Abstract

We describe some optical modulation/demodulation schemes that consist of shifting the optical frequency spectrum of mode-locked-laser pulses and interferometrically combining the shifted pulses. Several forms of frequency shift produced by electrooptic phase modulators driven at frequencies commensurable with the pulse repetition rate are considered. Analyses are made in terms of superposition of phase-modulation sidebands, and expressions are also obtained for the pulse intensity waveform in the time domain as the product of an envelope function times the original pulse waveform. Computed examples are presented.

© 1970 Optical Society of America

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References

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  1. S. E. Harris [Appl. Opt. 5, 1639 (1966); Proc. IEEE 54, 1401 (1966)] reviewed the application of internal time-varying perturbation to the problem of laser mode control and stabilization. See papers referred to therein and L. E. Hargrove, U. S. Patent3,412,251, issued 19November1968.
    [CrossRef] [PubMed]
  2. M. A. Duguay, L. E. Hargrove, K. B. Jefferts, Appl. Phys. Lett. 9, 287 (1966).
    [CrossRef]
  3. C. G. B. Garrett, M. A. Duguay, Appl. Phys. Lett. 9, 374 (1966).
    [CrossRef]
  4. D. J. Blattner, U.S. Patent3,353,896, issued 21November1967.
  5. M. Gottlieb, M. Garbuny, Appl. Opt. 7, 2238 (1968).
    [CrossRef] [PubMed]
  6. M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 477 (1968).
    [CrossRef]
  7. This technique is disclosed in J. S. Courtney-Pratt, L. E. Hargrove, U.S. Patent3,435,230, issued 25March1969.
  8. L. E. Hargrove, J. Opt. Soc. Amer. 59,1680 (L) (1969).
    [CrossRef]
  9. L. E. Hargrove, Robert L. Rosenberg, Appl. Phys. Lett., 16, 74 (1969).
    [CrossRef]
  10. J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
    [CrossRef]

1969 (2)

L. E. Hargrove, J. Opt. Soc. Amer. 59,1680 (L) (1969).
[CrossRef]

L. E. Hargrove, Robert L. Rosenberg, Appl. Phys. Lett., 16, 74 (1969).
[CrossRef]

1968 (3)

J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
[CrossRef]

M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 477 (1968).
[CrossRef]

M. Gottlieb, M. Garbuny, Appl. Opt. 7, 2238 (1968).
[CrossRef] [PubMed]

1966 (3)

Blattner, D. J.

D. J. Blattner, U.S. Patent3,353,896, issued 21November1967.

Courtney-Pratt, J. S.

This technique is disclosed in J. S. Courtney-Pratt, L. E. Hargrove, U.S. Patent3,435,230, issued 25March1969.

Duguay, M. A.

M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 477 (1968).
[CrossRef]

J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
[CrossRef]

M. A. Duguay, L. E. Hargrove, K. B. Jefferts, Appl. Phys. Lett. 9, 287 (1966).
[CrossRef]

C. G. B. Garrett, M. A. Duguay, Appl. Phys. Lett. 9, 374 (1966).
[CrossRef]

Garbuny, M.

Garrett, C. G. B.

C. G. B. Garrett, M. A. Duguay, Appl. Phys. Lett. 9, 374 (1966).
[CrossRef]

Giordmaine, J. A.

J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
[CrossRef]

Gottlieb, M.

Hansen, J. W.

J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
[CrossRef]

M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 477 (1968).
[CrossRef]

Hargrove, L. E.

L. E. Hargrove, J. Opt. Soc. Amer. 59,1680 (L) (1969).
[CrossRef]

L. E. Hargrove, Robert L. Rosenberg, Appl. Phys. Lett., 16, 74 (1969).
[CrossRef]

M. A. Duguay, L. E. Hargrove, K. B. Jefferts, Appl. Phys. Lett. 9, 287 (1966).
[CrossRef]

This technique is disclosed in J. S. Courtney-Pratt, L. E. Hargrove, U.S. Patent3,435,230, issued 25March1969.

Harris, S. E.

Jefferts, K. B.

M. A. Duguay, L. E. Hargrove, K. B. Jefferts, Appl. Phys. Lett. 9, 287 (1966).
[CrossRef]

Rosenberg, Robert L.

L. E. Hargrove, Robert L. Rosenberg, Appl. Phys. Lett., 16, 74 (1969).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

L. E. Hargrove, Robert L. Rosenberg, Appl. Phys. Lett., 16, 74 (1969).
[CrossRef]

M. A. Duguay, L. E. Hargrove, K. B. Jefferts, Appl. Phys. Lett. 9, 287 (1966).
[CrossRef]

C. G. B. Garrett, M. A. Duguay, Appl. Phys. Lett. 9, 374 (1966).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 477 (1968).
[CrossRef]

J. A. Giordmaine, M. A. Duguay, J. W. Hansen, IEEE J. Quantum Electron. QE4, 252 (1968).
[CrossRef]

J. Opt. Soc. Amer. (1)

L. E. Hargrove, J. Opt. Soc. Amer. 59,1680 (L) (1969).
[CrossRef]

Other (2)

This technique is disclosed in J. S. Courtney-Pratt, L. E. Hargrove, U.S. Patent3,435,230, issued 25March1969.

D. J. Blattner, U.S. Patent3,353,896, issued 21November1967.

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Figures (15)

Fig. 1
Fig. 1

Intensities of optical modes for gaussian distribution of mode amplitudes, a = 5.

Fig. 2
Fig. 2

Optical pulse intensity waveform that corresponds to the spectrum shown in Fig. 1 (a = 5).

Fig. 3
Fig. 3

Diagram representing a sinusoidal applied electric field with frequency ω phased such that the optical pulses go through the electrooptic medium during negative-going zero points of the applied field (solid line) or with frequency ω/2 and during successive zero points (dashed line) or during successive extrema (dotted line).

Fig. 4
Fig. 4

Intensities of optical frequency components after phase modulation, as indicated by the dashed line in Fig. 3 at frequency ω/2 with index z = 10, and the original optical mode intensities for comparison (open circles).

Fig. 5
Fig. 5

Intensities of optical frequency components of interferometrically combined pulses for the conditions indicated for Fig. 4.

Fig. 6
Fig. 6

Optical pulse intensity waveform that corresponds to the spectrum in Fig. 5 (a = 5, z = 10, and P = 0).

Fig. 7
Fig. 7

Frequency components of the signal which results from detection of the pulse train represented in Fig. 6, along with the original components for comparison (open circles).

Fig. 8
Fig. 8

Intensities of optical frequency components resulting from phase modulation at frequency ϕ = (7/2) ω with modulation index z = 16, along with the input components for comparison (open circles). Note that only the m-positive portion of the symmetrical spectrum is shown and that the values for Z = 16 are magnified by a factor of ten relative to the input.

Fig. 9
Fig. 9

Intensities of optical frequency components of interferometrically combined pulses for conditions indicated in Fig. 8.

Fig. 10
Fig. 10

Optical pulse intensity that corresponds to the spectrum shown in Fig. 9, compared with the original pulse shape (dashed).

Fig. 11
Fig. 11

Frequency components of the signal which result from detection of a train of pulses like the one shown in Fig. 10 along with the original components for comparison (open circles).

Fig. 12
Fig. 12

Intensities of optical frequency components (a) before frequency shifting, (b) resulting from upshifting, and (c) resulting from downshifting, for a = 5, ϕ = ω, and z = 10.

Fig. 13
Fig. 13

Intensities of optical frequency components of interferometrically combined pulses, for the conditions indicated in Fig. 12.

Fig. 14
Fig. 14

Optical pulse intensity that corresponds to the spectrum shown in Fig. 13, compared with the original pulse shape (dashed).

Fig. 15
Fig. 15

Frequency components of the signal which results from detection of a train of pulses like the one shown in Fig. 14, along with the original components for comparison (open circles).

Equations (53)

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Ω 0 = 2 π M c / 2 L
ω N = 2 π N c / 2 L ,
ω ω 1
A ( t ) = m = + A m exp [ i ( Ω 0 + m ω ) t ] ,
A m = exp ( m 2 / a 2 ) .
I ( t ) = 1 2 I 0 + l = 1 I l cos l ω t ,
I l = 2 m , l = + A m A m + l .
ϕ = ω / 2 .
ϕ = ( P + 1 2 ) ω
h = 1 2 A ( t ) exp { i z sin [ ( P + 1 2 ) ω t ] } + 1 2 A ( t + 2 π / ω ) exp { i z sin [ ( P + 1 2 ) ω ( t + 2 π / ω ) ] } .
h = p , s = + J 2 p ( z ) A s ( 2 P + 1 ) p exp [ i ( Ω 0 + s ω ) t ] ,
H = 1 2 H 0 + l = 1 H l cos l ω t ,
H l = 2 p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + ( 2 P + 1 ) ( P k ) + l .
H = 1 2 { 1 + cos [ 2 z sin ( P + 1 2 ) ω t ] } I ( t ) .
z ϕ = z ( P + 1 2 ) ω ,
2 z ϕ = 2 z ( P + 1 2 ) ω .
2 z sin ( P + 1 2 ) ω 2 z ( P + 1 2 ) ω
sin ϕ ϕ .
z max = π / [ 2 sin ( P + 1 2 ) ( π / 2 a ) ]
z max a / ( P + 1 2 )
ϕ = P ω
r = 1 2 A ( t ) exp ( i z sin P ω t ) + 1 2 A ( t ) exp ( i z sin P ω t ) .
r = p , s = + J 2 p ( z ) A s 2 P p exp [ i ( Ω 0 + s ω ) t ] ,
R l = 2 p , k , n = + J 2 ( z ) J 2 k ( z ) A n A n .
R = 1 2 [ 1 + cos ( 2 z sin P ω t ) ] I ( t ) .
g = 1 2 A ( t ) exp [ i z cos ( P + 1 2 ) ω t ] + 1 2 A ( t ) exp [ i z cos ( P + 1 2 ) ω t ] .
g = p , s = + ( 1 ) p J 2 p ( z ) A s ( 2 P + 1 ) p exp [ i ( Ω 0 + s ω ) t ] ,
G l = 2 p , k , n = + ( 1 ) p + k J 2 ( z ) J 2 k ( z ) A n A n ,
G = 1 2 { 1 + cos [ 2 z cos ( P + 1 2 ) ω t ] } I ( t ) .
G t = 0 = 1 2 ( 1 + cos 2 z ) I t = 0 .
exp ( i z sin ϕ t ) = p = + J p ( z ) exp ( i p ϕ t )
h = 1 2 m , p = + J p ( z ) A m exp ( i { Ω 0 + [ m + ( P + 1 2 ) p ] ω } t ) × ( 1 + exp { 2 π i [ ( Ω 0 / ω ) + m + ( P + 1 2 ) p ] } ) .
h = p , m = + J 2 p ( z ) A m exp ( i { Ω 0 + [ m × ( 2 P + 1 ) p ] ω } t ) .
h = p , s = + J 2 p ( z ) A s ( 2 P + 1 ) p exp [ i ( Ω 0 + s ω ) t ] ,
h h * = p , s , k , n = + J 2 p ( z ) J 2 k ( z ) A s ( 2 P + 1 ) p A n ( 2 P + 1 ) k exp [ i ( s n ) ω t ] .
( h h * ) l = p , k , n = + J 2 p ( z ) J 2 k ( z ) A n ( 2 P + 1 ) p + l A n ( 2 P + 1 ) k exp ( i l ω t ) + p , k , n = + J 2 p ( z ) J 2 k ( z ) A n ( 2 P + 1 ) p l A n ( 2 P + 1 ) k exp ( i l ω t ) .
( h h * ) l = p , k , n = + J 2 p ( z ) J 2 k ( z ) A n + ( 2 P + 1 ) ( k p ) + l A n exp ( i l ω t ) + p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + ( 2 P + 1 ) ( p k ) + l exp ( i l ω t ) .
( h h * ) l = 2 p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + ( 2 P + 1 ) ( p k ) + l cos l ω t .
H = 1 2 H 0 + l = 1 H l cos l ω t ,
H l = 2 p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + ( 2 P + 1 ) ( p k ) + l ,
h = cos [ z sin ( P + 1 2 ) ω t ] A ( t ) ,
H = cos 2 [ z sin ( P + 1 2 ) ω t ] A ( t ) A * ( t ) = 1 2 { 1 + cos [ 2 z sin ( P + 1 2 ) ω t ] } I ( t ) ,
J p ( z ) = ( 1 ) p J p ( z ) ,
r = 1 2 p , n = + [ 1 + ( 1 ) p ] J p ( z ) A m exp { i [ Ω 0 + ( m + P p ) ω ] t } .
r = p , s = + J 2 p ( z ) A s 2 P p exp [ i ( Ω 0 + s ω ) t ] ,
r r * = p , s , k , n = + J 2 p ( z ) J 2 k ( z ) A s 2 P p A n 2 P k exp [ i ( s n ) ω t ] .
( r r * ) l = p , k , n = + J 2 p ( z ) J 2 k ( z ) A n 2 P p + l A n 2 P k exp ( i l ω t ) + p , k , n = + J 2 p ( z ) J 2 k ( z ) A n 2 P p l A n 2 P k exp ( i l ω t ) .
( r r * ) l = p , k , n = + J 2 p ( z ) J 2 k ( z ) A n + 2 P ( k p ) + l A n exp ( i l ω t ) + p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + 2 P ( p k ) + l exp ( i l ω t ) .
( r r * ) l = 2 p , k , n = + J 2 p ( z ) J 2 k ( z ) A n A n + 2 P ( p k ) + l cos l ω t ,
r = cos ( z sin ω t ) A ( t ) ,
exp ( i z cos ϕ t ) = p = + i p J p ( z ) exp ( i p ϕ t ) ,
g = cos [ z cos ( P + 1 2 ) ω t ] A ( t ) ,
G = 1 2 { 1 + cos [ 2 z cos ( P + 1 2 ) ω t ] } I ( t ) ,

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