Abstract

The operator description of Fourier optics is extended and applied to holography. The existing lens models for ideal holographic processes appear as a self-evident intermediate result; generalization to include apertures, recording-material modulation transfer function, and extended source effects is straightforward. The extended source effect is generally shown to be equivalent to a modification of the actual holographic apertures. The final result is a compact expression for the description of the holographically reconstructed field distribution at an arbitrary plane. A useful, comprehensive list of operator relations is given in two appendixes.

© 1981 Optical Society of America

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References

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  1. M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [Crossref]
  2. A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
    [Crossref]
  3. J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1969).
    [Crossref]
  4. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  5. R. F. VanLigten, “Influence of photographic film on wavefront reconstruction, I: plane wavefronts,” “II: cylindrical wavefronts,” J. Opt. Soc. Am. 56, 1–9, 1009–1114 (1966).
    [Crossref]
  6. W. Lukosz, “Equivalent-lens theory of holographic imaging,” J. Opt. Soc. Am. 58, 1084–1091 (1968).
    [Crossref]
  7. W. T. Cathey, “Effect of reference and illuminating sources size in holography,” J. Opt. Soc. Am. 69, 273–277 (1979).
    [Crossref]
  8. F. P. Carlson and R. E. Francois, “Generalized linear processors for coherent optical computers,” Proc. IEEE 65, 10–17 (1977).
    [Crossref]
  9. R. E. Francois and F. P. Carlson, “Iterative Fourier approach for describing linear multiple plane, coherent optical processors,” Appl. Opt. 18, 2775–2782 (1979).
    [Crossref] [PubMed]
  10. W. T. Cathey, “Comparison of single-lens and two-lens coherent imaging of complex distributions,” J. Opt. Soc. Am. 56, 1015–1017 (1966).
    [Crossref]
  11. F. Mandelkorn, “Simple lens-system models for holographic techniques,” J. Opt. Soc. Am. 63, 1119–1124 (1973).
    [Crossref]
  12. M. Nazarathy and J. Shamir, “Effect of wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. (1981).
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  14. J. Shamir, “Holography,” in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).
  15. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).
  16. S. G. Sandoval and J. O. Castaneda, “Quasi-Fourier transform of an object from a Fresnel hologram,” Appl. Opt. 18, 950–951 (1979).
    [Crossref]
  17. J. T. Winthrop and C. R. Worthington, “Fresnel-transform representation of holograms and hologram classification,” J. Opt. Soc. Am. 56, 1362–1368 (1966).
    [Crossref]
  18. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  19. M. Nazarathy and J. Shamir, “Fourier optics without the Kirchhoff integral,” presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.
  20. Equation (15) of Ref. 5, Part I, essentially states in our notation: ui∝ ℛ[−l]θMTFℛ[l]uo apart from some phase distortion, shift, and scaling. It is evident that for any value of l(except zero), one of the FPO’s is virtual and nonrealizable.
  21. G. Bonnet, “Introduction a l’optique métaxiale,” Ann. Telecommun. 33, 143–165 (1978).
  22. K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
    [Crossref]

1981 (1)

M. Nazarathy and J. Shamir, “Effect of wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. (1981).

1980 (1)

1979 (3)

1978 (1)

G. Bonnet, “Introduction a l’optique métaxiale,” Ann. Telecommun. 33, 143–165 (1978).

1977 (1)

F. P. Carlson and R. E. Francois, “Generalized linear processors for coherent optical computers,” Proc. IEEE 65, 10–17 (1977).
[Crossref]

1973 (1)

1969 (1)

1968 (1)

1966 (4)

Bonnet, G.

G. Bonnet, “Introduction a l’optique métaxiale,” Ann. Telecommun. 33, 143–165 (1978).

Carlson, F. P.

R. E. Francois and F. P. Carlson, “Iterative Fourier approach for describing linear multiple plane, coherent optical processors,” Appl. Opt. 18, 2775–2782 (1979).
[Crossref] [PubMed]

F. P. Carlson and R. E. Francois, “Generalized linear processors for coherent optical computers,” Proc. IEEE 65, 10–17 (1977).
[Crossref]

Castaneda, J. O.

Cathey, W. T.

Francois, R. E.

R. E. Francois and F. P. Carlson, “Iterative Fourier approach for describing linear multiple plane, coherent optical processors,” Appl. Opt. 18, 2775–2782 (1979).
[Crossref] [PubMed]

F. P. Carlson and R. E. Francois, “Generalized linear processors for coherent optical computers,” Proc. IEEE 65, 10–17 (1977).
[Crossref]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Lukosz, W.

Mandelkorn, F.

Nazarathy, M.

M. Nazarathy and J. Shamir, “Effect of wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. (1981).

M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[Crossref]

M. Nazarathy and J. Shamir, “Fourier optics without the Kirchhoff integral,” presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Sandoval, S. G.

Shamir, J.

M. Nazarathy and J. Shamir, “Effect of wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. (1981).

M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[Crossref]

J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1969).
[Crossref]

M. Nazarathy and J. Shamir, “Fourier optics without the Kirchhoff integral,” presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

J. Shamir, “Holography,” in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).

Vander Lugt, A.

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[Crossref]

VanLigten, R. F.

Winthrop, J. T.

Wolf, K. W.

K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[Crossref]

Worthington, C. R.

Ann. Telecommun. (1)

G. Bonnet, “Introduction a l’optique métaxiale,” Ann. Telecommun. 33, 143–165 (1978).

Appl. Opt. (3)

Isr. J. Technol. (1)

M. Nazarathy and J. Shamir, “Effect of wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. (1981).

J. Opt. Soc. Am. (7)

Proc. IEEE (2)

F. P. Carlson and R. E. Francois, “Generalized linear processors for coherent optical computers,” Proc. IEEE 65, 10–17 (1977).
[Crossref]

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[Crossref]

Other (8)

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

M. Nazarathy and J. Shamir, “Fourier optics without the Kirchhoff integral,” presented at the Eleventh Convention of the Institute of Electrical and Electronics Engineers, Israel, Tel-Aviv, 1979.

Equation (15) of Ref. 5, Part I, essentially states in our notation: ui∝ ℛ[−l]θMTFℛ[l]uo apart from some phase distortion, shift, and scaling. It is evident that for any value of l(except zero), one of the FPO’s is virtual and nonrealizable.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. Shamir, “Holography,” in Optical Transforms, H. Lipson, ed. (Academic, London, 1972).

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978).

K. W. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1

The holographic process: (a) recording, (b) illumination (reconstruction).

Fig. 2
Fig. 2

Flow diagram for the holographic process.

Fig. 3
Fig. 3

Tilted plane wave.

Fig. 4
Fig. 4

Tilted spherical wave as G Q factor: (a) diverging wave, (b) converging wave.

Fig. 5
Fig. 5

Equivalence between tilted spherical wave and (a) decentered lens S Q, (b) centered lens and prism G Q.

Fig. 6
Fig. 6

Analysis of off-axis free propagation. The brackets indicate the extent of the input and output effective stops.

Fig. 7
Fig. 7

Lensless holographic configuration: (a) recording, (b) illumination.

Fig. 8
Fig. 8

Equivalent model for the holographic process: (a) primary image formation, (b) secondary image formation. θeq may be constructed alternatively as (c) two shifted lenses corresponding, respectively, to the reference and illumination field distributions, or (d) a single centered lens and prism.

Fig. 9
Fig. 9

Structure of equivalent transparencies for a nonideal holographic process (with apertures and extended sources of ideal recording material): (a) lens and prism with effective apertures, (b) two shifted lenses with Fresnel-transform source pupils and actual hologram apertures.

Fig. 10
Fig. 10

Equivalent models for recording material MTF (no apertures and extended sources): (a) real holographic system degraded by finite-bandwidth MTF, (b) lens model with mask, (c) equivalent ideal hologram with mask.

Fig. 11
Fig. 11

VanLigten’s equivalent model for the film MTF.

Fig. 12
Fig. 12

Observation plane misfocusing represented in the object space: O, actual object plane; I′, given observation plane (not coinciding with the image plane I, which is conjugate to O); O′, an object plane that would project an image on the actual observation plane I′.

Fig. 13
Fig. 13

Flow diagram showing the holographic image degradation that is due to misfocusing, extended sources, apertures, and recording-material response. For simplicity, lateral shifts (and the accompanying G factors) are ignored and operator labels are not explicitly written. The rectangular blocks represent linear-shift-invariant filters the impulse responses of which are written inside the blocks. The triangle represents a scaler.

Equations (206)

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ρ = x ˆ x + y ˆ y .
u 2 ( ρ ) = T [ a ] u 1 ( ρ ) .
u 2 ( ρ 2 ) = T ρ 2 [ a ] u ( ρ 1 ) ,
S [ m ] u ( ρ ) = ( ρ - m ) .
G ρ [ s ] = exp j k s · ρ
G [ s ] u ( ρ ) = exp ( j k s · ρ ) u ( ρ ) ,
δ ρ [ m ] = δ ( ρ - m ) ,
h u ( ρ ) = - d ρ h ( ρ - ρ ) u ( ρ ) ,
h * u ( ρ ) = - d ρ h ( ρ + ρ ) u * ( ρ ) .
* g ( ρ ) = g * ( ρ ) ,
v o = T o u o
v r = T r u r ,
v e = T e u e .
u i = T i v i ,
v i = θ v e
I = P r v o + v r 2 ,
I eff = I .
I eff = - 1 I = ( - 1 ) - 1 I = m I ,
m = - 1
θ P e I eff ,
θ P e V [ b ] I eff - γ / 2 ,
θ P e m P r v r + v o 2 .
θ = θ o + θ p + θ c ,
θ o P e m P r ( v r 2 + v o 2 ) ;
θ p P e m P r ( * v r ) v o ;
θ c P e m P r v r * v o .
u i p T i v e P e m ( * v r ) T o u o ,
u i c T i v e P e m v r * T o u o .
u i p = p u o ,
u i c = c u o ,
p = T i ( T e u e ) P e m P r ( * T r u r ) T o
c = T i ( T e u e ) P e m P r ( T r u r ) * T o .
s = λ ν ,
ν = f x x ˆ + f y y ˆ
u ( ρ 2 ) = A r 12 exp ( j k r 12 ) ,
r 12 = [ ( ρ 1 - ρ 2 ) 2 + z 12 2 ] 1 / 2 = ( R 1 2 + ρ 2 2 - 2 ρ 1 · ρ 2 ) 1 / 2 ,
R 1 = ( z 12 2 + ρ 1 2 ) 1 / 2
r 12 = R 1 [ 1 + ρ 2 2 2 R 1 2 - ρ 1 · ρ 2 R 1 2 - 1 8 ( ρ 2 2 R 1 2 - 2 ρ 1 · ρ 2 R 1 2 ) 2 + ] .
R 1 ρ 2 max ,
r 12 R 1 + ρ 2 2 2 R 1 - ρ 1 · ρ 2 R 1 - ( ρ 1 · ρ 2 ) 2 2 R 1 3 .
ρ 1 max R 1 .
u ( ρ 2 ) = A R 1 exp ( j k R 1 ) Q ρ 2 [ 1 R 1 ] G ρ 2 [ - ρ 1 R 1 ] .
s = - ρ 1 R 1
u ( ρ ) G ρ [ s ] Q ρ [ 1 R ] ,
u ( ρ ) S [ ρ s ] Q ρ [ 1 R ] ,
s = - ρ s R .
ρ = ρ 2 , ρ s = ρ 1 , R = R 1 > 0 ;
ρ = ρ 1 , ρ s = ρ 2 , R = - R 1 < 0.
L ρ [ f , m ] ,
L ρ [ f , 0 ] = Q ρ [ - 1 f ] .
L [ f , m ] = S [ m ] L [ f , 0 ] = S [ m ] Q ρ [ - 1 f ] .
L [ f , m ] = G ρ [ m f ] Q ρ [ - 1 f ] .
f = - R , m = ρ s .
u 2 ( ρ 2 ) = - d ρ 1 h ( ρ 2 - ρ 1 ) u ( ρ 1 ) ,
h ( ρ 2 - ρ 1 ) = z 12 j λ r 12 2 exp ( j k r 12 ) .
h ( ρ 2 - ρ 1 ) = z 12 j λ R o 2 exp ( j k R o ) Q ρ 0 × [ - 1 / R o ] Q ρ 1 [ 1 / R o ] Q ρ 2 [ 1 / R o ] G ρ 2 [ - ρ 1 / R o ] .
u 2 ( ρ 2 ) = z 12 j λ R o 2 exp ( j k R o ) Q ρ o [ - 1 / R o ] Q ρ 2 [ 1 / R o ] × - d ρ 1 G ρ 2 [ - ρ 1 / R o ] Q ρ 1 [ 1 / R o ] u 1 ( ρ 1 ) .
u 2 ( ρ 2 ) = [ R o ] u 1 ( ρ 1 ) ,
[ R o ] = z 12 j λ R o 2 exp j k R o Q ρ o [ - 1 / R o ] Q [ 1 / R o ] × V [ 1 / λ R o ] Q [ 1 / R o ] .
u ˜ = V [ 1 / λ d ] Q [ 1 / d ] u .
[ d ] u Q [ 1 / d ] V [ 1 / λ d ] Q [ 1 / d ] u = Q [ 1 / d ] u ˜ .
[ R o ] = z 12 R o exp j k R o Q ρ o [ - 1 / R o ] - 1 Q [ - λ 2 R o ] ,
[ R o ] = z 12 j λ R o 2 exp j k R o Q ρ o [ - 1 / R o ] Q [ 1 / R o ] .
T i = [ d i ]
T 0 = [ d 0 ] ,
p = [ d i ] v e P e m P r ( * v r ) [ d o ] ,
c = [ d i ] v e P e m P r v r * [ d o ] ,
v r [ d r ] u r ,
v e [ d e ] u e .
v r = [ d r ] δ ρ [ ρ r ] S [ ρ r ] Q [ 1 / d r ] G [ s r ] Q [ 1 / d r ]
v e = [ d e ] δ ρ [ ρ e ] S [ ρ e ] Q [ 1 / d e ] G [ s e ] Q [ 1 / d e ] ,
s e = - ρ e d e ,             s r = - ρ r d r .
p [ d i ] θ eq p [ d o ] ,
c [ d i ] θ eq c [ - d o ] * ,
θ eq p = v e v r * ( S [ ρ e ] Q [ 1 / d e ] ) ( S [ ρ r ] Q [ - 1 / d r ] ) ,
θ eq c = v e v r ( S [ ρ e ] Q [ 1 / d e ] ) ( S [ ρ r ] Q [ 1 / d r ] ) .
θ ep p G [ s p ] Q [ 1 / f p ] ,
θ eq c G [ s c ] Q [ 1 / f c ] ,
1 f p = 1 d e - 1 d r ,
1 f c = 1 d e + 1 d r ,
s p = s e - s r ,
s c = s e + s r .
u r = S [ ρ r ] u r ,
u e = S [ ρ e ] u e .
v r = [ d r ] S [ ρ r ] u r = S [ ρ r ] [ d r ] u r ,
v r S [ ρ r ] Q [ 1 / d r ] u ˜ r .
v e S [ ρ e ] Q [ 1 / d e ] u ˜ e .
v r ( S [ ρ r ] Q [ 1 / d r ] ) ( S [ ρ r ] u ˜ r )
v r Q [ 1 / d r ] G [ s r ] ( S [ ρ r ] u ˜ r ) .
v e Q [ 1 / d e ] G [ s e ] ( S [ ρ e ] u ˜ r ) .
p [ d i ] A e p Q [ 1 / d e ] G [ s e ] m A r p Q [ - 1 / d r ] G [ - s r ] [ d o ] ,
c [ d i ] A e c Q [ 1 / d e ] G [ s e ] m A r c Q [ 1 / d r ] G [ s r ] [ - d o ] * ,
A e p = A e c = ( S [ ρ e ] u ˜ e ) P e ,
A r c = ( S [ ρ r ] u ˜ r ) P r ,             A r p = ( S [ ρ r ] u ˜ r * ) P r .
p [ d i ] Q [ 1 / f p ] G [ s p ] A e p A r p [ d o ] ,
c [ d i ] Q [ 1 / f c ] G [ s c ] A e c A r c [ - d o ] * .
θ eq p = Q [ 1 / f p ] G [ s p ] A e p A r p
θ eq c = Q [ 1 / f c ] G [ s c ] A e c A r c .
θ eq p = P e ( S [ ρ e ] Q [ 1 / d e ] u ˜ e ) P r ( S [ ρ r ] Q [ - 1 / d r ] u ˜ r * ) ,
θ eq c = P e [ S ( ρ e ) Q [ 1 / d e ] u ˜ e ) P r ( S [ ρ r ] Q [ 1 / d r ] u ˜ r ) .
p [ d i ] Q [ 1 / d e ] G [ s e ] m Q [ - 1 / d r ] G [ - s r ] [ d o ] ,
c [ d i ] Q [ 1 / d e ] G [ s e ] m Q [ 1 / d r ] G [ s r ] [ - d o ] * .
p [ d i ] B G [ s e ] Q [ - 1 / d r ] G [ - s r ] [ d o ] ,
B Q [ 1 / d e ] { G [ s e ] m } .
m G [ s e ] m
[ d e ] [ - d e ] 1 ,
B = [ d e ] [ - d e ] Q [ 1 / d e ] m .
B [ d e ] Q [ - 1 / d e ] V [ - 1 / λ d e ] m [ d e ] Q [ - 1 / d e ] V [ - 1 / λ d e ] { m }
B [ d e ] Q [ - 1 / d e ] { V [ - 1 / λ d e ] m } V [ - 1 / λ d e ] .
θ MTF { V [ - 1 / λ d e ] m } ,
θ MTF = { S [ ρ e ] V [ - 1 / λ d e ] } .
B [ d e ] θ MTF Q [ - 1 / d e ] V [ - 1 / λ d e ] .
B [ d e ] θ MTF [ - d e ] Q [ 1 / d e ] .
p [ d i + d e ] θ MTF [ - d e ] × Q [ 1 / d e ] G [ s e ] Q [ - 1 / d r ] G [ - s r ] [ d o ] ,
p [ d i + d e ] θ MTF [ - d e ] Q [ 1 / f p ] G [ s p ] [ d o ] ,
p [ d i - d e ] θ MTF [ d e ] Q [ 1 / f p ] G [ s p ] [ d o ] .
p Q [ 1 / d i ] ( V [ 1 / λ d i ] A e p ) Q [ - 1 / d i ] 1 ,
1 = [ d i ] Q [ 1 / d e ] G [ s e ] m A r p Q [ - 1 / d r ] G [ - s r ] [ d o ] .
1 [ d i + d e ] θ MTF [ - d e ] A r p Q [ 1 / f p ] G [ s p ] [ d o ] .
1 Q [ 1 / ( d i + d e ) ] { V [ 1 / λ ( d i + d e ) ] θ MTF } Q [ - 1 / ( d i + d e ) ] [ d i + d e ] × [ - d e ] A r p Q [ 1 / f p ] G [ s p ] [ d o ] .
1 Q [ 1 / ( d i + d e ) ] { V [ d e / ( d i + d e ) ] m } Q [ - 1 / ( d i + d e ) ] 2 ,
2 [ d i ] A r p Q [ 1 / f p ] G [ s p ] [ d o ]
2 Q [ 1 / d i ] { V [ 1 / λ d i ] A r p } Q [ - 1 / d i ] 3 ,
3 [ d i ] Q [ 1 / f p ] G [ s p ] [ d o ]
3 G [ s p ] S [ s p d i ] [ d i ] Q [ 1 / f p ] [ d o ] G [ s p ] S [ s p d i ] 4 ,
4 [ d i ] Q [ 1 / f p ] [ d o ]
4 Q [ 1 / ( d i + f p ) ] V [ f p / ( d i + f p ) ] × [ ( 1 / d i + 1 / f p ) - 1 ] [ d o ]
4 Q [ 1 / ( d i + f p ) ] V [ f p / ( d i + f p ) ] [ Δ ] ,
Δ = ( 1 / d i + 1 / f p ) - 1 + d o
3 G [ s p ] S [ s p d i ] Q [ 1 / ( d i + f p ) ] V [ f p / ( d i + f p ) ] [ Δ ] .
3 G [ s p / M ] Q [ 1 / ( d i + f p ) ] S [ ρ i p ] V [ 1 / M ] [ Δ ] ,
p Q [ 1 / d i ] { V [ 1 / λ d i ] A e p } Q [ - 1 / d i ] × Q [ 1 / ( d i + d e ) ] { V [ d e / ( d i + d e ) ] m } Q [ - 1 / ( d i + d e ) ] × Q [ 1 / d i ] { V [ 1 / λ d i ] A r p } Q [ - 1 / d i ] 3 .
{ V [ 1 / λ d i ] A e p } = { V [ 1 / λ d i ] P e } { G [ ρ e / d i ] V [ - d e / d i ] Q [ 1 / d e ] u ˜ e } .
{ V [ 1 / λ d i ] A r p } = { V [ 1 / λ d i ] P r } { G [ ρ r / d i ] V [ - d r / d i ] Q [ 1 / d r ] u ˜ r } ,
= V [ - 1 ] * .
p Q [ 1 / d i ] { V [ 1 / λ d i ] P e } { G [ ρ e / d i ] V [ - d e / d i ] Q [ 1 / d e ] u ˜ e } Q [ 1 / ( d i + d e ) - 1 / d i ] { V [ d e / ( d i + d e ) ] m } Q [ 1 / d i - 1 / ( d i + d e ) ] { V [ 1 / λ d i ] P r } { G [ ρ r / d i ] V [ - d r / d i ] Q [ 1 / d r ] u ˜ r } Q [ 1 / ( d i + f p ) - 1 / d i ] G [ s p / M ] × S [ ρ i p ] V [ 1 / M ] [ Δ ] .
T 2 T 1 P , { T 2 T 1 P } , T 2 { T 1 P } .
T 2 T 1 P g
T 2 { T 1 P } g ,
G Q G Q g
G { Q g } ,
V [ b 1 ] V [ b 2 ] = V [ b 1 b 2 ] ,
S [ m 1 ] S [ m 2 ] = S [ m 1 + m 2 ] ,
V - 1 [ b ] = V [ 1 b ] ,
S - 1 [ m ] = S [ - m ] ,
V [ b ] g = { V [ b ] g } V [ b ] ,
g V [ b ] = V [ b ] { V [ 1 b ] g } ,
S [ m ] g = { S [ m ] g } S [ m ] ,
g S [ m ] = S [ m ] { S [ - m ] g } ,
S [ m ] V [ b ] = V [ b ] S [ b m ] ,
V [ b ] S [ m ] = S [ m b ] V [ b ] ,
Q [ a 1 ] Q [ a 2 ] = Q [ a 1 + a 2 ] ,
G [ s 1 ] G [ s 2 ] = G [ s 1 + s 2 ] ,
Q ρ - 1 [ a ] = Q ρ [ - a ] = ( * Q ρ [ a ] ) ,
G ρ - 1 [ s ] = G ρ [ - s ] = ( * G ρ [ s ] ) ,
V [ b ] Q [ a ] = Q [ b 2 a ] V [ b ] ,
Q [ a ] V [ b ] = V [ b ] V [ a b 2 ] ,
V [ b ] G [ s ] = G [ b s ] V [ b ] ,
G [ s ] V [ b ] = V [ b ] G [ s b ] ,
{ S [ m ] Q ρ [ a ] } = ( Q m [ a ] ) G p [ - a m ] Q ρ [ a ] ,
G p [ s ] Q ρ [ a ] = ( Q s [ - 1 / a ] ) S [ - s / a ] Q ρ [ a ] ,
{ S [ m ] G ρ [ s ] } = ( G m [ - s ] ) G ρ [ s ] .
- 1 = V [ - 1 ] = V [ - 1 ] ,
= V [ - 1 ] ,
V [ b ] = 1 b 2 V [ 1 b ] ,
V [ b ] = 1 b 2 V [ 1 b ] ,
S [ m ] = G [ - λ m ] ,
G [ s ] = S [ s λ ] ,
{ G ρ [ s ] } = δ ρ [ s λ ] ,
Q [ a ] = i λ a Q [ - λ 2 / a ] ,
{ Q ρ [ a ] } = j λ a Q ρ [ - λ 2 / a ] ,
g = { g } ,
g = ( g ) ,
V [ b ] g = b 2 { V [ b ] g } V [ b ] ,
g V [ b ] = 1 b 2 V [ b ] { V [ b - 1 ] g } ,
S [ m ] g = g S [ m ] = ( S [ m ] g ) ,
G [ s ] g = { G [ s ] g } G [ s ] ,
* * = 1 ,
* g = { * g } * ,
* V [ b ] = V [ b ] * ,
* S [ m ] = S [ m ] * ,
* Q [ a ] = Q [ - a ] * ,
* G [ s ] = G [ - s ] * ,
* = - 1 * = V [ - 1 ] * = V [ - 1 ] * ,
* g = { * g } * ,
S [ m 2 ] δ [ m 1 ] = δ [ m 1 + m 2 ] ,
S [ m ] δ [ o ] = δ [ m ] ,
V [ b ] δ [ m ] = 1 b δ [ m b ] ,
δ [ m ] g ( ρ ) = δ [ m ] g ( m ) ,
δ [ m ] = S [ m ] ,
{ δ [ m ] } = G [ - λ m ] ,
lim a a j λ Q [ a ] = δ [ o ] .
[ o ] = 1 ,
lim d [ d ] = lim d V [ 1 λ d ] ,
[ d 1 ] [ d 2 ] = [ d 1 + d 2 ] ,
- 1 [ d ] = [ - d ] ,
V [ b ] [ d ] = [ d / b 2 ] V [ b ] ,
[ d ] V [ b ] = V [ b ] [ b 2 d ] ,
[ d ] S [ m ] = S [ m ] [ d ] ,
{ [ d ] G ρ [ s ] } = ( exp j k d Q s [ - d ] ) G ρ [ s ] ,
[ d ] G [ s ] = ( exp j k d Q s [ - d ] ) G [ s ] S [ s d ] [ d ] ,
G [ s ] [ d ] = ( exp - j k d Q s [ d ] ) [ d ] G [ s ] S [ - s d ] .
{ [ d ] Q [ 1 q ] } Q [ 1 q + d ] ,
[ d ] Q [ 1 q ] Q [ 1 q + d ] V [ q q + d ] [ ( 1 d + 1 q ) - 1 ] .
[ d ] Q [ - 1 d ] Q [ 1 d ] V [ 1 λ d ] ,
[ d ] g Q [ 1 d ] { V [ 1 λ d ] g } Q [ - 1 d ] [ d ] .