Abstract

A systematic approach to the analysis of almost-periodic vertical-grating-assisted codirectional couplers with nonuniform duty ratios is presented. The Poisson sum formula is used to expand rigorously a nonuniform rectangular grating into a quasi-Fourier series, and local grating parameters such as period, width, and duty ratio can be defined unambiguously in such a procedure. On the basis of this expansion the coupled-local-mode formulation is the most natural extension for the analysis of such a nonuniform grating-assisted codirectional coupler filter. By transformation of the coupled-local-mode equations into the Zakharov–Shabat system the Gel’fand–Levintan–Marchenko inverse-scattering method is then used to synthesize special grating-assisted codirectional coupler filters. The design is illustrated by two typical examples: One is a third-order Butterworth filter, and the other is a linear filter.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
    [CrossRef]
  2. I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
    [CrossRef]
  3. W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
    [CrossRef]
  4. H. Sakata, “Analysis of wavelength-selective photodetectors based on grating-assisted forward coupling,” J. Lightwave Technol. 11, 560–566 (1993).
    [CrossRef]
  5. Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
    [CrossRef]
  6. R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. 14, 843–847 (1978).
    [CrossRef]
  7. H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
    [CrossRef] [PubMed]
  8. M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  9. M. Matsuhara, K. O. Hill, A. Watanabe, “Optical-waveguide filters: synthesis,” J. Opt. Soc. Am. 65, 804–809 (1975).
    [CrossRef]
  10. G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  11. K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
    [CrossRef]
  12. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Elsevier, New York, 1969).
  13. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 48, 109–126 (1976).
    [CrossRef]
  14. P. N. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 466.
  15. A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
    [CrossRef]
  16. V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  17. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.
  18. C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
    [CrossRef]
  19. G. H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A 11, 2027–2037 (1994).
    [CrossRef]

1997 (1)

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

1996 (1)

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

1994 (1)

1993 (2)

H. Sakata, “Analysis of wavelength-selective photodetectors based on grating-assisted forward coupling,” J. Lightwave Technol. 11, 560–566 (1993).
[CrossRef]

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

1992 (1)

1991 (1)

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

1987 (2)

M. Yamada, K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
[CrossRef] [PubMed]

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

1985 (2)

G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
[CrossRef]

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

1978 (1)

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. 14, 843–847 (1978).
[CrossRef]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 48, 109–126 (1976).
[CrossRef]

1975 (1)

1972 (1)

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1963 (1)

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

Alferness, R. C.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. 14, 843–847 (1978).
[CrossRef]

Buhl, L. L.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Burrus, C. A.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Chien, M. D.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Clarke, D. P.

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

Coldren, L. A.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Cross, P. S.

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. 14, 843–847 (1978).
[CrossRef]

Demeester, P.

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

DenBaars, S. P.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Feshbach, H.

P. N. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 466.

Fish, G. A.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Henry, C. H.

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

Hill, K. O.

Ishimaru, A.

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

Jan, Y. H.

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

Johnson, L. F.

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

Kim, I.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Koch, T. L.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 48, 109–126 (1976).
[CrossRef]

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

Koren, U.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Logan, R. A.

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Elsevier, New York, 1969).

Marcuse, D.

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

Matsuhara, M.

Miller, B. I.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Moerman, I.

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

Morse, P. N.

P. N. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 466.

Newkirk, M. A.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Raybon, G.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Sakata, H.

H. Sakata, “Analysis of wavelength-selective photodetectors based on grating-assisted forward coupling,” J. Lightwave Technol. 11, 560–566 (1993).
[CrossRef]

H. Sakata, “Sidelobe suppression in grating-assisted wavelength-selective couplers,” Opt. Lett. 17, 463–465 (1992).
[CrossRef] [PubMed]

Sakuda, K.

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shin, S. Y.

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Elsevier, New York, 1969).

Song, G. H.

Van Daele, P.

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

Vanderbauwhede, W.

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

Watanabe, A.

Winick, K. A.

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

Yamada, M.

Young, M. G.

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 48, 109–126 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. H. Henry, L. F. Johnson, R. A. Logan, D. P. Clarke, “Determination of the refractive index of InGaAsP epitaxial layers by mode line luminescence spectroscopy,” IEEE J. Quantum Electron. 21, 1887–1892 (1985).
[CrossRef]

R. C. Alferness, P. S. Cross, “Filter characteristics of codirectionally coupled waveguides with weighted coupling,” IEEE J. Quantum Electron. 14, 843–847 (1978).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

I. Kim, R. C. Alferness, L. L. Buhl, U. Koren, B. I. Miller, M. G. Young, M. A. Newkirk, M. D. Chien, T. L. Koch, G. Raybon, C. A. Burrus, “InGaAs/InGaAsP MQW optical amplifier integrated with grating-assisted vertical-coupler noise filter,” IEEE Photon. Technol. Lett. 5, 1319–1321 (1993).
[CrossRef]

Y. H. Jan, G. A. Fish, L. A. Coldren, S. P. DenBaars, “Demonstration of InP–InGaAsP vertical grating-assisted codirectional coupler filters and receivers with tapered coupling coefficient distributions,” IEEE Photon. Technol. Lett. 9, 994–996 (1997).
[CrossRef]

IRE Trans. Antennas Propag. (1)

A. Ishimaru, “Theory of unequally spaced arrays,” IRE Trans. Antennas Propag. 10, 691–702 (1963).
[CrossRef]

J. Lightwave Technol. (3)

K. A. Winick, “Design of grating-assisted waveguide couplers with weighted coupling,” J. Lightwave Technol. 9, 1481–1492 (1991).
[CrossRef]

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating-assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

H. Sakata, “Analysis of wavelength-selective photodetectors based on grating-assisted forward coupling,” J. Lightwave Technol. 11, 560–566 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

W. Vanderbauwhede, I. Moerman, P. Van Daele, P. Demeester, “Wavelength tuning by quantum well electrorefraction in a grating-assisted vertical coupler filter with InGaAsP/InAlGaAs MQW waveguide,” Opt. Quantum Electron. 28, 583–590 (1996).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (3)

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1975), Chap. 2.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Elsevier, New York, 1969).

P. N. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 466.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

(a) Waveguide structure and the refractive-index profile of a nonuniform vertical GACC with a varying grating width w n and varying positions z n . (b) Reference waveguide structure and refractive-index profile for the GACC shown in (a). The original grating layer is taken to be the same as the cover layer, i.e., with a refractive index n 1. This configuration corresponds to a nonsynchronous codirectional coupler without a grating overlay. (c) Waveguide structure and the refractive-index profile for the local modes; this configuration represents a combination of the reference index profile and the zeroth-order index perturbation that is due to the nonuniformity of the duty ratios μ.

Fig. 2
Fig. 2

Coupling coefficient κ plotted as a function of the local duty ratio μ for a first-order (m = 1) grating.

Fig. 3
Fig. 3

Synthesized filter response (points) compared with the desired filter response (solid curve) of a third-order Butterworth GACC filter designed by use of the coupled-local-mode equations and the GLM technique.

Fig. 4
Fig. 4

Local grating width w plotted as a function of the grating position z for a third-order Butterworth GACC filter.

Fig. 5
Fig. 5

Local grating period Λ plotted as a function of the grating position z for a third-order Butterworth GACC filter.

Fig. 6
Fig. 6

Response of the linear GACC filter designed by use of the coupled-local-mode equations and the GLM technique: the synthesized response (points) compared with the desired response (solid curve).

Fig. 7
Fig. 7

Local grating width w plotted as a function of the grating position z for a linear GACC filter.

Fig. 8
Fig. 8

Local grating period Λ plotted as a function of the grating position z for a linear GACC filter.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

Δn2x, z=n=1N Δng2 uz-znwn0<x<g0elsewhere,
n=- fn=m=-- fnexpi2mπndn,
Δn2z=Δng2m=-- uz-znwnexpi2mπndn.
n=nz=z/Λ0+νz,
1Λzdndz=1Λ0+dνdz.
Δn2z=Δng2m=-- uz-zwzexpi2mπnzΛzdz.
Δn2z=m=- Δnm2zexpim 2πΛ0 z,
Δnm2z=Δng2- uzwz-z×expi2mπνz-zΛz-z×exp-im2π/Λ0zdz,
dw/dz  1,  dΛ/dz  1
νz-zνz-dνdz z=νz+zΛ0-zΛz.
Δnm2zΔng2expi2mπνzΛz- uzwz×exp-im2π/Λzzdz.
Δn2z=m=- Δnˆm2zexpim 0z2πΛzdz.
Δnˆm2zΔnm2zexp-i2mπνz=Δng2Λz- uzwzexp-im2π/Λzzdz.
Δnˆm2z=Δng2μzm=0Δng2mπ sinmπμzm0,
μz=wzΛz.
n02x, zn¯2x+Δn02z=n12+Δng2μz0<x<gn¯2xelsewhere,
Ej=ejx, βjzexpi 0z βjzdz,  Hj=hjx, βjzexpi 0z βjzdz,
d2dx2+k02n¯2x+Δnˆ02x, z eyj=βj2zeyj,  hxj=-βjωμ eyj,  hzj=-iωμdeyjdx,
Et=j=1,2 ajzejx, βjzexpi 0z βjzdz,
da1dz=C11a1+C12a2 expi 0zβ2z-β1zdz,  da2dz=C22a2+C21a1 expi 0zβ1z-β2zdz,
Cjk=14-hxjeykz-eyjhxkzdx+i ω04 expi 0z2πΛzdz0g Δnˆ12eyjeykdx.
12-zetj×htkdx=δjk.
da1dz=iκza2 exp-2i 0z δzdz,  da2dz=iκza1 exp2i 0z δzdz,
δz=12β1z-β2z-2πΛz
κz=Δng2ω04π sinπμz0g ey1μzey2μzdx.
ã1z=a1zexpi 0z δzdz,  ã2z=a2zexp-i 0z δzdz,
dã1/dz=iδzã1+iκzã2,  dã2/dz=-iδzã2+iκzã1,
ξ=12β10λ-β20λ-2πΛ0.
ν1z=a1zexpiξz,  ν2z=a2zexp-iξz.
dν1z, ξ/dz-iξν1z, ξ=qzν2z, ξ,  dν2z, ξ/dz+iξν2z, ξ=-q*zν1z, ξ,
qz=iκzexpiθz,
θz=2πνz-0zΔβ1z-Δβ2zdz.
rξ=ν2L, ξ/ν1L, ξ,
Λz=1Λ0+12πdθdz+Δβ1z-Δβ2z2π-1.
Tξ=ν2L, ξν10, ξ2.
|rξ|2=Tξ1-Tξ.
Tξ=Tmax1+ξ/ξc6,
rξ=NξDξ,
Nξ=Tmax1/2,  Dξ=1-Tmax1/2+iξ/ξc3.
ξ12dβ10dλ-dβ20dλλ=λ0λ-λ0,
Tξ=Tmax1-ξ/ξc/2|ξ/ξc|10elsewhere,

Metrics