Abstract

Following conversion of a time signal to intensity transmittance variations on film, a lens can be used in collimated coherent light to project a light intensity pattern that is an estimate of the power spectrum of the data contained on the film. To record a general biplanar signal on a film a transmittance bias level is required. The effect of this bias is defined as noise, and signal-to-noise ratios are computed for single frequency signals. Both the unapodized aperture and a cosine-shaped apodization function are considered. The effect of the size of the aperture used to record the signal on film is shown, as is the effect of the recording film speed variations. Film thickness variations reduce the performance of the analyzer; this effect is also computed.

© 1974 Optical Society of America

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References

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  1. C. E. Thomas, Appl. Opt. 5, 1782 (1966).
    [CrossRef] [PubMed]
  2. H. Stark et al., Appl. Opt. 8, 2165 (1969).
    [CrossRef] [PubMed]
  3. E. B. Felstead, Appl. Opt. 10, 1185 (1971).
    [CrossRef] [PubMed]
  4. G. B. Brandt, A. K. Rigler, Appl. Opt. 9, 2554 (1970).
    [CrossRef] [PubMed]
  5. K. R. Hessel, Appl. Opt. 10, 206 (1971).
    [CrossRef]
  6. R. O. Harger, Appl. Opt. 4, 383 (1965).
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  7. A. Vander Lugt, R. H. Mitchel, J. Opt. Soc. Am. 57, 372 (1967).
    [CrossRef]

1971 (2)

1970 (1)

1969 (1)

1967 (1)

1966 (1)

1965 (1)

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Figures (5)

Fig. 1
Fig. 1

Optical spectrum analyzer.

Fig. 2
Fig. 2

SNR vs cycles in aperture for various modulation factors F.

Fig. 3
Fig. 3

Variation of cycles in aperture vs modulation factor F aperture not apodized.

Fig. 4
Fig. 4

Number of cycles in aperture vs modulation factor F apodized aperture.

Fig. 5
Fig. 5

Minimum analyzer performance for single frequency stationary signal in aperture.

Tables (1)

Tables Icon

Table I Expected SNR Limitations Due to Film Thickness Variations for Normal Distribution and Variance of 0.01

Equations (29)

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f ( x ) = [ 1 2 + F 2 cos ω 0 ( x - τ ) ] [ U ( x + A 2 ) - U ( x - A 2 ) ] ,
G ( ω ) = A 2 sin ω ( A / 2 ) ω ( A / 2 ) + F A 4 exp ( - j ω 0 τ ) sin ( ω - ω 0 ) ( A / 2 ) ( ω - ω 0 ) ( A / 2 ) F A 4 exp ( + j ω 0 τ ) sin ( ω + ω 0 ) ( A / 2 ) ( ω + ω 0 ) ( A / 2 )
G 1 ( ω , ω 0 , τ , F ) = F A 4 exp ( - j ω 0 τ ) sin ( ω - ω 0 ) ( A / 2 ) ( ω - ω 0 ) ( A / 2 ) + F A 4 exp ( + j ω 0 τ ) sin ( ω + ω 0 ) ( A / 2 ) ( ω + ω 0 ) ( A / 2 )
G 2 ( ω ) = A 2 sin ω ( A / 2 ) ω ( A / 2 )
SNR ( ω 0 , F ) = ω L ω H - ( π / ω 0 ) ( π / ω 0 ) G 1 G 1 * d τ d ω / ( G 1 G 2 * + G 2 G 1 * + G 2 G 2 * ) d τ d ω .
SNR min ( ω 0 ) = Min { ω L ω H G 1 ( τ ) G 1 ( τ ) * d ω ω L ω H [ G 1 ( τ ) G 2 * + G 2 G 1 * ( τ ) + G 2 G 2 * ] d ω } ,
G 1 ( ω , ω 0 , τ , F ) = F A 8 exp ( - j ω 0 τ ) sin ( ω - ω 0 ) ( A / 2 ) ( ω - ω 0 ) ( A / 2 ) + F A 8 exp ( + j ω 0 τ ) sin ( ω + ω 0 ) ( A / 2 ) ( ω + ω 0 ) ( A / 2 ) + F A 16 exp ( - j ω 0 τ ) sin [ ω - ω 0 - ( 2 π / A ) ] ( A / 2 ) [ ω - ω 0 - ( 2 π / A ) ] ( A / 2 ) + F A 16 exp ( + j ω 0 τ ) sin [ ω + ω 0 - ( 2 π / A ) ] ( A / 2 ) [ ω + ω 0 - ( 2 π / A ) ] ( A / 2 ) + F A 16 exp ( - j ω 0 τ ) sin [ ω - ω 0 + ( 2 π / A ) ] ( A / 2 ) [ ω - ω 0 + ( 2 π / A ) ] ( A / 2 ) + F A 16 exp ( + j ω 0 τ ) sin [ ω + ω 0 + ( 2 π / A ) ] ( A / 2 ) [ ω + ω 0 + ( 2 π / A ) ] ( A / 2 ) .
G 2 ( ω ) = A 4 sin ω ( A / 2 ) ω ( A / 2 ) + A 8 sin [ ω - ( 2 π / A ) ] ( A / 2 ) [ ω - ( 2 π / A ) ] ( A / 2 ) + A 8 sin [ ω + ( 2 π / A ) ] ( A / 2 ) [ ω + ( 2 π / A ) ] ( A / 2 ) .
E ( y ) = K y / s y / s + w / s f ( t ) d t ,
E ( t ) = - [ u ( t + w s - ξ ) - u ( t - ξ ) ] f ( ξ ) d ξ .
G w ( ω ) = w [ sin ( w / 2 ) ω / ω ( w / 2 ) ]
w 2 { [ sin 2 ω ( w / 2 ) ] / [ ω ( w / 2 ) ] 2 }
[ sin 2 ω H ( w / 2 ) ] / [ ω H ( w / 2 ) ] 2 1
f H = ( ω H s ) / ( 2 π ) .
[ ω ( w / 2 ) sin ω ( w / 2 ) ] 2 .
Δ N 1 implies Δ s s 2 / ( f t A ) .
1 2 + F 2 exp ( j b cos ω 0 x ) cos ω 0 x .
b = [ 2 π ( i - m ) / λ ] a ,
exp ( j b cos ω 0 x ) = f = - J f ( b ) ( j ) f exp ( j f ω 0 x ) ,
F ( ω ) = - ( A / 2 ) ( A / 2 ) [ 1 2 + F 2 f = - J f ( b ) ( j ) f exp ( j f ω 0 x ) cos ω 0 x ] × exp ( - j ω x ) d x ,
F ( ω ) = A 2 sin ω ( A / 2 ) ω ( A / 2 ) + F A 4 f = - J f ( b ) ( j ) f sin ( ω - ω 0 - f ω 0 ) ( A / 2 ) ( ω - ω 0 - f ω 0 ) ( A / 2 ) + F A 4 f = - J f ( b ) ( j ) f sin ( ω + ω 0 - f ω 0 ) ( A / 2 ) ( ω + ω 0 - f ω 0 ) ( A / 2 ) .
- 10 log 10 ( b 2 / 4 ) > N d B .
exp [ j ϕ ( x ) ] 1 + j ϕ ( x )
P ( ω ) = - ( A / 2 ) ( A / 2 ) - ( A / 2 ) ( A / 2 ) ( 1 2 + F 2 cos ω 0 x ) [ 1 + j ϕ ( x ) ] × ( 1 2 + F 2 cos ω 0 y ) [ 1 - j ϕ ( y ) ] exp [ - j ω ( x - y ) ] d x d y ,
E { P ( ω ) } = - ( A / 2 ) ( A / 2 ) - ( A / 2 ) ( A / 2 ) ( 1 2 + F 2 cos ω 0 x ) × ( 1 2 + F 2 cos ω 0 y ) exp [ - j ω ( x - y ) ] d x d y + ( 1 2 + F 2 cos ω 0 x ) ( 1 2 + F 2 cos ω 0 y ) E { - j ϕ ( y ) } × exp [ - j ω ( x - y ) ] d x d y + ( 1 2 + F 2 cos ω 0 x ) ( 1 2 + F 2 cos ω 0 y ) × E { j ϕ ( x ) } exp [ - j ω ( x - y ) ] d x d y + ( 1 2 + F 2 cos ω 0 x ) ( 1 2 + F 2 cos ω 0 y ) × R ϕ ( x - y ) exp [ - j ω ( x - y ) ] d x d y .
P n ( ω ) = σ 2 - ( A / 2 ) ( A / 2 ) - ( A / 2 ) ( A / 2 ) ( 1 2 + F 2 cos ω 0 x ) × ( 1 2 + F 2 cos ω 0 y ) exp ( - α x - y ) exp [ - j ω ( x - y ) ] d x d y .
P n ( ω ) σ 2 [ 1 + α - exp ( - α ) ] 2 ω 2 + F σ 2 [ exp ( - α ) - 1 ] 4 ω 2 + F 2 σ 2 8 [ 1 α + exp ( - α ) - 1 α 2 ]
SNR = F 2 / [ 16 P n ( ω ) ]
σ 2 = [ 2 π ( i - m ) / λ ] 2 σ t 2

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