Abstract

The measurement accuracy of laser telemeters have been analyzed, taking into account the statistical properties of the propagation, detection, and measure process. For an arbitrary waveform of the light intensity modulation envelope, the optimum filter response and the resulting accuracy are found. As special cases, the optical radar and the sine wave modulation techniques have been considered, and their accuracy and optimum mode of operation are evaluated and discussed.

© 1971 Optical Society of America

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References

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  1. A. Sona, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, in press), Chap. 6.
  2. P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
    [CrossRef]
  3. K. D. Froome, R. H. Bradsell, J. Sci. Instrum. 38, 458 (1961).
    [CrossRef]
  4. W. J. Rundle, J. Quantum Electron. QE-5, 342 (1969).
    [CrossRef]
  5. C. O. Alley et al., Science 167, 368 (1970).
    [CrossRef] [PubMed]
  6. AGA Geodimeter model 8 and Spectra-Physics Geodolite model 3G.
  7. A. A. Vuylsteke, J. Appl. Phys. 34, 1615 (1963).
    [CrossRef]
  8. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).
  9. V. S. Pugachev, Theory of Random Functions (Pergamon, Oxford, 1965).
  10. S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
    [CrossRef]
  11. E. Gatti, V. Svelto, Nucl. Instrum. Methods 39, 309 (1966).
    [CrossRef]
  12. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).
  13. Being h(t) = hf(t)*hd(t), a constraint is implied on the detector impulse response hd(t) such that the effective filter response hf(t) can be realized. Assuming an adequate speed of response for the detector, the distinction between h(t) and hf(t) can be overlooked.
  14. D. A. Gedcke, W. J. McDonald, Nucl. Instrum. Methods 58, 253 (1968).
    [CrossRef]
  15. The effect has been noted also in semiconductor lasers.

1970

C. O. Alley et al., Science 167, 368 (1970).
[CrossRef] [PubMed]

1969

S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
[CrossRef]

W. J. Rundle, J. Quantum Electron. QE-5, 342 (1969).
[CrossRef]

1968

D. A. Gedcke, W. J. McDonald, Nucl. Instrum. Methods 58, 253 (1968).
[CrossRef]

1966

E. Gatti, V. Svelto, Nucl. Instrum. Methods 39, 309 (1966).
[CrossRef]

1965

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

1963

A. A. Vuylsteke, J. Appl. Phys. 34, 1615 (1963).
[CrossRef]

1961

K. D. Froome, R. H. Bradsell, J. Sci. Instrum. 38, 458 (1961).
[CrossRef]

Alley, C. O.

C. O. Alley et al., Science 167, 368 (1970).
[CrossRef] [PubMed]

Bender, P. L.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

Bradsell, R. H.

K. D. Froome, R. H. Bradsell, J. Sci. Instrum. 38, 458 (1961).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Donati, S.

S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
[CrossRef]

Froome, K. D.

K. D. Froome, R. H. Bradsell, J. Sci. Instrum. 38, 458 (1961).
[CrossRef]

Gatti, E.

S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
[CrossRef]

E. Gatti, V. Svelto, Nucl. Instrum. Methods 39, 309 (1966).
[CrossRef]

Gedcke, D. A.

D. A. Gedcke, W. J. McDonald, Nucl. Instrum. Methods 58, 253 (1968).
[CrossRef]

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

McDonald, W. J.

D. A. Gedcke, W. J. McDonald, Nucl. Instrum. Methods 58, 253 (1968).
[CrossRef]

Owens, J. C.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Pugachev, V. S.

V. S. Pugachev, Theory of Random Functions (Pergamon, Oxford, 1965).

Rundle, W. J.

W. J. Rundle, J. Quantum Electron. QE-5, 342 (1969).
[CrossRef]

Sona, A.

A. Sona, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, in press), Chap. 6.

Svelto, V.

S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
[CrossRef]

E. Gatti, V. Svelto, Nucl. Instrum. Methods 39, 309 (1966).
[CrossRef]

Vuylsteke, A. A.

A. A. Vuylsteke, J. Appl. Phys. 34, 1615 (1963).
[CrossRef]

Adv. Electron. Electron Phys.

S. Donati, E. Gatti, V. Svelto, Adv. Electron. Electron Phys. 26, 251 (1969).
[CrossRef]

J. Appl. Phys.

A. A. Vuylsteke, J. Appl. Phys. 34, 1615 (1963).
[CrossRef]

J. Geophys. Res.

P. L. Bender, J. C. Owens, J. Geophys. Res. 70, 2461 (1965).
[CrossRef]

J. Quantum Electron.

W. J. Rundle, J. Quantum Electron. QE-5, 342 (1969).
[CrossRef]

J. Sci. Instrum.

K. D. Froome, R. H. Bradsell, J. Sci. Instrum. 38, 458 (1961).
[CrossRef]

Nucl. Instrum. Methods

E. Gatti, V. Svelto, Nucl. Instrum. Methods 39, 309 (1966).
[CrossRef]

D. A. Gedcke, W. J. McDonald, Nucl. Instrum. Methods 58, 253 (1968).
[CrossRef]

Science

C. O. Alley et al., Science 167, 368 (1970).
[CrossRef] [PubMed]

Other

AGA Geodimeter model 8 and Spectra-Physics Geodolite model 3G.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953).

Being h(t) = hf(t)*hd(t), a constraint is implied on the detector impulse response hd(t) such that the effective filter response hf(t) can be realized. Assuming an adequate speed of response for the detector, the distinction between h(t) and hf(t) can be overlooked.

The effect has been noted also in semiconductor lasers.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

V. S. Pugachev, Theory of Random Functions (Pergamon, Oxford, 1965).

A. Sona, in Laser Handbook, F. T. Arecchi, E. O. Schulz-DuBois, Eds. (North-Holland, Amsterdam, in press), Chap. 6.

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Figures (5)

Fig. 1
Fig. 1

The multiplicative factor D(μ) of the time variance in the PM telemeter against the dark noise parameter μ. Full line refers to the optimum result, i.e., the filter impulse response is varied with ρ according to Eq. (12). Dotted line represents the result obtained with the filter which is optimum at μ = 0.2.

Fig. 2
Fig. 2

The waveforms of the optimum filter impulse response h(t) for different values of the dark noise parameter μ (amplitudes not to the same scale) in a PM telemeter.

Fig. 3
Fig. 3

The effect of simple approximations of the optimum filter impulse response. The multiplicative factor D(μ) of Eq. (16) is plotted for a short circuited delay line shaping (SCDL) with a shaping time Ts (dotted line) and for a constant fraction pulse height (CFPH) at the 10% amplitude level (full line) together with the optimum result.

Fig. 4
Fig. 4

The multiplicative factor E(χ) of the time variance in the SWM telemeter [Eq. (24)] plotted against the dark noise parameter χ [Eq. (22)]. Dotted line refers to the result obtained with the correlation method of measure.

Fig. 5
Fig. 5

The optimum filter response h(t) for some values of χ. Since h(t) is periodic, the diagram has been limited to one period. Amplitudes are not to the same scale.

Equations (33)

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I ( t ) = A s ( t ) β exp [ - 2 α ( t ) L ] ,
I ¯ ( t ) = A s ( t ) β exp [ - 2 α ( t ) L ] ¯ .
f ( t ) = exp [ - 2 α ( t ) L ] / exp [ - 2 α ( t ) L ] ¯ - 1 ,
I ( t ) = I ¯ ( t ) [ 1 + f ( t ) ] .
S ¯ r ( t ) = η I ¯ ( t ) .
Δ S r ( t ) = S ¯ r ( t ) f ( t ) + R ( t ) + Q ( t ) .
K r ( t , t ) = Δ S r ( t ) Δ S r ( t ) ¯ .
K r ( t , t ) = f ( t ) f ( t ) ¯ S ¯ r ( t ) S ¯ r ( t ) + R ( t ) R ( t ) ¯ + Q ( t ) Q ( t ) ¯ .
K r ( t , t ) = K f ( t , t ) [ S ¯ r ( t ) S ¯ r ( t ) + S ¯ r ( t ) δ ( t - t ) ] + [ S ¯ r ( t ) + ρ ] δ ( t - t ) .
S ¯ h ( t ) = - S ¯ r ( t - τ ) h ( τ ) d τ ,
K h ( t , t ) = - - K r ( t - τ , t - τ ) h ( τ ) h ( τ ) d τ d τ .
τ 2 ( t ) = K h ( t , t ) / [ d S ¯ h ( t ) / d t ] 2 .
- K r ( t , t ) h ( T m - t ) d t = d d t S ¯ r ( t ) ,
h ( T m - t ) = [ d S ¯ r ( t ) / d t ] / [ ( 1 + f 2 ) S ¯ r ( t ) + ρ ] .
S ¯ h ( T m ) = 0 ,             [ d S ¯ h ( t ) / d t ] t = T m > 0 ,
h 2 ( T m ) = 1 / - [ d S ¯ r ( t ) / d t ] 2 ( 1 + f 2 ) S ¯ r ( t ) + ρ d t .
S ¯ r ( t ) = N ( 2 π σ ) 1 2 exp [ ( t - τ 0 ) 2 / 2 σ 2 ] .
h 2 = ( 1 + f 2 ) σ 2 N D ( μ ) ,
D ( μ ) = 1 / - + [ ξ 2 / ( 2 π ) 1 2 ] exp ( - ξ 2 / 2 ) 1 + μ ( 2 π ) 1 2 exp ( ξ 2 / 2 ) d ξ
μ = ρ σ / N ( 1 + f 2 ) .
D ( μ ) = D ( μ 0 ) + ( μ - μ 0 ) D ( μ 0 ) ,
h 2 = [ ( 1 + f 2 ) σ 2 / N ] D 0 + ( ρ σ 3 / N 2 ) D ( μ 0 ) ,
s ( t ) = 1 + m sin Ω t ,
S ¯ r ( t ) = ( N / T ) [ 1 + m sin Ω ( t - τ 0 ) ] ,
h ( t ) cos Ω ( T m - t ) χ + sin Ω ( T m - t ) ,
χ = ( 1 / m ) { 1 + [ ρ T / ( 1 + f 2 ) N ] } .
h 2 = [ ( 1 + f 2 ) / m Ω 2 N ] E ( χ ) ,
E ( χ ) = 2 π / 0 2 π cos 2 ξ χ + sin ξ d ξ .
0 T m S r ( t ) cos Ω t d t ,
h 2 = 1 + f 2 Ω 2 ( m 2 / 2 ) N + ρ T Ω 2 ( m 2 / 2 ) N 2 .
r 2 ( T m ) = { [ K r ( t , t ) * * h ( t ) h ( t ) ] / [ S ¯ ˙ r ( t ) * h ( t ) ] 2 } t = t = T m .
[ K r ( t , t ) * * η ( t ) h 0 ( t ) K r ( t , t ) * * h 0 ( t ) h 0 ( t ) ] t = t = T m = [ S ¯ ˙ r ( t ) * η ( t ) S r ( t ) * h 0 ( t ) ] t = T m .
[ K r ( t , t ) * h 0 ( t ) ] t = T m = S ¯ ˙ r ( t ) ,

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