Abstract

A coherent optical system design is proposed for simultaneously heterodyning, correlating, and beam forming on a large number of input channels. Techniques are described with application to two- as well as one-dimensional arrays. To accommodate the former, one direction in the optical aperture must serve both as a time and space dimension. The particular technique proposed resolves the apparent ambiguity. The method uses the phase of the correlation amplitude, after heterodyning, to effect the beam forming operation. Although the full theoretical correlation processing gain against background noise is not realized, very substantial gains and fine time and angle resolutions can be achieved. Because of its inherent physical simplicity, particularly as contrasted with electronic methods of similar capacity, the coherent optical correlator–beam former has significant advantages for many applications.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
    [CrossRef]
  2. G. E. Stanford, J. Acoust. Soc. Amer. 42, 1166 (1967).
    [CrossRef]
  3. D. C. Beste, E. N. Leith, IEEE Trans. Aerospace Electron. Systems AES-2, 376 (1966).
    [CrossRef]
  4. G. L. Turin, IRE Trans. Inform. Theory IT-6, 319 (1960), see Eq. (40).

1967

G. E. Stanford, J. Acoust. Soc. Amer. 42, 1166 (1967).
[CrossRef]

1966

D. C. Beste, E. N. Leith, IEEE Trans. Aerospace Electron. Systems AES-2, 376 (1966).
[CrossRef]

1960

G. L. Turin, IRE Trans. Inform. Theory IT-6, 319 (1960), see Eq. (40).

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Beste, D. C.

D. C. Beste, E. N. Leith, IEEE Trans. Aerospace Electron. Systems AES-2, 376 (1966).
[CrossRef]

Cutrona, L. J.

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Leith, E. N.

D. C. Beste, E. N. Leith, IEEE Trans. Aerospace Electron. Systems AES-2, 376 (1966).
[CrossRef]

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Palermo, C. J.

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Porcello, L. J.

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

Stanford, G. E.

G. E. Stanford, J. Acoust. Soc. Amer. 42, 1166 (1967).
[CrossRef]

Turin, G. L.

G. L. Turin, IRE Trans. Inform. Theory IT-6, 319 (1960), see Eq. (40).

IEEE Trans. Aerospace Electron. Systems

D. C. Beste, E. N. Leith, IEEE Trans. Aerospace Electron. Systems AES-2, 376 (1966).
[CrossRef]

IRE Trans. Inform. Theory

G. L. Turin, IRE Trans. Inform. Theory IT-6, 319 (1960), see Eq. (40).

E.g., L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[CrossRef]

J. Acoust. Soc. Amer.

G. E. Stanford, J. Acoust. Soc. Amer. 42, 1166 (1967).
[CrossRef]

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 (2)

Fig. 1
Fig. 1

Spectrum analysis with line receiving array.

Fig. 2
Fig. 2

Coherent heterodyner–correlator–beam former.

Equations (26)

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

i = 1 N s ( x - x i t ) δ ( y - y i ) i = 1 N S ( f x ) × exp [ - 2 π j ( f x x i t + f y y i ) ] ,
x i t = y i sin θ / c ,
i = 1 N S ( f x ) exp [ - 2 π j ( f x x i t + f y y i ) ] = i = 1 N S ( f x ) × exp [ - 2 π j y i ( f x sin θ / c + f y ) ] .
f y = - f x sin θ / c = - [ k / ( 2 π ) ] sin θ = - [ k y / ( 2 π ) ] ,
i = 1 N φ ( x - x i t ) δ ( y - y i ) i = 1 N S ( f x ) 2 × exp [ - 2 π j ( f x x i t + f y y i ) ] ,
i = 1 N u i ( x , y ) = i = 1 N [ s ( x - x i t + x i p - x i g ) × cos 2 π f 0 ( x - x i t + x i p - x i g ) + n i ( x + x i p ) ] δ ( y - y i g ) ,
i = 1 N U i ( f x , f y ) = 1 2 i = 1 N exp ( - 2 π j f y y i g ) [ S ( f x - f 0 ) + S ( f x + f 0 ) ] exp [ - 2 π j f x ( x i t - x i p + x i g ) ] ,
i = 1 N U i ( f x + f h , f y ) = 1 2 i = 1 N exp ( - 2 π j f y y i g ) S ( f x - f 0 + f h ) × exp [ - 2 π j ( f x + f h ) ( x i t - x i p + x i g ) ] ,
s ( x ) cos 2 π ( f 0 - f h ) x = s ( x ) , for f 0 = f h ,
S ( f x ) .
1 2 S ( f x ) 2 i = 1 N exp ( - 2 π j f y y i g ) exp [ - 2 π j ( f x + f 0 ) × ( x i t - x i p + x i g ) ] = A 2 / 2 rect [ f x / W ] i = 1 N exp ( - 2 π j f y y i g ) × exp [ - 2 π j ( f x + f 0 ) ( x i t - x i p + x i g ) ] ,
i = 1 N φ i ( x - x i t + x i p - x i g , y - y i g ) = W [ A 2 / 2 ] i = 1 N sinc [ W ( x - x i t + x i p - x i g ) ] × exp [ - 2 π j f 0 ( x i t - x i p + x i g ) ] δ ( y - y i g ) ,
i = 1 N δ ( x - x i g ) δ ( y - y i g ) ,
b ( f x , f y ) = [ A 2 / 2 ] i = 1 N exp ( - 2 π j f y y i g ) rect [ f x / W ] × exp [ - 2 π j ( f x + f 0 ) ( x i t - x i p + x i g ) ] .
f 0 x i g = n i ,             n i an integer .
x i t - x i p = l x i g + m y i g ,
( f x + f 0 ) ( l + 1 ) x i g = n i x , n i x an integer , [ m ( f x + f 0 ) + f y ] y i g = n i y , n i y an integer ,
[ ( l + 1 ) f x + l f 0 ] x i g = n i x , [ m ( f x + f 0 ) + f y ] y i g = n i y .
f x = - l ( l + 1 ) f 0 ,
f y = - m ( f x + f 0 ) = - m ( l + 1 ) f 0 ,
Δ f x = Δ l ( l max + 1 ) 2 f 0 = 5 3 × 10 - 3 × 200 ( 1.05 ) 2 0.3 line mm - 1 .
( Δ f x ) min = ( Δ x 4 ) min λ f = 10 - 5 λ = 10 - 5 5 × 10 - 4 = 0.02 line min - 1 .
( x i t - x i p ) max ( x i g ) max = 0.25 5 = 0.05.
f x = ( n i x / n i - l ) l + 1 f 0 .
Δ x P 4 λ f Δ f x 5 × 10 - 4 mm × 10 3 mm × 0.2 line mm - 1 = 0.1 mm .
( Δ x P 4 ) min λ f / A = 5 × 10 - 4 × 10 3 / 50 = 0.01 mm .

Metrics