A theoretical analysis is made of Gaussian beam propagation in media where the refractive index and/or the gain constant varies quadratically with the distance from the optic axis. Use of a complex beam parameter simplifies the analysis. A differential equation for the complex beam parameter is deduced from the wave equation for the complex beam parameter is deduced from the wave equation, and various of its solutions are discussed. This includes a discussion of light propagation in media with a gain profile, which are found capable of supporting stationary beams of constant diameter.
© 1965 Optical Society of America
This paper considers the propagation of optical modes or Gaussian beams of light of the kind produced by laser oscillators or used in microwave beam waveguides. It discusses the propagation of these beams through optical systems like lenses, gas lenses, and quite general lenslike media including media exhibiting a nonuniform loss or gain profile. An example of a medium with a nonuniform gain profile is the medium of a gas laser amplifier where the gain decreases with the distance from the axis of the laser tube.
The laws governing the propagation of the light beam will be formulated in terms of a complex valued beam parameter. This complex parameter gives information on the width of the beam (spot size), and on the curvature of the phase front in each beam cross section of interest. The advantage of using a complex beam parameter is apparent from the recent literature. Following the initial proposal by Collins various circle diagrams for Gaussian beams were introduced,, which are (or can be regarded as) plotted in the complex plane of a beam parameter of this kind. It was also found that the laws of beam propagation in homogeneous media can be written in a very simple form if a complex notation is used.– It will be seen that the use of a complex parameter also simplifies the analysis of beam propagation in lenslike media.
First, the paper recalls the laws of propagation for spherical waves in lossless lenslike media in the geometrical optics approximation. The variation of the phase fronts of these waves along the optic axis is described in terms of the elements of the ray matrix. These matrix elements can be obtained by tracing paraxial rays through the medium. Then Gaussian beams in lossless lenslike media are considered, and it is found that the law for the transformation of the curvature radius of spherical waves is formally the same as the transformation law for the complex beam parameter (ABCD law).
The paper continues by discussing wave propagation in lenslike media where loss or gain varies with the square of the distance from the optic axis. Assuming paraxial waves of small wavelengths, a parabolic equation is obtained from the scalar wave equation for these media. This parabolic equation is a generalization of the parabolic equation for a homogeneous medium which is used in diffraction theory. Light beams with a Gaussian amplitude profile are obtained as a solution of this parabolic equation together with a differential equation of first order. It can be transformed into a linear differential equation of second order which has the form of the paraxial ray equation for the medium. The connection between the beam parameter equation and the differential equation used by Tien et al. is studied and several solutions are discussed. Beam propagation in media with a gain profile is investigated in greater detail.
The paper is concluded by a brief discussion of analog electric circuits which correspond to the various lenslike media. The input and output admittances of these circuits (or impedances in the dual circuits) correspond to the complex parameters of the beam in the input and output planes of the optical system as in the lumped circuit analogs described by Deschamps and Mast.
Propagation of Spherical Waves in the Geometrical Optics Approximation
Some well-known properties of spherical waves are recalled here for later comparison with the laws obeyed by Gaussian beams. Consider first the curvature of two phase fronts of a spherical wave propagating in the z direction. Call the radii of curvature of the two phase fronts R1 and R2, respectively, and z the distance between the intersections of these phase fronts with the optic axis. It is quite obvious from Fig. 1 that the curvature radii are related by
If a spherical wave passes through a lens of focal length f, it is transformed into another spherical wave. For an incoming wave with a radius of curvature R1 at the lens the wave going out from the lens has a curvature radius R2 given by
Consider now paraxial light rays passing through a medium whose refractive index near the optic axis is described by,, Fig. 2 there is schematically indicated a ray path through a lenslike medium and an input and an output plane. The quantities of interest are also indicated. They are the ray position x2 and the ray slope x2′ in the output plane and the corresponding ray parameters x1 and x1′ in the input plane. The solution of the paraxial ray equation yields a linear relation between the input and output parameters which is conveniently written in matrix form,
The passage of rays through lenses and lens combinations can also be described in this matrix form, and the elements A, B, C, and D of the ray matrix are known for several optical structures. The matrix elements are determined by the over-all focal length of the optical structure of interest and by the position of its principal planes.,
The rays associated with a spherical wave are perpendicular to the wavefront. The position x and slope x′ of a paraxial ray are related to the radius of curvature R of the wavefront by
Propagation of Gaussian Beams, ABCD Law
The properties of Gaussian beams of light in homogeneous media are well known. Near the optic axis the field distribution E(r,z) of a fundamental Gaussian mode is described by
As the light beam propagates in space it expands due to diffraction, but the transverse field distribution remains Gaussian. The law of expansion is
It is convenient to combine the beam parameters w and R in a complex-valued parameter q defined by9) and (10) can then be written as one equation in the simple form– 13) the relation between the input parameter q1 and the output parameter q2 can then be written as
The parameters of a Gaussian beam are transformed when it passes through an ideal thin lens. The beam radii remain unchanged, i.e., the beam radius w1 immediately to the left of the lens will be equal to the beam radius w2 immediately to the right of it. The curvature radii R1 and R2 of the corresponding phase fronts are transformed in the same way as the curvature radii of spherical waves, i.e., they are related by (2). Therefore, the lens transformation law for the complex beam parameter is ,
The laws for the transformation of Gaussian beams are seen to be formally the same as the transformation laws obeyed by the spherical waves of geometrical optics. Equation (15) is formally the same as (1), and (16) is formally the same as (2). The complex beam parameter q plays a role corresponding to the one played by the radius R of the spherical waves and might be called a complex curvature radius.
Like other optical structures a lenslike medium can be regarded as composed of a set of thin lenses. The number of lenses required for a fair representation of the medium may, of course, be very large. But the fact remains that (15) and (16) can be used to trace a Gaussian beam through each lens element, while (1) and (2) are used to trace a spherical wave. As a result of ray tracing one has the law (7) for the transformation of a spherical wave by the optical structure. Because of the formal equivalence discussed above, the transformation law for Gaussian beams must be It allows one to trace Gaussian beams through any optical structure for which the elements A, B, C, and D of the ray matrix (or the focal length and the principal planes) are known. Its application to some optical structures of interest has been given in [ref. 3].
Almost-Plane Waves in Lenslike Media
Pierce starts from the wave equation to derive the propagation laws for Gaussian beams in free space and in sequences of lenses. A modification of his results was used by Tien et al. for their study of beam propagation in various lenslike media. The present derivation of the propagation laws for Gaussian beams in quite general lenslike media starts also with the scalar wave equation*
The main interest here is in waves that propagate primarily in the z direction, or almost-plane waves. Accordingly, one writes21) becomes the usual exp (−jk0z). The function ψ is assumed to vary so slowly with z that its second derivative can be neglected. If one inserts (20), (21), and (22) into the wave equation (18) and neglects ψ″ one obtains equation (23) is a generalized form of the parabolic equation for lossless homogeneous media which was used by Leontovich et al. in their analyses of various diffraction phenomena. This parabolic equation takes into account the transverse diffusion of the field, but it neglects longitudinal diffusion. It yields good results for paraxial waves whose wavelengths are much smaller than their transverse dimensions.
A wave with a Gaussian transverse field distribution is an exact solution of Eq. (23). To determine the parameters of the Gaussian beam one can write25) into (23) one gets
Equation (28) is a nonlinear differential equation of the Ricatti type. Its connection to the paraxial ray equation, and several of its solutions will be discussed below.
Beam Parameter Equation and Paraxial Ray Equation
Solutions of (28) describe the propagation properties of Gaussian beams. These solutions can also be constructed from solutions of the paraxial ray equation. The connection between these two equations is made by28) one arrives at equation (4) except that one has, in general, complexvalued k0 and k2 instead of the real n0 and n2 which describe lossless media. In media with loss or gain the variable x is also complex-valued, and its physical meaning is not so clear as its meaning as a ray position in lossless media. However, the general solutions of (30) can be combined according to (29) to yield solutions of (28) for the complex beam parameter which is a measure for the width and the phase-front curvature of Gaussian beams. Equation (29) is essentially the ABCD law (17) extended in validity to media with loss or gain.
The example of a medium with k0 = const and k2 = const will help to illustrate the preceding remarks. For k2 ≠ 0 such a medium can support Gaussian beams whose diameter does not change along the optic axis. The Q parameter of this kind of a beam is obtained from (28) by postulating that Q′ = 0. One finds30). The general solution of this equation is 32) and its derivative with respect to z into (29). After some rearranging one arrives at
Refractive Indices and Gain Constants
For a more detailed study of beam propagation the complex propagation constant is separated into its real and imaginary part36) with (19) and gets 35) the power of a plane wave grows like
Near the optic axis the propagation constant of a lenslike medium changes generally as in
It is not difficult to relate the above gain and propagation constants to the complex constants used in the beam parameter equations. It follows from (20) that, near the axis, the complex propagation constant is
Keeping in mind that the gain per wavelength is small, one can write for the product k0k2
Solution for Beams in Free Space
In this section the solution of the beam parameter equation for Gaussian beams in free space is discussed briefly. This will serve to connect the present method to the published literature on the subject, and to demonstrate that the parabolic equation (23) leads to the same results as the theory of Fresnel diffraction.28), the beam parameter equation becomes 48) is, clearly, 15).14). As is well known, the real and imaginary parts of the phase shift can then be separated by using the identity 8), and the real part predicts the same additional phase shift as (11).
The Lossless Lenslike Medium,, of Gaussian beam propagation in lenslike media. A familiar result for media with constant n0 and n2 is the guided beam of constant diameter 2wm whose parameter qm is obtained from (52) by putting q′ = 0.  have studied the propagation of Gaussian beams through various media, each characterized by refractive indices n0(z) and n2(z) with a certain z dependence of practical interest. They have based their work on a differential equation for the beam radius w. In order to see the connection between their differential equation and the beam parameter equation (52) one introduces in the latter the definition of the q parameter given in (12). By comparing the imaginary parts of (52) one then obtains  One then compares the real parts of (52), eliminates R by means of (54), and arrives at equation (12) of Tien et al. It is interesting to note that the differential equation (52) for the complex beam parameter q has a simpler form than the differential equation (55) for the real beam radius w.
Medium with a Gain Profile
In this section the propagation of Gaussian beams in a medium with a quadratic gain profile is studied in greater detail. This is a medium where the gain decreases with the distance from the axis according to (42). The plasma medium of various gas laser tubes is of that nature.
For simplicity it is assumed here that the gain constants are independent of z, i.e., α0 = const and α2 = const. The refractive index of the medium is taken to be that of free space, β0 = 2π/λ and β2 = 0. With these assumptions (46) becomes31) it follows that a medium with a gain profile is capable of supporting a beam of constant diameter even though there is no refractive index variation. The q parameter of this stationary beam is computed from (31) and (56) as 57) that the radius of curvature Rm of these wavefronts is
For several practical laser tubes it is reasonable to assume that the gain constant has its largest value at the tube axis, decreases quadratically, and is zero at the tube walls. If one calls the tube radius r0 one can, then, write for the gain constant42) one has 58) and (59) become
The reported gain of a He–Xe amplifier tube of 4-mm diam (r0 = 2 mm) at a wavelength of 3.5 μ is about 100 dB/m. From this value one computes with (39) a gain constant on the axis of approximately62) one calculates for Rm and wm the values
For the stationary beam the formula (27) for the complex phase shift becomes
A stationary beam is obtained only when the light is so injected into the medium that the incoming beam matches the stationary beam in diameter and phase-front curvature. If these matching conditions are not satisfied, the beam will generally fluctuate along z in diameter and curvature. To study this situation in more detail one has to evaluate the constant γ given in (33). For the medium with a gain profile one gets34) together with (47) to yield the result
Equation (69) can be so cast as to show directly the fluctuations of both the wavefront curvature and the beam diameter. In order to do that one defines two angles δ and φ by69) in the form 72) leads to an expression for the fluctuation of the curvature radius R2 of the beam phase front 73) indicate the fluctuations of the beam radius w2
Electric Circuit Analogs
The equations (15) and (16) for the complex beam parameter have the same form as the well-known formulas of the input impedance of a series circuit of two impedances and of a parallel circuit of two impedances. On this basis Deschamps and Mast have discussed lumped electric circuits which are analogs of systems of lenses. Two dual circuit analogs of a system of lenses are shown in Fig. 3.
The first is an admittance analog. Here the admittance Y of each section is related to the complex beam parameter in the corresponding beam cross section by
Below the admittance analog there is shown the dual impedance analog. In the latter all admittances from above are replaced by impedance
The analog of a lenslike medium is a distributed electric circuit or transmission line. The voltage V on such a line obeys the differential equationEquation (77) corresponds to the paraxial ray equation (30) if one puts
The author would like to thank J. P. Gordon, J. W. Kluver, W. H. Louisell, and P. K. Tien for helpful comments and stimulating discussions.
|*||A treatment of the exact vector problem has to start from Maxwell’s equations, where one obtains this form of the wave equation for the electric field E in a medium of constant permeability u and space-dependent dielectric constant ∊ by neglecting the term grad div|
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