Abstract

A general method which allows precise measurements of the frequency spectrum of a phase modulated wave down to a few cycles per second is presented. The momenta of the distribution of the instantaneous frequency (time derivative of the phase) can also be easily obtained. An experiment is described in which the study of the phase correlation induced by a laboratory generated thermal turbulence in a laser beam is described.

© 1970 Optical Society of America

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References

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  1. M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.
  2. R. B. Errick, J. R. Meyer-Arendt, Appl. Opt. 5, 981 (1966).
    [CrossRef]
  3. A. L. Buck, Appl. Opt. 6, 703 (1967).
    [CrossRef] [PubMed]
  4. P. Burlamacchi, A. Consortini, L. Ronchi, Appl. Opt. 6, 1273 (1967).
    [CrossRef] [PubMed]
  5. M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 7, 1121 (1968).
    [CrossRef] [PubMed]
  6. M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.
  7. M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.
  8. M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
    [CrossRef] [PubMed]
  9. B. Daino, Quantum Electronics Conference, Miami, 1968.
  10. B. Daino, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.
  11. M. Subramanian, J. A. Collinson, Bell Syst. Tech. J. 44, 543 (1965).
  12. L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
    [CrossRef]
  13. J. E. Rowe, Signal and Noise in Communication System (D. Van Nostrand Company, Inc., Princeton, N. J., 1965), Chap. 4.
  14. M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 8, 1111 (1969).
    [CrossRef] [PubMed]

1969 (1)

1968 (2)

1967 (2)

1966 (2)

R. B. Errick, J. R. Meyer-Arendt, Appl. Opt. 5, 981 (1966).
[CrossRef]

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

1965 (2)

M. Subramanian, J. A. Collinson, Bell Syst. Tech. J. 44, 543 (1965).

L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
[CrossRef]

Bertolotti, M.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 8, 1111 (1969).
[CrossRef] [PubMed]

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

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Buck, A. L.

Burlamacchi, P.

Carnevale, M.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 8, 1111 (1969).
[CrossRef] [PubMed]

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

M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 7, 1121 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Chabot, A.

L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
[CrossRef]

Collinson, J. A.

M. Subramanian, J. A. Collinson, Bell Syst. Tech. J. 44, 543 (1965).

Consortini, A.

Crosignani, B.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 8, 1111 (1969).
[CrossRef] [PubMed]

M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 7, 1121 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Daino, B.

B. Daino, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

B. Daino, Quantum Electronics Conference, Miami, 1968.

Di Porto, P.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 8, 1111 (1969).
[CrossRef] [PubMed]

M. Carnevale, B. Crosignani, P. Di Porto, Appl. Opt. 7, 1121 (1968).
[CrossRef] [PubMed]

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Errick, R. B.

Goldestein, L.

L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
[CrossRef]

Meyer-Arendt, J. R.

Miles, P. A.

L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
[CrossRef]

Muzii, L.

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

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Ronchi, L.

Rowe, J. E.

J. E. Rowe, Signal and Noise in Communication System (D. Van Nostrand Company, Inc., Princeton, N. J., 1965), Chap. 4.

Sette, D.

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

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

Subramanian, M.

M. Subramanian, J. A. Collinson, Bell Syst. Tech. J. 44, 543 (1965).

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

M. Subramanian, J. A. Collinson, Bell Syst. Tech. J. 44, 543 (1965).

Italian Phys. Soc. Symp. (1)

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, L. Muzii, Italian Phys. Soc. Symp., 1966.

J. IEEE (1)

L. Goldestein, P. A. Miles, A. Chabot, J. IEEE 53, 1172 (1965).
[CrossRef]

Other (5)

J. E. Rowe, Signal and Noise in Communication System (D. Van Nostrand Company, Inc., Princeton, N. J., 1965), Chap. 4.

M. Bertolotti, M. Carnevale, B. Crosignani, P. Di Porto, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

B. Daino, Quantum Electronics Conference, Miami, 1968.

B. Daino, URSI Symp. Electromagnetic Waves, Stresa, 24–29 June 1968.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Fringe pattern recorded by the film.

Fig. 4
Fig. 4

Fourier transform of the fringe pattern.

Fig. 5
Fig. 5

Light distribution on each line at a given temperature gradient.

Fig. 6
Fig. 6

Light distribution on each line in different experimental conditions for studying the saturation zone.

Fig. 7
Fig. 7

Fourier transform of Pν(f).

Fig. 8
Fig. 8

Behavior of 〈Δφ(O)Δφ(τ)〉 as a function of τ.

Equations (21)

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V i ( t , x ) = A i ( t ) exp j [ ω t + m i x + φ i ( t ) ]
m i = ω sin θ i / c ω θ i / c .
I ( t , x ) = [ i = 1 N V i ( t , x ) ] [ i = 1 N V i * ( t , x ) ] = l = 1 N K = 1 N A k A l exp j [ ( m k m l ) x + ( φ k φ l ) ] .
I ( t , x ) = N A 2 + A 2 k = 1 N 1 l = 1 N k { exp j [ ( m l m l + k ) x + ( φ l φ l + k ) ] + c . c . } ,
F ( p , q ) = C + + I ( x , t ) exp j ( p x + q t ) d x d t ,
F ( p , q ) = C A 2 k = 1 N 1 l = 1 N K δ [ p ( m l m l + k ) ] + exp j [ ( φ l φ l + k ) + q t ] d t + C A 2 k = 1 N 1 l = 1 N K δ [ p + ( m l m l + k ) ] + exp { j [ ( φ l φ l + k ) + q t ] } d t .
P i = Δ m i j , i j ,
a k , l ( p = m l m l + k , q ) = C a 2 × + exp j [ ( φ l φ l + k ) + q t ] d t ,
a k , l [ p = ( m l m l + k ) , q ] = a k , l * ( p = m l m l + k , q ) .
l , k ( q ) = C 2 | a k , l ( p = m l m l + k , q ) | 2 = C 2 A 4 | + exp j { [ φ l ( t ) φ l + k ( t ) ] + q t } d t | 2 ·
ν ( t ) = exp j [ φ l ( t ) φ l + k ( t ) ]
f = υ / q ,
φ ˙ = ( 1 / 2 π ) ( d / d t ) φ ( t ) .
P ν ( f ) / + P ν ( f ) d f = p ( φ ˙ ) ,
q n ( q ) d q
q n ( q ) d q = 0 φ ˙ n p ( φ ˙ ) d φ ˙ = 0 φ ˙ n ,
0 = ( q ) d q .
ν ( t + τ ) ν * ( t ) = exp { [ Δ φ 2 ( t ) Δ φ ( t + τ ) Δ φ ( t ) ] } .
ν ( t + τ ) ν * ( t ) = + P ν ( f ) exp ( 2 π j f τ ) d f ,
ν ( t + τ ) ν * ( t ) = exp [ 2 + P φ ( f ) sin 2 ( π f τ ) d f ] .
I ( t , x ) = h = 1 N A h 2 ( t ) + k = 1 N 1 l = 1 N K { A k ( t ) A l + k ( t ) exp j [ ( m l m l + k ) x + ( φ l φ l + k ) ] + c . c . } .

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