Abstract

An investigation of the chromatic properties of waveguide lenses is described. In general, the focal length of mode-index and Fresnel zone (diffractive) lenses will be a function of wavelength. As a result, these lenses will have high optical quality over only a relatively small wavelength range. A method is given to correct for this chromatic dispersion by forming a hybrid mode-index/diffractive doublet. Using this approach, the various lens parameters can be chosen so that the chromatic dispersion of the diffractive element will cancel the dispersion of the refractive element. In certain conditions, the power of the diffractive element in the hybrid achromatic doublet can be shown to go to zero producing an achromatic mode-index lens. It was found that, with typical waveguide materials, a 10-mm focal length, f/5 hybrid mode-index/diffractive lens can be made that has a usable wavelength range of ~80 nm. This is over an order of magnitude improvement compared with that obtained with conventional mode-index and diffractive lens types.

© 1991 Optical Society of America

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1991 (1)

1989 (8)

D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical Performance of Holographic Kinoforms,” Appl. Opt. 28, 976–983 (1989).
[CrossRef] [PubMed]

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

G. T. Sincerbox, “Challenges for the Use of Holographic Elements in Optical Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 80–91 (1989).

W. Zhou, V. M. Ristic, “Anisotropic Aberrations in Planar Waveguide Lenses,” IEEE J. Quantum Electron. QE-25, 749–754 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

D. Faklis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).

G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).

1988 (3)

1987 (2)

G. C. Righini, G. Molesini, “Design of Acircular Refractive Lenses for Integrated Optical Circuits,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 185–194 (1987).

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
[CrossRef]

1986 (5)

1985 (1)

P. Gidon, S. Valette, P. Mottier, “Integrated Lenses on Silicon Nitride Waveguides,” Opt. Eng. 24, 235–240 (1985).

1983 (4)

Z. D. Yu, “Waveguide Optical Planar Lenses in LiNbO3—Theory and Experiments,” Opt. Commun. 47, 248–250 (1983).
[CrossRef]

T. Suhara, T. Shiono, H. Nishihara, J. Koyama, “Integrated-Optic Fourier Processor Using an Acoustooptic Deflector and Fresnel Lenses in an As2S3 Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 624–630 (1983).
[CrossRef]

S. Valette et al., “Integrated Optical Spectrum Analyzer Using Planar Technology on Oxidized Silicon Substrate,” Electron. Lett. 19, 883–885 (1983).
[CrossRef]

S. Forouhar, R.-X. Lu, W. S. C. Chang, R. L. Davis, S.-K. Yao, “Chirped Grating Lenses on Nb2O5 Transition Waveguides,” Appl. Opt. 22, 3128–3132 (1983).
[CrossRef] [PubMed]

1982 (3)

1981 (4)

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

T. H. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE 69, 79–92 (1981).
[CrossRef]

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

G. M. Morris, “Diffraction Theory for an Achromatic Fourier Transformation,” Appl. Opt. 20, 2017–2025 (1981).
[CrossRef] [PubMed]

1980 (1)

W. S. C. Chang, P. R. Ashley, “Fresnel Lenses in Optical Waveguides,” IEEE J. Quantum Electron. QE-16, 744–753 (1980).
[CrossRef]

1979 (10)

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

G. C. Righini, V. Russo, S. Sottini, “A Family of Perfect Aspherical Geodesic Lenses for Integrated Optical Circuits,” IEEE J. Quantum Electron. QE-15, 1–4 (1979).
[CrossRef]

B. Chen, O. G. Ramer, “Diffraction-Limited Geodesic Lens for Integrated Optic Circuit,” IEEE J. Quantum Electron. QE-15, 853–860 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1979).

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

H. Madjidi-Zolbanine, C. Froehly, “Holographic Correction of Both Chromatic and Spherical Aberrations of Single Glass Lenses,” Appl. Opt. 18, 2385–2393 (1979).
[CrossRef] [PubMed]

S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, C. M. Oania, “Guided-Wave Optical Thin-Film Luneburg Lenses: Fabrication Technique and Properties,” Appl. Opt. 18, 4067–4079 (1979).
[CrossRef] [PubMed]

D. Kassai, E. Marom, “Aberration-Corrected Rounded-Edge Geodesic Lenses,” J. Opt. Soc. Am. 69, 1242–1248 (1979).
[CrossRef]

S. Sottini, V. Russo, G. C. Righini, “General Solution of the Problem of Perfect Geodesic Lenses for Integrated Optics,” J. Opt. Soc. Am. 69, 1248–1254 (1979).
[CrossRef]

1978 (3)

G. C. Righini, V. Russo, S. Sottini, “Signal Processing in Integrated Optics Employing Geodesic Lenses,” Proc. Soc. Photo-to-Opt. Instrum. Eng. 164, 20–26 (1978).

S. K. Yao, D. E. Thompson, “Chirp-Grating Lens for Guided-Wave Optics,” Appl. Phys. Lett. 33, 635–637 (1978).
[CrossRef]

G. Hatakoshi, S. Tanaka, “Grating Lenses for Integrated Optics,” Opt. Lett. 2, 142–144 (1978).
[CrossRef] [PubMed]

1977 (8)

1974 (3)

L. P. Boivin, “Thin-Film Laser-to-Fiber Coupler,” Appl. Opt. 13, 391–395 (1974).
[CrossRef] [PubMed]

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, G. Smolinsky, “Two-Layered Construction of Integrated Optical Circuits and Formation of Thin-Film Prisms, Lenses, and Reflectors,” Appl. Phys. Lett. 24, 547–549 (1974).
[CrossRef]

F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

1973 (1)

1972 (1)

1971 (2)

1970 (1)

1961 (1)

1959 (1)

A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. USSR 6, 126–133 (1959).

1955 (1)

L. Ronchi, “Geometrical Optics of Toroidal Junctions in Configuration Lenses,” Opt. Acta 2, 64–80 (1955).
[CrossRef]

1954 (1)

K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Double Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
[CrossRef]

Aagard, R. L.

R. L. Aagard, L. T. Shepherd, R. P. Ulmer, “Electron-Beam and Photolithographic Fabrication of Guided-Wave Optical Components,” J. Appl. Phys. 48, 4149–4151 (1977).
[CrossRef]

Anderson, D. B.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1979).

S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, C. M. Oania, “Guided-Wave Optical Thin-Film Luneburg Lenses: Fabrication Technique and Properties,” Appl. Opt. 18, 4067–4079 (1979).
[CrossRef] [PubMed]

Ashley, P. R.

W. S. C. Chang, P. R. Ashley, “Fresnel Lenses in Optical Waveguides,” IEEE J. Quantum Electron. QE-16, 744–753 (1980).
[CrossRef]

August, R. R.

S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, C. M. Oania, “Guided-Wave Optical Thin-Film Luneburg Lenses: Fabrication Technique and Properties,” Appl. Opt. 18, 4067–4079 (1979).
[CrossRef] [PubMed]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1979).

Barnoski, M. K.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

Boivin, L. P.

Boyd, J. T.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1979).

Bradley, J. C.

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

Buralli, D. A.

Chang, W. S. C.

S. Forouhar, R.-X. Lu, W. S. C. Chang, R. L. Davis, S.-K. Yao, “Chirped Grating Lenses on Nb2O5 Transition Waveguides,” Appl. Opt. 22, 3128–3132 (1983).
[CrossRef] [PubMed]

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

W. S. C. Chang, P. R. Ashley, “Fresnel Lenses in Optical Waveguides,” IEEE J. Quantum Electron. QE-16, 744–753 (1980).
[CrossRef]

Chen, B.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

B. Chen, O. G. Ramer, “Diffraction-Limited Geodesic Lens for Integrated Optic Circuit,” IEEE J. Quantum Electron. QE-15, 853–860 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

Davis, R. L.

S. Forouhar, R.-X. Lu, W. S. C. Chang, R. L. Davis, S.-K. Yao, “Chirped Grating Lenses on Nb2O5 Transition Waveguides,” Appl. Opt. 22, 3128–3132 (1983).
[CrossRef] [PubMed]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of Optical-Waveguide Lens Technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1979).

Delavaux, J.

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

Faklis, D.

D. Faklis, G. M. Morris, “Broadband Imaging with Holographic Lenses,” Opt. Eng. 28, 592–598 (1989).

Forouhar, S.

S. Forouhar, R.-X. Lu, W. S. C. Chang, R. L. Davis, S.-K. Yao, “Chirped Grating Lenses on Nb2O5 Transition Waveguides,” Appl. Opt. 22, 3128–3132 (1983).
[CrossRef] [PubMed]

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

Froehly, C.

Fujiwara, S.

George, N.

Gidon, P.

P. Gidon, S. Valette, P. Mottier, “Integrated Lenses on Silicon Nitride Waveguides,” Opt. Eng. 24, 235–240 (1985).

Goodman, J. W.

For example, see J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), pp. 90–96, 120–125.

Hamilton, M. C.

M. C. Hamilton, D. A. Wille, W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” in Ultrasonics Symposium, J. deKlerk, B. McAvoy, Eds. (IEEE, New York, 1976), pp. 218–222.

Harris, J. H.

Haruna, M.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Hatakoshi, G.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

G. Hatakoshi, S. Tanaka, “Grating Lenses for Integrated Optics,” Opt. Lett. 2, 142–144 (1978).
[CrossRef] [PubMed]

Hirsch, P. M.

Inoue, H.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

Ishimara, H.

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

Ito, S.

K. Tatsumi, T. Nakaguchi, S. Ito, “Wide Field Angle Bi-aspherical Waveguide Lens in LiNbO3 Fabricated by Proton-Exchange,” Electron. Lett. 24, 546–548 (1988).
[CrossRef]

Jordan, J. A.

Joseph, T. R.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

Kassai, D.

Kellner, A. L.

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

Klienhans, W. A.

Kobayashi, K.

Koyama, J.

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Integrated-Optic Disk Pickup Device,” IEEE/OSA J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

T. Suhara, T. Shiono, H. Nishihara, J. Koyama, “Integrated-Optic Fourier Processor Using an Acoustooptic Deflector and Fresnel Lenses in an As2S3 Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 624–630 (1983).
[CrossRef]

T. Suhara, K. Kobayashi, H. Nishihara, J. Koyama, “Graded-Index Fresnel Lenses for Integrated Optics,” Appl. Opt. 21, 1966–1971 (1982).
[CrossRef] [PubMed]

Kunz, K. S.

K. S. Kunz, “Propagation of Microwaves Between a Parallel Pair of Double Curved Conducting Surfaces,” J. Appl. Phys. 25, 642–653 (1954).
[CrossRef]

Lee, J. Y.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

Lesem, L. B.

Lin, Z.

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

Lu, R.-X.

Madjidi-Zolbanine, H.

Malarky, E. C.

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

Marom, E.

Martin, R. J.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, G. Smolinsky, “Two-Layered Construction of Integrated Optical Circuits and Formation of Thin-Film Prisms, Lenses, and Reflectors,” Appl. Phys. Lett. 24, 547–549 (1974).
[CrossRef]

R. Ulrich, R. J. Martin, “Geometrical Optics in Thin Film Light Guides,” Appl. Opt. 10, 2077–2085 (1971).
[CrossRef] [PubMed]

Mergerian, D.

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

Miceli, W. J.

M. C. Hamilton, D. A. Wille, W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” in Ultrasonics Symposium, J. deKlerk, B. McAvoy, Eds. (IEEE, New York, 1976), pp. 218–222.

Miyamoto, K.

Molesini, G.

G. C. Righini, G. Molesini, “Design of Acircular Refractive Lenses for Integrated Optical Circuits,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 185–194 (1987).

Morque, A.

S. Valette, A. Morque, P. Mottier, “High-Performance Integrated Fresnel Lenses on Oxidized Silicon Substrate,” Electron. Lett. 18, 13–15 (1982).
[CrossRef]

Morris, G. M.

Mottier, P.

P. Gidon, S. Valette, P. Mottier, “Integrated Lenses on Silicon Nitride Waveguides,” Opt. Eng. 24, 235–240 (1985).

S. Valette, A. Morque, P. Mottier, “High-Performance Integrated Fresnel Lenses on Oxidized Silicon Substrate,” Electron. Lett. 18, 13–15 (1982).
[CrossRef]

Naito, K.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

Nakaguchi, T.

K. Tatsumi, T. Nakaguchi, S. Ito, “Wide Field Angle Bi-aspherical Waveguide Lens in LiNbO3 Fabricated by Proton-Exchange,” Electron. Lett. 24, 546–548 (1988).
[CrossRef]

Nishihara, H.

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Integrated-Optic Disk Pickup Device,” IEEE/OSA J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
[CrossRef] [PubMed]

T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
[CrossRef] [PubMed]

T. Suhara, T. Shiono, H. Nishihara, J. Koyama, “Integrated-Optic Fourier Processor Using an Acoustooptic Deflector and Fresnel Lenses in an As2S3 Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 624–630 (1983).
[CrossRef]

T. Suhara, K. Kobayashi, H. Nishihara, J. Koyama, “Graded-Index Fresnel Lenses for Integrated Optics,” Appl. Opt. 21, 1966–1971 (1982).
[CrossRef] [PubMed]

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

Norris, J. A.

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Formation of Negative-Index-Change Waveguide Lenses in LiNbO3 by Using Ion Milling,” Opt. Lett. 13, 1141–1143 (1988).
[CrossRef] [PubMed]

Oania, C. M.

Pautienus, R. P.

D. Mergerian, E. C. Malarky, R. P. Pautienus, J. C. Bradley, A. L. Kellner, “Advances in Integrated Optical Spectrum Analyzers,” Proc. Soc. Photo-Opt. Instrum. Eng. 269, 129–135 (1981).

Ramer, O. G.

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

B. Chen, O. G. Ramer, “Diffraction-Limited Geodesic Lens for Integrated Optic Circuit,” IEEE J. Quantum Electron. QE-15, 853–860 (1979).
[CrossRef]

M. K. Barnoski, B. Chen, T. R. Joseph, J. Y. Lee, O. G. Ramer, “Integrated-Optic Spectrum Analyzer,” IEEE Trans. Circuits Syst. CAS-26, 1113–1124 (1979).
[CrossRef]

Righini, G. C.

G. C. Righini, G. Molesini, “Design of Acircular Refractive Lenses for Integrated Optical Circuits,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 185–194 (1987).

G. C. Righini, V. Russo, S. Sottini, “A Family of Perfect Aspherical Geodesic Lenses for Integrated Optical Circuits,” IEEE J. Quantum Electron. QE-15, 1–4 (1979).
[CrossRef]

S. Sottini, V. Russo, G. C. Righini, “General Solution of the Problem of Perfect Geodesic Lenses for Integrated Optics,” J. Opt. Soc. Am. 69, 1248–1254 (1979).
[CrossRef]

G. C. Righini, V. Russo, S. Sottini, “Signal Processing in Integrated Optics Employing Geodesic Lenses,” Proc. Soc. Photo-to-Opt. Instrum. Eng. 164, 20–26 (1978).

G. C. Righini, V. Russo, S. Sottini, G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Windows,” Appl. Opt. 12, 1477–1481 (1973).
[CrossRef] [PubMed]

G. C. Righini, V. Russo, S. Sattini, G. Toraldo di Francia, “Thin Film Geodesic Lens,” Appl. Opt. 11, 1442–1443 (1972).
[CrossRef] [PubMed]

Ristic, V. M.

W. Zhou, V. M. Ristic, “Anisotropic Aberrations in Planar Waveguide Lenses,” IEEE J. Quantum Electron. QE-25, 749–754 (1989).
[CrossRef]

Riva-Sanseverino, S.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, G. Smolinsky, “Two-Layered Construction of Integrated Optical Circuits and Formation of Thin-Film Prisms, Lenses, and Reflectors,” Appl. Phys. Lett. 24, 547–549 (1974).
[CrossRef]

Rogers, J. R.

Ronchi, L.

L. Ronchi, “Geometrical Optics of Toroidal Junctions in Configuration Lenses,” Opt. Acta 2, 64–80 (1955).
[CrossRef]

Russo, V.

G. C. Righini, V. Russo, S. Sottini, “A Family of Perfect Aspherical Geodesic Lenses for Integrated Optical Circuits,” IEEE J. Quantum Electron. QE-15, 1–4 (1979).
[CrossRef]

S. Sottini, V. Russo, G. C. Righini, “General Solution of the Problem of Perfect Geodesic Lenses for Integrated Optics,” J. Opt. Soc. Am. 69, 1248–1254 (1979).
[CrossRef]

G. C. Righini, V. Russo, S. Sottini, “Signal Processing in Integrated Optics Employing Geodesic Lenses,” Proc. Soc. Photo-to-Opt. Instrum. Eng. 164, 20–26 (1978).

G. C. Righini, V. Russo, S. Sottini, G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Windows,” Appl. Opt. 12, 1477–1481 (1973).
[CrossRef] [PubMed]

G. C. Righini, V. Russo, S. Sattini, G. Toraldo di Francia, “Thin Film Geodesic Lens,” Appl. Opt. 11, 1442–1443 (1972).
[CrossRef] [PubMed]

Sattini, S.

Shepherd, L. T.

R. L. Aagard, L. T. Shepherd, R. P. Ulmer, “Electron-Beam and Photolithographic Fabrication of Guided-Wave Optical Components,” J. Appl. Phys. 48, 4149–4151 (1977).
[CrossRef]

Shiono, T.

T. Suhara, T. Shiono, H. Nishihara, J. Koyama, “Integrated-Optic Fourier Processor Using an Acoustooptic Deflector and Fresnel Lenses in an As2S3 Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 624–630 (1983).
[CrossRef]

Shubert, R.

Sincerbox, G. T.

G. T. Sincerbox, “Challenges for the Use of Holographic Elements in Optical Storage,” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 80–91 (1989).

Smith, W. J.

For example, see W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 334–340.

Smolinsky, G.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, G. Smolinsky, “Two-Layered Construction of Integrated Optical Circuits and Formation of Thin-Film Prisms, Lenses, and Reflectors,” Appl. Phys. Lett. 24, 547–549 (1974).
[CrossRef]

Sottini, S.

G. C. Righini, V. Russo, S. Sottini, “A Family of Perfect Aspherical Geodesic Lenses for Integrated Optical Circuits,” IEEE J. Quantum Electron. QE-15, 1–4 (1979).
[CrossRef]

S. Sottini, V. Russo, G. C. Righini, “General Solution of the Problem of Perfect Geodesic Lenses for Integrated Optics,” J. Opt. Soc. Am. 69, 1248–1254 (1979).
[CrossRef]

G. C. Righini, V. Russo, S. Sottini, “Signal Processing in Integrated Optics Employing Geodesic Lenses,” Proc. Soc. Photo-to-Opt. Instrum. Eng. 164, 20–26 (1978).

G. C. Righini, V. Russo, S. Sottini, G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Windows,” Appl. Opt. 12, 1477–1481 (1973).
[CrossRef] [PubMed]

Southwell, W. H.

Spaulding, K. E.

Sprague, R. A.

R. A. Sprague, “A Review of Acousto-Optic Signal Correlators,” Opt. Eng. 16, 467–474 (1977).

Stone, T. W.

Suhara, T.

T. Suhara, H. Nishihara, “Integrated-Optic Pickup Devices Using Waveguide Holographic Components (Extended Paper),” Proc. Soc. Photo-Opt. Instrum. Eng. 1136, 92–99 (1989).

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Integrated-Optic Disk Pickup Device,” IEEE/OSA J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
[CrossRef] [PubMed]

T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
[CrossRef] [PubMed]

T. Suhara, T. Shiono, H. Nishihara, J. Koyama, “Integrated-Optic Fourier Processor Using an Acoustooptic Deflector and Fresnel Lenses in an As2S3 Waveguide,” IEEE/OSA J. Lightwave Technol. LT-1, 624–630 (1983).
[CrossRef]

T. Suhara, K. Kobayashi, H. Nishihara, J. Koyama, “Graded-Index Fresnel Lenses for Integrated Optics,” Appl. Opt. 21, 1966–1971 (1982).
[CrossRef] [PubMed]

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

Swanson, G. J.

G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).

Sweatt, W. C.

Tanaka, S.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

G. Hatakoshi, S. Tanaka, “Grating Lenses for Integrated Optics,” Opt. Lett. 2, 142–144 (1978).
[CrossRef] [PubMed]

Tatsumi, K.

K. Tatsumi, T. Nakaguchi, S. Ito, “Wide Field Angle Bi-aspherical Waveguide Lens in LiNbO3 Fabricated by Proton-Exchange,” Electron. Lett. 24, 546–548 (1988).
[CrossRef]

Taylor, H. F.

H. F. Taylor, “Application of Guided-Wave Optics in Signal Processing and Sensing,” Proc. IEEE 75, 1524–1535 (1987).
[CrossRef]

Thompson, D. E.

S. K. Yao, D. E. Thompson, “Chirp-Grating Lens for Guided-Wave Optics,” Appl. Phys. Lett. 33, 635–637 (1978).
[CrossRef]

Tien, P. K.

P. K. Tien, S. Riva-Sanseverino, R. J. Martin, G. Smolinsky, “Two-Layered Construction of Integrated Optical Circuits and Formation of Thin-Film Prisms, Lenses, and Reflectors,” Appl. Phys. Lett. 24, 547–549 (1974).
[CrossRef]

Toraldo di Francia, G.

Tsai, C. S.

Tudorovskii, A. I.

A. I. Tudorovskii, “An Objective with a Phase Plate,” Opt. Spectrosc. USSR 6, 126–133 (1959).

Turpin, T. H.

T. H. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE 69, 79–92 (1981).
[CrossRef]

Ulmer, R. P.

R. L. Aagard, L. T. Shepherd, R. P. Ulmer, “Electron-Beam and Photolithographic Fabrication of Guided-Wave Optical Components,” J. Appl. Phys. 48, 4149–4151 (1977).
[CrossRef]

Ulrich, R.

Umegaki, S.

G. Hatakoshi, H. Inoue, K. Naito, S. Umegaki, S. Tanaka, “Optical Waveguide Lenses,” Opt. Acta 26, 961–968 (1979).
[CrossRef]

Ura, S.

S. Ura, T. Suhara, H. Nishihara, J. Koyama, “Integrated-Optic Disk Pickup Device,” IEEE/OSA J. Lightwave Technol. LT-4, 913–918 (1986).
[CrossRef]

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

Vahey, D. W.

D. W. Vahey, V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–134 (1977).
[CrossRef]

Valette, S.

P. Gidon, S. Valette, P. Mottier, “Integrated Lenses on Silicon Nitride Waveguides,” Opt. Eng. 24, 235–240 (1985).

S. Valette et al., “Integrated Optical Spectrum Analyzer Using Planar Technology on Oxidized Silicon Substrate,” Electron. Lett. 19, 883–885 (1983).
[CrossRef]

S. Valette, A. Morque, P. Mottier, “High-Performance Integrated Fresnel Lenses on Oxidized Silicon Substrate,” Electron. Lett. 18, 13–15 (1982).
[CrossRef]

Van Rooy, D. L.

Veldkamp, W. B.

G. J. Swanson, W. B. Veldkamp, “Diffractive Optical Elements for use in Infrared Systems,” Opt. Eng. 28, 605–608 (1989).

Vu, T. Q.

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Planar Waveguide Lenses in GaAs by Using Ion Milling,” Appl. Phys. Lett. 54, 1098–1100 (1989).
[CrossRef]

T. Q. Vu, J. A. Norris, C. S. Tsai, “Formation of Negative-Index-Change Waveguide Lenses in LiNbO3 by Using Ion Milling,” Opt. Lett. 13, 1141–1143 (1988).
[CrossRef] [PubMed]

Welford, W. T.

For example, see W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986), pp. 35–40.

Wille, D. A.

M. C. Hamilton, D. A. Wille, W. J. Miceli, “An Integrated Optical RF Spectrum Analyzer,” in Ultrasonics Symposium, J. deKlerk, B. McAvoy, Eds. (IEEE, New York, 1976), pp. 218–222.

Wood, V. E.

D. W. Vahey, V. E. Wood, “Focal Characteristics of Spheroidal Geodesic Lenses for Integrated Optical Processing,” IEEE J. Quantum Electron. QE-13, 129–134 (1977).
[CrossRef]

Yao, S. K.

Yao, S.-K.

Youmans, B. R.

Yu, Z. D.

Z. D. Yu, “Waveguide Optical Planar Lenses in LiNbO3—Theory and Experiments,” Opt. Commun. 47, 248–250 (1983).
[CrossRef]

Zang, D. Y.

Zernike, F.

F. Zernike, “Luneburg Lens for Optical Waveguide Use,” Opt. Commun. 12, 379–381 (1974).
[CrossRef]

Zhou, S.

Z. Lin, S. Zhou, W. S. C. Chang, S. Forouhar, J. Delavaux, “A Generalized Two-Dimensional Coupled-Mode Analysis of Curved and Chirped Periodic Structures in Open Dielectric Waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 881–891 (1981).

Zhou, W.

W. Zhou, V. M. Ristic, “Anisotropic Aberrations in Planar Waveguide Lenses,” IEEE J. Quantum Electron. QE-25, 749–754 (1989).
[CrossRef]

Appl. Opt. (19)

L. P. Boivin, “Thin-Film Laser-to-Fiber Coupler,” Appl. Opt. 13, 391–395 (1974).
[CrossRef] [PubMed]

K. E. Spaulding, G. M. Morris, “Achromatic Waveguide Input/Output Coupler Design,” Appl. Opt. 30, 1096–1112 (1991).
[CrossRef] [PubMed]

G. C. Righini, V. Russo, S. Sattini, G. Toraldo di Francia, “Thin Film Geodesic Lens,” Appl. Opt. 11, 1442–1443 (1972).
[CrossRef] [PubMed]

G. C. Righini, V. Russo, S. Sottini, G. Toraldo di Francia, “Geodesic Lenses for Guided Optical Windows,” Appl. Opt. 12, 1477–1481 (1973).
[CrossRef] [PubMed]

D. Y. Zang, C. S. Tsai, “Titanium-Indiffused Proton-Exchanged Waveguide Lenses in LiNbO3 for Optical Information Processing,” Appl. Opt. 25, 2264–2271 (1986).
[CrossRef] [PubMed]

T. Suhara, S. Fujiwara, H. Nishihara, “Proton-Exchanged Fresnel Lenses in Ti:LiNbO3 Waveguides,” Appl. Opt. 25, 3379–3383 (1986).
[CrossRef] [PubMed]

R. Ulrich, R. J. Martin, “Geometrical Optics in Thin Film Light Guides,” Appl. Opt. 10, 2077–2085 (1971).
[CrossRef] [PubMed]

D. Y. Zang, C. S. Tsai, “Titanium-Indiffused Proton-Exchanged Waveguide Lenses in LiNbO3 for Optical Information Processing,” Appl. Opt. 25, 2264–2271 (1986).
[CrossRef] [PubMed]

S. K. Yao, D. B. Anderson, R. R. August, B. R. Youmans, C. M. Oania, “Guided-Wave Optical Thin-Film Luneburg Lenses: Fabrication Technique and Properties,” Appl. Opt. 18, 4067–4079 (1979).
[CrossRef] [PubMed]

W. H. Southwell, “Planar Optical Waveguide Lens Design,” Appl. Opt. 21, 1985–1988 (1982).
[CrossRef] [PubMed]

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Other (8)

T. Suhara, H. Ishimara, S. Ura, H. Nishihara, “Integration of Detection Optics for Magnetooptical Disk Pickup,” in Technical Digest, Seventh International Conference on Integrated Optics and Optical Fiber Communication, Kobe, Japan (1989), Vol. 4, pp. 80–81.

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For example, see H. Nishihara, M. Haruna, T. Suhara, Optical Integrated Circuits (McGraw-Hill, New York, 1989), pp. 11–15.

For example, see W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986), pp. 35–40.

Corning 7059 characteristics: “Corning Material Information Data-Sheet MI-7059-89. SiO2 Characteristics,” in CRC Handbook of Laser Science and Technology, Vol. 4: Optical Materials Part 2: Properties, M. J. Weber, Ed. (CRC Press, Boca Raton, FL, 1986), p. 31.

For example, see W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), pp. 334–340.

For example, see J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), pp. 90–96, 120–125.

Super-Oslo is a trademark of Sinclair Optics, 6780 Palmyra Rd., Fairport, NY 14450.

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Figures (11)

Fig. 1
Fig. 1

Top and side views of standard types of waveguide lens: (a) geodesic, (b) mode index, (c) Luneburg, and (d) diffractive.

Fig. 2
Fig. 2

Effective mode index for the TE0 mode as a function of wavelength for various thicknesses t of a step-index Corning 7059 waveguide on an oxidized silicon substrate.

Fig. 3
Fig. 3

Relative index dispersion of the TE0 mode as a function of the effective mode index for a step-index Corning 7059 waveguide on an oxidized silicon substrate. The design wavelength is 0.6328 μm. Each point on the curve corresponds to a different waveguide thickness. Points A and B are an example of a pair of indices which would give an achromatic mode-index lens corresponding to waveguide thicknesses of 0.35 and 0.66 μm, respectively.

Fig. 4
Fig. 4

Possible configuration for hybrid mode-index/diffractive achromat.

Fig. 5
Fig. 5

Comparison of longitudinal chromatic aberration (wavelength dependent focal length errors) for conventional mode-index and diffractive lenses and a hybrid mode-index/diffractive achromat. The simulated waveguide consisted of a 0.5-μm layer of Corning 7059 glass on an oxidized silicon substrate giving a TE0 mode index of 1.484 at λ0 = 0.6328 μm. The mode-index lens elements were formed by reducing the thickness to 0.3 μm to give a mode-index difference of ΔN0) = −0.024. All three lenses were designed to be f/5 and have a nominal focal length of 10 mm.

Fig. 6
Fig. 6

Notation used for point spread function calculation: s is the distance from the exit pupil to the paraxial image plane, ξm is the halfwidth of the exit pupil, and d is the defocus amount. W(ξ) is the difference between the reference circle and the aberrated wavefront.

Fig. 7
Fig. 7

Strehl ratio as a function of wavelength error for conventional mode-index (solid curve) and diffractive (dotted curve) lenses and the hybrid mode-index/diffractive achromat (dashed curve). The simulated waveguide consisted of a 0.5-μm layer of Corning 7059 glass on an oxidized silicon substrate giving a mode index of 1.484 at λ0 = 0.6328 μm for the TE0 mode. The mode-index lens elements were formed by reducing the thickness to 0.3 μm to give a mode-index difference of ΔN0) = −0.024. All three lenses were designed to be f/5 and have a nominal focal length of 10 mm. The line at a Strehl ratio of 0.8 indicates the generally accepted limit for near diffraction-limited performance.

Fig. 8
Fig. 8

Spot size (FWHM) as a function of wavelength error for conventional mode-index (solid curve) and diffractive (dotted curve) lenses and the hybrid mode-index/diffractive achromat (dashed curve). The simulated waveguide consisted of a 0.5-μm layer of Corning 7059 glass on an oxidized silicon substrate giving a mode index of 1.484 at λ0 = 0.6328 μm for the TE0 mode. The mode-index lens elements were formed by reducing the thickness to 0.3 μm to give a mode-index difference of ΔN0) = −0.024. All three lenses were designed to be f/5 and have a nominal focal length of 10 mm.

Fig. 9
Fig. 9

Comparison of the Strehl ratio calculated using the defocus-only assumption (solid curve) with results of the exact ray trace (solid circles) for a hybrid mode-index/diffractive achromat. The simulated waveguide consisted of a 0.5-μm layer of Corning 7059 glass on an oxidized silicon substrate. The mode-index lens element was formed by reducing the thickness to 0.3 μm. The lens was designed to be f/5 and have a nominal focal length of 10 mm. The exact ray trace data were generated using the program Super-Oslo.

Fig. 10
Fig. 10

Comparison of the FWHM spot size calculated using a defocus-only assumption (solid curve) with results of an exact ray trace (solid circles) for a hybrid mode-index/diffractive achromat. The simulated waveguide consisted of a 0.5-μm layer of Corning 7059 glass on an oxidized silicon substrate. The mode-index lens element was formed by reducing the thickness to 0.3 μm. The lens was designed to be f/5 and have a nominal focal length of 10 mm. The exact ray trace data were generated using the program Super-Oslo.

Fig. 11
Fig. 11

Achromatic wavelength range of hybrid mode-index/diffractive achromatic lens as a function of the background mode index for a series of different ΔN values. The simulated waveguide was Corning 7059 glass on an oxidized silicon substrate, and the nominal wavelength was λ0 = 0.6328 μm. The desired mode indices were obtained by choosing the appropriate thickness of the waveguide layer. The lenses were designed to be f/5 and have a nominal focal length of 10 mm. For each configuration, the power of the mode-index and Fresnel components was chosen using Eqs. (18) and (19) to satisfy the achromatic lens condition. The solid points on the curves indicate the locus of points where the lens is purely of the mode-index type.

Tables (1)

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Table I Material Dispersion Characteristics Used in Calculatlons

Equations (27)

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Φ mi ( λ ) = C [ N L ( λ ) - N B ( λ ) ] ,
F mi ( λ ) = N B ( λ ) Φ mi ( λ ) = N B ( λ ) C [ N L ( λ ) - N B ( λ ) ] ,
d F mi ( λ ) d λ = Δ N ( λ ) d N B ( λ ) d λ - Δ [ d N ( λ ) d λ ] N B ( λ ) C Δ N ( λ ) 2 ,
Δ N ( λ ) N L ( λ ) - N B ( λ ) ,
Δ [ d N ( λ ) d λ ] d N L ( λ ) d λ - d N B ( λ ) d λ .
1 N L ( λ 0 ) d N L ( λ 0 ) d λ = 1 N B ( λ 0 ) d N B ( λ 0 ) d λ ,
Φ d ( λ ) = ( λ λ 0 ) Φ d 0 ,
F d ( λ ) = N B ( λ ) Φ d ( λ ) = λ 0 λ N B ( λ ) N B 0 F d 0 ,
d F d ( λ ) d λ = λ 0 F d 0 N B 0 λ d N B ( λ ) d λ - N B ( λ ) λ 2 .
d F d ( λ 0 ) d λ = F d 0 N B 0 [ d N B ( λ 0 ) d λ - N B ( λ 0 ) λ 0 ] .
d N B ( λ 0 ) d λ = N B ( λ 0 ) λ 0 .
Φ T ( λ ) Φ mi ( λ ) + Φ d ( λ ) .
Φ T ( λ ) = C Δ N ( λ ) + ( λ λ 0 ) Φ d 0 .
F T ( λ ) = N B ( λ ) Φ T ( λ ) = N B ( λ ) C Δ N ( λ ) + ( λ λ 0 ) Φ d 0 .
d F T ( λ ) d λ = [ C Δ N ( λ ) + ( λ λ 0 ) Φ d 0 ] d N B ( λ ) d λ - [ C Δ ( d N ( λ ) d λ ) + Φ d 0 λ 0 ] N B ( λ ) [ C Δ N ( λ ) + ( λ λ 0 ) Φ d 0 ] 2 .
[ C Δ N ( λ 0 ) + Φ d 0 ] d N B ( λ 0 ) d λ { C Δ [ d N ( λ 0 ) d λ ] + Φ d 0 λ 0 } N B ( λ 0 ) = 0.
Φ T 0 = Φ T ( λ 0 ) = C Δ N ( λ 0 ) + Φ d 0 .
Φ mi 0 = C Δ N ( λ 0 ) = Φ T 0 Δ N ( λ 0 ) N B ( λ 0 ) { d N B ( λ 0 ) d λ - N B ( λ 0 ) λ 0 Δ [ d N ( λ 0 ) d λ ] - Δ N ( λ 0 ) λ 0 } ,
Φ d 0 = Φ T 0 N B ( λ 0 ) { N B ( λ 0 ) Δ [ d N ( λ 0 ) d λ ] - d N B ( λ 0 ) d λ Δ N ( λ 0 ) Δ [ d N ( λ 0 ) d λ ] - Δ N ( λ 0 ) λ 0 } .
F T ( λ 1 ) = F T 0 = N B ( λ 1 ) C 2 Δ N ( λ 1 ) + ( λ 1 λ 0 ) Φ d 0 , 2 ,
F T ( λ 2 ) = F T 0 = N B ( λ 2 ) C 2 Δ N ( λ 2 ) + ( λ 2 λ 0 ) Φ d 0 , 2 .
Φ mi 0 , 2 = C 2 Δ N ( λ 0 ) = Δ N ( λ 0 ) F T 0 [ λ 2 N B ( λ 1 ) - λ 1 N B ( λ 2 ) λ 2 Δ N ( λ 1 ) - λ 1 Δ N ( λ 2 ) ] ,
Φ d 0 , 2 = λ 0 F T 0 [ N B ( λ 2 ) N L ( λ 1 ) - N B ( λ 1 ) N L ( λ 2 ) λ 2 Δ N ( λ 1 ) - λ 1 Δ N ( λ 2 ) ] .
U ( x ) = A - ξ m ξ m exp { - i 2 π λ [ x ξ N B s + W ( ξ ) ] } d ξ ,
W ( ξ ) a 1 ( ξ ξ m ) 2 ,
a 1 N B d 2 ( ξ m s ) 2 .
U ( 0 ) = - ξ m ξ m exp [ - i 2 π λ W ( ξ ) ] d ξ - ξ m ξ m d ξ ,

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