Abstract

An examination of the conditions under which the normalized coherence function γ(x1,x2,τ), at two points x1 and x2 in an optical field, is reducible to the product of a function of x1, x2 and a function of τ leads to the concept of cross spectral purity. Spectrally pure beams of light are characterized by the property that their coherence function is so reducible and, operationally, by the fact that superposition does not affect the spectral distribution. Light beams that do not have this property are called spectrally impure, and it is shown to be characteristic of such beams that the interference fringes exhibit a detailed periodic coloring. A measure of the departure from cross-spectral purity is introduced and evaluated in some special cases. It is shown by an example that spectrally pure and spectrally impure beams of light may be derived from the same source with similar optical components. Moreover, these beams may be identical as regards their intensity, their spectral distribution, and their degree of coherence, and differ only as regards their state of spectral purity.

© 1961 Optical Society of America

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References

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  1. F. Zernike, Physica 1, 201 (1934).
    [Crossref]
  2. H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).
  3. E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).
  4. E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).
  5. H. H. Hopkins, J. Opt. Soc. Am. 47, 508 (1957).
    [Crossref]
  6. R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).
  7. L. Mandel, Proc. Phys. Soc. (London) 71, 1037 (1958).
    [Crossref]
  8. D. Gabor, J. Inst. Electr. Engrs. (London) 93, Part III, 429 (1936).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959).
  10. In this representation a1 and a2 may be complex functions.
  11. W. P. Alford and A. Gold, Am. J. Phys. 56, 481 (1958).
    [Crossref]
  12. L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
    [Crossref]
  13. G. B. Parrent, J. Opt. Soc. Am. 49, 787 (1959).
    [Crossref]

1961 (1)

1959 (1)

1958 (2)

W. P. Alford and A. Gold, Am. J. Phys. 56, 481 (1958).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 71, 1037 (1958).
[Crossref]

1957 (2)

H. H. Hopkins, J. Opt. Soc. Am. 47, 508 (1957).
[Crossref]

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

1955 (1)

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

1954 (1)

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

1951 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).

1936 (1)

D. Gabor, J. Inst. Electr. Engrs. (London) 93, Part III, 429 (1936).

1934 (1)

F. Zernike, Physica 1, 201 (1934).
[Crossref]

Alford, W. P.

W. P. Alford and A. Gold, Am. J. Phys. 56, 481 (1958).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959).

Gabor, D.

D. Gabor, J. Inst. Electr. Engrs. (London) 93, Part III, 429 (1936).

Gold, A.

W. P. Alford and A. Gold, Am. J. Phys. 56, 481 (1958).
[Crossref]

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

Hopkins, H. H.

H. H. Hopkins, J. Opt. Soc. Am. 47, 508 (1957).
[Crossref]

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).

Mandel, L.

L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 71, 1037 (1958).
[Crossref]

Parrent, G. B.

Twiss, R. Q.

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

Wolf, E.

L. Mandel and E. Wolf, J. Opt. Soc. Am. 51, 815 (1961).
[Crossref]

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959).

Zernike, F.

F. Zernike, Physica 1, 201 (1934).
[Crossref]

Am. J. Phys. (1)

W. P. Alford and A. Gold, Am. J. Phys. 56, 481 (1958).
[Crossref]

J. Inst. Electr. Engrs. (London) (1)

D. Gabor, J. Inst. Electr. Engrs. (London) 93, Part III, 429 (1936).

J. Opt. Soc. Am. (3)

Physica (1)

F. Zernike, Physica 1, 201 (1934).
[Crossref]

Proc. Phys. Soc. (London) (1)

L. Mandel, Proc. Phys. Soc. (London) 71, 1037 (1958).
[Crossref]

Proc. Roy. Soc. (London) (4)

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).

E. Wolf, Proc. Roy. Soc. (London) A225, 96 (1954).

E. Wolf, Proc. Roy. Soc. (London) A230, 246 (1955).

R. Hanbury Brown and R. Q. Twiss, Proc. Roy. Soc. (London) A243, 291 (1957).

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959).

In this representation a1 and a2 may be complex functions.

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Figures (7)

Fig. 1
Fig. 1

The principle of the superposition experiment.

Fig. 2
Fig. 2

Illustrating the notation for the propagation law of the coherence function.

Fig. 3
Fig. 3

Illustrating the spectral densities before and after superposition.

Fig. 4
Fig. 4

The difference of the spectral densities as a function of path difference.

Fig. 5
Fig. 5

The integrated square difference of the spectral densities as a function of path difference (with K 12 γ 12 ( η 0 ) = 1 2).

Fig. 6
Fig. 6

Illustrating the formation of spectrally impure light beams from spectrally pure sources.

Fig. 7
Fig. 7

Illustrating the formation of spectrally pure light beams which are otherwise similar to those emerging in Fig. 6.

Tables (1)

Tables Icon

Table I Some estimates of μ and γI II(0).

Equations (62)

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Γ ( x 1 , x 2 , τ ) = V ( x 1 , t + τ ) V * ( x 2 , t ) ,
ϕ ( x 1 , x 2 , ν ) = lim θ A ( x 1 , ν ) A * ( x 2 , ν ) θ [ I ( x 1 ) I ( x 2 ) ] ,
V ( x 3 , t ) = a 1 ( x 1 , x 3 ) V ( x 1 , t - η 1 ) + a 2 ( x 2 , x 3 ) V ( x 2 , t - η 2 ) ,
Γ 33 ( τ ) = a 1 2 Γ 11 ( τ ) + a 2 2 Γ 22 ( τ ) + a 1 a 2 * Γ 12 ( τ + η 2 - η 1 ) + a 2 a 1 * Γ 21 ( τ + η 1 - η 2 ) .
I 3 = a 1 2 I 1 + a 2 2 I 2 + 2 R [ a 1 a 2 * Γ 12 ( η 2 - η 1 ) ] ,
γ 33 ( τ ) = γ 11 ( τ ) + 1 2 K 12 γ 12 ( τ + η 2 - η 1 ) + 1 2 K 12 * γ 21 ( τ + η 1 - η 2 ) 1 + R [ K 12 γ 12 ( η 2 - η 1 ) ] ,
K 12 = 2 ( I 1 I 2 ) a 1 a 2 * a 1 2 I 1 + a 2 2 I 2 .
ϕ 33 ( ν ) = ϕ 11 ( ν ) + R [ K 12 ϕ 12 ( ν ) exp 2 π i ν ( η 2 - η 1 ) ] 1 + R [ K 12 γ 12 ( η 2 - η 1 ) ] ,
ϕ 33 ( ν ) - ϕ 11 ( ν ) = R [ K 12 { ϕ 12 ( ν ) exp 2 π i ν ( η 2 - η 1 ) - γ 12 ( η 2 - η 1 ) ϕ 11 ( ν ) } ] 1 + R [ K 12 γ 12 ( η 2 - η 1 ) ] .
γ 12 ( η 0 + Δ η ) = γ 12 ( η 0 ) exp ( 2 π i ν 0 Δ η ) .
R [ K 12 exp 2 π i ν 0 Δ η { ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) - γ 12 ( η 0 ) ϕ 11 ( ν ) } ] = 0 ,
ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) = γ 12 ( η 0 ) ϕ 11 ( ν ) .
γ 12 ( τ + η 0 ) = γ 12 ( η 0 ) γ 11 ( τ ) .
γ 11 ( τ ) γ 11 ( 0 ) ,
γ 12 ( τ + η 0 ) γ 12 ( η 0 ) ,
Γ 13 ( τ + η 3 ) = a 1 * Γ 11 ( τ + η 1 + η 3 ) + a 2 * Γ 12 ( τ + η 2 + η 3 ) ,
γ 13 ( τ + η 3 ) = 1 ( I 1 I 3 ) [ a 1 * I 1 γ 11 ( τ + η 1 + η 3 ) + a 2 * ( I 1 I 2 ) γ 12 ( τ + η 1 + η 3 + η 0 + Δ η ) ] .
γ 13 ( τ + η 3 ) = γ 11 ( τ + η 1 + η 3 ) ( I 1 I 3 ) [ a 1 * I 1 + a 2 * ( I 1 I 2 ) × γ 12 ( η 0 ) exp 2 π i ν 0 Δ η ] .
γ 13 ( τ - η 1 ) = γ 11 ( τ ) γ 13 ( - η 1 ) ,
a ( ξ , x 1 ) V ( ξ , t - η ( ξ , x 1 ) ) d ξ ,
Γ ( x 1 , x 2 , τ ) = Σ Σ d ξ d ξ a ( ξ , x 1 ) a ( ξ , x 2 ) × Γ ( ξ , ξ , τ + η ( ξ , x 2 ) - η ( ξ , x 1 ) ) .
η ( ξ , x 2 ) - η ( ξ , x 1 ) 1 / Δ ν ,
γ ( x 1 , x 2 , τ ) = 1 [ I ( x 1 ) I ( x 2 ) ] Σ Σ d ξ d ξ a ( ξ , x 1 ) × a ( ξ , x 2 ) [ I ( ξ ) I ( ξ ) ] γ ( ξ , ξ , τ ) × exp 2 π i ν 0 [ η ( ξ , x 2 ) - η ( ξ , x 1 ) ] .
γ ( ξ , ξ , τ ) = γ ( ξ , ξ , η 0 ) γ ( ξ , ξ , τ - η 0 ) ,
γ ( x 1 , x 2 , τ ) = γ ( ξ , ξ , τ ) Σ Σ d ξ d ξ a ( ξ , x 1 ) a ( ξ , x 2 ) [ I ( ξ ) I ( ξ ) ] γ ( ξ , ξ , 0 ) exp 2 π i ν 0 [ η ( ξ , x 2 ) - η ( ξ , x 1 ) ] [ I ( x 1 ) I ( x 2 ) ] .
γ ( x 1 , x 2 , τ ) = γ ( x 1 , x 2 , 0 ) γ ( ξ , ξ , τ ) .
γ 12 ( τ ) = γ 12 ( 0 ) γ 11 ( τ ) ,
ϕ 33 ( ν ) - ϕ 11 ( ν ) = R [ K 12 exp 2 π i ν 0 Δ η { ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) - γ 12 ( η 0 ) ϕ 11 ( ν ) } ] 1 + R [ K 12 exp ( 2 π i ν 0 Δ η ) γ 12 ( η 0 ) ] ,
ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) - γ 12 ( η 0 ) ϕ 11 ( ν ) = B ( ν ) , arg B ( ν ) = α ( ν ) , arg γ 12 ( η 0 ) = β , arg K 12 = δ , }
ϕ 33 ( ν ) - ϕ 11 ( ν ) = K 12 B ( ν ) cos ( 2 π ν 0 Δ η + α ( ν ) + δ ) 1 + K 12 γ 12 ( η 0 ) cos ( 2 π ν 0 Δ η + β + δ ) .
2 π ν 0 Δ η = ( n + 1 2 ) π - α ( ν ) - δ , n = 0 , ± 1 , ± 2 , etc . ,
0 ϕ 33 ( ν ) d ν = 0 ϕ 11 ( ν ) d ν = 1 ,
d d ( Δ η ) [ ϕ 33 ( ν ) - ϕ 11 ( ν ) ] = 0 ,
sin ( 2 π ν 0 Δ η + α ( ν ) + δ ) = K 12 γ 12 ( η 0 ) sin ( β - α ( ν ) ) .
0 B ( ν ) 2 d ν = γ 12 ( η 0 ) 2 0 ψ 12 ( ν ) - ϕ 11 ( ν ) 2 d ν ,
1 - μ = γ 12 ( η 0 ) 2 0 ψ 12 ( ν ) - ϕ 11 ( ν ) 2 d ν 0 [ ψ 12 ( ν ) 2 + ϕ 11 2 ( ν ) ] d ν ,
0 B ( ν ) 2 d ν = 0.
0 ψ 12 ( ν ) - ϕ 11 ( ν ) 2 d ν = 0 [ ψ 12 ( ν ) 2 + ϕ 11 2 ( ν ) ] d ν - 2 R 0 ϕ 11 ( ν ) ψ 12 ( ν ) d ν
ρ = 2 R 0 ϕ 11 ( ν ) ψ 12 ( ν ) d ν 0 [ ψ 12 ( ν ) 2 + ϕ 11 2 ( ν ) ] d ν .
1 - μ = γ 12 ( η 0 ) 2 ( 1 - ρ ) ,
ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) = ϕ 12 ( ν ) exp ( i α ) .
B ( ν ) = [ ϕ 12 ( ν ) - γ 12 ( η 0 ) ϕ 11 ( ν ) ] exp ( i β ) ,
arg B ( ν ) = β             or             β + π according as ϕ 12 ( ν ) γ 12 ( η 0 ) ϕ 11 ( ν ) or ψ 12 ( ν ) ϕ 11 ( ν ) . }
ϕ 12 ( ν ) exp ( 2 π i ν η 0 ) = γ 12 ( η 0 ) ψ ( ν ) .
γ 12 ( τ + η 0 ) = γ 12 ( η 0 ) λ ( τ ) ,
ϕ 33 ( ν ) - ϕ 11 ( ν ) = ψ ( ν ) - ϕ 11 ( ν ) 1 + 1 / [ K 12 γ 12 ( η 0 ) cos ( 2 π ν 0 Δ η + β + δ ) ] .
A ( x , ν ) = Σ i a ( x , x i , ν ) A ( x i , ν )
A ( x , ν ) = Σ i a ( x , x i , ν ) A ( x i , ν ) .
ϕ ( x , x , ν ) = ϕ 11 ( ν ) I ( x ) i a ( x , x i , ν ) 2 I ( x i ) ,
ϕ ( x , x , ν ) = ϕ 11 ( ν ) I ( x ) i a ( x , x i , ν ) 2 I ( x i ) ,
ϕ ( x , x , ν ) = ϕ 11 ( ν ) [ I ( x ) I ( x ) ] i a ( x , x i , ν ) × a * ( x , x i , ν ) I ( x i ) .
ϕ ( x , x , ν ) = ϕ 11 ( ν ) [ I ( x ) I ( x ) ] i I ( x i ) a ( x , x i , ν ) × a ( x , x i , ν ) exp 2 π i ν Δ η i ( ν ) .
ϕ ( x , x , ν ) exp ( - 2 π i ν Δ η ) = ϕ 11 ( ν ) [ I ( x ) I ( x ) ] i I ( x i ) × a ( x , x i , ν ) a ( x , x i , ν ) .
V I ( t ) = V 1 ( t ) + V 2 ( t )
V II ( t ) = V 3 ( t ) + V 2 ( t ) ,
I I = I II = I 1 + I 2 .
γ I , I ( τ ) = γ II II ( τ ) = I 1 γ 11 ( τ ) + I 2 γ 22 ( τ ) I 1 + I 2
γ I II ( τ ) = I 2 γ 22 ( τ ) I 1 + I 2 ,
ϕ II ( ν ) = ϕ II II ( ν ) = I 1 ϕ 11 ( ν ) + I 2 ϕ 22 ( ν ) I 1 + I 2 ,
ϕ I II ( ν ) = I 2 ϕ 22 ( ν ) I 1 + I 2 .
1 - μ = ( I 2 I 1 + I 2 ) 2 × I 1 2 0 [ ϕ 22 ( ν ) - ϕ 11 ( ν ) ] 2 d ν 0 { I 1 2 [ ϕ 11 2 ( ν ) + ϕ 22 2 ( ν ) ] + 2 I 2 2 ϕ 22 2 ( ν ) } d ν = ( I 2 I 1 + I 2 ) 2 × [ 1 - 2 I 2 2 2 I 2 2 + I 1 2 + I 1 2 0 ϕ 11 2 ( ν ) d ν / 0 ϕ 22 2 ( ν ) d ν ] ,
μ = I 1 2 + 2 I 1 I 2 ( I 1 + I 2 ) 2 + 2 I 2 4 ( I 1 + I 2 ) 2 × 1 2 I 2 2 + I 1 2 + I 1 2 0 ϕ 11 2 ( ν ) d ν / 0 ϕ 22 2 ( ν ) d ν .