Abstract

The analysis of structural information of planar holograms (i.e., “surface” holograms in integrated optics) is presented by using Ewald’s construction and a grating vector uncertainty theorem. The influences of the geometry and the material constants of the holographic system, the modulation transfer function, and dimensions of the hologram are given. A new approach to the transfer of information, connected with multimode operation, is presented.

© 1981 Optical Society of America

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References

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  1. D. B. Anderson, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–274 (1977).
    [Crossref]
  2. G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).
  3. See, for example, R. Shubert and J. H. Harris, “Optical guided-wave focusing and diffraction,” J. Opt. Soc. Am. 61, 154–160 (1970).
    [Crossref]
  4. See, for example, P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
    [Crossref]
  5. W. Lukosz and A. Wutrich, “Hologram recording and read-out with evanescent field of guide waves,” Opt. Commun. 19, 232–235 (1976).
    [Crossref]
  6. Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
    [Crossref]
  7. According to D. M. MacKay, “Quantal aspects of scientific information,” Phil. Mag. 41, 289 (1950), each scientific measurement (such as holographic detection of optical fields) a priori defines the discrete set of so-called degrees of freedom (they determine the structural information capacity), which are a posteriori connected with a discrete and limited set of the total numbers that result from the experiment. All the configurations of these numbers (signal-to-noise ratio included) define the metric information.
  8. P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. 2, 393–400 (1962).
    [Crossref]
  9. V. V. Aristov and V. Sh. Shektman, “Properties of three-dimensional holograms,” Sov. Phys. Usp. 14, 263–277 (1971).
    [Crossref]
  10. V. V. Aristov, “Optical memory of three-dimensional holograms,” Opt. Commun. 3, 194–196 (1971).
    [Crossref]
  11. S. Kusch and R. Guther, “Theoretical considerations on the bit capacity of volume holograms,” Exp. Tech. Phys. 22, 37–51 (1974).
  12. T. Jannson, “Structural information in volume holography,” Opt. Appl. IX, 169–177 (1979).
  13. T. Jannson, “Shannon number of an image and structural information capacity in volume holography,” Opt. Acta, accepted for publication.
  14. V. I. Sukhanov and Yu. N. Denisyuk, “On the relationship between spatial frequency spectra of a three-dimensional object and its three-dimensional hologram,” Opt. Spectrosc. 28, 63–66 (1970).
  15. M. R. B. Forshaw, “Explanation of the ‘venetian blind’ effect in holography, using the Ewald sphere concept,” Opt. Commun. 8, 201–206 (1973).
    [Crossref]
  16. For the analysis of Shannon number in plane holography, see A. Macovski, “Hologram information capacity,” J. Opt. Soc. Am. 60, 21–29 (1970); W. Lukosz, “Optical systems with resolving powers exceeding the classical limit. II,” J. Opt. Soc. Am. 57, 932–940 (1967); T. Jannson, “Impulse response and Shannon number of holographic optical systems,” Opt. Commun. 10, 232–237 (1974). For volume holography, see Ref. 13.
    [Crossref]
  17. E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
    [Crossref]
  18. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  19. See, for example, Ref. 4, p. 388.
  20. T. Jannson and J. Sochacki, “Primary aberrations of thin planar surface lenses,” J. Opt. Soc. Am. 70, 1079–1084 (1980).
    [Crossref]
  21. In the 3D case, the problem of the scattering of a plane wave incident upon an arbitrary phase structure was solved in the weak diffraction approximation (first Born approximation) by Wolf17; see also Ref. 12.
  22. For a grating, with one dimension unlimited, the spread of the K vector was also introduced by J. W. Goodman in “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
    [Crossref]
  23. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 8, Section 5.
  24. Note that if |cos 2γ/cos Φ| > 1 and |tan Φ/tan 2γ| > 1, then the arccos {·} are treated as complex functions, so α1 and α2 equal 0 or π, depending on the sign of the argument of a given arccos function.
  25. R. Ulrich, “Theory of the prism-film coupler by plane wave analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [Crossref]
  26. D. Gabor, “Communication theory and physics,” Phil. Mag. 41, 1161–1187 (1950).
  27. Analogously, in color holography, the radii of the Ewald circles (or Ewald spheres, in the 3D case) are different for different colors and equal rm′=2πnm′/λ0m, where λ0m is a wavelength in vacuum corresponding to the m th color and nm′=n˙m′(λ0m) is the photosensitive material refractive index for the m th color.
  28. See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. I.
  29. It should be noted that the resolution power of a Fourier transformer may be characterized by the typical parameter 1/Δfx, where Δfx is the minimal resolution interval in the 1D object Fourier space, so that ρ is connected with this parameter only by the relation ρ= (λ1dFΔfx)−1(thus Δfx= 2/T).
  30. In structural information analysis we ignore the effects of mode coupling and scattering noise [for the 3D case, see, e.g., H. Nomura and T. Okoshi, “Capacity limitation of volume hologram memory,” Electron. Commun. Jpn. 58, 108–115 (1975)]. However, the considerations of these effects are necessary for metric information analysis.
  31. T. Jannson and R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik, accepted for publication.

1980 (1)

1979 (1)

T. Jannson, “Structural information in volume holography,” Opt. Appl. IX, 169–177 (1979).

1977 (2)

See, for example, P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

D. B. Anderson, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–274 (1977).
[Crossref]

1976 (2)

W. Lukosz and A. Wutrich, “Hologram recording and read-out with evanescent field of guide waves,” Opt. Commun. 19, 232–235 (1976).
[Crossref]

Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
[Crossref]

1975 (1)

In structural information analysis we ignore the effects of mode coupling and scattering noise [for the 3D case, see, e.g., H. Nomura and T. Okoshi, “Capacity limitation of volume hologram memory,” Electron. Commun. Jpn. 58, 108–115 (1975)]. However, the considerations of these effects are necessary for metric information analysis.

1974 (1)

S. Kusch and R. Guther, “Theoretical considerations on the bit capacity of volume holograms,” Exp. Tech. Phys. 22, 37–51 (1974).

1973 (2)

M. R. B. Forshaw, “Explanation of the ‘venetian blind’ effect in holography, using the Ewald sphere concept,” Opt. Commun. 8, 201–206 (1973).
[Crossref]

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

1971 (3)

V. V. Aristov and V. Sh. Shektman, “Properties of three-dimensional holograms,” Sov. Phys. Usp. 14, 263–277 (1971).
[Crossref]

V. V. Aristov, “Optical memory of three-dimensional holograms,” Opt. Commun. 3, 194–196 (1971).
[Crossref]

For a grating, with one dimension unlimited, the spread of the K vector was also introduced by J. W. Goodman in “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[Crossref]

1970 (4)

1969 (2)

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[Crossref]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

1962 (1)

1950 (2)

According to D. M. MacKay, “Quantal aspects of scientific information,” Phil. Mag. 41, 289 (1950), each scientific measurement (such as holographic detection of optical fields) a priori defines the discrete set of so-called degrees of freedom (they determine the structural information capacity), which are a posteriori connected with a discrete and limited set of the total numbers that result from the experiment. All the configurations of these numbers (signal-to-noise ratio included) define the metric information.

D. Gabor, “Communication theory and physics,” Phil. Mag. 41, 1161–1187 (1950).

Anderson, D. B.

D. B. Anderson, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–274 (1977).
[Crossref]

Aristov, V. V.

V. V. Aristov and V. Sh. Shektman, “Properties of three-dimensional holograms,” Sov. Phys. Usp. 14, 263–277 (1971).
[Crossref]

V. V. Aristov, “Optical memory of three-dimensional holograms,” Opt. Commun. 3, 194–196 (1971).
[Crossref]

Denisyuk, Yu. N.

V. I. Sukhanov and Yu. N. Denisyuk, “On the relationship between spatial frequency spectra of a three-dimensional object and its three-dimensional hologram,” Opt. Spectrosc. 28, 63–66 (1970).

Forshaw, M. R. B.

M. R. B. Forshaw, “Explanation of the ‘venetian blind’ effect in holography, using the Ewald sphere concept,” Opt. Commun. 8, 201–206 (1973).
[Crossref]

Gabor, D.

D. Gabor, “Communication theory and physics,” Phil. Mag. 41, 1161–1187 (1950).

Goodman, J. W.

For a grating, with one dimension unlimited, the spread of the K vector was also introduced by J. W. Goodman in “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[Crossref]

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 8, Section 5.

Guther, R.

S. Kusch and R. Guther, “Theoretical considerations on the bit capacity of volume holograms,” Exp. Tech. Phys. 22, 37–51 (1974).

Harris, J. H.

Janicki, R.

T. Jannson and R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik, accepted for publication.

Jannson, T.

T. Jannson and J. Sochacki, “Primary aberrations of thin planar surface lenses,” J. Opt. Soc. Am. 70, 1079–1084 (1980).
[Crossref]

T. Jannson, “Structural information in volume holography,” Opt. Appl. IX, 169–177 (1979).

T. Jannson, “Shannon number of an image and structural information capacity in volume holography,” Opt. Acta, accepted for publication.

T. Jannson and R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik, accepted for publication.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Koyama, Y.

Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
[Crossref]

Kusch, S.

S. Kusch and R. Guther, “Theoretical considerations on the bit capacity of volume holograms,” Exp. Tech. Phys. 22, 37–51 (1974).

Lukosz, W.

W. Lukosz and A. Wutrich, “Hologram recording and read-out with evanescent field of guide waves,” Opt. Commun. 19, 232–235 (1976).
[Crossref]

MacKay, D. M.

According to D. M. MacKay, “Quantal aspects of scientific information,” Phil. Mag. 41, 289 (1950), each scientific measurement (such as holographic detection of optical fields) a priori defines the discrete set of so-called degrees of freedom (they determine the structural information capacity), which are a posteriori connected with a discrete and limited set of the total numbers that result from the experiment. All the configurations of these numbers (signal-to-noise ratio included) define the metric information.

Macovski, A.

Marcuse, D.

See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. I.

Nishihara, H.

Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
[Crossref]

Nomura, H.

In structural information analysis we ignore the effects of mode coupling and scattering noise [for the 3D case, see, e.g., H. Nomura and T. Okoshi, “Capacity limitation of volume hologram memory,” Electron. Commun. Jpn. 58, 108–115 (1975)]. However, the considerations of these effects are necessary for metric information analysis.

Okoshi, T.

In structural information analysis we ignore the effects of mode coupling and scattering noise [for the 3D case, see, e.g., H. Nomura and T. Okoshi, “Capacity limitation of volume hologram memory,” Electron. Commun. Jpn. 58, 108–115 (1975)]. However, the considerations of these effects are necessary for metric information analysis.

Righini, G. C.

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

Russo, V.

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

Shektman, V. Sh.

V. V. Aristov and V. Sh. Shektman, “Properties of three-dimensional holograms,” Sov. Phys. Usp. 14, 263–277 (1971).
[Crossref]

Shubert, R.

Sochacki, J.

Sottini, S.

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

Suhura, T.

Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
[Crossref]

Sukhanov, V. I.

V. I. Sukhanov and Yu. N. Denisyuk, “On the relationship between spatial frequency spectra of a three-dimensional object and its three-dimensional hologram,” Opt. Spectrosc. 28, 63–66 (1970).

Tien, P. K.

See, for example, P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Toraldo di Francia, G.

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

Ulrich, R.

van Heerden, P. J.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[Crossref]

Wutrich, A.

W. Lukosz and A. Wutrich, “Hologram recording and read-out with evanescent field of guide waves,” Opt. Commun. 19, 232–235 (1976).
[Crossref]

Appl. Opt. (1)

Appl. Phys. (1)

G. C. Righini, V. Russo, S. Sottini, and G. Toraldo di Francia, “Geodesic lenses for guided optical waves,” Appl. Phys. 12, 1477–1481 (1973).

Bell. Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Electron. Commun. Jpn. (1)

In structural information analysis we ignore the effects of mode coupling and scattering noise [for the 3D case, see, e.g., H. Nomura and T. Okoshi, “Capacity limitation of volume hologram memory,” Electron. Commun. Jpn. 58, 108–115 (1975)]. However, the considerations of these effects are necessary for metric information analysis.

Exp. Tech. Phys. (1)

S. Kusch and R. Guther, “Theoretical considerations on the bit capacity of volume holograms,” Exp. Tech. Phys. 22, 37–51 (1974).

IEEE J. Quantum Electron. (1)

D. B. Anderson, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–274 (1977).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Appl. (1)

T. Jannson, “Structural information in volume holography,” Opt. Appl. IX, 169–177 (1979).

Opt. Commun. (5)

M. R. B. Forshaw, “Explanation of the ‘venetian blind’ effect in holography, using the Ewald sphere concept,” Opt. Commun. 8, 201–206 (1973).
[Crossref]

E. Wolf, “Three-dimensional structure determination of semitransparent objects from holographic data,” Opt. Commun. 1, 153 (1969).
[Crossref]

W. Lukosz and A. Wutrich, “Hologram recording and read-out with evanescent field of guide waves,” Opt. Commun. 19, 232–235 (1976).
[Crossref]

Another scheme of planar hologram recording is preferred by T. Suhura, H. Nishihara, and Y. Koyama, “Waveguide holograms: a new approach to hologram integration,” Opt. Commun. 19, 353–358 (1976); see also Ref. 5.
[Crossref]

V. V. Aristov, “Optical memory of three-dimensional holograms,” Opt. Commun. 3, 194–196 (1971).
[Crossref]

Opt. Spectrosc. (1)

V. I. Sukhanov and Yu. N. Denisyuk, “On the relationship between spatial frequency spectra of a three-dimensional object and its three-dimensional hologram,” Opt. Spectrosc. 28, 63–66 (1970).

Phil. Mag. (2)

According to D. M. MacKay, “Quantal aspects of scientific information,” Phil. Mag. 41, 289 (1950), each scientific measurement (such as holographic detection of optical fields) a priori defines the discrete set of so-called degrees of freedom (they determine the structural information capacity), which are a posteriori connected with a discrete and limited set of the total numbers that result from the experiment. All the configurations of these numbers (signal-to-noise ratio included) define the metric information.

D. Gabor, “Communication theory and physics,” Phil. Mag. 41, 1161–1187 (1950).

Proc. IEEE (1)

For a grating, with one dimension unlimited, the spread of the K vector was also introduced by J. W. Goodman in “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[Crossref]

Rev. Mod. Phys. (1)

See, for example, P. K. Tien, “Integrated optics and new wave phenomena in optical waveguides,” Rev. Mod. Phys. 49, 361–420 (1977).
[Crossref]

Sov. Phys. Usp. (1)

V. V. Aristov and V. Sh. Shektman, “Properties of three-dimensional holograms,” Sov. Phys. Usp. 14, 263–277 (1971).
[Crossref]

Other (9)

T. Jannson, “Shannon number of an image and structural information capacity in volume holography,” Opt. Acta, accepted for publication.

See, for example, Ref. 4, p. 388.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 8, Section 5.

Note that if |cos 2γ/cos Φ| > 1 and |tan Φ/tan 2γ| > 1, then the arccos {·} are treated as complex functions, so α1 and α2 equal 0 or π, depending on the sign of the argument of a given arccos function.

In the 3D case, the problem of the scattering of a plane wave incident upon an arbitrary phase structure was solved in the weak diffraction approximation (first Born approximation) by Wolf17; see also Ref. 12.

Analogously, in color holography, the radii of the Ewald circles (or Ewald spheres, in the 3D case) are different for different colors and equal rm′=2πnm′/λ0m, where λ0m is a wavelength in vacuum corresponding to the m th color and nm′=n˙m′(λ0m) is the photosensitive material refractive index for the m th color.

See, for example, D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. I.

It should be noted that the resolution power of a Fourier transformer may be characterized by the typical parameter 1/Δfx, where Δfx is the minimal resolution interval in the 1D object Fourier space, so that ρ is connected with this parameter only by the relation ρ= (λ1dFΔfx)−1(thus Δfx= 2/T).

T. Jannson and R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik, accepted for publication.

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Figures (4)

Fig. 1
Fig. 1

Geometry of planar hologram recording. Here n1 and n2 are constant modal indices in the planar hologram region and outside, and nM describes the modal index modulation. The hologram is recorded by the interference of a signal planar beam with complex amplitude U(x,y) and a linear reference wave with wave vector βr.

Fig. 2
Fig. 2

Illustration of Ewald’s construction for planar holograms. The Ewald circles with radii r = βr, primary and conjugate, are loci of K vectors that may be recorded during exposure by a linear reference wave with wave vector βr. The elementary cells have rectangular shape, with sizes 2π/Tx and 2π/Ty. The angle δ0 denotes the angular selectivity. The sizes of the spread of the K vector are 4π/Tx and 4π/Ty.

Fig. 3
Fig. 3

Permissible contours of Ewald’s circle in the case of many reference beams. R0 = 2πf0, Rs = 2πfs, where f0 = MTF cutoff frequency and f s = 3 / λ 1. Lx= Lx1 + Lx2 and Ly = Ly1 + Ly2 are projections of permissible arcs. The hatched area is the permissible part of ( K x , K y) space.

Fig. 4
Fig. 4

Normalized structural information capacity NISH1/2/10−3 (mm−1) for fs = 0 versus angle of inclination Φ of the reference beam in the hologram (see Fig. 3), for different shape coefficients η = 1, 5, 10, and for two values of cutoff frequency f 0 ( 1 ) = 1000 lines / mm lines/mm and f 0 ( 2 ) = 3000 lines / mm lines/mm. The quantity SH is the area of the planar hologram and the angle Φ is connected with the angle of incidence Φ0 (see Fig. 1) by the relation n2 sin Φ0 = n1 sin Φ; λ1 = 0.4 μm.

Tables (1)

Tables Icon

Table 1 Structural Information Parameters for Some Phase Structuresa

Equations (16)

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U ( x , y ) = - + A ( β x ) e j ( y β y + x β x ) d β x ,
β y = { β m 2 - β x 2             for β x β m j β x 2 - β m 2             for β x > β m .
[ 2 x 2 + 2 y 2 + k 0 2 n m 2 ( x , y ) ] U ( x , y ) = 0.
T x T y sinc [ ( β x - β r x - K x ) T x / 2 ] × sinc [ ( β y - β r y - K y ) T y / 2 ] ,
Δ K x T x 2 π ,             Δ K y T y 2 π ,
Ω 0 = Δ K x Δ K y = ( 2 π ) 2 ( T x T y ) - 1 .
( N G ) max = 2 π S H / λ 1 2 ,
N G = { π S H 2 ( f 0 2 - f s 2 ) _ for f 0 f s 0 for f 0 < f s ,
N I = ( 2 π ) - 1 ( L x T x + L y T y ) ,
L x = [ 2 r ( 1 - cos α 1 ) + 2 r sin ( 2 γ ) cos α 2 ] | γ ( R s ) γ ( R 0 ) ,
α 1 = Re { arccos [ cos 2 γ cos Φ ] } ,             α 2 = Re { arccos [ tan Φ tan 2 γ ] } ,
N I = ( 2 π ) - 1 S H 1 / 2 η - 1 / 2 ( L x + η L y ) .
G = ( N G ) max ( N I ) max = S H λ 1 L H = S H 1 / 2 η - 1 / 2 2 λ 1 ( η + 1 ) .
F x = { ( 2 π ) - 1 Δ ω x x for x T x ( 2 π ) - 1 Δ ω x T x for x > T x ,
N G D ( Φ ) = m = 1 G D N I ( m ) ( Φ ) ,
f D 2 = 2 n ¯ 2 λ 0 Δ n 1 T x 2 + 1 T y 2 ,