Abstract

We present details of the construction and performance of an optomechanical amplitude modulator. The heart of the modulator is an oscillating small mirror which makes it a low-power low-voltage device. The modulation frequency is fixed, but, in our model, it can be changed from 10 to 20 kHz by simple adjustments, and the modulator has high transmission. Because it is an all-reflective device, it is achromatic with respect to modulation depth. Calculations for uniform and Gaussian beams are compared to experimental results obtained with He–Ne and CO2 lasers. The results are similar to those achievable with other types of modulator.

© 1986 Optical Society of America

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References

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  1. D. Vincent, G. Otis, “Télémètre à détection cohérente avec un laser CO2 à ondes entretenues: étude et conception,” Can. J. Phys. 61, 318 (1983).
    [CrossRef]
  2. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 7.1.6.
  3. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Sect. 3.915(4).
  4. K. Sassen, G. C. Dodd, “Lidar Crossover Function and Misalignment Effects,” Appl. Opt. 21, 3162 (1982).
    [CrossRef] [PubMed]
  5. Ref. 2, Sec. 26.3.24.
  6. Ref. 2, Secs. 26.3.26 and 26.3.27. (Note that x2 in 26.3.27 should read x2=R2-1-r.)
  7. Data sheet on off-axis parabolas available from Space Optics Research Laboratories.
  8. Data sheet on scanner L45 available from American Time Products Division of Frequency Control Products, Inc., Woodside, NY.
  9. Data sheet available from Scitec Instruments, Arcade Chambers, Victoria Road, Aldershot, Hampshire, U.K.
  10. Data sheet available from Bentham Instruments Ltd., 2 Boulton Rd., Reading, Berkshire, England RG2 ONH.
  11. D. Vincent, G. Otis, “Achromatic Opto-Mechanical Modulator,” DREV Report 4361/85, Appendix B; D. Vincent, “Amplitude Modulation with a Mechanical Chopper,” Appl. Opt. 25, 1035 (1986).
    [CrossRef] [PubMed]

1983 (1)

D. Vincent, G. Otis, “Télémètre à détection cohérente avec un laser CO2 à ondes entretenues: étude et conception,” Can. J. Phys. 61, 318 (1983).
[CrossRef]

1982 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 7.1.6.

Dodd, G. C.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Sect. 3.915(4).

Otis, G.

D. Vincent, G. Otis, “Télémètre à détection cohérente avec un laser CO2 à ondes entretenues: étude et conception,” Can. J. Phys. 61, 318 (1983).
[CrossRef]

D. Vincent, G. Otis, “Achromatic Opto-Mechanical Modulator,” DREV Report 4361/85, Appendix B; D. Vincent, “Amplitude Modulation with a Mechanical Chopper,” Appl. Opt. 25, 1035 (1986).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Sect. 3.915(4).

Sassen, K.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 7.1.6.

Vincent, D.

D. Vincent, G. Otis, “Télémètre à détection cohérente avec un laser CO2 à ondes entretenues: étude et conception,” Can. J. Phys. 61, 318 (1983).
[CrossRef]

D. Vincent, G. Otis, “Achromatic Opto-Mechanical Modulator,” DREV Report 4361/85, Appendix B; D. Vincent, “Amplitude Modulation with a Mechanical Chopper,” Appl. Opt. 25, 1035 (1986).
[CrossRef] [PubMed]

Appl. Opt. (1)

Can. J. Phys. (1)

D. Vincent, G. Otis, “Télémètre à détection cohérente avec un laser CO2 à ondes entretenues: étude et conception,” Can. J. Phys. 61, 318 (1983).
[CrossRef]

Other (9)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 7.1.6.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Sect. 3.915(4).

Ref. 2, Sec. 26.3.24.

Ref. 2, Secs. 26.3.26 and 26.3.27. (Note that x2 in 26.3.27 should read x2=R2-1-r.)

Data sheet on off-axis parabolas available from Space Optics Research Laboratories.

Data sheet on scanner L45 available from American Time Products Division of Frequency Control Products, Inc., Woodside, NY.

Data sheet available from Scitec Instruments, Arcade Chambers, Victoria Road, Aldershot, Hampshire, U.K.

Data sheet available from Bentham Instruments Ltd., 2 Boulton Rd., Reading, Berkshire, England RG2 ONH.

D. Vincent, G. Otis, “Achromatic Opto-Mechanical Modulator,” DREV Report 4361/85, Appendix B; D. Vincent, “Amplitude Modulation with a Mechanical Chopper,” Appl. Opt. 25, 1035 (1986).
[CrossRef] [PubMed]

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Figures (22)

Fig. 1
Fig. 1

Optical layout of the modulator.

Fig. 2
Fig. 2

Mechanical layout: (a) top view; (b) side view.

Fig. 3
Fig. 3

Geometries used for computation: x, horizontal axis; y, vertical axis. (a) Uniform beam with iris, (b) uniform beam with slit, (c) Gaussian beam with iris, (d) Gaussian beam with slit.

Fig. 4
Fig. 4

Loci (ρ,δ) for γ = ½ at t = 0 uniform beam with iris.

Fig. 5
Fig. 5

Loci (ρ,δ) for γ = ½ at t = 0, Gaussian beam with iris.

Fig. 6
Fig. 6

Best uniform case with iris; (δ,ρ,α) = (1.3,0,1.9) in 2f mode, (1.01,0.81,0.8) in 1f mode. The left trace is in the time domain with linear vertical scale, and the right trace is the symmetrical power spectrum with the vertical scale in decibels.

Fig. 7
Fig. 7

Best uniform case with slit; (δ,ρ,α) = (1.3,0,1.9) in 2f mode, (1,1,1) in 1f mode. Scales as in Fig. 6.

Fig. 8
Fig. 8

Best Gaussian case with iris; (δ,ρ,α) = (2.3,0,3) in 2f mode, (2.5,2.4,1.2) in 1f mode. Scales as in Fig. 6.

Fig. 9
Fig. 9

Best Gaussian case with slit; (δ,ρ,α) = (2.2,0,3) in 2f mode, (2.5,2.5,1.3) in 1f mode. Scales as in Fig. 6.

Fig. 10
Fig. 10

Photographs of the modulator: (a) top view; (b) exterior view.

Fig. 11
Fig. 11

Power spectrum of the monitor signal from TBM (synchronization output).

Fig. 12
Fig. 12

Uniform beam setup: LA, laser (He–Ne or CO2); T, expanding telescope (28× at 0.633 μm, 5× at 10.6 μm); M, modulator; L1, focusing lens (f = 10 cm at 0.633 μm, f = 12.5 cm at 10.6 μm); D, detector (Si photodiode at 0.633 μm, room-temperature photoelectromagnetic HgCdTe at 10.6 μm).

Fig. 13
Fig. 13

Experimental dependence of deflection angle on monitor signal.

Fig. 14
Fig. 14

Best experimental result for a uniform beam with iris (He–Ne laser).

Fig. 15
Fig. 15

Experimental relation between peak V m and d e for the 1f-mode uniform beam with iris.

Fig. 16
Fig. 16

Experimental relation between ρ and d e for the 1f-mode uniform beam with iris.

Fig. 17
Fig. 17

Experimental relation between peak V m and d e for the 2f-mode uniform beam with iris when δ ≥ 1 and ρ = 0.

Fig. 18
Fig. 18

Calculated 2f-mode uniform beam with iris for (δ,ρ,α) = (1.05,0,1.7): the upper trace is in the time domain with linear vertical scale, and the lower trace is the symmetrical power spectrum with the vertical scale in decibels.

Fig. 19
Fig. 19

Gaussian beam setup: (a) with He–Ne laser at 0.633 μm where L2 has a focal length of −25 cm; (b) with CO2 laser where the telescope T has a 5× magnification.

Fig. 20
Fig. 20

Gaussian beam with iris (He–Ne laser).

Fig. 21
Fig. 21

Gaussian beam with iris (CO2 laser).

Fig. 22
Fig. 22

AM signal at 20 kHz with random carrier around 1.6 MHz in a cw range finder.

Tables (3)

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Table I Definitions of Symbols

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Table II Calculated Results

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Table III Experimental Results

Equations (25)

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X = 2 F θ ,
θ L ½ F - 1 D TBM ;
R L ~ F θ max ,
X = X 0 + X m sin ω m t
u + ρ + α sin ω m t ,
γ ( δ , u ) = [ R L 2 ( ϕ L - sin ϕ L ) + R I 2 ( ϕ I - sin ϕ I ) ] / 2 π R L 2 = π - 1 Re ( cos - 1 Z 1 + δ 2 cos - 1 Z 2 - Z 1 1 - Z 1 2 - δ 2 Z 2 1 - Z 2 2 ) ,
Z 1 = ( 1 - δ 2 + u 2 ) / 2 u Z 2 = ( δ 2 - 1 + u 2 ) / 2 δ u , cos - 1 Z = { π + j cosh - 1 ( - Z ) if Z - 1 , cos - 1 Z if - 1 Z 1 , j cosh - 1 ( Z ) if Z 1 ,
γ ( δ , u ) = { ϕ - θ + [ ( S + X ) sin θ + ( S - X ) sin ϕ ] / R L } / π = π - 1 Re { cos - 1 ( u - δ ) - cos - 1 ( u + δ ) + ( u + δ ) [ 1 - ( u + δ ) 2 ] 1 / 2 - ( u - δ ) [ 1 - ( u - δ ) 2 ] 1 / 2 }
γ ( δ , u ) = π - 1 / 2 - δ δ erf ( δ 2 - v 2 ) 1 / 2 exp [ - ( v - u ) 2 ] d v ,
γ ( δ , u ) = exp ( - u 2 - δ 2 ) n = 0 δ n + 1 I n + 1 ( 2 δ u ) / u n + 1 ,
γ ( δ , 0 ) = 1 - exp ( - δ 2 ) ,
γ ( δ , u ) = ½ [ erf ( δ - u ) + erf ( δ + u ) ]
P opt ( dB ) = ½ [ 3 + a 1 , 20 ( dB ) ] ,
D opt ( dB ) = 10 log 10 [ n 0 n 1 , 2 ( a n / a 1 , 2 ) ½ ] ,
γ ( δ , u ) = 1 2 π - x exp ( - r 2 / 2 ) d r
= ½ ( 1 + erfx ) ,
x = ( δ 2 1 + u 2 ) 1 / 3 - [ 1 - ( 1 + 2 u 2 ) 9 ( 1 + u 2 ) 2 ] [ 1 + 2 u 2 9 ( 1 + u 2 ) 2 ] 1 / 2 if δ 1 / 2 ,
x = ( 2 δ 2 - 1 ) 1 / 2 - 2 u             if δ 5 / 2 .
δ = ( 1 + ρ 2 ) 1 / 2 [ 1 - ( 1 + 2 ρ 2 ) 9 ( 1 + ρ 2 ) 2 ] 3 / 2             if δ 1 / 2 ,
δ = ( ρ 2 + ½ ) 1 / 2             if δ 5 / 2 .
θ m = 7.0 V m ( mrad V peak - 1 ) .
X m = F θ m = 1.05 V m ( mm V peak - 1 ) .
α ( 1 f ) = 2 × 1.05 V m d e = 0.76.
α ( 2 f ) = 1.7
P opt ( dB ) = ½ a 1 , 2 ( dB ) + 10 log ( β s p p - 1 ) ,

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