Abstract

The fundamental invariant of an optical system is the number N of degrees of freedom of the message it can transmit. The spatial bandwidth of the system can be increased over the classical limit by reducing one of the other constituent factors of N. As examples of this invariance theorem N=const. established in Part I of this series [ J. Opt. Soc. Am. 56, 1463 ( 1966)], we discuss (a) a system whose spatial-bandwidth increase is achieved by a proportional reduction of its temporal bandwidth, and (b) the airborne synthetic-aperture, terrain-mapping radar, whose spatial resolution comes from exploitation of the temporal degrees of freedom of the received signal. The increase of the spatial bandwidth beyond the classical limit is, however, limited by the appearance of evanescent waves.

The number of degrees of freedom of the object wave field stored in a hologram is discussed. The storage capacity of the photographic plate, which is proportional to its size times its spatial cutoff frequency, is fully exploited only by single-sideband Fraunhofer but not by single-sideband Fresnel holograms.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
    [Crossref]
  2. In this paper kx, ky denote radian spatial frequencies, kx/2π, ky/2π the corresponding proper spatial frequencies (in lines/mm); ω denotes a radian temporal frequency, and ν=ω/2 the corresponding temporal frequency. In Eq. (1.2) both the object field and the aperture of the system are assumed to be rectangular.
  3. A. Bachl and W. Lukosz, J. Opt. Soc. Am. 57, 163 (1967).
    [Crossref]
  4. W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963).
    [Crossref]
  5. L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
    [Crossref]
  6. C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
    [Crossref]
  7. L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).
  8. E. N. Leith, L. J. Cutrona, and L. J. Porcello, J. Opt. Soc. Am. 56, 1419A (1966).
    [Crossref]
  9. In the experiments reported in Ref. 4, no coherent background was used. The receiver integrated the intensity in the image plane. In this case the gratings M and M′ have to be inserted in or very near to the object and image planes, respectively.
  10. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962);J. Opt. Soc. Am. 53, 1377 (1963);J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  11. In quantum mechanics this is the condition for the validity of the WKB approximation [E. E. Merzbacher, Quantum Mechanics (John Wiley & Sons, New York, 1961), Ch. 7].
  12. If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx = r0λ0/Dx. The resolution is Δx0 = Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
    [Crossref]
  13. J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
    [Crossref]
  14. G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
    [Crossref]
  15. This assumption involves no loss of generality of our subsequent considerations. The effect of an off-axis angle of the reference beam is an apparent lateral shift of the object and of the light source illuminating the object.
  16. A. Lohmann, Opt. Acta 3, 97 (1956).
    [Crossref]
  17. G. B. Parrent and G. O. Reynolds, J. Opt. Soc. Am. 56, 1400 (1966).
    [Crossref]

1967 (1)

1966 (3)

1965 (2)

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

1963 (1)

W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963).
[Crossref]

1962 (3)

C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
[Crossref]

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962);J. Opt. Soc. Am. 53, 1377 (1963);J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx = r0λ0/Dx. The resolution is Δx0 = Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
[Crossref]

1961 (1)

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

1956 (1)

A. Lohmann, Opt. Acta 3, 97 (1956).
[Crossref]

Bachl, A.

Cutrona, L. J.

E. N. Leith, L. J. Cutrona, and L. J. Porcello, J. Opt. Soc. Am. 56, 1419A (1966).
[Crossref]

If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx = r0λ0/Dx. The resolution is Δx0 = Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
[Crossref]

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).

Hall, G. O.

If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx = r0λ0/Dx. The resolution is Δx0 = Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
[Crossref]

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

Leith, E. N.

E. N. Leith, L. J. Cutrona, and L. J. Porcello, J. Opt. Soc. Am. 56, 1419A (1966).
[Crossref]

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962);J. Opt. Soc. Am. 53, 1377 (1963);J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).

Lohmann, A.

A. Lohmann, Opt. Acta 3, 97 (1956).
[Crossref]

Lukosz, W.

A. Bachl and W. Lukosz, J. Opt. Soc. Am. 57, 163 (1967).
[Crossref]

W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
[Crossref]

W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963).
[Crossref]

Merzbacher, E. E.

In quantum mechanics this is the condition for the validity of the WKB approximation [E. E. Merzbacher, Quantum Mechanics (John Wiley & Sons, New York, 1961), Ch. 7].

Parrent, G. B.

Porcello, L. J.

E. N. Leith, L. J. Cutrona, and L. J. Porcello, J. Opt. Soc. Am. 56, 1419A (1966).
[Crossref]

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).

Rawcliffe, R. D.

C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
[Crossref]

Reynolds, G. O.

Ruina, J. P.

C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
[Crossref]

Sherwin, C. W.

C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
[Crossref]

Stroke, G. W.

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

Upatnieks, J.

Vivian, W. E.

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).

Winthrop, J. T.

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Worthington, C. R.

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Appl. Phys. Letters (1)

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

IRE Trans. Military Electron. (3)

If the emitting antenna has the diameter Dx, the width of the illuminated ground field is Lx = r0λ0/Dx. The resolution is Δx0 = Dx/2, according to L. J. Cutrona and G. O. Hall, IRE Trans. Military Electron. 6, 119 (1962).
[Crossref]

L. J. Cutrona, W. E. Vivian, E. N. Leith, and G. O. Hall, IRE Trans. Military Electron. 6, 127 (1961).
[Crossref]

C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, IRE Trans. Military Electron. 6, 111 (1962).
[Crossref]

J. Opt. Soc. Am. (5)

Opt. Acta (1)

A. Lohmann, Opt. Acta 3, 97 (1956).
[Crossref]

Phys. Letters (1)

J. T. Winthrop and C. R. Worthington, Phys. Letters 15, 124 (1965).
[Crossref]

Z. Naturforsch. (1)

W. Lukosz, Z. Naturforsch. 18a, 436 (1963);W. Lukosz and M. Marchand, Opt. Acta 10, 241 (1963).
[Crossref]

Other (5)

In this paper kx, ky denote radian spatial frequencies, kx/2π, ky/2π the corresponding proper spatial frequencies (in lines/mm); ω denotes a radian temporal frequency, and ν=ω/2 the corresponding temporal frequency. In Eq. (1.2) both the object field and the aperture of the system are assumed to be rectangular.

In the experiments reported in Ref. 4, no coherent background was used. The receiver integrated the intensity in the image plane. In this case the gratings M and M′ have to be inserted in or very near to the object and image planes, respectively.

L. J. Cutrona, E. N. Leith, L. J. Porcello, and W. E. Vivian, Proc. 9th AGARD Symposium on Opto-Electronic Components and Devices, Paris, September 1965 (MIT Press, Cambridge, Mass., 1965).

In quantum mechanics this is the condition for the validity of the WKB approximation [E. E. Merzbacher, Quantum Mechanics (John Wiley & Sons, New York, 1961), Ch. 7].

This assumption involves no loss of generality of our subsequent considerations. The effect of an off-axis angle of the reference beam is an apparent lateral shift of the object and of the light source illuminating the object.

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 (4)

F. 1
F. 1

Coherent waves with different temporal frequencies are produced by diffraction of one wave of frequency ν0 by a grating M moved with constant velocity v(υxy). The frequencies of the diffracted waves of order (j,l) are shown, for (a) a line-grating moved with υ x / d x = Δ ν ̂, and (b) a two-dimensional grating moved with υ x / d x = 1 3 υ y / d y = Δ ν ̂. (These waves, with frequencies separated by at least Δ ν ̂ from each other, are used to illuminate the object; they transmit information about different spatial frequency bands in different temporal-frequency channels of the optical system.)

F. 2
F. 2

Principle of an optical arrangement that increases the spatial bandwidth of a system by sacrificing temporal bandwidth. OP object, IP image plane; S optical system; M, M′ gratings in conjugate planes moved with conjugate velocities υ and υ′, respectively. Here M and M′ are assumed to produce only the diffraction orders 0 and ±1. The temporal-frequency filter of bandwidth Δ ν ̂ behind M′ is not shown.

F. 3
F. 3

Geometry of airborne ground-mapping radar.7 The aircraft P in the altitude z moving with the velocity υx in the x direction, carries a side-looking antenna which illuminates the indicated ground field and receives the reflected signals.

F. 4
F. 4

Single-sideband Fresnel holography. O object (x0 ≤ 0); H hologram plate (x≥0); ur on-axis reference beam.

Equations (105)

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

N = 2 N x , y N t .
N x , y = ( 1 + L x Δ k x / 2 π ) ( 1 + L y Δ k y / 2 π ) ,
N x , y = L x L y Δ k x Δ k y / 4 π 2 = S W .
N t = 2 ( 1 + T Δ ν ) ,
N t = 2 T Δ ν .
Ŵ Δ ν ̂ = W Δ ν ,
M ( x , y ) = j , l = 0 , ± 1 , m j , l exp [ 2 π i ( j x / d x + l y / d y ) ] .
exp ( 2 π i ν 0 t ) M ( x υ x t , y υ y t ) = j , l = 0 , ± 1 , m j , l exp { 2 π i [ j υ x / d x + l υ y / d y ( ν 0 + j υ x / d x + l υ y / d y ) ] } .
k x ( s ) = 2 π j / d x , k y ( s ) = 2 π l / d y
ν j , l = ν 0 + j υ x / d x + l υ y / d y .
k x ( s ) = 2 π j / d x , k y ( s ) = 0 ,
ν j = ν 0 + j υ x / d x .
υ x / d x Δ ν ̂ .
M ( x ) = 1 + c x cos 2 π x / d x ,
M ( x , y ) = ( 1 + c x cos 2 π x / d x ) ( 1 + c y cos 2 π y / d y ) ,
υ x / d x = υ y / 3 d y Δ ν ̂ .
M ( x , y ) = j , l = 0 , ± 1 , m j , l exp [ 2 π i ( j x / d x + l y / d y ) ] .
A ( x , y ; t ) = ( 2 π ) 3 + a ( k x , k y ; ω ) × exp i ( k x x + k y y ω t ) d k x d k y d ω .
U ( x , y , z = 0 ; t ) = exp ( i ω 0 t ) A ( x , y ; t ) .
U ( x , y , z ; t ) = ( 2 π ) 3 + u ( k x , k y ; ω ) × exp i ( k x x + k y y + k z z ω t ) d k x d k y d ω
k z = [ ( ω / c ) 2 k x 2 k y 2 ] 1 2 ,
u ( k x , k y ; ω ) = a ( k x , k y ; ω ω 0 ) .
U M ( x , y , z = z 0 ; t ) = U ( x , y , z = z 0 ; t ) M ( x υ x t , υ y t ) .
u M ( k x , k y ; ω ) = j , l = 0 , ± 1 , m j , l u ( k x 2 π j / d x , k y 2 π l / d y ; ω 2 π j υ x / d x 2 π l υ y / d y ) × exp i z 0 { [ ( ω 2 π j υ x / d x 2 π l υ y / d y ) 2 c 2 ( k x 2 π j / d x ) 2 ( k y 2 π l / d y ) 2 ] 1 2 [ ( ω / c ) 2 k x 2 k y 2 ] 1 2 } .
u ( k x , k y ; ω ) = u M ( k x , k y ; ω ) f ( k x , k y ) .
u M ( k x , k y ; ω ) = p , q = 0 , ± 1 , f p , q ( k x , k y ) u ( k x 2 π p / d x , k y 2 π q / d y ) × exp i z 0 { [ ( ω 2 π p υ x / d x 2 π q υ y / d y ) 2 c 2 ( k x 2 π p / d x ) 2 ( k y 2 π q / d y ) 2 ] 1 2 [ ( ω / c ) 2 k x 2 k y 2 ] 1 2 } ,
f p , q ( k x , k y ) = j , l = 0 , ± 1 , m j , l m p j , q l f ( k x 2 π j / d x , k y 2 π l / d y ) .
| p υ x / d x + q υ y / d y | Δ ν ̂
û ( k x , k y ; ω ) = f ̂ ( k x , k y ) u ( k x , k y ; ω ) .
f ̂ ( k x , k y ) = j , l = 0 , ± 1 , m ̂ j , l f ( k x 2 π j / d x , k y 2 π l / d y ) ,
m ̂ j , l = m j , l m j , l ,
F ̂ ( x , y ) = F ( x , y ) M ̂ ( x , y ) ,
M ̂ ( x , y ) = ( d x d y ) 1 0 d x 0 d y M ( x ¯ , y ¯ ) M ( x ¯ + x , y ¯ + y ) d x ¯ d y ¯ = i , l = 0 , ± 1 , m ̂ j , l exp [ 2 π i ( j x / d x + l y / d y ) ]
u s ( t ) = u 0 exp [ i ω 0 ( t t P O ) ] .
t P O = 2 c 1 [ ( x x 0 ) 2 + y 0 2 + z 2 ] 1 2 , 2 r 0 / c + ( x x 0 ) 2 / r 0 c ,
r 0 = ( y 0 2 + z 2 ) 1 2 .
u r ( t ) = u r exp ( i ω 0 t ) .
u r * ( t ) u s ( t ) + u r ( t ) u s * ( t )
x ¯ = υ ¯ x t = ( υ ¯ x / υ x ) x .
u r * ( x ¯ ) u s ( x ¯ ) + u r ( x ¯ ) u s * ( x ¯ ) = u r * u 0 exp i φ ( x ¯ ) + c . c .
φ ( x ¯ ) = 2 k 0 r 0 + ( υ x / υ ¯ x ) 2 k 0 ( x ¯ x ¯ 0 ) 2 / r 0 .
φ ( x ) = φ ( x ) + ( d φ / d x ) | x = x ( x x ) + 1 2 ( d 2 φ / d x 2 ) | x = x ( x x ) 2 + .
k x ( x ) = ( d φ / d x ) | x = x .
δ x δ k x 2 π .
d k x ( x ) / d x ( 1 / 2 π ) [ k x ( x ) ] 2 .
υ x k x ( x ) = ω ( t ) = υ ¯ x k ¯ x ( x ¯ ) .
k x ( x ) = 2 k 0 ( x x 0 ) / r 0 .
Δ k x = 2 k 0 L x / r 0 .
Δ x 0 = 2 π / Δ k x = λ 0 r 0 / 2 L x
υ x Δ k x = Δ ω x = υ ¯ x Δ k ¯ x .
n x = 2 s x Δ k x / 2 π = 2 s x / Δ x 0 , n t = 2 T Δ ω x / 2 π = 2 T Δ ν x , n x ¯ = 2 s ¯ x Δ k ¯ x / 2 π ; n x = n t = n x ¯ .
u ( t ) = u 0 exp i φ ( t ) ,
φ ( t ) = ω 0 t + 1 2 ω 0 t 2 .
w ( t ) = d φ ( t ) / d t = ω 0 + ω 0 t
1 / τ Δ ν x = υ x / Δ x 0 ,
u r ( t ) = u ( t ) .
u s ( t ) = u ( t + τ t P O ) .
u r * ( t ) u s ( t ) = | u 0 | 2 exp i Δ φ ( t ) ,
Δ φ ( t ) = φ ( t + τ t P O ) φ ( t ) = ω 0 ( τ t P O ) [ t + 1 2 ( τ t P O ) ] + ω 0 ( τ t P O ) .
t P O τ 2 ( sin γ ) ( y 0 y 0 ) / c .
ω = υ ¯ y k ¯ y = 2 ω 0 ( sin γ ) ( y 0 y 0 ) / c .
Δ y 0 = c / 2 ν 0 τ sin γ .
Δ ν = υ ¯ y Δ k ¯ y / 2 π = 2 ν 0 ( sin γ ) L y / c .
n t = 2 T eff Δ ν = 2 n τ Δ ν ,
τ Δ ν = L y / Δ y 0 .
n s x / Δ x 0 ,
n t n x , y = 2 ( s x / Δ x 0 ) ( L y / Δ y 0 ) .
u s ( x , y ; t ) = u 0 exp i [ ω 0 t φ ( x , y ) ] ,
φ ( x , y ) = k 0 [ ( x x 0 ) 2 + ( y y 0 ) 2 + z 0 2 ] 1 2 ,
u r ( x , y ; t ) = u r exp ( i ω 0 t ) .
I ( x , y ) = [ u r ( x , y ; t ) + u s ( x , y ; t ) ] * × [ u r ( x , y ; t ) + u s ( x , y ; t ) ] .
u r * ( x , y ; t ) u s ( x , y ; t ) = u r * u 0 exp i φ ( x , y )
k x = d φ / d x = k 0 ( x x 0 ) / [ ( x x 0 ) 2 + ( y y 0 ) 2 + z 0 2 ] 1 2 k 0 ( x x 0 ) / | z 0 | , k y = d φ / d y = k 0 ( y y 0 ) / [ ( x x 0 ) 2 + ( y y 0 ) 2 + z 0 2 ] 1 2 k 0 ( y y 0 ) / | z 0 | ,
0 k x k 0 L x / z 0
L m = z 0 k x / k 0
sin α m = k x / k 0 .
k 0 ( x 0 ) / z 0 k x k 0 ( L x x 0 ) / z 0
Δ k x ( x 0 ) = k 0 L x / z 0 .
k 0 ( x 0 ) / z 0 k x k x = k 0 L m / z 0
Δ k x ( x 0 ) = k 0 ( L m + x 0 ) / z 0
( n x ) max = 2 L x k x / 2 π
Δ k x ( x ) = k x k 0 x / z 0 .
n x = 2 ( 2 π ) 1 0 L x Δ k x ( x ) d x = 2 L x ( L m 1 2 L x ) / z 0 λ 0 ,
n x = 2 ( 2 π ) 1 0 L m Δ k x ( x 0 ) d | x 0 | .
η = n x / ( n x ) max = 1 1 2 L x / L m .
L x = k x ( z 0 ) min / k 0
η = 1 1 2 ( z 0 ) min / z 0 .
( n x , y ) max = 2 1 2 ( L x k x / π ) ( L y k y / π )
( n x , y ) max = ( n x ) max n y ,
n y = L y k y / π .
k 0 ( 1 2 L y + y 0 ) / z 0 k y k 0 ( 1 2 L y y 0 ) / z 0
Δ k y ( y 0 ) = k 0 L y / z 0 .
k y k y k 0 ( 1 2 L y y 0 ) / z 0 ,
k 0 ( 1 2 L y + y 0 ) / z 0 k y k y .
Δ k y ( y 0 ) = k 0 ( L m + 1 2 L y | y 0 | ) / z 0
n x , y = n x n y ,
n y = ( 2 π ) 1 ( L m + 1 2 L y ) ( L m + 1 2 L y ) Δ k y ( y 0 ) d y 0 .
η = n x , y / ( n x , y ) max = n x / ( n x ) max .
k x 2 + k y 2 k 2
Δ k x / 2 π = 2 / λ ;
W = ( 2 π ) 2 ( k x 2 + k y 2 k 2 ) d k x d k y = π / λ 2 .
[ k x k x ( s ) ] 2 + [ k y k y ( s ) ] 2 k 2
[ k x ( s ) ] 2 + [ k y ( s ) ] 2 k 2 ,
Δ k ̂ x / 2 π = 2 ( 1 + sin α x ) / λ ,
Δ k x / 2 π = 4 / λ .