## Focusing of partially coherent light into planar waveguides

Optics Express, Vol. 12, Issue 19, pp. 4511-4522 (2004)

http://dx.doi.org/10.1364/OPEX.12.004511

Acrobat PDF (232 KB)

### Abstract

Edge coupling of a focused partially coherent Gaussian Schell-model beam into a planar dielectric waveguide is examined. The incident field is decomposed into a sum of coherent modes that are expressed as a discrete superposition of plane-wave components. A model based on the rigorous diffraction theory of gratings is used to replace the waveguide with a corresponding periodic multilayer structure to determine the coupling efficiencies. Numerical simulations are presented for single and multimode planar waveguides and for a graded index waveguide. The results are compared with the predictions of the overlap integral method.

© 2004 Optical Society of America

## 1 Introduction

1. P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. **MTT-29**, 107–114 (1981). [CrossRef]

11. T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science **17**, 75–83 (1982). [CrossRef]

18. E. Wolf, “New theory of partial coherence in the spacefrequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

## 2 Coherent mode representation of spatially partially coherent fields

18. E. Wolf, “New theory of partial coherence in the spacefrequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

*λ*and

*ϕ*(

**r**) are the eigenvalues and the eigenfunctions, respectively, of the Fredholm integral equation of the second kind:

*D*is the domain of integration. We note that the dependence of quantities such as the modal eigenfunctions and eigenvalues on the angular frequency

*ω*is left implicit here and in what follows.

18. E. Wolf, “New theory of partial coherence in the spacefrequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. **72**, 343–351 (1982). [CrossRef]

*λ*

_{n}are real and non-negative, and they may be understood to represent the energy distribution between the modes. The eigenfunctions

*ϕ*(

**r**) are orthonormal, i.e.,

*δ*

_{mn}denotes the Kronecker delta symbol.

## 3 Field representation in the waveguide

*l*= 1,…,

*L*,

*x*

_{0}= 0, and

*x*

_{L}=

*d*

_{g}. We replace this real structure with an artificially periodic structure illustrated in Fig. 1, with a period

*d*=

*d*

_{s}+

*d*

_{g}+

*d*

_{c}, where the subscripts s, g, and c denote substrate, guide, and cover respectively. In addition, we demand that

*n*

_{s}and

*n*

_{c}of the substrate and cover. If we let

*d*

_{c}→ ∞ and

*d*

_{s}→ ∞, the periodic structure reduces to the real non-periodic waveguide described by Eq. (4). Graded-index waveguides fabricated, for example, by ion-exchange techniques can be modelled to any degree of accuracy by choosing a sufficient large value for

*L*.

*U*(which is

*E*

_{y}in TE polarization and

*H*

_{y}in TM polarization). To find the solutions of the Maxwell equations in the waveguide structure separate the variables as

*a*,

*b*and

*γ*are constants and

*X*(

*x*) is a Bloch wave function that satisfies the condition

*α*

_{0}is a pseudoperiodicity constant. The periodicity leads to a discrete set of possible values of

*γ*and the associated functions

*X*(

*x*). The solutions may be divided in two categories: The homogenous eigenmodes for which 0 <

*γ*

^{2}<

*k*

^{2}

*γ*

^{2}< 0, where

*k*= 2

*π*/

*λ*and

*λ*is the wavelength of light in vacuum. Obviously, the latter class of modes, whose field amplitudes grow exponentially in the

*z*direction, are non-physical. The modes with exponentially decaying field amplitude have no contribution in a waveguide with an infinite length but they play an important role in the vicinity of the structural discontinuities.

*γ*

_{m}and the corresponding eigenfunctions

*X*

_{m}(

*x*) for all homogenous modes as well as for a sufficient number of inhomogenous modes. The field in the waveguide structure (4) illustrated in Fig. 1 must satisfy the Helmholtz equation in each homogenous region, the appropriate electromagnetic boundary conditions at all interfaces, and the pseudo-periodicity condition (7). An algebraic eigenvalue equation can be obtained [19

19. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta **28**, 413–428 (1981). [CrossRef]

20. R. H. Morf, “Exponentially convergent and numerically efficient soltution of Maxwell’s equations for lamellar gratings,”J. Opt. Soc. Am. A **12**, 1043–1056 (1995). [CrossRef]

21. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978) [CrossRef]

*d*→ ∞ supports the following types of modes [26–30

26. J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A **1**, 742–753 (1984). [CrossRef]

*γ*of the guided modes form a discrete spectrum, but the spectra of the all other types of modes are continuous, which makes the treatment of non-periodic waveguide discontinuity problems more difficult than the solution of the grating diffraction problem.

## 4 Boundary value problem

*X*

_{m}(

*x*) and the eigenvalues

*γ*

_{m}using one of the methods discussed in the previous section, it remains to solve the amplitudes of the reflected and transmitted fields.

*z*= 0. The periodicity of the geometry discretizes the known angular spectrum

*A*

_{m}(

*α*) of the incident field denoted by

*x*,

*z*) and the unknown angular spectrum

*R*(

*α*) of the reflected field

*x*,

*z*) in the half-space

*z*≤ 0. This leads to the use of Rayleigh expansions, which give the incident and reflected fields in the half-space

*z*≤ 0 in the form

*A*

_{m}and

*R*

_{m}represent the sampled values of the angular spectrum of the incident and the reflected fields, respectively,

*n*is the refractive index of the medium in the half-space

*z*< 0. In Eq. (9) the summation of the incident field is restricted to the index

*M*to take account only the homogenous plane waves with real-valued

*r*

_{m}, since we assume that sources are located sufficiently far away from the examined boundary.

*b*= 0 because we assume that there are no sources in the positive half-space that would give rise to modal fields propagating in the negative

*z*direction:

*a*

_{m}are the unknown amplitudes of the

*m*th mode propagating in the positive

*z*direction and

*γ*

_{m}are the propagation constants, which can be determined by methods presented in Sect. 3.

*a*

_{m}and the coefficients

*R*

_{m}of the angular spectrum of the reflected field, we require the continuity of the electric field and its

*z*derivative at the boundary between the half-space

*z*< 0 and the waveguide:

**A**,

**R**, and

**a**are

*A*

_{m},

*R*

_{m}, and

*a*

_{m}, respectively, and

**Γ**and

**r**are diagonal matrices with the elements

*γ*

_{m}and

*r*

_{m}, respectively. The elements of the matrix

**P**are obtained from the equation

*P*

_{mq}are the solved eigenvector coefficients of the eigenvalue problem discussed in Section 3.

*R*

_{m}and

*a*

_{m}may be solved from Eqs. (16) and (17). For example, the amplitudes

*a*

_{m}of the waveguide modes are obtained by eliminating

**R**, which yields

*x*,

*z*) by the coherent eigenfunctions

*ϕ*

_{n}(

*x*,

*z*). Repeating this procedure for each eigenfunction separately we obtain the unknown amplitudes

*a*

_{m}and

*R*

_{m}corresponding to each incident coherent mode. We note that the waveguide modes

*X*

_{m}(

*x*) and the eigenvalues

*γ*

_{m}need to be solved only once regardless of the number of coherent modes.

*w*

_{0}and

*σ*

_{0}denote the beam 1/e

^{2}width and coherence width at the beam-waist plane, respectively, and

*x̅*is the beam center position. The coherent modes of the incident beam (20) can be solved analytically, which gives the eigenfunctions and eigenvalues of Eq. (2) in the form [31

31. E. Wolf, “Coherent-mode representation of Gaussian-Schell model sources and of their radiation fields,” J. Opt. Soc. Am. **72**, 923–928 (1982). [CrossRef]

*H*

_{n}is a Hermite polynomial of the order

*n*. The geometry of the waveguide and the incident beam are illustrated in Fig. 2.

## 5 Input coupling efficiency

*P*is given by the

*z*component of the time-averaged Poynting vector as

*n*, has to be weighted by the modal coefficients

*λ*

_{n}) in Eq. (22). In view of Eq. (25), the power carried by the incoming partially coherent field is

*m*th coherent mode is given by

*X*

_{m}are defined piecewise as sinusoidal, exponential, and hyperbolic functions, the integrals can be evaluated analytically for Gaussian illumination.

## 6 Optimization of coupling efficiency

*w*, the coherence width

*σ*, and the beam center position

*x̅*. The determination of the efficiency is a rather simple task by using the overlap integral method [32

32. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (Wiley, New York, 1991). [CrossRef]

*x̅*and the half-width

*w*. In practice the latter scan can be performed with a positive lens. The beam half-width at the focal plane of the lens is

*β*= (1 +1/

*δ*

^{2})

^{-1/2}and

*δ*=

*w*

_{0}/

*σ*

_{0}is the global degree of spatial coherence of the GSM beam [17]. The coherence width at the focal plane is

*w*

_{F}≥

*λ*/2 is made, which is rather optimistic for practical measurements. The coupling efficiencies are calculated by the overlap integral method (29) and the quasi-rigorous method for two values of the global degree of coherence, namely

*δ*= 1 and

*δ*= 1/2, in TE polarization. The number of significant coherent modes retained in the calculations depends on the value of

*δ*. In the following examples the ratio of the normalized modal weights is chosen to be

*λ*

_{N}/

*λ*

_{1}< 10

^{-3}, where

*N*is the order of the highest mode. For comparison, results are also provided for the fully coherent Gaussian beam that is obtained in the limit

*δ*→ ∞.

21. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978) [CrossRef]

### 6.1 Single mode waveguide

*n*

_{s}= 1.55,

*n*

_{g}= 1.97, and

*n*

_{c}= 1 with

*d*

_{g}= 0.21

*μ*m. The wavelength of the light in all calculations is

*λ*= 0.6328

*μ*m.

*w*and the offset

*x̅*in the cases

*δ*→ ∞,

*δ*= 1, and

*δ*= 1/2 are presented for both methods in Table 1. As an example, the results in the case

*δ*= 1 are illustrated in Figs. 3(a) and (b), and the convergence of the coupling efficiency for the optimum parameters is shown in Fig. 4 with several values of the period

*d*as a function of the number of retained eigenmodes. We emphasize that the convergence of the results must be confirmed for every configuration.

### 6.2 Three-mode waveguide

*d*

_{g}= 0.7

*μ*m. The coupling efficiencies are presented in Table 2.

*w*are no longer correctly predicted by the overlap integral method when the value of

*δ*is decreased. Interestingly, the coupling efficiencies into the fundamental mode of the three-mode waveguide are higher than in the case of single-mode waveguide.

### 6.3 Graded index waveguide

*n*

_{s}= 1.5232,

*n*

_{c}= 1 and the guiding layer thickness

*d*

_{g}= 5.0

*μ*m. The refractive index distribution of the structure is shown in Fig. 5 where the number of layers [Eq. (4)] is chosen to be

*L*= 50. The optimum coupling efficiencies into waveguide modes

*m*= 1 and

*m*= 3 are presented in Tables 3 and 4, respectively.

*δ*→ ∞, i.e., when the incident field reduces to a conventional fully coherent Gaussian beam. This result is easily understood if we consider the transversal mode profiles of the GSM beam and the overlap integral method. The best coupling efficiency is obtained for the lowest-order coherent mode,

*n*= 1, as it resembles most the fundamental waveguide mode

*m*= 1. The fully coherent Gaussian beam is obtained when the coherence length

*σ*→ ∞ and thus only one modal coefficient is non-zero. The more the value of

*δ*decreases the greater number of coherent modes have to be retained to model the incident beam. The higher order coherent modes, however, are Hermite-Gaussian modes, which have weaker coupling efficiencies.

## 7 Conclusions

## Acknowledgments

## References and links

1. | P. Gelin, M. Petenzi, and J. Citerne, “Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,” IEEE Trans. Microwave Theory Technol. |

2. | B. M. A. Rahman and J. B. Davies, “Finite-element analysis of optical and microwave waveguide problems,” IEEE Trans. Microwave Theory Technol. |

3. | V. Ramaswamy and P. G. Suchoski Jr., “Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. |

4. | C. N. Capsalis, J. G. Fikioris, and N. K. Uzunoglu, “Scattering from an abruptly terminated dielectric-slab waveguide,” IEEE J. Lightwave Technol. |

5. | B. M. A. Rahman and J. B. Davies, “Analyses of optical waveguide discontinuities,” J. Lightwave Technol. |

6. | N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, “Scattering from an abruptly terminated single-mode-fiber waveguide,” J. Opt. Soc. Am. |

7. | K. Hirayama and M. Koshiba, “Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,” IEEE Trans. Microwave Theory Technol. |

8. | S. S. Patrick and K. J. Webb, “A variational vector finite difference analysis for dielectric waveguides,” IEEE Trans. Microwave Theory Technol. |

9. | K. Hirayama and M. Koshiba, “Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,” J. Lightwave Technol. |

10. | S. S. A. Obayya, “Novel finite element analysis of optical waveguide discontinuity problems,” J. Lightwave Technol. |

11. | T. Hosono, T. Hinata, and A. Inoue, “Numerical analysis of the discontinuities in slab dielectric waveguides,” Radio Science |

12. | P. Vahimaa and J. Turunen, in |

13. | P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, “Guided-mode propagation through an ion-exchanged graded-index boundary,” Opt. Commun. |

14. | Ph. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. |

15. | E. Silberstein, Ph. Lalanne, J-P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A |

16. | J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, “Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer’s star product,” Opt. Commun. |

17. | L. Mandel and E. Wolf, |

18. | E. Wolf, “New theory of partial coherence in the spacefrequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. |

19. | L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta |

20. | R. H. Morf, “Exponentially convergent and numerically efficient soltution of Maxwell’s equations for lamellar gratings,”J. Opt. Soc. Am. A |

21. | K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. |

22. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

23. | Ph. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method in TM polarization,” J. Opt. Soc. Am. A |

24. | G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A |

25. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic strucutures,” J. Opt. Soc. Am. A |

26. | J. Chilwell and I. Hodgkinson, “Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A |

27. | L. M. Walpita, “Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,” J. Opt. Soc. Am. A |

28. | K.-H. Schlereth and M. Tacke, “The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,” J. Quantum Electron. |

29. | R. E. Smith, S. N. Houde-Walter, and G. W. Forbes, “Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,” Opt. Lett. |

30. | X. Wang, Z. Wang, and Z. Huang, “Propagation constant of a planar dielectric waveguide with arbitrary refractive-index variation,” Opt. Lett. |

31. | E. Wolf, “Coherent-mode representation of Gaussian-Schell model sources and of their radiation fields,” J. Opt. Soc. Am. |

32. | B. E. A. Saleh and M. C. Teich, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.1940) Diffraction and gratings : Diffraction

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 10, 2004

Revised Manuscript: September 10, 2004

Published: September 20, 2004

**Citation**

Toni Saastamoinen, Markku Kuittinen, Pasi Vahimaa, Jari Turunen, and Jani Tervo, "Focusing of partially coherent light into planar waveguides," Opt. Express **12**, 4511-4522 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4511

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### References

- P. Gelin, M. Petenzi, and J. Citerne, �??Rigorous analysis of the scattering of surface waves in an abruptly ended slab electric waveguide,�?? IEEE Trans. Microwave Theory Technol. MTT-29, 107�??114 (1981). [CrossRef]
- B. M. A. Rahman and J. B. Davies, �??Finite-element analysis of optical and microwave waveguide problems,�?? IEEE Trans. Microwave Theory Technol. MTT-32, 20�??28 (1984). [CrossRef]
- V. Ramaswamy and P. G. Suchoski Jr., �??Power loss at step discontinuity in an asymmetrical dielectric slab waveguide,�?? J. Opt. Soc. Am. 1, 754�??759 (1984). [CrossRef]
- C. N. Capsalis, J. G. Fikioris and N. K. Uzunoglu, �??Scattering from an abruptly terminated dielectric-slab waveguide,�?? IEEE J. Lightwave Technol. LT-3, 408�??415 (1985). [CrossRef]
- B. M. A. Rahman and J. B. Davies, �??Analyses of optical waveguide discontinuities,�?? J. Lightwave Technol. LT-6, 52�??57 (1988). [CrossRef]
- N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, �??Scattering from an abruptly terminated single-mode-fiber waveguide,�?? J. Opt. Soc. Am. 4, 2150�??2157 (1987). [CrossRef]
- K. Hirayama and M. Koshiba, �??Analysis of discontinuities in an open dielectric slabe waveguide by combination of finite and boundary elements,�?? IEEE Trans. Microwave Theory Technol. MTT-37, 761�??768 (1989). [CrossRef]
- S. S. Patrick and K. J. Webb, �??A variational vector finite difference analysis for dielectric waveguides,�?? IEEE Trans. Microwave Theory Technol. MTT-40, 692�??698 (1992). [CrossRef]
- K. Hirayama and M. Koshiba, �??Rigorous analysis of coupling between laser and passive waveguide in multilayer slab waveguide,�?? J. Lightwave Technol. LT-11, 1353�??1358 (1993). [CrossRef]
- S. S. A. Obayya, �??Novel finite element analysis of optical waveguide discontinuity problems,�?? J. Lightwave Technol. 22, 1420�??1425 (2004). [CrossRef]
- T. Hosono, T. Hinata, and A. Inoue, �??Numerical analysis of the discontinuities in slab dielectric waveguides,�?? Radio Science 17, 75�??83 (1982). [CrossRef]
- P. Vahimaa and J. Turunen, in Diffractive Optics and Micro-Optics, Vol 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 69.
- P. Vahimaa, M. Kuittinen, J. Turunen, J. Saarinen, R.-P. Salmio, E. Lopez Lago, and J. Liñares, �??Guided-mode propagation through an ion-exchanged graded-index boundary,�?? Opt. Commun. 147, 247�??253 (1998). [CrossRef]
- Ph. Lalanne and E. Silberstein, �??Fourier-modal methods applied to waveguide computational problems,�?? Opt. Lett. 25, 1902�??1904 (2000). [CrossRef]
- E. Silberstein, Ph. Lalanne, J-P. Hugonin, and Q. Cao, �??Use of grating theories in integrated optics,�?? J. Opt. Soc. Am. A 18, 2865�??2875, (2001). [CrossRef]
- J. Tervo, M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, �??Efficient Bragg waveguide-grating analysis by quasi-rigorous approach based on Redheffer�??s star product,�?? Opt. Commun. 198, 265�??272 (2001). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- E. Wolf, �??New theory of partial coherence in the spacefrequency domain. Part I: Spectra and cross-spectra of steady-state sources,�?? J. Opt. Soc. Am. 72, 343�??351 (1982). [CrossRef]
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, �??The dielectric lamellar diffraction grating,�?? Opt. Acta 28, 413�??428 (1981). [CrossRef]
- R. H. Morf, �??Exponentially convergent and numerically efficient soltution of Maxwell�??s equations for lamellar gratings,�?? J. Opt. Soc. Am. A 12, 1043�??1056 (1995). [CrossRef]
- K. Knop, �??Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,�?? J. Opt. Soc. Am. 68, 1206�??1210 (1978). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, �??Stable implementation of the rigorous coupled wave analysis for surface-relief gratings: enhanced transmittance matrix approach,�?? J. Opt. Soc. Am. A 12, 1077�??1086 (1995). [CrossRef]
- Ph. Lalanne and G. M. Morris, �??Highly improved convergence of the coupled-wave method in TM polarization,�?? J. Opt. Soc. Am. A 13, 779�??784 (1996). [CrossRef]
- G. Granet and B. Guizal, �??Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,�?? J. Opt. Soc. Am. A 13, 1019�??1023 (1996). [CrossRef]
- L. Li, �??Use of Fourier series in the analysis of discontinuous periodic strucutures,�?? J. Opt. Soc. Am. A 13, 1870�??1876 (1996). [CrossRef]
- J. Chilwell and I. Hodgkinson, �??Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection from prism-loaded waveguides,�?? J. Opt. Soc. Am. A 1, 742�??753 (1984). [CrossRef]
- L. M. Walpita, �??Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix,�?? J. Opt. Soc. Am. A 2, 595�??602 (1985). [CrossRef]
- K.-H. Schlereth and M. Tacke, �??The complex propagation constant of multilayer waveguides: an algorithm for a personal computer,�?? J. Quantum Electron. 26, 627�??630 (1990). [CrossRef]
- R. E. Smith, S. N. Houde-Walter, and G.W. Forbes, �??Numerical determination of planar waveguide modes using the analyticity of the dispersion relation,�?? Opt. Lett. 16, 1316�??1318 (1991). [CrossRef] [PubMed]
- X. Wang, Z. Wang, and Z. Huang, �??Propagation constant of a planar dielectric waveguide with arbitrary refractive-index variation,�?? Opt. Lett. 18, 805�??807 (1992). [CrossRef]
- E. Wolf, �??Coherent-mode representation of Gaussian-Schell model sources and of their radiation fields,�?? J. Opt. Soc. Am. 72, 923�??928 (1982). [CrossRef]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991). [CrossRef]

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