Abstract

A method to determine the amplitude and phase fluctuations in each point of a wavefront is presented. The basic idea of the method is the possibility of investigating the amplitude and phase at each point of a wavefront by superposing a spherical reference wavefront that has its origin in the point under investigation. The information on the fluctuations is extracted in the form of an electrical signal. Application to the study of phase fluctuations produced by atmospheric turbulence is considered.

© 1971 Optical Society of America

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References

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  1. M. Bertolotti, F. Gori, G. Guattari, B. Daino, Appl. Opt. 7, 1961 (1968).
    [CrossRef] [PubMed]
  2. M. Bertolotti, F. Gori, G. Guattari, B. Daino, Symposium on Application of Coherent Light, Florence (September 1968).
  3. L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).
  4. See for example, H. E. Rowe, Signal and Noise in Communication Systems (Van Nostrand, Princeton, N. J., 1965).
  5. A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
    [CrossRef]
  6. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1960).
  7. M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 143.
  8. M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
    [CrossRef] [PubMed]
  9. M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 9, 510 (1970).
    [CrossRef]
  10. P. Burlamacchi, A. Consortini, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 149.
  11. M. Bertolotti, M. Carnevale, B. Daino, D. Sette, Appl. Opt. 9, 962 (1970).
    [CrossRef] [PubMed]

1970 (2)

1969 (1)

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

1968 (3)

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Appl. Opt. 7, 1961 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
[CrossRef] [PubMed]

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).

Bertolotti, M.

M. Bertolotti, M. Carnevale, B. Daino, D. Sette, Appl. Opt. 9, 962 (1970).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 9, 510 (1970).
[CrossRef]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
[CrossRef] [PubMed]

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Appl. Opt. 7, 1961 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 143.

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Symposium on Application of Coherent Light, Florence (September 1968).

Burlamacchi, P.

P. Burlamacchi, A. Consortini, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 149.

Carnevale, M.

Consortini, A.

P. Burlamacchi, A. Consortini, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 149.

Daino, B.

Enloe, L. H.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).

Gori, F.

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Appl. Opt. 7, 1961 (1968).
[CrossRef] [PubMed]

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Symposium on Application of Coherent Light, Florence (September 1968).

Guattari, G.

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Appl. Opt. 7, 1961 (1968).
[CrossRef] [PubMed]

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Symposium on Application of Coherent Light, Florence (September 1968).

Jakes, W. C.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).

Macovski, A.

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

Muzii, L.

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 9, 510 (1970).
[CrossRef]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 143.

Rowe, H. E.

See for example, H. E. Rowe, Signal and Noise in Communication Systems (Van Nostrand, Princeton, N. J., 1965).

Rubinstein, C. B.

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).

Sette, D.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1960).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

A. Macovski, Appl. Phys. Lett. 14, 166 (1969).
[CrossRef]

Bell System Tech. J. (1)

L. H. Enloe, W. C. Jakes, C. B. Rubinstein, Bell System Tech. J. 47, 1875 (1968); A. B. Larsen, Bell System Tech. J. 48, 2507 (1969).

Other (5)

See for example, H. E. Rowe, Signal and Noise in Communication Systems (Van Nostrand, Princeton, N. J., 1965).

M. Bertolotti, F. Gori, G. Guattari, B. Daino, Symposium on Application of Coherent Light, Florence (September 1968).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1960).

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 143.

P. Burlamacchi, A. Consortini, URSI Symposium on Electromagnetic Waves, Stresa, 24–29 June 1968, published in Alta Frequenza38, (May1969), p. 149.

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Figures (2)

Fig. 1
Fig. 1

Schematic arrangement for the measurement of phase and amplitude fluctuations of a wavefront. The wavefront under examination comes from points on the x axis and is made to interfere on the η axis with the field from a light source coming from a point on the α axis by mean of a beam-splitter B.

Fig. 2
Fig. 2

Schematic arrangement for the measurement of phase and amplitude fluctuations induced by turbulence on a laser beam.

Equations (28)

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V 1 ( x , t ) = A ( x , t ) e i [ ω t + ϕ ( x , t ) ]
V 1 η ( η , t ) = c 1 - + A ( x , t ) e i [ ω t + ϕ ( x , t ) ] e i k ( x - n ) 2 / 2 R d x ,
V R = A R e i [ ω t + ϕ R ] e i k ( α - η ) 2 / 2 R
I ( α , t ) = - + V 1 η + V R 2 d η = cos t + I ( t ) + 2 c 3 R e [ V 1 η ( α ) V R * ] = cos t + I ( t ) + 2 c 3 A R A ( α , t ) cos [ ϕ ( α , t ) - ϕ R ]
I ( t ) = c 2 - + A 2 ( x , t ) d x ,
I ( t ) + 2 c 3 A R A ( α , t ) cos [ ϕ ( α , t ) - ϕ R ] .
I 1 = A ( α , t ) cos [ ϕ ( α , t ) - ϕ 0 ] I 2 = A ( α , t ) sin [ ϕ ( α , t ) - ϕ 0 ] .
I 1 2 + I 2 2 = A 2 ( α , t )
I 2 I 1 = tan [ ϕ ( α , t ) - ϕ 0 ] .
Δ ϕ ( α , t ) = ϕ ( α , t ) - ϕ 0 = tan - 1 I 2 I 1
I ( α ) = cos t + 2 c 3 A R - + A ( α ) cos Δ ϕ ( α ) p ( A , Δ ϕ ) d A d ( Δ ϕ ) .
p ( A , Δ ϕ ) d A d ( Δ ϕ ) = p ( A ) d A p ( Δ ϕ ) d ( Δ ϕ ) ,
I ( α ) = cos t + 2 c 3 A R A ( α ) - + cos Δ ϕ p ( Δ ϕ ) d ( Δ ϕ ) .
p ( Δ ϕ ) d ( Δ ϕ ) = 1 σ ( 2 π ) ½ e - ( Δ ϕ ) 2 / 2 σ 2 d ( Δ ϕ )
I ( α ) = cos t + 2 c 3 A R A ( α ) e - σ 2 / 2 .
A ( α ) = A 0 + Δ A ( α ) .
I ( α ) = cos t + 2 c 3 A R A 0 e - σ 2 / 2 .
log { I ( α ) - I ( ) 2 A R A 0 c 3 } = - σ 2 ( α ) 2 .
V R = V 1 ( 0 , t ) e i k ( α - η ) 2 / 2 R .
I ( α ) = cos t + 2 c 3 R e V 1 ( α ) V 1 * ( 0 ) = cos t + 2 c 3 R e Γ ( α )
I ( α , t ) = cos t + I ( t ) + 2 c 3 A R A ( α , t ) cos [ Δ ϕ ( α ) + ( ω - ω R ) t ] .
I Ω ( α , t ) = 2 A R A ( α , t ) cos [ ϕ ( α , t ) - ϕ R ] .
I ( α ) = cos t + 2 c 3 - + A R ( 0 ) A ( α ) cos Δ ϕ ( α ) × p [ A ( 0 ) , A ( α ) , Δ ϕ ( α ) ] d [ A ( 0 ) ] d [ A ( α ) ] d [ Δ ϕ ( α ) ] ,
p [ A ( 0 ) , A ( α ) , Δ ϕ ( α ) ] d [ A ( 0 ) ] d [ A ( α ) ] d [ Δ ϕ ( α ) ]
I ( α ) = cos t + 2 c exp { - [ D ϕ + D log A + 2 D ϕ log A ] / 2 }
D ϕ = [ ϕ R - ϕ ( α ) ] 2 D log A = [ ln A ( α ) A ( 0 ) ] 2 D ϕ log A = [ ( ϕ R - ϕ ( α ) ) log A ( α ) ]
D ϕ ( α ) = σ 2 ( α x ) .
I ( α , t ) I ( α , t + τ ) = exp { - [ Δ ϕ 2 ( α , t ) - Δ ϕ ( α , t + τ ) Δ ϕ ( α , t ) ] }

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