Abstract

Spurious background fringes have been observed in time-resolved holograms recorded with a pulsed laser source and a rapidly deflected reference beam. These fringes were determined to be a time smear effect due to the small change in the angle of incidence of the reference beam during the exposure time. By analyzing the phenomenon theoretically, equations were derived that could be used to describe the observed fringe formation. A numerical solution was obtained for a realistic laser pulse time profile and an analytical solution was obtained using a simple mathematical model of the pulse profile. These solutions allow an a priori determination of bounds of the exposure parameters that would reduce the time smear effects to acceptable levels.

© 1989 Optical Society of America

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References

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  1. R. G. Racca, J. M. Dewey, “Time-Resolved Holography for the Study of Shock Waves,” Proc. Soc. Photo-Opt. Instrum. Eng. 1032, 578–586 (1989).
  2. J. M. Dewey, D. K. Walker, “A Multiply Pulsed Double-Pass Laser Schlieren System for Recording the Movement of Shocks and Particle Tracers Within a Shock Tube,” J. Appl. Phys. 46, 3454–3458 (1975).
    [CrossRef]
  3. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  4. C. C. Aleksoff, “Temporally Modulated Holography,” Appl. Opt. 10, 1329–1341 (1971).
    [CrossRef] [PubMed]

1989 (1)

R. G. Racca, J. M. Dewey, “Time-Resolved Holography for the Study of Shock Waves,” Proc. Soc. Photo-Opt. Instrum. Eng. 1032, 578–586 (1989).

1975 (1)

J. M. Dewey, D. K. Walker, “A Multiply Pulsed Double-Pass Laser Schlieren System for Recording the Movement of Shocks and Particle Tracers Within a Shock Tube,” J. Appl. Phys. 46, 3454–3458 (1975).
[CrossRef]

1971 (1)

Aleksoff, C. C.

Dewey, J. M.

R. G. Racca, J. M. Dewey, “Time-Resolved Holography for the Study of Shock Waves,” Proc. Soc. Photo-Opt. Instrum. Eng. 1032, 578–586 (1989).

J. M. Dewey, D. K. Walker, “A Multiply Pulsed Double-Pass Laser Schlieren System for Recording the Movement of Shocks and Particle Tracers Within a Shock Tube,” J. Appl. Phys. 46, 3454–3458 (1975).
[CrossRef]

Racca, R. G.

R. G. Racca, J. M. Dewey, “Time-Resolved Holography for the Study of Shock Waves,” Proc. Soc. Photo-Opt. Instrum. Eng. 1032, 578–586 (1989).

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Walker, D. K.

J. M. Dewey, D. K. Walker, “A Multiply Pulsed Double-Pass Laser Schlieren System for Recording the Movement of Shocks and Particle Tracers Within a Shock Tube,” J. Appl. Phys. 46, 3454–3458 (1975).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

J. M. Dewey, D. K. Walker, “A Multiply Pulsed Double-Pass Laser Schlieren System for Recording the Movement of Shocks and Particle Tracers Within a Shock Tube,” J. Appl. Phys. 46, 3454–3458 (1975).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. G. Racca, J. M. Dewey, “Time-Resolved Holography for the Study of Shock Waves,” Proc. Soc. Photo-Opt. Instrum. Eng. 1032, 578–586 (1989).

Other (1)

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

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Figures (6)

Fig. 1
Fig. 1

Diagram of the optical system used to record spatial frequency multiplexed sequences of holograms.

Fig. 2
Fig. 2

Holographic image of shock tube test section recorded with the mirror spinning during exposure. The profile of the wedge-shaped model is overlaid in black on the print to enhance contrast.

Fig. 3
Fig. 3

Holographic image of shock tube test section recorded with the mirror stationary during exposure. The profile of the wedge-shaped model is overlaid in black on the print to enhance contrast.

Fig. 4
Fig. 4

Profiles of laser pulse irradiance vs. time used in the examples discussed. The solid curve shows the digitized irradiance as recorded through a photodetector. The dashed curve shows the analytical function (the square of a parabola) used to approximate the measured irradiance profile.

Fig. 5
Fig. 5

Graph of hologram irradiance modulation vs. lateral displacement given by the numerical model using a digitized laser pulse irradiance profile.

Fig. 6
Fig. 6

Graph of hologram irradiance modulation vs. lateral displacement given by the analytical model based on a parabolic laser pulse amplitude profile.

Equations (19)

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object wave : U O ( x , y )
reference wave : U R ( x , y ) = a R exp ( i 2 π f y y ) ,
I ( x , y ) = | U O + a R exp ( i 2 π f y y ) | 2 = | U O | 2 + a R 2 + a R U O exp ( i 2 π f y y ) + a R U O * exp ( i 2 π f y y ) .
t ( x , y ) = t b + β [ | U O | 2 + a R 2 + a R U O exp ( i 2 π f y y ) + a R U O * exp ( i 2 π f y y ) ] .
U c ( x , y ) = a c exp ( i 2 π f y y ) .
U I ( x , y ) = ( t b + β | U O | 2 ) a c exp ( i 2 π f y y ) + β a c a R U O + β a c a R U O * exp ( i 4 π f y y ) .
I O I ( x , y ) = β 2 a c 2 a R 2 | U O | 2 .
object wave : A ( t ) U O ( x , y )
reference wave : A ( t ) a R exp ( i 2 π sin ( θ 0 + ω t ) λ y ) .
I ( x , y , t ) = | A ( t ) U O + A ( t ) a R exp ( i 2 π sin ( θ 0 + ω t ) λ y ) | 2 = A 2 ( t ) | U O | 2 + A 2 ( t ) a R 2 + A 2 ( t ) a R U O exp ( i 2 π sin ( θ 0 + ω t ) λ y ) + A 2 ( t ) a R U O * exp ( i 2 π sin ( θ 0 + ω t ) λ y ) .
t ( x , y ) = t b + β Δ [ | U O | 2 Δ / 2 Δ / 2 A 2 ( t ) d t + a R 2 Δ / 2 Δ / 2 A 2 ( t ) d t + a R U O Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π sin ( θ 0 + ω t ) λ y ) d t + a R U O * Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π sin ( θ 0 + ω t ) λ y ) d t ] .
sin ( θ 0 + ω t ) = sin θ 0 cos ( ω t ) + cos θ 0 sin ( ω t ) sin θ 0 + ω t cos θ 0 .
t ( x , y ) = t b + β Δ [ | U O | 2 Δ / 2 Δ / 2 A 2 ( t ) d t + a R U O exp ( i 2 π sin θ 0 λ y ) Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t + a R U O * exp ( i 2 π sin θ 0 λ y ) Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t ] .
U c ( x , y ) = a c exp ( i 2 π sin θ 0 λ y ) .
U I ( x , y ) = ( t b + β Δ | U O | 2 Δ / 2 Δ / 2 A 2 ( t ) d t ) a c exp ( i 2 π sin θ 0 λ y ) + β a c a R Δ U O Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t + β a c a R Δ U O * exp ( i 4 π sin θ 0 λ y ) Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t .
I O I ( x , y ) = | β a c a R Δ U O Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t | 2 = β 2 a c 2 a R 2 | U O | 2 | 1 Δ Δ / 2 Δ / 2 A 2 ( t ) exp ( i 2 π ω t cos θ 0 λ y ) d t | 2 .
A ( t ) = { 30 ( 1 4 t 2 Δ 2 ) for Δ 2 t Δ 2 , 0 elsewhere
I O I ( x , y ) = β 2 a c 2 a R 2 | U O | 2 225 ( γ y ) 6 × [ 3 γ y cos γ y + ( 1 3 ( γ y ) 2 ) sin γ y ] 2 ,
γ = π ω Δ cos θ 0 λ .

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