Abstract

We discuss the use of holographic photopolymer materials for use as dense wavelength division multiplexing filters in the C-band of the optical communication spectrum. An edge-illuminated hologram configuration is described that effectively extends the grating length to achieve narrow band filters operating near 1550 nm in photopolymers that are 100–200-μm thick. This configuration enables the formation of apodized and cascaded filter systems. Rouard’s method is used to examine the properties of both apodization and cascaded gratings and indicates the potential for narrow spectral bandwidths (<0.2 nm) and high side-lobe suppression (<-30 dB). Initial experimental results with a commercially available photopolymer are provided that verify narrowband spectral-transmittance properties (<0.6 nm) and the ability to apodize the index profile. The primary limitation of the design is the absorption of existing photopolymer materials. Optimizing the polymer chemistry for filter design at 1550 nm may solve this problem.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.
  2. N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).
  3. R. Ramaswami, K. N. Sivarajan, Optical Networks, A Practical Perspective, 2nd ed., (Morgan Kaufmann, Los Altos, Calif., 2002), Chap. 3.
  4. R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).
  5. J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
    [CrossRef]
  6. P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
    [CrossRef]
  7. H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2946 (1969).
    [CrossRef]
  8. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  9. D. C. O’Shea, Elements of Modern Optical Design, (Wiley Series in Pure and Applied Optics Wiley-Interscience, New York, 1985), pp. 232–234.
  10. J. N. Latta, R. V. Pole, “Design techniques for forming 488 nm holographic lenses with reconstruction at 633 nm,” Appl. Opt. 18, 2418–2421 (1979).
    [CrossRef] [PubMed]
  11. M. P. Rouard, “Etudes des propertietes optiques des lames metal-liques tres minces,” Ann. Phys. (Paris) ser. II 7, 291–384 (1937).
  12. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: application of Rouard’s method,” J. Opt. Soc. Am. A 2, 863–871 (1985).
    [CrossRef]
  13. L. A. Weller-Brophy, D. G. Hall, “Analysis of waveguide gratings: a comparison of the results of Rouard’s method and coupled-wave theory,” J. Opt. Soc. Am. A 4, 60–65 (1987).
    [CrossRef]
  14. D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
    [CrossRef]
  15. D. A. Waldman, H.-Y. S. Li, “Determination of low-transverse shrinkage in slant fringe gratings of a cationic ring-opening volume hologram recording material,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 354–372 (1997).
    [CrossRef]
  16. D. A. Waldman, H.-Y. S. Li, E. Cetin, “Holographic recording properties in thick films of ULSH-500 photopolymer,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 89–103 (1998).
    [CrossRef]
  17. T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1978).
    [CrossRef]
  18. L. Eldada, “Advances in telecom and datacom optical components,” Opt. Eng. 40, 1165–1178 (2001).
    [CrossRef]

2001 (1)

L. Eldada, “Advances in telecom and datacom optical components,” Opt. Eng. 40, 1165–1178 (2001).
[CrossRef]

2000 (2)

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

1997 (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

1987 (1)

1985 (1)

1979 (1)

1978 (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1978).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2946 (1969).
[CrossRef]

1937 (1)

M. P. Rouard, “Etudes des propertietes optiques des lames metal-liques tres minces,” Ann. Phys. (Paris) ser. II 7, 291–384 (1937).

Boffi, P.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

Castanon, G.

G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.

Cetin, E.

D. A. Waldman, H.-Y. S. Li, E. Cetin, “Holographic recording properties in thick films of ULSH-500 photopolymer,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 89–103 (1998).
[CrossRef]

Chen, R. T.

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

Chong, T. H.

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

Choudhary, S.

G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.

Dhal, P. K.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

Dout, D. G.

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

Eldada, L.

L. Eldada, “Advances in telecom and datacom optical components,” Opt. Eng. 40, 1165–1178 (2001).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Frascolla, C.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

Hall, D. G.

Hamdan, A. A.

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

Horner, M. G.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

Hoshida, T.

G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.

Ingwall, R. T.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2946 (1969).
[CrossRef]

Kolb, E. S.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

Kubota, T.

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1978).
[CrossRef]

Latta, J. N.

Li, H.-Y. S.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, E. Cetin, “Holographic recording properties in thick films of ULSH-500 photopolymer,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 89–103 (1998).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, “Determination of low-transverse shrinkage in slant fringe gratings of a cationic ring-opening volume hologram recording material,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 354–372 (1997).
[CrossRef]

Liu, J.

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

Makeda, N.

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

Martinelli, M.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

Minns, R. A.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design, (Wiley Series in Pure and Applied Optics Wiley-Interscience, New York, 1985), pp. 232–234.

Piccinin, D.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

Pole, R. V.

Qiao, J.

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

Ramaswami, R.

R. Ramaswami, K. N. Sivarajan, Optical Networks, A Practical Perspective, 2nd ed., (Morgan Kaufmann, Los Altos, Calif., 2002), Chap. 3.

Rouard, M. P.

M. P. Rouard, “Etudes des propertietes optiques des lames metal-liques tres minces,” Ann. Phys. (Paris) ser. II 7, 291–384 (1937).

Schild, H. G.

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

Sivarajan, K. N.

R. Ramaswami, K. N. Sivarajan, Optical Networks, A Practical Perspective, 2nd ed., (Morgan Kaufmann, Los Altos, Calif., 2002), Chap. 3.

Ubaldi, M. C.

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

Vassilieva, O.

G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.

Waldman, D. A.

D. A. Waldman, H.-Y. S. Li, “Determination of low-transverse shrinkage in slant fringe gratings of a cationic ring-opening volume hologram recording material,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 354–372 (1997).
[CrossRef]

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, E. Cetin, “Holographic recording properties in thick films of ULSH-500 photopolymer,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 89–103 (1998).
[CrossRef]

Weller-Brophy, L. A.

Zhao, F.

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

Ann. Phys. (Paris) ser. II (1)

M. P. Rouard, “Etudes des propertietes optiques des lames metal-liques tres minces,” Ann. Phys. (Paris) ser. II 7, 291–384 (1937).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram grating,” Bell Syst. Tech. J. 48, 2909–2946 (1969).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

N. Makeda, A. A. Hamdan, T. H. Chong, D. G. Dout, “Polarization independent, linear-tuned interference filter with constant transmission characteristics over 1530–1570-nm tuning range,” IEEE Photon. Technol. Lett. 9, 783–784 (1997).

J. Qiao, F. Zhao, J. Liu, R. T. Chen, “Dispersion enhanced volume hologram for dense wavelength division demultiplexer,” IEEE Photon. Technol. Lett. 12, 1070–1072 (2000).
[CrossRef]

P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, M. Martinelli, “1550 nm volume holography for optical communication devices,” IEEE Photon. Technol. Lett. 12, 1355–1357 (2000).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

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

Opt. Acta (1)

T. Kubota, “Characteristics of thick hologram grating recorded in absorptive medium,” Opt. Acta 25, 1035–1053 (1978).
[CrossRef]

Opt. Eng. (1)

L. Eldada, “Advances in telecom and datacom optical components,” Opt. Eng. 40, 1165–1178 (2001).
[CrossRef]

Other (7)

D. A. Waldman, R. T. Ingwall, P. K. Dhal, M. G. Horner, E. S. Kolb, H.-Y. S. Li, R. A. Minns, H. G. Schild, “Cationic ring-opening photopolymerimization methods for volume hologram recording,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 127–141 (1996).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, “Determination of low-transverse shrinkage in slant fringe gratings of a cationic ring-opening volume hologram recording material,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 354–372 (1997).
[CrossRef]

D. A. Waldman, H.-Y. S. Li, E. Cetin, “Holographic recording properties in thick films of ULSH-500 photopolymer,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 89–103 (1998).
[CrossRef]

D. C. O’Shea, Elements of Modern Optical Design, (Wiley Series in Pure and Applied Optics Wiley-Interscience, New York, 1985), pp. 232–234.

G. Castanon, O. Vassilieva, S. Choudhary, T. Hoshida, “Requirement of filter characteristics for 40 Gbit/s-based DWDM systems,” in Proc. 27th Eur. Conf. on Opt. Comm. (ECOC’01—Amsterdam) (IEEE, New York2001), pp. 60–61.

R. Ramaswami, K. N. Sivarajan, Optical Networks, A Practical Perspective, 2nd ed., (Morgan Kaufmann, Los Altos, Calif., 2002), Chap. 3.

R. Kashyap, Fiber Bragg Gratings (Academic, San Diego, Calif., 1999).

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) Conventional reflection volume hologram, (b) edge-illuminated hologram configuration. The hologram thickness for the conventional hologram is d and has a value of L for the edge-illuminated hologram. EI and ER represent incident and reflected electric fields, respectively.

Fig. 2
Fig. 2

(a) Edge-illuminated holographic filter used in a fiber optic system. L is the length of the grating. (b) Edge-illuminated grating and Gaussian beam mode formed by the micro lens coupler.

Fig. 3
Fig. 3

(a) Representation of Rouard’s method of grating analysis, (b) replacement of a thin-film layer 1 by a single interface described by complex reflectivity ρ1.

Fig. 4
Fig. 4

(a) Gaussian apodized index modulation for an edge-illuminated hologram. The average index (1.4956) corresponds to the measured value for Aprilis ULSH 500-7A. The grating is 20 mm long. (b) Frequency response for the Gaussian apodized index modulation. Solid curve is the reflectance for a normally incident reconstruction beam and the dashed curve is the reflectance when the reconstruction beam is incident at an angle of θ = 5.74 × 10-4 radians. This angle corresponds to the divergence angle of the Gaussian beam in a typical edge-illuminated holographic filter.

Fig. 5
Fig. 5

Diffraction efficiency as a function of the reconstruction angle at 514.5 nm (cross) and 1550 nm (circle) for an unslanted transmission hologram with a 1.2-μm period in a 100-μm thick sample of ULSH 500-7A photopolymer. The solid curve shows the theoretical values of diffraction efficiency with Kogelnik’s two-wave coupled-wave model with attenuation as a function of depth into the material.

Fig. 6
Fig. 6

(a) Experimental system for measuring the spectral bandwidth of an edge-illuminated hologram in ULSH 500-7A with high material absorption. The angle of incidence θin = 8.5° will change the interaction length within the grating. (b) Transmittance spectrum with an interaction length of 1.3 mm.

Fig. 7
Fig. 7

Diffraction efficiency as a function of position (circle) for a grating with an apodized index modulation formed by use of the Gaussian profiles of the exposing beams. Also shown is a polynomial function fitted to the measured data (solid curve) and an actual Gaussian apodization function (dashed curve).

Fig. 8
Fig. 8

(a) Schematic of a cascaded DFB filter implemented with an edge-illuminated hologram. Each grating has a length of Lgr and a phase gap of Lph between them. (b) Simulated transmittance function of the 200-μm long DFB filter with a λ/4 phase gap (solid curve) and with an additional phase delay of λ/20 between grating elements (dotted curve).

Equations (11)

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

Lη= λBπΔntanh-1η,
LS= λB2nΔλ2-ΔnλB21/2,
LS= λB2nΔλ.
θ= λπnω,
ZR= 2ωθ.
T22ω=2θZR=2λLπn1/2,
ρ1= r0+r1 exp-2jφ11+r0r1 exp-2jφ1,φ1= 2πλ n1d cosθ1,
ρm= rm+ρm-1 exp-2jφm1+rmρm-1 exp-2jφm.
ρ1= r0+r1 exp-2jφ11+r0r1 exp-2jφ1exp-αd.
Ax=exp-xεL2.
nx=n0+ΔnxcosK · x, Δnx=AxΔn0,

Metrics