## Waves, rays, and the method of stationary phase

Optics Express, Vol. 10, Issue 16, pp. 740-751 (2002)

http://dx.doi.org/10.1364/OE.10.000740

Acrobat PDF (457 KB)

### Abstract

If one employs a diffraction-integral approach to wave propagation and diffraction, the connection between waves and *conventional* geometrical and diffracted rays is provided by the Method of Stationary Phase (MSP). However, conventional ray methods break down in focal regions because of the coalescense of stationary points. Then one may use the MSP to express the focused field in terms of *aperture-plane Point-Spread-Function (PSF)* rays. A tutorial review of these two ray techniques is given, and a number of applications are discussed with emphasis on the physical interpretation. Examples include focusing in free space or through a plane interface, plane-wave diffraction by a circular aperture, and diffraction of a Gaussian beam by a circular aperture followed by transmission into a biaxial crystal.

© 2002 Optical Society of America

## 1 Introduction

*shortest*time, from which rectilinear propagation in free space, the reflection law, and Snell’s law readily follow. In 1924 Rubinowicz [1

1. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. Lpz. **73**, 339–364(1924). [CrossRef]

*boundary-diffracted rays*, and in the 1950s Keller ([2], [3

3. J.B. Keller, “Diffraction by an aperture,” J. Appl. Phys. **28**, 426–444 (1957). [CrossRef]

4. J.B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. **52**, 116–130 (1962). [CrossRef] [PubMed]

*geometrical theory of diffraction*. According to Keller’s theory, a diffracted ray satisfies

*Fermat’s principle of edge diffraction*, which states that an edge-diffracted ray from a source point

*S*to an observation point

*P*is the curve that has stationary optical path length among all curves from

*S*to

*P*with one point on the edge.

*conventional*rays and

*aperture-plane Point-Spread-Function (PSF)*rays in section 2, using the diffraction of a converging spherical wave through a circular aperture as an example. In section 3 we illustrate the use of aperture-plane PSF rays in connection with focusing through a plane interface between two isotropic media. In section 4 we study the diffraction of a plane wave by a circular aperture with particular emphasis on the axial caustic, the Poisson spot, and the axial interference pattern. In section 5 we study a Gaussian beam that is diffracted through a circular aperture and subsequently transmitted through a plane interface into a biaxially anisotropic medium. Finally, in section 6 we give some concluding remarks.

## 2 Conventional rays and aperture-plane Point-Spread-Function (PSF) rays

### 2.1 Conventional rays

*conventional*rays and

*aperture-plane Point-Spread-Function (PSF)*rays, we consider a scalar converging spherical wave with focus on the

*z*axis that is incident upon a circular aperture

*A*of radius

*a*, centered at

*x*=

*y*= 0 in the plane

*z*= 0 (Fig. 1). To determine the field in the half-space

*z*> 0 we use the first Rayleigh-Sommerfeld diffraction formula combined with a Kirchhoff-type of approximation. The latter implies that the field in the aperture plane is taken to be equal to the incident field inside the aperture and equal to zero outside the aperture. Provided the observation distance

*R*

_{2}in Fig. 1 is much larger than the wavelength λ, the focused field is given by the diffraction integral (Eq. (11.29a) in [5] with

*u*

_{0}(

*x*,

*y*) = 1)

*conventional*geometrical and diffracted rays we assume that the observation point

*P*

_{2}(

*x*

_{2},

*y*

_{2},

*z*

_{2}) neither lies in the vicinity of the focus at

*P*

_{1}(0, 0,

*z*

_{1}) nor in the vicinity of the geometrical shadow boundary in Fig. 1. Then the stationary point

*P*

_{s}(

*x*

_{s},

*y*

_{s}) in the aperture plane

*z*= 0, which is determined by the requirement that

*x*

_{s},

*y*

_{s}), does not lie near the aperture boundary. A simple calculation shows that the stationary point

*P*

_{s}lies on the geometrical ray that passes through the focus and the observation point, as shown in Fig. 1. Consider first the case in which the stationary point lies inside the aperture, but not close to the aperture boundary. Then the contribution

*u*

_{IS}of the stationary point is given by the asymptotic formula (lowest-order term of Eq. (9.7a) in [5])

*R*

_{jS}(

*j*= 1,2) is the value of

*R*

_{j}at the stationary point. Thus, as expected, the contribution from the interior stationary point is equal to the geometrical-optics field. We emphasize that this result is valid only at observation points far from both the focus and the shadow boundary. Note the well-known phase change of

*π*on passage through the focus contained in (7).

*u*

_{IB}from the aperture boundary to the integral in (1) can be obtained via a change to polar integration variables followed by integration by parts with respect to the radial variable. For observation points on the

*z*axis this gives (Eq. (11.54b) with

*u*

_{0}(

*a*) = 1 in [5])

*u*

_{0}(0) =

*u*

_{0}(

*a*) = 1)

*z*axis. This phenomenon is intimately connected with the existence of an

*axial caustic*, which we discuss in more detail in section 4.

*i*λ to compensate for the fact that here the incident field is a converging spherical wave, whereas the result in Eq. (12.54a) pertains to an incident converging dipole wave)

*x*) = sin(

*x*)/

*x*and

*off*the

*z*axis but

*well inside*the lit cone in Fig. 1, there are only two critical points on the aperture boundary, namely those for which the distance

*R*

_{2}from the aperture boundary to the observation point is a maximum or a minimum. The secondary waves emitted by these two boundary points are associated with

*diffracted rays*, and their contributions can be obtained using the MSP in a similar manner as explained in section 4 for a normally incident plane wave. The total field now consists of three terms, the geometrical-optics contribution given by (7), and the contributions

*u*from the two diffracted rays, given by

_{IDRfoc}*well outside*the lit cone in Fig. 1, the stationary point lies outside the aperture. Hence it does not contribute to the diffracted field, which now is determined entirely by the contributions from the two critical points on the aperture boundary given by (14).

### 2.2 Aperture-plane Point-Spread-Function (PSF) rays

*kR*

_{2}≫ 1 was used to obtain the former result, whereas the latter result was obtained directly from the first Rayleigh-Sommerfeld formula without any approximations.

*u*

^{i}(

*x*,

*y*, 0) =

*δ*(

*x*-

*x*

_{0})

*δ*(

*y*-

*y*

_{0}), so that the field

*h*(

*x*

_{2}-

*x*,

*y*

_{2}-

*y*,

*z*

_{2}) becomes equal to the field at the observation point (

*x*

_{2},

*y*

_{2},

*z*

_{2}) that is due to a point source at the point (

*x*

_{0},

*y*

_{0}) in the aperture plane. Hence we call

*h*(

*x*

_{2}-

*x*,

*y*

_{2}-

*y*,

*z*

_{2}) the aperture-plane Point-Spread-Function (PSF). In (17) the aperture-plane PSF is expressed as an angular spectrum of plane waves. By applying a MSP formula analogous to that in (4) to the double integral in (17), we find that only one plane wave contributes, namely that which is directed from the point source at

*P*(

*x*,

*y*, 0) in the aperture plane to the observation point

*P*

_{2}(

*x*

_{2},

*y*

_{2},

*z*

_{2}). The asymptotic contribution

*h*

_{S}(

*x*

_{2}-

*x*,

*y*

_{2}-

*y*,

*z*

_{2}) obtained in this manner may be considered to be due to an aperture-plane PSF ray. It is given by

6. H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. Lpz. **60**, 481–500 (1919). [CrossRef]

*h*

_{S}(

*x*

_{2}-

*x*,

*y*

_{2}-

*y*,

*z*

_{2}) is approximately equal to

*h*

_{Exact}(

*x*

_{2}-

*x*,

*y*

_{2}-

*y*,

*z*

_{2}) when

*kR*

_{2}≫ 1. Thus, for propagation in free space the asymptotic result in (20) is superfluous. But for wave propagation in inhomogeneous media, the aperture-plane PSF integral can

*not*be evaluated analytically, whereas a very accurate approximation to it can be obtained using the MSP for double integrals [7 – 17

7. H. Ling and S.W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A **1**, 559–567 (1984). [CrossRef]

*conventional*rays and

*aperture-plane PSF*rays is that a conventional ray starts at the source of the incident field, whereas an aperture-plane PSF ray starts in the aperture plane

*z*= 0. Thus,

*an aperture-plane PSF ray is a conventional ray that connects an integration point P*(

*x*,

*y*, 0)

*in the aperture plane with the observation point P*

_{2}(

*x*

_{2},

*y*

_{2},

*z*

_{2}).

*z*= 0 and the observation point is not homogeneous, a significant saving of computing time can be obtained by using aperture-plane PSF rays, as illustrated in section 3.

## 3 Focusing through a plane interface

*z*=

*z*

_{0}<

*z*

_{1}there is an interface with a medium with a different refractive index. Thus, the wave number in the half-space

*z*<

*z*

_{0}of the incident field is

*k*

_{1}, and the wave number in the half-space

*z*>

*z*

_{0}of the transmitted field is

*k*

_{2}. The exact solution to the problem of reflection and refraction of an electromagnetic field at plane interface between two different media can be expressed in terms of a vectorial aperture-plane PSF, which appears in the form of an angular spectrum of plane waves. If the field in the plane

*z*= 0 is generated by an aperture current that produces a convergent spherical wave polarized in the

*x*direction, the exact solution for the transmitted electric field is given by ([7

7. H. Ling and S.W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A **1**, 559–567 (1984). [CrossRef]

*A*(

*x*′,

*y*′) = - 1/2

*R*

_{1}and ϕ(

*x*′,

*y*′) =

*R*

_{1}and Eqs. (16.31h, j, l, m) and (16.32e, g) in [5])

**k**

^{t}=

*k*

_{x}

**ê**

_{x}+

*k*

_{y}

**ê**

_{y}+

*k*

_{z2}

**ê**

_{z}, and with

*T*

^{TE}and

*T*

^{TM}being the Fresnel transmission coefficients given by Eqs. (15.40b, d) in [5].

*x*′ ,

*y*′) in the aperture plane

*z*= 0 emits a secondary diverging spherical wave which is weighted by the aperture field exp(-

*ik*

_{i}

*R*

_{1})/

*R*

_{1}. This secondary wave produces an angular spectrum of plane waves (given by (23)) which are incident on the plane interface at

*z*=

*z*

_{0}, and each of these plane waves is multiplied by the transmission coefficient

*T*

^{TE}or

*T*

^{TM}on refraction at the interface.

12. D. Jiang and J.J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. **163**, 55–71 (1999). [CrossRef]

13. V. Dhayalan and J.J. Stamnes, “Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. **39**, 6332–6340 (2001). [CrossRef]

*x*,

*y*,

*z*) in the second medium due to a point source at (

*x*′,

*y*′,0) in the aperture plane is provided by that particular plane wave emitted by the source at (

*x*′,

*y*′, 0), which after refraction through the interface is directed towards (

*x*,

*y*,

*z*). Thus, through the use of aperture-plane PSF rays we obtain both a valuable insight into the diffraction process and a significant reduction in the computing time.

## 4 Plane-wave diffraction by a circular aperture

*z*= 0 (Fig. 2). We let

*kR*

_{2}≫ 1 so that the diffracted field is given by (15) with

*u*

^{i}(

*x*,

*y*, 0) = 1 and with the aperture-plane PSF given by (20). Further, we introduce polar co-ordinates and integration variables given by

*A*as the integral over the entire aperture plane minus the integral over the area outside the aperture area to obtain

*u*

_{IS}in (27) we recognize as the geometrical-optics contribution. In (29) we may integrate by parts with respect to

*ρ*to obtain the contribution due to the aperture boundary as

*r*≠ 0 the phase function

*f*(

*ϕ*) in (31) has two stationary points, one for

*ϕ*=

*β*and another for

*ϕ*=

*β*+

*π*. These correspond to the minimum and maximum values of

*R*

_{2}. The asymptotic result is (lowest-order term of Eq. (8.8a) in [5])

*ϕ*, and we have an

*axial caustic*([18

18. P. Wolfe, “A new approach to edge diffraction,” SIAM J. Appl. Math. **15**, 1434–1469 (1967). [CrossRef]

*r*→ 0 all points on the aperture boundary become stationary because the secondary waves they emit are in phase at any axial observation point.

^{2}(

*ka*

^{2}/4

*z*

_{2}). The axial intensity distribution results from interference between the geometrical-optics wave and the boundary-diffracted waves. Because of the symmetry all secondary waves emitted by the aperture boundary are in phase, as noted above, and therefore the boundary-diffracted field is just as strong as the geometrical-optics field, giving full contrast of the axial interference pattern, i.e. axial zeros.

*u*

_{ID}where

*u*

_{ID}is given in (29). On the axis the field then becomes

*I*= cos

^{2}(

*θ*), where

*θ*is the angle subtended by the disk at the axial observation point. This is the famous Poisson spot. In 1818 Poisson observed that Fresnel’s wave theory of light predicted a bright spot at the center of the shadow of a small circular disk, and he used this to dispute Fresnel’s theory. But when Arago later performed the experiment, he found the prediction to be correct. This bright spot is caused by the

*axial caustic*, i.e. by the infinite number of diffracted rays issued by the boundary whose contributions are identical.

19. N. Sergienko, J.J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, and A. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A: Pure Appl. Opt. **1**, 552–559 (1999). [CrossRef]

20. J.J. Stamnes and N. Sergienko, “Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation,” J. Opt. A: Pure Appl. Opt. **2**, 365–371 (2000). [CrossRef]

## 5 Transmission of a truncated Gaussian beam into a biaxial crystal

*x*direction and has its main propagation direction in the

*z*direction. The beam waist is

*σ*

_{0}in the plane

*z*= 0, and in this plane there is a circular aperture of radius

*a*. After passing through this aperture the diffracted beam is transmitted through a plane interface at

*z*=

*z*

_{0}, which separates the isotropic medium of the incident beam from a biaxial crystal with one of its principal axes along the interface normal. The principal indices of refraction of the crystal are

*n*

_{1},

*n*

_{2}, and

*n*

_{3}, and the index of refraction of the isotropic medium is

*n*

^{(1)}. To simplify matter, we take the distance

*z*

_{0}from the aperture plane to the interface to be so small that we may neglect depolarization effects on transmission through the interface. Then a scalar theory is adequate, and the transmitted field can be expressed as [17]

14. J.J. Stamnes and G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A **18**, 3119–3129 (2001). [CrossRef]

*n*. Therefore we let

*n*

_{1}=

*n*

_{2}=

*n*

_{3}=

*n*to obtain from (38)

*KIO*

_{3}with principal refractive indices of

*n*

_{1}= 1.700,

*n*

_{2}= 1.828, and

*n*

_{3}= 1.832. The interface with the crystal is taken to be at a distance of

*z*

_{0}= 1.0 mm behind the aperture plane

*z*= 0. The beam waist in the aperture plane is taken to be

*σ*

_{0}= 1.0 mm, and the wavelength of the incident beam is taken to be λ

^{(1)}= 0.633

*μ*m. The solid curve in Fig. 3 (a) is computed from (38) with an aperture radius of

*a*= 5 mm. Since

*a*≫

*σ*

_{0}, the field at the aperture boundary is so small that it gives negligible diffraction effects. Thus, the solid curve in Fig. 3 (a) shows the axial intensity of a non-truncated transmitted Gaussian beam. The dashed curve in Fig. 3 (a) is computed from the same equation as the solid curve but for an aperture radius of

*a*= 2 mm. Now we see strong interference effects due to the boundary-diffracted waves.

*a*= 2 mm, whereas the dashed curve shows the axial intensity computed from (44). Note that according to (44), the axial intensity maxima of a truncated Gaussian beam that is transmitted into an

*isotropic*medium, decay

*monotonically*with

*z*, as illustrated in the dashed curve in Fig. 3 (b). However, for a truncated Gausian beam that is transmitted into a

*biaxially anisotropic*medium, the axial intensity maxima, shown by the solid curve in Fig. 3 (b),

*increase*with

*z*in the region

*z*≤ 4 m. To explain this unexpected behavior let us first consider the case in which the incident field is a plane wave propagating along the

*z*axis and the second medium is isotropic. Then at every observation point on the

*z*axis all boundary-diffracted waves are in phase, so that they interfere constructively to give a diffracted field of the same strength as the geometrical-optics field. Hence we get axial intensity zeros, as shown by (36). For an incident Gaussian beam all boundary-diffracted waves are still in phase on the axis, but the field amplitude at the aperture boundary is now exp(-

*a*

^{2}

*z*because each edge diffracted wave decays as 1/

*R*, where

*R*, but now the various boundary-diffracted waves are not in phase on the

*z*axis because the phase velocity depends on the direction of propagation. However, as

*z*increases the boundary-diffracted waves get more in phase on the axis, and this explains the growth of the maxima of the on-axis intensity in the region

*z*≤ 4 m in Fig. 3 (b).

## 6 Conclusions

*non-uniform*MSP for double integrals apply. This means that we have avoided to discuss complications that arise in caustic regions, where we have coalescence of interior stationary points and associated

*diffraction catastrophes*([21

21. M.V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Progress in Optics , vol. **XVIII**, E. Wolf (ed.) (North-Holland, Amsterdam, 1980). [CrossRef]

22. J.J. Stamnes, “Diffraction, asymptotics, and catastrophes,” Opt. Acta **29**, 823–842 (1982). [CrossRef]

*uniform*asymptotic methods are required to get quantitative descriptions of the diffraction phenomena (see e.g. sections 8.2 and 9.2 in [5]). Alternatively, as discussed in the paper, we may use aperture-plane PSF rays to avoid complications in caustic regions due to the coalescence of stationary points.

## References and links

1. | A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Ann. Phys. Lpz. |

2. | J.B. Keller, “The geometrical theory of diffraction,” |

3. | J.B. Keller, “Diffraction by an aperture,” J. Appl. Phys. |

4. | J.B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. |

5. | J.J. Stamnes, “Waves in Focal Regions” (Adam Hilger, Bristol and Boston, 1986). |

6. | H. Weyl, “Ausbreitung elektromagnetischer Wellen über einem ebenen Leiter,” Ann. Phys. Lpz. |

7. | H. Ling and S.W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A |

8. | J.J. Stamnes and D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. |

9. | D. Jiang and J.J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. |

10. | V. Dhayalan and J.J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. |

11. | J.J. Stamnes and D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. |

12. | D. Jiang and J.J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. |

13. | V. Dhayalan and J.J. Stamnes, “Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. |

14. | J.J. Stamnes and G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A |

15. | G. Sithambaranathan and J.J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A |

16. | J.J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A |

17. | G. Sithambaranathan and J.J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” accepted by Opt. Commun. (2002). |

18. | P. Wolfe, “A new approach to edge diffraction,” SIAM J. Appl. Math. |

19. | N. Sergienko, J.J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, and A. Friberg, “Asymptotic methods for evaluation of diffractive lenses,” J. Opt. A: Pure Appl. Opt. |

20. | J.J. Stamnes and N. Sergienko, “Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation,” J. Opt. A: Pure Appl. Opt. |

21. | M.V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Progress in Optics , vol. |

22. | J.J. Stamnes, “Diffraction, asymptotics, and catastrophes,” Opt. Acta |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(080.0080) Geometric optics : Geometric optics

(080.1510) Geometric optics : Propagation methods

(080.2720) Geometric optics : Mathematical methods (general)

(260.1440) Physical optics : Birefringence

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Focus Issue: Rays in wave theory

**History**

Original Manuscript: June 18, 2002

Revised Manuscript: August 1, 2002

Published: August 12, 2002

**Citation**

Jakob Stamnes, "Waves, rays, and the method of stationary phase," Opt. Express **10**, 740-751 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-740

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### References

- A. Rubinowicz, �??Zur Kirchho.schen Beugungstheorie,�?? Ann. Phys. Lpz. 73, 339-364 (1924 ). [CrossRef]
- J.B. Keller, �??The geometrical theory of diffraction,�?? Proc. Symp. on Microwave Optics, McGill University Press, Montreal, 1953.
- J.B. Keller, �??Diffraction by an aperture,�?? J. Appl. Phys. 28, 426-444 (1957). [CrossRef]
- J.B. Keller, �??Geometrical theory of diffraction,�?? J. Opt. Soc. Am. 52, 116-130 (1962). [CrossRef] [PubMed]
- J.J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol and Boston, 1986).
- H. Weyl, �??Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter,�?? Ann. Phys. Lpz. 60, 481-500 (1919). [CrossRef]
- H. Ling and S.W. Lee, �??Focusing of electromagnetic waves through a dielectric interface,�?? J. Opt. Soc. Am. A 1, 559-567 (1984). [CrossRef]
- J.J. Stamnes and D. Jiang, �??Focusing of two-dimensional electromagnetic waves through a plane interface,�?? Pure Appl. Opt. 7, 603-625 (1998). [CrossRef]
- D. Jiang and J.J. Stamnes, �??Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,�?? Pure Appl. Opt. 7, 627-641 (1998). [CrossRef]
- V. Dhayalan and J.J. Stamnes, �??Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,�?? Pure Appl. Opt. 7, 33-52 (1998). [CrossRef]
- J.J. Stamnes and D. Jiang, �??Focusing of electromagnetic waves into a uniaxial crystal,�?? Opt. Commun. 150, 251-262 (1998). [CrossRef]
- D. Jiang and J.J. Stamnes, �??Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,�?? Opt. Commun. 163, 55-71 (1999). [CrossRef]
- V. Dhayalan and J.J. Stamnes, �??Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface,�?? Appl. Opt. 39, 6332-6340 (2001). [CrossRef]
- J.J. Stamnes and G. Sithambaranathan, �??Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,�?? J. Opt. Soc. Am. A 18, 3119-3129 (2001). [CrossRef]
- G. Sithambaranathan and J.J. Stamnes, �??Transmission of a Gaussian beam into a biaxial crystal,�?? J. Opt. Soc. Am. A 18, 1662-1669 (2001). [CrossRef]
- J.J. Stamnes and V. Dhayalan, �??Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,�?? J. Opt. Soc. Am. A 18, 1670-1677 (2001). [CrossRef]
- G. Sithambaranathan and J.J. Stamnes, �??Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,�?? accepted by Opt. Commun. (2002).
- P. Wolfe, �??A new approach to edge diffraction,�?? SIAM J. Appl. Math. 15, 1434-1469 (1967). [CrossRef]
- N. Sergienko, J.J. Stamnes, V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, and A. Friberg, �??Asymptotic methods for evaluation of di.ractive lenses,�?? J. Opt. A: Pure Appl. Opt. 1, 552-559 (1999). [CrossRef]
- J.J. Stamnes and N. Sergienko, �??Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation,�?? J. Opt. A: Pure Appl. Opt. 2, 365-371 (2000). [CrossRef]
- M.V. Berry and C. Upstill, �??Catastrophe optics: morphologies of caustics and their diffraction patterns,�?? Progress in Optics, vol. XVIII, E. Wolf (ed.) (North-Holland, Amsterdam, 1980). [CrossRef]
- J.J. Stamnes, �??Diffraction, asymptotics, and catastrophes,�?? Opt. Acta 29, 823-842 (1982). [CrossRef]

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