Abstract

A general analysis is presented of an interferometric acoustooptic signal processor for simultaneous direction finding and spectrum analysis. Topics include amplitude window weighting function, scaling factor, error tolerance analysis, linear periodic vs aperiodic arrays, and comparison between microwave and optical antenna array far-field patterns. Finally a specific design example is given of synthesizing an optimum array.

© 1983 Optical Society of America

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References

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  1. D. L. Hecht, Opt. Eng. 16, 461 (1977).
    [CrossRef]
  2. L. B. Lambert, M. Arm, A. Aimette, “Electro-optical Signal Processors for Phased Array Antennas,” in Optical and Electro-optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), Chap. 38, pp. 715–748.
  3. R. A. Coppock, R. F. Croce, “Wideband Optical Channelizer for Simultaneous Frequency and Direction Finding,” in Proceedings, Acousto-Optic Bulk Wave Devices (SPIE, Monterey, 1979), pp. 124–129.
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  5. J. P. Y. Lee, Appl. Opt. 20, 595 (1981).
    [CrossRef] [PubMed]
  6. B. D. Steinberg, Principles of Aperture and Array System Design (Wiley, New York, 1976).

1981 (1)

1977 (1)

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

Aimette, A.

L. B. Lambert, M. Arm, A. Aimette, “Electro-optical Signal Processors for Phased Array Antennas,” in Optical and Electro-optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), Chap. 38, pp. 715–748.

Arm, M.

L. B. Lambert, M. Arm, A. Aimette, “Electro-optical Signal Processors for Phased Array Antennas,” in Optical and Electro-optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), Chap. 38, pp. 715–748.

Coppock, R. A.

R. A. Coppock, R. F. Croce, “Wideband Optical Channelizer for Simultaneous Frequency and Direction Finding,” in Proceedings, Acousto-Optic Bulk Wave Devices (SPIE, Monterey, 1979), pp. 124–129.

Croce, R. F.

R. A. Coppock, R. F. Croce, “Wideband Optical Channelizer for Simultaneous Frequency and Direction Finding,” in Proceedings, Acousto-Optic Bulk Wave Devices (SPIE, Monterey, 1979), pp. 124–129.

Goodman, J. W.

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

Hecht, D. L.

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

Lambert, L. B.

L. B. Lambert, M. Arm, A. Aimette, “Electro-optical Signal Processors for Phased Array Antennas,” in Optical and Electro-optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), Chap. 38, pp. 715–748.

Lee, J. P. Y.

Steinberg, B. D.

B. D. Steinberg, Principles of Aperture and Array System Design (Wiley, New York, 1976).

Appl. Opt. (1)

Opt. Eng. (1)

D. L. Hecht, Opt. Eng. 16, 461 (1977).
[CrossRef]

Other (4)

L. B. Lambert, M. Arm, A. Aimette, “Electro-optical Signal Processors for Phased Array Antennas,” in Optical and Electro-optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), Chap. 38, pp. 715–748.

R. A. Coppock, R. F. Croce, “Wideband Optical Channelizer for Simultaneous Frequency and Direction Finding,” in Proceedings, Acousto-Optic Bulk Wave Devices (SPIE, Monterey, 1979), pp. 124–129.

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

B. D. Steinberg, Principles of Aperture and Array System Design (Wiley, New York, 1976).

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Figures (5)

Fig. 1
Fig. 1

Fourier transforming configuration.

Fig. 2
Fig. 2

Configuration of a linear aperiodic antenna array.

Fig. 3
Fig. 3

Acoustooptic modulator configuration for a linear aperiodic antenna array.

Fig. 4
Fig. 4

Normalized intensity patterns of the optimum array (base line = 10.5λ at 12.5 GHz; element locations: 0, 3.71, 6.23, and 10.5, w = LN/60).

Fig. 5
Fig. 5

Simulated intensity pattern of the optimum array (photodetector array cell-to-cell center spacing = 13 μm, F = 0.15 m, S.F. = 4.43 × 10−3, w = LN/60).

Tables (1)

Tables Icon

Table I Optimum 2-D Plot Showing the Peak Sidelobe to Mainlobe Values for an Interferometer of 10.5 Wavelengths (D4 = 10.5λ)

Equations (33)

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E ( x , y , z , t ) = E x ( x , y , z , t ) = E 0 ( x 0 , y 0 ) cos ( 2 π ν t k z + ϕ i ) = Re { E 0 ( x 0 , y 0 ) exp [ j ( 2 π ν t k z + ϕ i ) ] } ,
U 1 ( x 1 , y 1 , t ) = E 0 ( x 0 , y 0 ) j λ F T ( x 0 , y 0 , t ) × exp [ j 2 π λ F ( x 0 x 1 + y 0 y 1 ) ] d x 0 d y 0 = K 1 E 0 ( x 0 , y 0 ) T ( x 0 , y o , t ) × exp [ j 2 π ( u x 0 + υ y 0 ) ] d x 0 d y 0 ,
K 1 = 1 j λ F u = x 1 λ F υ = y 1 λ F ,
V in ( t ) = A ( t ) cos ( 2 π f t ) .
S ( x 0 υ s t ) = A ( x 0 υ s t ) exp [ α ( f ) τ ( x 0 D + 1 2 ) ] × cos [ 2 π f / υ s ( x 0 υ s t ) ] .
T ( x 0 , y 0 , t ) = rect ( x 0 D ) rect ( y 0 L ) exp { j ψ ( x 0 υ s t ) × exp [ α ( f ) τ ( x 0 D + 1 2 ) ] × cos [ 2 π f / υ s ( x 0 υ s t ) + ϕ 0 ] } ,
T ( x 0 , y 0 , t ) = rect ( x 0 D ) rect ( y 0 L ) exp { j ψ B ( x 0 υ s t ) × exp [ α ( f ) τ ( x 0 D + 1 2 ) ] × exp j [ 2 π f / υ s ( x 0 υ s t ) + ϕ 0 ] } rect ( x 0 D ) rect ( y 0 L ) { 1 + j ψ B ( x 0 υ s t ) exp [ α ( f ) × τ ( x 0 D + 1 2 ) ] exp j [ 2 π f / υ s ( x 0 υ s t ) + ϕ 0 ] } for ψ B max 1 ,
V in ( t ) = g ( t ) = A ( t ) / ϕ ( t ) ¯ ,
U 1 ( x 1 , y 1 , t ) = K B λ F g ( x 0 υ s t ) w ( x 0 , y 0 ) × exp [ j 2 π λ F ( x 0 x 1 + y 0 y 1 ) ] d x 0 d y 0 ,
g ( x 0 υ s t ) = A ( x 0 υ s t ) exp [ j ϕ ( t ) x 0 ( x 0 υ s t ) ] , w ( x 0 , y 0 ) = E 0 ( x 0 , y 0 ) rect ( x 0 D ) rect ( y 0 L ) × exp [ α ( f ) τ ( x 0 D + 1 2 ) ]
V n ( t ) = E n ( f R , Ө 0 ) A ( t τ n ) cos [ 2 π f R ( t τ n ) ] ,
τ n = d n c sin Ө 0 = d n λ R f R sin Ө 0 , n = 1,2,3 , , N .
V IF , n = A n ( t ) cos [ 2 π ( f i t β n ) ] E n ( f R , Ө 0 ) ,
S n ( x 0 υ s t ) = A ( x 0 υ s t ) cos { 2 π [ f i / υ s ( x 0 υ s t ) β n ] } · exp [ α ( f ) τ ( x 0 D + 1 2 ) ] E ( f R , Ө 0 ) × rect ( x 0 D ) rect ( y 0 l n w ) .
T ( x 0 , y 0 , t ) = n = 1 N rect ( x 0 D ) rect ( y 0 l n w ) · { 1 + j ψ B ( x 0 υ s t ) exp [ α ( f ) τ ( x 0 D + 1 2 ) ] E n ( f R , Ө 0 ) · exp j [ 2 π f i / υ s ( x 0 υ s t ) 2 π β n + ϕ 0 ] } .
U 1 ( x 1 , y 1 , t ) = K 2 f ( υ ) g ( x 0 υ s t ) w ( x 0 , y 0 ) · exp [ j 2 π λ F ( x 0 x 1 + y 0 y 1 ) ] d x 0 d y 0 ,
K 2 = N K B E ( f R , Ө 0 ) λ F exp ( j ϕ 0 ) ; f ( υ ) = 1 N n = 1 N a n ( f R , Ө 0 ) exp [ j 2 π ( υ l n + β n ) ] ;
w ( x 0 , y 0 ) = rect ( x 0 D ) rect ( y 0 w ) exp [ α ( f ) τ ( x 0 D + 1 2 ) ] × exp [ ( 2 T x x 0 D ) 2 ( 2 T y y 0 w ) 2 ] ,
w ( x 0 , y 0 ) = rect ( x 0 D ) rect ( y 0 w ) ,
U 1 ( u , υ , t ) = K B E ( f R , Ө 0 ) D w λ F exp ( j 2 π f i t ) × sinc D ( u 1 λ s ) sinc υ w n = 1 N × exp [ j 2 π ( υ l n + f R τ n ) ] ,
sinc ( υ w ) = sin [ π ( d n λ R w l n ) ] π d n λ R w l n 0.9.
w l n λ R 4 d n .
υ l n = f R τ n , n = 1,2 , , N ( d n / λ R ) / ( l n / λ ) = y 1 F sin Ө 0 = S . F .
ϕ = sin 1 ( y 1 F ) y 1 , F for y 1 F 1 = S . F . sin Ө 0 = S . F . Ө 0 for Ө 0 1.
Ө 0 = sin 1 ( y 1 F · S . F . ) .
f ( υ ) = 1 N n = 1 N ( a n + δ a n ) exp ( j δ ϕ n ) · exp [ j 2 π ( υ l n + f R τ n ) ] ,
E [ f ( υ ) f * ( υ ) ] = | exp ( j δ ϕ ) ¯ | 2 f 0 ( υ ) f * 0 ( υ ) + a 2 [ 1 | exp ( j δ ϕ ) ¯ | 2 ] + σ a 2 N ,
| exp ( j δ ϕ ) ¯ | 2 = exp ( σ ϕ 2 ) .
E ( δ ϕ n ) = E ( δ ϕ m ) = 0 , all n , m , E ( δ ϕ n δ ϕ m ) = 0 , n m , σ ϕ 2 = E ( δ ϕ 2 ) 1.
f ( υ ) f * ( υ ) = 1 N 2 n = 1 N m = 1 N a n a m exp [ j ( 2 π l n m υ + δ n m ) ] .
E ( 2 π υ ) 2 = σ ϕ 2 ( n = 1 N a n ) 2 n = 1 N a n 2 l n 2 [ n = 1 N a n n = 1 N a n l n 2 ( n = 1 N a n l n ) 2 ] 2 .
σ υ = σ ϕ n = 1 N a n ( n = 1 N a n 2 l n 2 ) 1 / 2 2 π [ n = 1 N a n n = 1 N a n l n 2 ( n = 1 N a n l n ) 2 ] .
σ y 1 F = Δ ϕ 2 π σ ϕ n = 1 N a n [ n = 1 N a n 2 l n 2 ] 1 / 2 [ n = 1 N a n n = 1 N a n l n 2 ( n = 1 N a n l n 2 ) 2 ] ,

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