}&={\text {curl}}\left( i\frac{\omega }{c} \mu {\varvec{H}}\right) =\left( k^{2}f+g_{xz}+ikg_{x}-f_{yy}-2ikf_{z} -f_{zz}\right) {\text{e}}^{ikz}{\varvec{e}}_{x} \\&\quad +\left( f_{xy}+g_{yz}+ikg_{y}\right) {\text{e}}^{ikz}{\varvec{e}}_{y}+\left( f_{xz}+ikf_{x}-g_{xx}-g_{yy}\right) {\text{e}}^{ikz}{\varvec{e}}_{z}. \end{aligned}$$

(5.6)

In view of (5.5), the latter equation can be simplified to

$$\begin{aligned} k^{2}{\varvec{E}}&=k^{2}\left( f{\varvec{e}}_{x}+g{\varvec{e}} _{z}\right) {\text{e}}^{ikz}\\&=\left( k^{2}f-f_{xx}-f_{yy}-2ikf_{z}-f_{zz}\right) {\text{e}}^{ikz} {\varvec{e}}_{x}\\&\quad+\left( k^{2}g-g_{xx}-g_{yy}-2ikg_{z}-g_{zz}\right) {\text{e}}^{ikz}{\varvec{e}}_{z}. \end{aligned}$$

Finally, under the imposed conditions $\left| f_{zz}\right| \ll 2\left| kf_{z}\right|$ and $\left| g_{zz}\right| \ll 2\left| kg_{z}\right| ,$ we arrive to the paraxial wave equations,

$$\begin{aligned} 2ik\frac{\partial F}{\partial z}+\frac{\partial ^{2}F}{\partial x^{2}} +\frac{\partial ^{2}F}{\partial y^{2}}=0, \end{aligned}$$

(5.7)

for the transversal and longitudinal components, $F=\left\{ f,g\right\} ,$ of the complex electric field, respectively. [The corresponding magnetic field can be evaluated by (5.4).] Once again, these components are related by (5.5), which implies that

$$\begin{aligned} \frac{\partial }{\partial z}\left( {\text{e}}^{ikz}g\right) =-f_{x}{\text{e}}^{ikz}, \end{aligned}$$

(5.8)

and, integrating by parts,

$$\begin{aligned} {\text{e}}^{ikz}g=-\int f_{x}{\text{e}}^{ikz}\ {\text{d}}z=-\frac{1}{ik}f_{x}{\text{e}}^{ikz}+\frac{1}{ik}\int f_{xz}{\text{e}}^{ikz}\ {\text{d}}z\approx -\frac{1}{ik}f_{x}{\text{e}}^{ikz}, \end{aligned}$$

provided that $\left| k\right| \gg 1.$ In paraxial approximation, it is a custom to write

$$\begin{aligned} g\approx -\frac{1}{ik}f_{x}=-\frac{1}{ik}\frac{\partial f}{\partial x} \end{aligned}$$

(5.9)

for the small longitudinal component of electric field that automatically satisfies (5.7). More details can be found in [9, 91]. A general solution is a superposition of two waves of the form (5.3).

Note The paraxial wave equations (5.7) for transversal and longitudinal components, $F=\left\{ f(x,y,z),g(x,y,z)\right\} ,$ can be solved by the Fresnel integral,

$$\begin{aligned} F(x,y,z)&=\frac{k}{2\pi i z}\iint _{{\mathbb {R}}^{2}}\exp \left( \frac{ik}{2z}\left[ (x-\xi )^{2}+(y-\eta )^{2}\right] \right)\\&\quad\times F_{0}(\xi ,\eta )\ {\text{d}}\xi {\text{d}}\eta , \end{aligned}$$

(5.10)

subject to proper “initial” data, $F_{0}=\left\{ f_{0}(x,y),g_{0}(x,y)\right\} ,$ which are related as follows,

$$\begin{aligned} g_{0}+\frac{1}{ik}\frac{\partial f_{0}}{\partial x}+\frac{1}{2k^{2}}\left( \frac{\partial ^{2}g_{0}}{\partial x^{2}}+\frac{\partial ^{2}g_{0}}{\partial y^{2}}\right) =0, \end{aligned}$$

(5.11)

in view of “divergence” condition (5.5). [When $k\gg 1,$ one formally gets (5.9).]

In fact, Eq. (5.11) is the 2D inhomogeneous Helmholtz equation [129, 145]:

$$\begin{aligned} \frac{\partial ^{2}g_{0}}{\partial x^{2}}+\frac{\partial ^{2}g_{0}}{\partial y^{2}}+2k^{2}g_{0}=2ik\frac{\partial f_{0}}{\partial x}, \end{aligned}$$

(5.12)

which can be solved exactly provided that function $f_{0}(x,y)$ is known. Under the Sommerfeld radiation condition,

$$\begin{aligned} \lim _{r\rightarrow \infty }r^{1/2}\left( \frac{\partial }{\partial r}-ik\sqrt{2}\right) g_{0}\left( r{\varvec{e}}\right) =0,\quad r=\sqrt{x^{2}+y^{2} }, \end{aligned}$$

(5.13)

uniformly in ${\varvec{e}}$, $\left| {\varvec{e}}\right| =1,$ one gets [35, 145]:

$$\begin{aligned} g_{0}(x,y)&=\frac{k}{2}\iint _{{\mathbb {R}}^{2}}H_{0}^{(1)}\left( k\sqrt{2\left[ (x-\zeta )^{2}+(y-\vartheta )^{2}\right] }\right) \\&\quad\times\frac{\partial f_{0} }{\partial \zeta }(\zeta ,\vartheta )\ {\text{d}}\zeta {\text{d}}\vartheta , \end{aligned}$$

(5.14)

where $H_{0}^{(1)}(z)$ is a Hankel function [109].

Appendix 2: From Maxwell to nonlinear paraxial optics

In a more general case (of a weakly inhomogeneous linear or nonlinear medium with a complex-valued dielectric permittivity $\varepsilon ;$ see, for example, Refs. [50] and [143] for more details), one can look for solutions of Eqs. (5.1), (5.2) as a superposition,

$$\begin{aligned} {\varvec{E}}={\varvec{E}}_{x}+{\varvec{E}}_{y}, \end{aligned}$$

(5.15)

of two “polarized” waves:

$$\begin{aligned} {\varvec{E}}_{x}&=f(x,y,z){\text{e}}^{ik(z)}{\varvec{e}}_{x}+g(x,y,z){\text{e}}^{ik(z)} {\varvec{e}}_{z}, \\ {\varvec{E}}_{y}&=h(x,y,z){\text{e}}^{ik(z)}{\varvec{e}}_{y}+l(x,y,z){\text{e}}^{ik(z)} {\varvec{e}}_{z}, \end{aligned}$$

(5.16)

where f, g, h, l, and k are some complex-valued functions. In a similar fashion,

$$\begin{aligned}&F_{xx}+F_{yy}+F_{zz}+2ik_{z}F_{z}+\left( \varepsilon \mu \frac{\omega ^{2} }{c^{2}}-k_{z}^{2}+ik_{zz}\right) F \\&\quad \quad =-\left\{ \begin{array}{l} \left( {\mathcal {E}}+\dfrac{\varepsilon _{z}}{\varepsilon }\left( g+l\right) \right) _{x}\\ \left( {\mathcal {E}}+\dfrac{\varepsilon _{z}}{\varepsilon }\left( g+l\right) \right) _{y} \end{array} \right. ,\quad {\mathcal {E}}=\frac{\varepsilon _{x}}{\varepsilon }f+\frac{\varepsilon _{y}}{\varepsilon }h \end{aligned}$$

(5.17)

and

$$\begin{aligned}&G_{xx}+G_{yy}+G_{zz}+2ik_{z}G_{z}+\left( \varepsilon \mu \frac{\omega ^{2} }{c^{2}}-k_{z}^{2}+ik_{zz}\right) G \\&\quad =-ik_{z}\left( \mathcal {E+}\frac{\varepsilon _{z} }{\varepsilon }G\right) -\left( \mathcal {E+}\frac{\varepsilon _{z} }{\varepsilon }G\right) _{z}, \end{aligned}$$

(5.18)

where, by definition,

$$\begin{aligned} F=\left\{ \begin{array}{l} f (x,y,z)\\ h(x,y,z) \end{array} \right. ,\qquad G=\left\{ \begin{array}{l} g (x,y,z)\\ l(x,y,z) \end{array} \right. . \end{aligned}$$

(5.19)

Here, it is convenient to rewrite the last equation (5.2) as a sum of two equations:

$$\begin{aligned} f_{x}+g_{z}+ik_{z}g+\frac{\varepsilon _{x}}{\varepsilon }f+\frac{\varepsilon _{z}}{\varepsilon }g=0,\quad h_{y}+l_{z}+ik_{z}l+\frac{\varepsilon _{y} }{\varepsilon }h+\frac{\varepsilon _{z}}{\varepsilon }l=0. \end{aligned}$$

(5.20)

We did not impose any conditions yet and Eqs. (5.15)–(5.20) are equivalent to the original Maxwell system (5.1)–(5.2) under consideration. For paraxial approximation, we may choose $k_{zz}=0,$ namely, $k(z)=kz,$ where k is a constant.

Let us first consider linear and nonlinear codimension 1D cases. When $h=l=f_{y} =g_{y}=\varepsilon _{y}=0,$ one can simplify to

$$\begin{aligned}&f_{xx}+f_{zz}+2ikf_{z}+\left( \varepsilon \mu \frac{\omega ^{2}}{c^{2}} -k^{2}\right) f+\left( \frac{\varepsilon _{x}}{\varepsilon }f+\dfrac{\varepsilon _{z}}{\varepsilon }g\right) _{x}=0, \end{aligned}$$

(5.21)

$$\begin{aligned}&g_{xx}+g_{zz}+2ikg_{z}+\left( \varepsilon \mu \frac{\omega ^{2}}{c^{2}} -k^{2}\right) g=-ik\left( \frac{\varepsilon _{x}}{\varepsilon }f\mathcal {+} \frac{\varepsilon _{z}}{\varepsilon }g\right) -\left( \frac{\varepsilon _{x} }{\varepsilon }f\mathcal {+}\frac{\varepsilon _{z}}{\varepsilon }g\right) _{z},\end{aligned}$$

(5.22)

$$\begin{aligned}&f_{x}+g_{z}+ikg+\frac{\varepsilon _{x}}{\varepsilon }f+\frac{\varepsilon _{z} }{\varepsilon }g=0. \end{aligned}$$

(5.23)

From the last equation,

$$\begin{aligned} f=-\frac{{\text{e}}^{-ikz}}{\varepsilon }\int (\varepsilon g{\text{e}}^{ikz})_{z}\ {\text{d}}x,\qquad g=-\frac{{\text{e}}^{-ikz}}{\varepsilon }\int {\text{e}}^{ikz}(\varepsilon f)_{x}\ {\text{d}}z. \end{aligned}$$

(5.24)

Thus Eqs. (5.21) and (5.22) can be thought of as certain integro-differential equations for complex-valued functions f and g, respectively. Integrating by parts,

$$\begin{aligned} g=-\frac{{\text{e}}^{-ikz}}{ik\varepsilon }\int (\varepsilon f)_{x}\ \mathrm{de}^{ikz} =-\frac{(\varepsilon f)_{x}}{ik\varepsilon }+\frac{{\text{e}}^{-ikz}}{ik\varepsilon }\int {\text{e}}^{ikz}(\varepsilon f)_{xz}\ {\text{d}}z\approx -\frac{(\varepsilon f)_{x} }{ik\varepsilon }. \end{aligned}$$

(5.25)

For large |k|,  it is also a custom to assume that $\left| f_{zz} \right| \ll 2\left| kf_{z}\right| ,$ $\left| f_{zz}\right| \ll 2\left| kf_{z}\right| ,$ and $|g|\ll |f|.$ As a result, one may concentrate on the study of scalar inhomogeneous paraxial wave equation of the form:

$$\begin{aligned} f_{xx}+2ikf_{z}+\left( \varepsilon \mu \frac{\omega ^{2}}{c^{2}}+\left( \frac{\varepsilon _{x}}{\varepsilon }\right) _{x}-k^{2}\right) f+\frac{\varepsilon _{x}}{\varepsilon }f_{x}=0. \end{aligned}$$

(5.26)

In a weakly inhomogeneous nonlinear medium, we expand the (complex-valued) permittivity $\varepsilon ,$

$$\begin{aligned} \varepsilon (x,z)=\varepsilon _{0}(z)+\varepsilon _{1}(z)x+\varepsilon _{2}(z)x^{2}+\cdots \left( +\lambda \left| f\right| ^{2}+\cdots \right) \end{aligned}$$

(5.27)

and neglect the higher order terms. In this approximation,

$$\begin{aligned} \frac{\varepsilon _{x}}{\varepsilon }=\frac{\varepsilon _{1}}{\varepsilon _{0} }+\left[ 2\frac{\varepsilon _{2}}{\varepsilon _{0}}-\left( \frac{\varepsilon _{1}}{\varepsilon _{0}}\right) ^{2}\right] x,\qquad \left( \frac{\varepsilon _{x}}{\varepsilon }\right) _{x}=2\frac{\varepsilon _{2} }{\varepsilon _{0}}-\left( \frac{\varepsilon _{1}}{\varepsilon _{0}}\right) ^{2}. \end{aligned}$$

(5.28)

and one arrives at a form of the paraxial wave equation (2.1) (or its nonlinear versions).

The corresponding linear and nonlinear codimension 2D cases, when one can concentrate on a certain dominant component of electric field once again, are similar. Further details are left to the reader.