## Stable aggregates of flexible elements give a stronger link between rays and waves

Optics Express, Vol. 10, Issue 16, pp. 728-739 (2002)

http://dx.doi.org/10.1364/OE.10.000728

Acrobat PDF (351 KB)

### Abstract

A recently proposed ray-based method for wave propagation is used to provide a meaningful criterion for the validity of rays in wave theory. This method assigns a Gaussian contribution to each ray in order to estimate the field. Such contributions are inherently flexible. By means of a simple example, it is shown that superior field estimates can result when the contributions are no longer forced to evolve like parabasal beamlets.

© 2002 Optical Society of America

## 1. Introduction

2. Yu. A. Kravtsov and Yu. I. Orlov, *Geometrical Optics of Inhomogeneous Media* (Springer, Berlin, 1990). [CrossRef]

3. Yu. A. Kravtsov and Yu. I. Orlov, *Caustics, Catastrophes and Wave Fields* (Springer, Berlin, 2^{nd} Edition, 1999). [CrossRef]

4. G.W. Forbes and M.A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A **18**, 1132–1145 (2001). [CrossRef]

7. G.W. Forbes, “Using rays better. IV. Refraction and reflection,” J. Opt. Soc. Am. A **18**, 2557–2564 (2001). [CrossRef]

## 2. Sums of Gaussian contributions

*U*considered here satisfies the Helmholtz equation.

*k*= 2

*π*/

*λ*and

*n*is the refractive index. We assume that this field propagates essentially (but not necessarily paraxially) towards larger

*z*. (That is, it is assumed that rays do not turn around in

*z*.) A one-parameter family of rays can be associated with this two-dimensional wave field. The rays can be written as

*x*=

*X*(

*ξ*;

*z*), where

*ξ*smoothly identifies the rays. The initial conditions for these rays, namely

*X*(

*ξ*;0) and

*∂X*/

*∂z*(

*ξ*;0), must be determined from the specified initial field

*U*(

*x*,0). The rays are then traced through the medium by using standard ray equations:

*P*and

*H*correspond, respectively, to

*n*(

*X*,

*z*) times the

*x*and

*z*components of a unit vector locally tangent to the ray. Hence

*L*, satisfies

*L*is the eikonal evaluated at [

*X*(

*ξ*;

*z*),

*z*].

*w*is a weight function and

*g*is the Gaussian contribution given by

*z*, the contribution in Eq. (6) is centered at

*x*=

*X*(

*ξ*;

*z*) and has a transverse phase that is linear and matches that of a plane wave propagating in the ray’s local direction. The width of the Gaussian is evidently determined by

*γ*. We now summarize two ways in which the form in Eq. (5) can be forced to satisfy Eq. (1) asymptotically:

### a) Gaussian beamlets

*wg*is forced to be an asymptotic solution of Eq. (1), i.e. to evolve like a parabasal Gaussian beamlet around the ray identified by

*ξ*. As discussed in the Appendix of Ref. 4

4. G.W. Forbes and M.A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A **18**, 1132–1145 (2001). [CrossRef]

*w*,

*γ*,

*P*, and

*H*are functions of

*ξ*and

*z*, and

*n*and its derivatives are evaluated at [

*X*(

*ξ*;

*z*),

*z*]. The beamlets that result from these expressions are asymptotic solutions of the wave equation, valid under what is known as the parabasal approximation, that is, the assumption that the beamlet remains well localized around its central ray. The field estimate for

*z*> 0 results from the sum of the independent beamlets.

### b) SAFE

4. G.W. Forbes and M.A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A **18**, 1132–1145 (2001). [CrossRef]

*ξ*can then be used to mix different terms that could not be combined in the beamlet approach. The simple form of the weight function that results was shown in Ref. 4

**18**, 1132–1145 (2001). [CrossRef]

**18**, 1132–1145 (2001). [CrossRef]

6. M.A. Alonso and G.W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A **18**, 1357–1370 (2001). [CrossRef]

*γ*is now a free parameter and not constrained by the likes of Eq. (8). This means that the Gaussian contributions are not forced to evolve (and disintegrate) like beamlets. Even though they don’t accurately satisfy Eq. (1) individually, they do so collectively. The form of the weight in Eq. (9) not only guarantees that the estimate in Eq. (5) satisfies the Helmholtz equation asymptotically, it also makes the estimate asymptotically insensitive to the width parameter

*γ*. That is, as shown in Ref. 4

**18**, 1132–1145 (2001). [CrossRef]

*k*or sending it to infinity:

*a ray family has an adequate connection to a wave field only when the width of the contributions can be varied over a significant range without disrupting the associated field estimate.*That is, the estimate must remain within the desired accuracy window. In cases when there are no caustics, the width of the contributions can even be made to vanish without changing the estimate significantly. In this case SAFE reduces to the most standard ray-based construction, where a ray is interpreted as a local conduit of power. Similarly, when there are no “momentum caustics” (i.e. neighboring rays are parallel), the contribution’s width can be sent to infinity, and we get another standard interpretation of a ray, namely a plane wave. In cases when these two options are valid, SAFE gives a continuous link between them. More generally, both extremes fail catastrophically, but SAFE remains valid for a wide range of intermediate values of

*γ*.

*a*

_{0}). For the beamlet summation case, on the other hand, one must also trace the width and weight of each beam. The bounds for the variation of

*γ*mentioned above [which follow from the explicit form of the right-hand side of Eq. (10) as presented in Ref. 4

**18**, 1132–1145 (2001). [CrossRef]

## 3. Example: Modes of a quadratic gradient-index waveguide

6. M.A. Alonso and G.W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A **18**, 1357–1370 (2001). [CrossRef]

*n*

_{0}and

*ν*both positive and real. The modes of this waveguide are Hermite-Gaussians of the form

*k¯*=

*kn*

_{0}/

*ν*and H

_{m}is the Hermite polynomial of order

*m*. When

*m*>

*M*= (

*k¯*-1)/2, the argument of the square root within the second exponential becomes negative. In this case, the positive imaginary root must be chosen, so the corresponding modes decay exponentially in

*z*, i.e. they are evanescent. For 0 ≤

*m*≥

*M*, the modes in Eq. (12) are homogeneous, and they linearly accumulate a phase as they propagate in

*z*.

### 3.1 Matching the initial condition and achieving insensitivity

6. M.A. Alonso and G.W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A **18**, 1357–1370 (2001). [CrossRef]

**U**

_{m}(

*x*,0) was proposed in Ref. 5

5. M.A. Alonso and G.W. Forbes, “Using rays better. II. Ray families to match prescribed wave fields,” J. Opt. Soc. Am. A **18**, 1146–1159 (2001). [CrossRef]

**U**

_{m}(

*x*,0), and

*ξ*, varies over an interval of length 2

*π*. It follows from Eqs. (2) and (11) that these rays propagate according to

*H*(

*ξ*;

*z*) takes for this example. The corresponding ray family is overlaid onto the intensity profile of the mode for

*m*= 8 in Fig. 1.

*z*, all rays are identical and have a uniform weight, i.e.

*a*

_{0}(

*ξ*) =

*c*. The optical length of each ray follows from Eqs. (4):

*X*and

*P*, this function is not periodic in

*ξ*. It can be seen from Eqs. (5), (6), and (9), however, that the values of

*X*

_{0}and

*P*

_{0}given after Eq. (13) guarantee that the field contribution

*wg*is periodic when the square root in Eq. (9) is chosen to vary continuously in

*ξ*. That is, by admitting only these discrete values for

*X*

_{0}and

*P*

_{0}, this “quantization” condition ensures that the resulting estimate is independent of the chosen interval.

5. M.A. Alonso and G.W. Forbes, “Using rays better. II. Ray families to match prescribed wave fields,” J. Opt. Soc. Am. A **18**, 1146–1159 (2001). [CrossRef]

*γ*is chosen to be

*γ*

_{0}=

*n*

_{0}

*ν*, the initial condition

**U**

_{m}(

*x*,0) is met exactly by the estimate that results upon substituting Eq. (6) and the leading order of Eq. (9) into Eq. (5) with

*z*= 0. Here, the ray family is the one discussed above. Although the match is exact, we claim that this construction gives a meaningful connection between rays and waves only if the estimate is insensitive to changes in

*γ*over a significant interval around

*γ*

_{0}. (We only consider here real values for

*γ*, although this parameter can be complex in general as discussed in Refs. 4–6

**18**, 1132–1145 (2001). [CrossRef]

*k*, but such observations are not useful in practice. However, we have derived an explicit insensitivity condition. In the present case, insensitivity demands that

*z*, the ray family can be represented by the parametric curve [

*X*(

*ξ*;

*z*),

*P*(

*ξ*;

*z*)], for varying

*ξ*, in the (

*x*,

*p*) plane referred to as phase space. It is seen from Eqs. (14) that, for the present example, this curve is always an ellipse enclosing an area of

*πX*

_{0}

*P*

_{0}=

*π*(2

*m*+ 1)/

*k*. The geometric criterion for insensitivity found in Ref. 4

**18**, 1132–1145 (2001). [CrossRef]

*γ*must be chosen to maximize the tightest local radius of curvature of the curve

*γ*. The value

*γ*

_{0}=

*n*

_{0}

*ν*turns the ellipse into a circle with a radius of

*γ*deforms the circle into an ellipse, although the enclosed area is fixed at

*π*(2

*m*+ 1). Clearly, a larger circle can withstand stronger deformations before its peak curvature becomes problematic.

*m*=8, the deformation of the circle and the corresponding variation of the field estimates (and resulting errors) associated with variation of

*γ*. Notice that the errors gather in

*x*around the caustics (at |

*x*| =

*X*

_{0}in this case) for larger

*γ*, and around the momentum caustics (at

*x*= 0) for smaller

*γ*. Although

*γ*is varied by a factor of four for the animation — hence the ray localization varies by a factor of two —the heights of the intensity peaks change by no more than about five percent. Of course, this insensitivity is far more pronounced for higher order modes (where it is usual to expect ray methods to apply).

### 3.2 Propagation

#### a) Gaussian beamlets

*z*> 0, we must calculate the evolution of the width and weight for each beamlet. The initial conditions to be used are

*γ*(

*ξ*;0) =

*γ*

_{0}and

*a*

_{0}=

*c*] which, as discussed in the previous subsection, make the continuous superposition in Eq. (5) match exactly the initial field. For this case, and by using Γ(

*ξ*;

*τ*) =

*γ*(

*ξ*;

*H*

_{0}

*τ*/

*n*

_{0}

*ν*)/

*γ*

_{0}, Eq. (8) becomes

*P*

_{0}/

*H*

_{0}= [(

*m*+ 1/2)/(

*M*-

*m*)]

^{1/2}is a measure of the non-paraxiality of the mode. Notice that

*ρ*is small only when

*m*≪

*M*. That is, only the low-order modes in a waveguide that supports many propagating modes are in fact paraxial. The solution to this equation is given by

*z*. In this simple example, the rays are known in closed form. In general, however, rays must be traced numerically, so we can only keep track of a finite, discrete set of them. To simulate this, we replace the integral in Eq. (5) by a sum at uniform intervals in

*ξ*of 2

*π*/

*N*. Fig. 3a shows, for

*m*= 8 and several values of

*N*, the evolution in

*z*of the sums of beamlets compared to the exact result. Notice that, with the chosen initial conditions, the initial field is well matched by the estimate even for a relatively small number of contributions. As

*z*increases, however, the numerical errors caused by the discretization become apparent. Further, it turns out that these are not the only sources of error. Even for very large

*N*, where the sum approaches the integral and the result retains a Hermite-Gaussian form (due to an intrinsic symmetry of this problem), the amplitude and phase go drastically astray. These variations are shown in Fig. 4. The amplitude (which should be constant) initially increases with

*z*, and then decays roughly as

*z*

^{-0.53}. The source of this fundamental error is explained by the diagram in Fig. 3b, which shows schematically the region of phase space occupied by each beamlet. As

*z*increases, these Gaussians rotate and shear linearly, reaching spatial spreads well beyond the parabasal regime. The rate of deterioration of this estimate is then dictated by the rate of shearing of each beamlet, which as seen from the second expression in Eq. (17) is given by

*ρ*

^{2}

*τ*/

*z*=

*γ*

_{0}

*πρ*

^{2})

^{-1}ray periods for the estimate to fail. That is, modes that are closer to the paraxial limit (

*ρ*→ 0) have longer-lived estimates. This makes sense because, in the paraxial limit, the medium in Eq. (11) becomes the optical analogue of the harmonic oscillator, the beamlets are analogous to coherent states, and the parabasal approximation made in the derivation of Eqs. (7) and (8) is exactly valid. Beyond the paraxial limit, the beamlets do not individually retain their Gaussian profile.

#### b) SAFE

*z*, the phase space curve is always the same ellipse, and rays just circulate around it. Therefore, the most convenient choice for

*γ*is always

*γ*

_{0}=

*n*

_{0}

*ν*. By changing the ray parameter to

*ξ*′ = ξ -

*n*

_{0}

*ν*

*z*/

*H*

_{0}, the estimate in Eq. (5) becomes independent of

*z*, except for a phase factor due to the last term in Eq. (15). This phase matches exactly the linear phase in Eq. (12), and the estimate matches, at all

*z*, the exact result. That is, for this example, SAFE can always match the exact result. Further, the result is equally insensitive at all

*z*, so the grip between rays and waves holds indefinitely. Even when a discrete number of contributions is used, the estimate is quite accurate, as seen from the initial frame of Fig. 3 (where

*z*= 0). This frame characterizes SAFE’s results at all

*z*. Notice also that the non-paraxiality of the problem has no impact on the validity of SAFE.

### 3.3 Discussion

*z*, it is nevertheless representative of a more general behavior for (at least) smooth waveguides in which

*n*is independent of

*z*. Any guided ray in such medium has a periodic trajectory, which in phase space describes a closed loop (symmetric around the

*x*axis) parametrized by

*z*. Rays belonging to different loops have different periods, meaning that the waveguide is anharmonic. [The only harmonic optical waveguide is the one in Eq. (11) within the paraxial approximation.] The ray families associated to the modes of the waveguide correspond to all the rays that compose one of these phase space loops (i.e. identical rays shifted in

*z*) with the “quantization” condition that the area enclosed by the loop is

*π*(2

*m*+ 1)/

*k*. A suitable value (or range of values) for

*γ*can be chosen from the geometric criterion for insensitivity, and the desired mode can then be constructed approximately. Its profile is independent of

*z*. If, on the other hand, we let each Gaussian propagate as a beam, the anharmonicity will cause it to stretch in phase space in a form similar to that in Fig. 3b, causing the estimate to deteriorate under propagation. Therefore, for smooth waveguides, it is always more effective to make

*γ*independent of

*z*instead of forcing it to evolve as the width of a Gaussian beamlet. This constancy of

*γ*gives an analogue to Heller’s “frozen Gaussian approximation” used in semiclassical quantum mechanics [8]. There are other options, however, considered in Refs. 4–6

**18**, 1132–1145 (2001). [CrossRef]

*γ*is allowed to depend on

*ξ*, and this can help to further enhance SAFE’s estimates for more complex waveguides.

*z*increases, the anharmonicity causes the resulting curve to get wrapped up in a spiral-like form, where each turn becomes increasingly similar to one of the natural loops. Again, the corresponding propagated beamlets would be highly stretched in phase space, causing errors in the estimate. SAFE, on the other hand, can handle such cases well, as seen in the example in Ref. 6

**18**, 1357–1370 (2001). [CrossRef]

## 4. Propagation through an interface

*z*=

*z*

_{1}and a homogeneous medium of refractive index

*n*

_{1}. The objective is to find the transmitted and reflected fields that are generated by the incidence of a waveguide mode at the interface. This problem has no closed-form solution, and an accurate analysis may appear to be beyond the reach of rays.

7. G.W. Forbes, “Using rays better. IV. Refraction and reflection,” J. Opt. Soc. Am. A **18**, 2557–2564 (2001). [CrossRef]

*U*

^{t}and

*U*

^{r}resulting from the incidence of

*U*

^{i}=

**U**

_{m}at the interface are estimated from the corresponding transmitted and reflected ray families by using the prescription in Eqs. (5), (6), and (9). By requiring that the boundary conditions, namely

*U*

^{i}

_{γ0}+

*U*

^{r}

_{γ0}=

*U*

^{t}

_{γ0}and

*z*=

*z*

_{1}, are asymptotically satisfied, it follows that the weights used in these estimates are given by the following intuitive relations:

*q*=

*n*

_{1}/

*n*

_{0}. Note that the first part of Eq. (21) follows from the fact that, due to Snell’s law,

*P*is conserved across a flat interface perpendicular to

*z*. In the last part of Eq. (21), we used Eq. (14) as well as

*n*

_{1}<

*P*

_{0}), the rays hitting the interface in the vicinity of the

*z*axis are totally reflected. As a result, the corresponding transmitted rays have a (positive) imaginary

*H*

_{1}, i.e. the transmitted field has evanescent components. Estimates of the resulting wavefield when the incident mode has

*m*= 8 and

*ρ*= 1 are shown in Fig. 5 as an animation where

*q*=

*n*

_{1}/

*n*

_{0}is changed in time.

*a*

_{1}can depend on

*z*and

*γ*as well as on

*ξ*. For the incident field we let

*a*

_{1}(

*ξ*,

*γ*

_{0};

*z*) = 0 since, with

*γ*=

*γ*

_{0}and

*a*

_{0}=

*c*, SAFE gives an exact reconstruction of

**U**

_{m}(so no correction is needed). As with

*a*

_{0}earlier, the values of

*a*

_{1}at

*z*

_{1}for the transmitted and reflected field estimates are found from the boundary conditions (as in Ref. 7

7. G.W. Forbes, “Using rays better. IV. Refraction and reflection,” J. Opt. Soc. Am. A **18**, 2557–2564 (2001). [CrossRef]

*h*(

*ξ*) =

*H*

_{1}(

*ξ*)/

*H*

_{0}. [A simple prescription for finding the functions in Eq. (23) at other values of

*γ*is given in Ref. 6

**18**, 1357–1370 (2001). [CrossRef]

*γ*can be found by minimizing the error estimates discussed below. We do not bother with this here, however.]

*a*

_{0}in order to improve the accuracy of the estimates, roughly doubling the number of significant digits. Alternatively, they can provide a valuable measure of error of the estimates. It was shown in Ref. 6

**18**, 1357–1370 (2001). [CrossRef]

*ε*

_{γ}^{r}for the transmitted and reflected field estimates are given by

*T*and

*R*are given by

*z*<0 is generally dominated by the incident field, the relative error in the total field is typically much less than the value indicated by the orange curve in Fig. 6. It is also interesting that the reflected field is no longer propagated without error by SAFE. This follows from the fact that

*ξ*) is not constant.

## 5. Concluding remarks

9. M.M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion **4**, 85–97 (1982). [CrossRef]

15. B.Z. Steinberg, E. Heyman, and L.B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A **8**, 41–59 (1991). [CrossRef]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | Yu. A. Kravtsov and Yu. I. Orlov, |

3. | Yu. A. Kravtsov and Yu. I. Orlov, |

4. | G.W. Forbes and M.A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A |

5. | M.A. Alonso and G.W. Forbes, “Using rays better. II. Ray families to match prescribed wave fields,” J. Opt. Soc. Am. A |

6. | M.A. Alonso and G.W. Forbes, “Using rays better. III. Error estimates and illustrative applications in smooth media,” J. Opt. Soc. Am. A |

7. | G.W. Forbes, “Using rays better. IV. Refraction and reflection,” J. Opt. Soc. Am. A |

8. | E.J. Heller, “Frozen Gaussians: A very simple semiclassical approximation,” J. Chem. Phys. |

9. | M.M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion |

10. | V.M. Babich and M.M. Popov, “Gaussian summation method (review),” Izvestiya Vysshikh Zavedenii, Radiofizika |

11. | M.J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik |

12. | A.N. Norris, “Complex point-source representation of real point sources and the Gaussian beam summation method,” J. Opt. Soc. Am. A |

13. | P.D. Einziger, S. Raz, and M. Shapira, “Gabor representation and aperture theory,” J. Opt. Soc. Am. A |

14. | P.D. Einziger and S. Raz, “Beam-series representation and the parabolic approximation: the frequency domain,” J. Opt. Soc. Am. A |

15. | B.Z. Steinberg, E. Heyman, and L.B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A |

16. | J.M. Arnold, “Phase-space localization and discrete representation of wave fields,” J. Opt. Soc. Am. A |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(080.2720) Geometric optics : Mathematical methods (general)

(350.7420) Other areas of optics : Waves

**ToC Category:**

Focus Issue: Rays in wave theory

**History**

Original Manuscript: June 19, 2002

Revised Manuscript: July 30, 2002

Published: August 12, 2002

**Citation**

Miguel Alonso and G. Forbes, "Stable aggregates of flexible elements give a stronger link between rays and waves," Opt. Express **10**, 728-739 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-728

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### References

- M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, Cambridge, 7th edition, 1999), pp. 116-129.
- Yu. A. Kravtsov andYu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, Berlin, 1990). [CrossRef]
- Yu. A. Kravtsov andYu. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, Berlin, 2nd Edition, 1999). [CrossRef]
- G.W. Forbes and M.A. Alonso, �??Using rays better. I. Theory for smoothly varying media,�?? J. Opt. Soc. Am. A 18, 1132-1145 (2001). [CrossRef]
- M.A. Alonso and G.W. Forbes, �??Using rays better. II. Ray families to match prescribed wave fields,�?? J. Opt. Soc. Am. A 18, 1146-1159 (2001). [CrossRef]
- M.A. Alonso and G.W. Forbes, �??Using rays better. III. Error estimates and illustrative applications in smooth media,�?? J. Opt. Soc. Am. A 18, 1357-1370 (2001). [CrossRef]
- G.W. Forbes, �??Using rays better. IV. Refraction and reflection,�?? J. Opt. Soc. Am. A 18, 2557-2564 (2001). [CrossRef]
- E.J. Heller, �??Frozen Gaussians: A very simple semiclassical approximation,�?? J. Chem. Phys. 75, 2923-2931 (1981). [CrossRef]
- M.M. Popov, �??A new method of computation of wave fields using Gaussian beams,�?? Wave Motion 4, 85-97 (1982). [CrossRef]
- V.M. Babich and M.M. Popov, �??Gaussian summation method (review),�?? Izvestiya Vysshikh Zavedenii, Radiofizika 32, 1447-1466 (1989).
- M.J. Bastiaans, �??The expansion of an optical signal into a discrete set of Gaussian beams,�?? Optik 57, 95-102 (1980).
- A.N. Norris, �??Complex point-source representation of real point sources and the Gaussian beam summation method,�?? J. Opt. Soc. Am. A 12, 2005-2010 (1986). [CrossRef]
- P.D. Einziger, S. Raz, and M. Shapira, �??Gabor representation and aperture theory,�?? J. Opt. Soc. Am. A 3, 508-522 (1986). [CrossRef]
- P.D. Einziger and S. Raz, �??Beam-series representation and the parabolic approximation: the frequency domain,�?? J. Opt. Soc. Am. A 5, 1883-1892 (1988). [CrossRef]
- B.Z. Steinberg, E. Heyman, and L.B. Felsen, �??Phase-space beam summation for time-harmonic radiation from large apertures,�?? J. Opt. Soc. Am. A 8, 41-59 (1991). [CrossRef]
- J.M. Arnold, �??Phase-space localization and discrete representation of wave fields,�?? J. Opt. Soc. Am. A 12, 111-123 (1995). [CrossRef]

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