Abstract

The dispersion of modulated or time-varying light is analyzed for an arbitrary disperser, general modulation, gaussian and incoherent illumination, and nonideal optical systems. The conditions for quasi-static response are examined in detail, and a stricter condition than that previously published is derived. The practical implications of this condition are discussed. The moments of the intensity distribution are derived for the general case. The analysis is related to specific dispersing elements.

© 1967 Optical Society of America

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References

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  1. S. E. Harris and A. E. Siegman, IRE Trans. Electron Devices ED-9, 322 (1962).
    [Crossref]
  2. S. E. Harris, J. Opt. Soc. Am. 54, 1147 (1964).
    [Crossref]
  3. J. R. Kerr, IEEE J. Quant. Electronics QE-2, 21 (1966).
    [Crossref]
  4. C. W. Barnes, J. Opt. Soc. Am. 56, 53 (1966).
    [Crossref]
  5. E. Bedrosian, Proc. IRE 50, 2071 (1962).
    [Crossref]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, N. Y.1959), Sec. 10.2.
  7. M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques (McGraw-Hill Book Company, New York, 1966), Sec. 5-4.
  8. Reference 6, Sec. 4.7.2.
  9. Reference 6, Sec. 8.6.
  10. Reference 6, Sec. 7.6.
  11. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

1966 (2)

J. R. Kerr, IEEE J. Quant. Electronics QE-2, 21 (1966).
[Crossref]

C. W. Barnes, J. Opt. Soc. Am. 56, 53 (1966).
[Crossref]

1964 (1)

1962 (2)

S. E. Harris and A. E. Siegman, IRE Trans. Electron Devices ED-9, 322 (1962).
[Crossref]

E. Bedrosian, Proc. IRE 50, 2071 (1962).
[Crossref]

Barnes, C. W.

Bedrosian, E.

E. Bedrosian, Proc. IRE 50, 2071 (1962).
[Crossref]

Bennett, W. R.

M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques (McGraw-Hill Book Company, New York, 1966), Sec. 5-4.

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, N. Y.1959), Sec. 10.2.

Harris, S. E.

S. E. Harris, J. Opt. Soc. Am. 54, 1147 (1964).
[Crossref]

S. E. Harris and A. E. Siegman, IRE Trans. Electron Devices ED-9, 322 (1962).
[Crossref]

Kerr, J. R.

J. R. Kerr, IEEE J. Quant. Electronics QE-2, 21 (1966).
[Crossref]

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

Schwartz, M.

M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques (McGraw-Hill Book Company, New York, 1966), Sec. 5-4.

Siegman, A. E.

S. E. Harris and A. E. Siegman, IRE Trans. Electron Devices ED-9, 322 (1962).
[Crossref]

Stein, S.

M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques (McGraw-Hill Book Company, New York, 1966), Sec. 5-4.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, N. Y.1959), Sec. 10.2.

IEEE J. Quant. Electronics (1)

J. R. Kerr, IEEE J. Quant. Electronics QE-2, 21 (1966).
[Crossref]

IRE Trans. Electron Devices (1)

S. E. Harris and A. E. Siegman, IRE Trans. Electron Devices ED-9, 322 (1962).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. IRE (1)

E. Bedrosian, Proc. IRE 50, 2071 (1962).
[Crossref]

Other (6)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, N. Y.1959), Sec. 10.2.

M. Schwartz, W. R. Bennett, and S. Stein, Communications Systems and Techniques (McGraw-Hill Book Company, New York, 1966), Sec. 5-4.

Reference 6, Sec. 4.7.2.

Reference 6, Sec. 8.6.

Reference 6, Sec. 7.6.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).

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Figures (2)

Fig. 1
Fig. 1

(a) Basic disperser configuration. (b) Amplitude distribution on screen for monochromatic carrier and discrete sidebands. (c) Normalized amplitude distribution U(x,ω).

Fig. 2
Fig. 2

Illustration of QS condition (12b). (a) Sidebands at x0. (b) QS net moving spot. (c) Resulting pulse at x0.

Equations (41)

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u ( x , t ) = 1 2 π - U ( x , ω ) e i ω t d ω .
S ( ω ) = - s ( t ) e - i ω t d t .
s ( t ) = 2 a ( t ) e i ϕ ( t ) e i ω 0 t ,
U ( x , ω ) = U [ x - K ( ω - ω 0 ) ] ,
u ( x , t ) = exp [ i ( ω 0 + x / K ) t ] · 1 2 π - U ( Ω ) e i Ω t d Ω exp [ i ( ω 0 + x / K ) t ] · u ( t ) .
e ( x , t ) = 2 2 π - S ( ω ) U [ ω - ( ω 0 + x / K ) ] e i ω t d ω .
e ( x , t ) = exp [ i ( ω 0 + x / K ) t ] 2 - a ( τ ) u ( t - τ ) × exp [ i ( ϕ ( τ ) - x τ / K ) ] d τ .
U ( x , ω ) = exp { - ( 1 / 4 σ 2 ) [ x - K ( ω - ω 0 ) ] 2 } ,
u ( x , t ) = [ σ / K ( π ) 1 2 ] exp [ i ( ω 0 + x / K ) t - σ 2 t 2 / K 2 ] u ( t ) = [ σ / K ( π ) 1 2 ] exp ( - σ 2 t 2 / K 2 ) .
2 π / Δ ω = Δ t K / σ .
e ( x , t ) = exp [ i ( ω 0 + x / K ) t ] × - u ( t - τ ) exp { i [ ϕ ( t ) + ϕ ( t ) ( τ - t ) + [ ϕ ( t ) ( τ - t ) 2 / 2 ] - x τ / K ] } d τ .
ϕ ( t ) ( Δ t ) 2 1.
ϕ ( t ) = δ cos ω m t ,
δ ω m 2 ( Δ t ) 2 1.
ω m Δ t 1 ,
ω m / Δ ω 1 ,
ϕ ( t ) / Δ ω Δ ω ,
δ ω m 2 / ( Δ ω ) 2 1 ,
e ( x , t ) = exp { i [ ω 0 t + x t / K - t ϕ ( t ) + ϕ ( t ) ] } × 2 - u ( t - τ ) exp { i τ [ ϕ ( t ) - x / K ] } d τ = - 2 e i ω 0 t e i ϕ ( t ) U [ ϕ ( t ) - x / K ] .
I ( x , t ) = 1 2 e ( x , t ) e * ( x , t ) = U [ ϕ ( t ) - x / K ] 2 .
δ required = 100 N 2 .
ω m / 2 π = 10 9 / 100 N 2 Hz ,
ω a max Δ ω = 2 π / Δ t ,
I ( t ) = 1 2 - e ( x , t ) e * ( x , t ) d x .
I ( t ) = 2 π - a 2 ( τ ) u ( t - τ ) 2 d τ ,
I ( t ) = a 2 ( t ) .
I ( t ) = 2 π a 2 ( t ) av - u ( t - τ ) 2 d τ = a 2 ( t ) av .
m ( t ) = 1 2 - x e ( x , t ) e * ( x , t ) d x .
R ( ω ) = - U ( y - K ω / 2 ) U * ( y + K ω / 2 ) d y .
m ( t ) = 2 π K - u ( t - τ ) 2 ϕ ( τ ) d τ
= K 2 π - Φ ( ω ) R ( ω ) e i ω t d ω ,
m ( t ) QS = K ϕ ( t ) .
σ 2 ( t ) = w ( t ) - m 2 ( t ) = 2 π K 2 - u ( t - τ ) 2 [ ϕ ( t ) ] 2 d τ + - y 2 U ( y ) 2 d y - m 2 ( t )
= K 2 2 π - { - [ ϕ ( t ) ] 2 e - i ω t d t } R ( ω ) e i ω t d ω + - y 2 U ( y ) 2 d y - m 2 ( t ) .
2 ω m / Δ ω 1 ,
σ 2 QS ( t ) = - y 2 U ( y ) 2 d y ,
a ( t ) = a n ( t ) + a m ( t ) ϕ ( t ) = ϕ n ( t ) + ϕ m ( t ) ,
K d x d ω = L d θ d ω = L d θ d n d n d ω = 2 L sin ( α / 2 ) cos θ d n d ω ,
K = L d θ / d ω = 2 π c L M / ω 2 d ,
( spectral width ) ( ω / M ) · d / s = ( F S R ) · d / s .
f ( x ) = exp [ i ω x 2 / 2 c L ] ,