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

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References

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IEEE J. Lightwave Technol. (1)

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]

IEEE Trans. Microwave Theory Technol. (4)

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]

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]

J. Lightwave Technol. (3)

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]

B. M. A. Rahman and J. B. Davies, �??Analyses of optical waveguide discontinuities,�?? J. Lightwave Technol. LT-6, 52�??57 (1988).
[CrossRef]

S. S. A. Obayya, �??Novel finite element analysis of optical waveguide discontinuity problems,�?? J. Lightwave Technol. 22, 1420�??1425 (2004).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (8)

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. 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]

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]

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]

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]

J. Quantum Electron. (1)

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]

Opt. Acta (1)

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]

Opt. Commun. (2)

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]

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]

Opt. Lett. (3)

OSA Technical Digest Series (1)

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.

Radio Science (1)

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

Other (2)

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

The periodic waveguide structure of period d used to model the real non-periodic structure of Eq. (4).

Fig. 2.
Fig. 2.

Coupling of a set of Gaussian Schell-model beams from a uniform medium into a periodically modulated waveguide used to model single-beam coupling into non-periodic waveguide.

Fig. 3.
Fig. 3.

Coupling efficiencies η 1 into the fundamental waveguide mode as a function of beam center position and beam width w when the global degree of coherence δ = 1. (a) Overlap integral method. (b) Quasi-rigorous method.

Fig. 4.
Fig. 4.

Convergence of the coupling efficiency into the fundamental mode of the single mode waveguide in the case δ = 1 as a function of the number of eigenmodes retained in the calculation. The curves represent the periods d = 80d g, d = 125d g, d = 150d g, and d = 175d g respectively, starting from the lowermost.

Fig. 5.
Fig. 5.

The refractive index distribution of the graded index waveguide as a function of the guiding layer thickness d g.

Tables (4)

Tables Icon

Table 1. Optimum coupling efficiencies of the GSM beam into a single mode waveguide with guiding layer thickness dg = 0.21 μm.

Tables Icon

Table 2. Optimum coupling efficiencies of the GSM beam into a three-mode waveguide with guiding layer thickness dg = 0.7 μm.

Tables Icon

Table 3. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 1 with guiding layer thickness dg = 5.0 μm.

Tables Icon

Table 4. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 3 with guiding layer thickness dg = 5.0 μm.

Equations (31)

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W ( r 1 , r 2 ) = n = 1 λ n ϕ n * ( r 1 ) ϕ n ( r 2 ) .
D W ( r 1 , r 2 ) ϕ n ( r 1 ) d 3 r 1 = λ n ϕ n ( r 2 ) ,
D ϕ m * ( r ) ϕ n ( r ) d 3 r = δ mn ,
n ( x ) = { n c when x > d g n l when x l x < x l + 1 , n s when x < 0
n max = max { n l }
U ( x , z ) = X ( x ) [ a exp ( z ) + b exp ( z ) ] .
X ( x + d ) = X ( x ) exp ( i α 0 d ) ,
{ guided modes : [ k max ( n s , n c ) ] 2 < γ 2 < ( k n max ) 2 evanescent modes : γ 2 < 0 radiation modes : otherwise .
E y in ( x , z ) = m = M M A m exp [ i ( α m x + r m z ) ] ,
E y r ( x , z ) = m = R m exp [ i ( α m x + r m z ) ] ,
α m = α 0 + 2 π m d ,
r m = { ( k 2 n 2 α m 2 ) 1 2 if α m k n i ( α m 2 k 2 n 2 ) 1 2 otherwise ,
E y t ( x , z ) = m = 1 a m exp ( i γ m z ) X m ( x ) ,
E y in ( x , z ) + E y r ( x , z ) = E y t ( x , z ) ,
z E y in ( x , z ) + z E y r ( x , z ) = z E y t ( x , z ) .
A + R = Pa ,
r ( A R ) = P Γ a ,
P m q = 1 d o d X m ( x ) exp ( i 2 π q x d ) d x .
a = 2 ( + rP ) 1 rA .
W y in ( x 1 , x 2 ) = exp [ ( x 1 x ¯ ) 2 + ( x 2 x ¯ ) 2 w 0 2 ] exp [ ( x 1 x 2 ) 2 2 σ 0 2 ] ,
ϕ n ( x , z ) = ( 2 c π ) 1 4 1 2 n n ! H n [ ( x x ¯ ) 2 c ] exp [ c ( x x ¯ ) 2 ] ,
λ n = ( π a + b + c ) 1 2 ( b a + b + c ) n ,
{ a = w 0 2 b = σ 0 2 2 c = ( a 2 + 2 a b ) 1 2 ,
η m = P m P in .
P = 1 2 0 d { E y ( x , z ) H x * ( x , z ) } d x ,
P in = d 2 ω μ 0 n = 1 λ n m = M M r m A m n 2 ,
P m = d 2 ω μ 0 { γ m } n = 1 λ n a m n 2 q = P m q 2 .
P r = d 2 ω μ 0 { γ m } n = 1 λ n q = R m n 2 .
η m = n = 1 λ n X m * ( x ) ϕ n ( x ) d x 2 X m ( x ) 2 d x n = 1 λ n ϕ n ( x ) 2 d x .
W F = λ f π w 0 β ,
σ F = σ 0 w F w 0 .

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