Abstract

The output of an acoustooptic spectrum analyzer contains both spatial and temporal frequencies. In a previous paper, VanderLugt predicted a difference in the temporal frequency as a function of spatial frequency for a cw signal, a pulse evolving into the acoustooptic cell, and an isolated pulse moving through the cell. We provide a more complete analysis for each of the three input signals. An analysis is made of the effects of sidelobe movement in the evolving pulse signal. This analysis, combined with a mixed transform analysis, leads to an accurate prediction of the temporal frequency behavior and to new interpretations of the results presented in the original paper. In particular, our analysis accurately predicts the temporal frequencies associated with the evolving pulse. Experimental results validate the methematical analysis for all three signal types.

© 1989 Optical Society of America

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References

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  1. A. VanderLugt, “Interferometric Spectrum Analyzer,” Appl. Opt. 20, 2770–2779 (1981).
    [CrossRef]
  2. J. P. Y. Lee, J. S. Wight, “Acoustooptic Spectrum Analyzer: Detection of Pulsed Signals,” Appl. Opt. 25, 193–198 (1986).
    [CrossRef] [PubMed]
  3. A. VanderLugt, A. M. Bardos, “Spatial and Temporal Spectra of Periodic Functions for Spectrum Analysis,” Appl. Opt. 23, 4269–4279 (1984).
    [CrossRef] [PubMed]
  4. A. M. Bardos, Harris Corp., Melbourne, FL, private communication.

1986 (1)

1984 (1)

1981 (1)

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Figures (11)

Fig. 1
Fig. 1

Interferometric spectrum analyzer.

Fig. 2
Fig. 2

Evolving pulse input signal.

Fig. 3
Fig. 3

Short pulse input signal.

Fig. 4
Fig. 4

Amplitude of the spectrum of the evolving pulse at two time instances.

Fig. 5
Fig. 5

Mixed transform of the evolving pulse.

Fig. 6
Fig. 6

Position of the reference probe in the signal spectrum.

Fig. 7
Fig. 7

Frequency resolution components of the measurement system.

Fig. 8
Fig. 8

Spectrum of the cw signal: reference probe at (a) Δf = 0.0 Hz and (b) Δf = 353 kHz.

Fig. 9
Fig. 9

Spectrum of the evolving pulse: reference probe at (a) Δf = 0.0 Hz and (b) Δf = 345 kHz.

Fig. 10
Fig. 10

Spectrum of the short pulse: reference probe at (a) Δf = 0.0 Hz, (b) Δf = 1.5 KHz, (c) Δf = 2.5 MHz, (d) Δf = 3.5 MHz.

Fig. 11
Fig. 11

Oscilloscope trace with probe at Δf = 345 kHz for a 20-μs pulse.

Tables (4)

Tables Icon

Table I Frequency Shift Corresponding to Position in Spatial Frequency Spectrum

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Table II Measured Values for cw Input Signal

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Table III Measured Values for 10-μs Pulse Input Signal

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Table IV Measured Values for 1-μs Pulse Input Signal

Equations (22)

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A ( x , t ) = s ( t x / υ ) exp [ j 2 π f c ( t x / υ ) ] ,
S ( α , t ) = L / 2 L / 2 s ( t x / υ ) exp [ j 2 π f c ( t x / υ ) ] exp ( j 2 π α x ) d x ,
S ( α , t ) = υ exp ( j 2 π α υ t ) t T t s ( u ) exp [ j 2 π ( α c α ) υ u ] d u ,
S ( α , t ) = υ exp ( j 2 π α υ t ) t T t exp [ j 2 π ( α α c ) υ u ] d u = L exp ( j 2 π f c t ) sinc [ ( α α c ) L ] .
S ( α , t ) = L / 2 L / 2 + υ t exp [ j 2 π f c ( t x / υ ) ] exp ( j 2 π α x ) d x = exp [ j 2 π ( α + α c ) υ t / 2 ] υ t × sinc [ ( α α c ) υ t ] ; 0 t T .
S ( α , t ) = L / 2 + υ t L / 2 + υ t + υ T 0 exp [ j 2 π f c ( t x / υ ) ] exp ( j 2 π α x ) d x = L 0 exp [ j 2 π α υ t ] sinc [ ( α α c ) L 0 ] ; T 0 t D ( T T 0 ) .
I ( α , t ) = | R ( α ) + S ( α , t ) | 2 = | R ( α ) | 2 + | S ( α , t ) | 2 + 2 Re [ R ( α ) S * ( α , t ) ] ,
n = ξ p λ s = ( Δ α ) λ F 2 λ F / υ t = ( Δ α ) υ t / 2 .
υ s = d ( 2 n ξ 0 ) d t = 2 n λ F υ t 2 .
f = υ s λ s = ( Δ α υ ) ξ 0 2 ξ 0 = Δ α υ 2 .
S ( α , f ) = s ( x , t ) exp [ j 2 π ( α x f t ) ] dxdt .
S ( α , f ) = S ( α , t ) exp ( j 2 π f t ) d t .
S ( α , f ) = L sinc [ ( α α c ) L ] exp [ j 2 π ( f α c υ ) t ] d x = L T sinc [ ( α α c ) L ] δ ( f α c υ ) .
S ( α , f ) = L 0 sinc [ ( α α c ) L 0 ] 0 T exp [ j 2 π ( f α υ ) t ] d t = L 0 T sinc [ ( α α c ) L 0 ] sinc [ ( f α υ ) T ] .
S ( α , f ) = 0 T υ t exp [ j π ( α + α c ) υ t ] × sinc [ ( α α c ) υ t ] exp ( j 2 π f t ) d t = T α α c [ sinc [ ( f α υ ) T ] exp [ j ϕ ( α , f ) ] sinc [ ( f α c υ ) T ] ] ,
| S ( α , f ) | 2 = | S ( α , t ) | 2 exp ( j 2 π f t ) d t = L 2 sinc 2 [ ( α α c ) L ] exp [ j 2 π f t ) d t = L 2 sinc 2 [ ( α α c ) L ] δ ( f ) .
| S ( α , f ) | 2 = | S ( α , t ) | 2 exp ( j 2 π f t ) d t = L 0 2 sinc 2 [ ( α α c ) L 0 ] 0 T exp ( j 2 π f t ) d t = L 0 2 sinc 2 [ ( α α c ) L 0 ] sinc ( f T ) ,
| S ( α , f ) | 2 = | S ( α , t ) | 2 exp ( j 2 π f t ) d t = 0 T ( υ t ) 2 sinc 2 [ ( α α c ) υ t ] exp ( j 2 π f t ) d t = T ( α α c ) 2 { 2 sinc ( f T ) exp ( j ϕ ) sinc [ ( f α υ + α c ) T ] exp ( j θ ) sinc [ ( f + α υ α c υ ) T ] } .
| S ( α c + Δ α , f ) | 2 = T ( Δ α ) 2 { 2 sinc ( f T ) exp [ j π ( f Δ α ) T ] sinc [ ( f Δ α υ ) T ] exp [ j π ( f + Δ α υ ) T ] sinc [ ( f + Δ α υ ) T ] } ,
S ( α , t ) = υ t cos [ 2 π ( α + α c ) υ t / 2 ] sinc [ ( α + α c ) υ t ] , 0 t T .
S ( α , t ) = 1 ( α α c ) cos [ 2 π ( α + α c ) υ t / 2 ] sin [ 2 π ( α α c ) υ t / 2 ] ,
S ( α , t ) = 1 Δ α cos [ 2 π ( α + Δ α / 2 ) υ t ] sin [ 2 π ( Δ α / 2 ) υ t ] .

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