Abstract

With the help of interference effects, two-beam and multiple-beam spectroscopy detect in the pairs of beams (bundles of rays selected by the optical system) phase correlations due to certain fluctuations in optically thin distributions of incoherent light sources. Originally spatial resolution along the line of sight was expected for multiple-beam spectroscopy because of the limited region of intersection for pairs of beams. Here more general analysis shows another mechanism of spatial resolution allowing use of broader overlapping beams. Thus a simpler two-beam spectroscopy configuration (to be discussed in more detail elsewhere) capable of making more efficient use of emitted light proves to offer the same localized measurement of spatially harmonic fluctuations in the appropriate light source distributions.

© 1981 Optical Society of America

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  1. P. S. Rostler (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-2014 (1974) (unpublished).
  2. B. D. Billard (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-10913 (1980) (unpublished).
    [PubMed]
  3. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 183.
  4. Ref. 3, p. 186.
  5. For a more detailed treatment, see also Ref. 2, pp. 162–177.
  6. Ref. 1, pp. 63–69.
  7. For a more detailed treatment, see Ref. 2, pp. 35–39.

Billard, B. D.

B. D. Billard (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-10913 (1980) (unpublished).
[PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 183.

Rostler, P. S.

P. S. Rostler (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-2014 (1974) (unpublished).

Other (7)

P. S. Rostler (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-2014 (1974) (unpublished).

B. D. Billard (to be submitted to Appl. Opt.); see also Ph.D. Thesis, U. California, Berkeley, LBL-10913 (1980) (unpublished).
[PubMed]

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 183.

Ref. 3, p. 186.

For a more detailed treatment, see also Ref. 2, pp. 162–177.

Ref. 1, pp. 63–69.

For a more detailed treatment, see Ref. 2, pp. 35–39.

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Figures (5)

Fig. 1
Fig. 1

An example of how spatial localization of spectroscopic information may be achieved using pairs of beams. Light from an optically thin distribution of incoherent sources within two beams, or bundles of rays selected by a system of masks, is analyzed for correlations. Correlations between the two beams must be due to sources common to both beams, located within the limited region of overlap.

Fig. 2
Fig. 2

Schematic of a two-beam spectroscopy system. Equal-amplitude orthogonal components of the initial polarization are transmitted by separate paths in the dual-path optical system. This serves to combine two beams entering the dual-path system into one beam entering the analyzer. The net phase correlations between the two beams result in a net polarization (corresponding to the initial polarization or to the component perpendicular to the initial polarization depending on phase) and an intensity difference in the light to the two detectors.

Fig. 3
Fig. 3

Relationship of the polarization components in the two-beam spectroscopy system. The dual-path optical system splits the initial polarization into two equal components A and B. Initially A and B are in phase, as shown. At the analyzer, if the equivalent phase relation between beams A and B is maintained or inverted, polarizations 1 or 2, respectively, result. The analyzer is oriented to split the combined beams into these two components for an intensity comparison by the two detectors (reprinted from Ref. 1).

Fig. 4
Fig. 4

Radii of curvature of the wave fronts obtained by assuming spherical wave fronts (centered on the image point r c) propagating beyond the lens. Radii of curvature ρ 0 in and ρin are obtained from the wave fronts at points r 0 in and rin, respectively. In this case ρ 0 in and ρin have positive signs indicating the wave fronts are diverging from the image point r c. A negative radius of curvature would indicate the wave front is converging on the image point (i.e., the image point is located beyond the wave front in the direction of propagation).

Fig. 5
Fig. 5

Determination of fields ξA and ξB reaching point r″. Spherical wave fronts centered at the image point r c are transformed by the two rotations representing the interferometer into wave fronts for beams A and B, which then propagate directly to the aperture. The same result is obtained by finding points r A and r B, which are transformed by the corresponding rotations into r″ and following the propagation of the field directly to these two points. The field reaching r A is equivalent to ξA and that reaching r B to ξB.

Equations (46)

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2 ξ ( r , t ) 1 c 2 2 t 2 ξ ( r , t ) = 4 π s ( r , t ) ,
ξ ( r , t ) = d 3 r 1 | r r | s ( r , t | r r | c ) .
ξ ( r , t ) = d 3 r ξ 0 ( r , t ; r ) ,
ξ 0 ( r , t ; r ) = 1 | r r | s ( r , t | r r | c )
ξ 0 ( r , t ; r ) = ρ 0 ρ ξ 0 ( r 0 , t ρ ρ 0 c ; r )
ξ 0 ( r in , t ; r ) = | ρ 0 in ρ in | ξ 0 ( r 0 in , t ρ in ρ 0 in c ; r ) ,
| ρ 0 in | = | r in r c | , | ρ in | = | r in r c | , sign ( ρ 0 in ) = sign [ ( r 0 in r c ) · ( r 0 in r ) ] , sign ( ρ in ) = sign [ ( r in r c ) · ( r in r ) ] ,
R A = R x A ( e ˆ , 2 θ ) , R B = R x B ( e ˆ , 2 θ ) .
ξ A , B ( r , t ; r ) = 1 2 ξ 0 ( R A , B r in , t ; r ) ,
r = R A , B r in
r A , B = R A , B 1 r
ξ A , B ( r , t ; r ) = 1 2 | ρ 0 in ρ A , B ( r ) | 1 | r 0 in r | × s [ r , t | r 0 in r | + ϕ 1 ( r 0 in ) ρ A , B ( r ) + ρ 0 in c ] ,
ξ ˜ A , B ( r , ω ; r ) = d t exp ( i ω t ) ξ A , B ( r , t ; r ) ,
ξ ˜ A , B ( r , ω ; r ) = d t exp ( i ω t ) | ρ 0 in | 2 | ρ A , B ( r ) | | r 0 in r | × s [ r , t | r 0 in r | + ϕ 1 ( r 0 in ) ρ A , B ( r ) + ρ 0 in c ] = | ρ 0 in | exp i ω c [ | r 0 in r | + ϕ 1 ( r 0 in ) ρ A , B ( r ) + ρ 0 in ] 2 | ρ A , B ( r ) | | r 0 in r | d t exp ( i ω t ) s ( r , t ) = | ρ 0 in | exp i ω c [ | r 0 in r | + ϕ 1 ( r 0 in ) ρ A , B ( r ) + ρ 0 in ] 2 | ρ A , B ( r ) | | r 0 in r | s ˜ ( r , ω )
ξ ˜ A , B ( r , ω ; r ) = ϕ A , B ( r , r , ω ) s ˜ ( r , ω ) .
ϕ A , B ( r , r , ω ) = | ρ 0 in | exp i ω c [ | r 0 in r | + ϕ 1 ( r 0 in ) ρ A , B ( r ) + ρ 0 in ] 2 | ρ A , B ( r ) | | r 0 in r | .
ϕ B ( r , r , ω ) = | ρ A ( r ) ρ B ( r ) | exp { i ω c [ ρ A ( r ) ρ B ( r ) ] } ϕ A ( r , r , ω ) .
ξ 1 = 1 2 ( ξ A + ξ B ) , ξ 2 = 1 2 ( ξ A ξ B ) .
I 1,2 ( t ; r ) = α d 2 r 0 d ω 2 π | f ( ω ) | 2 · 1 2 | ϕ A ( r , ω , r ) s ( r , ω ; t ) ± ϕ B ( r , ω , r ) s ( r , ω ; t ) | 2 .
Y ( t ) = d 2 r I 1 ( t ; r ) d 3 r I 2 ( t ; r ) = d 3 r 0 d ω π | f ( ω ) | 2 · α d 2 r Re [ ϕ * A ( r , ω , r ) s * ( r , ω ; t ) · ϕ B ( r , ω , r ) s ( r , ω ; t ) ] .
Y ( t ) = α d 3 r 0 d ω | f ( ω ) | 2 T ( r , ω ) S ( ω ; r , t ) ,
T ( r , ω ) = Re 1 π α d 2 r ϕ * A ( r , ω ; r ) ϕ B ( r , / ω ; r ) ,
S ( ω ; r , t ) = | s ( r , ω ; t ) | 2 ,
T ( r , ω ) = Re 1 π α d 2 r | ρ A ( r ) ρ B ( r ) | exp { i ω c [ ρ A ( r ) ρ B ( r ) ] } · | ϕ A ( r , ω , r ) | 2 = Re 1 π α d 2 r | ρ A ( r ) ρ B ( r ) | exp { i ω c [ ρ A ( r ) ρ B ( r ) ] } ( ρ 0 in ) 2 2 ρ A ( r ) 2 | r 0 in r | 2 = Re 1 π α d 2 r ( ρ 0 in ) 2 exp { i ω c [ ρ A ( r ) ρ B ( r ) ] } | 2 ρ A ( r ) ρ B ( r ) | | r 0 in r | 2 .
T ( r c , ω ) = 1 2 π ( ρ 0 in ) 2 | r 0 in r ( r c ) | 2 Re α d 2 r exp { i ω c [ ρ A ( r ) ρ B ( r ) ] } | ρ A ( r ) ρ B ( r ) | = 1 2 π ( ρ 0 in ) 2 | r 0 in r ( r c ) | 2 α d 2 r cos { ω c [ ρ A ( r ) ρ B ( r ) ] } | ρ A ( r ) ρ B ( r ) |
α d 2 r cos ω c [ ρ A ( r ) ρ B ( r ) ] | ρ A ( r ) ρ B ( r ) | = α d 2 r cos ω c ( | R A 1 r r c | | R B 1 r r c | ) | R A 1 r r c | · | R B 1 r r c | = α d 2 r cos ω c ( | r r c A | | r r c B | ) | r r c A | · | r r c B | ,
r c 0 = 1 2 ( r c A + r c B ) , r A B = 1 2 ( r c A r c B ) ,
| r r c A | | r r c B | 2 ( r r c 0 ) | r r c 0 | · r A B ,
| r r c A | · | r r c B | | r r c 0 | 2 { 1 [ ( r r c 0 ) · r A B | r r c 0 | 2 ] 2 } .
α d 2 r cos ω c [ ρ A ( r ) ρ B ( r ) ] | ρ A ( r ) ρ B ( r ) | = α d 2 r cos 2 ω c ( r r c 0 ) · r A B | r r c 0 | | r r c 0 | 2 = α d 2 υ cos [ 2 ω c υ ˆ · r A B ] υ 2 ,
v = r = r c 0 , υ ˆ v / υ .
α d 2 r cos ( 2 ω c r r · r A B ) | r | 2 = y 0 ( L / 2 ) y 0 + ( L / 2 ) d y × x 0 ( W / 2 ) x 0 + ( W / 2 ) d x cos [ 2 ω c x A B x + z A B z 0 ( x 2 + y 2 + z 0 2 ) 1 / 2 ] x 2 + y 2 + z 0 2 .
u = x / [ ( x 2 + y 2 + z 0 2 ) 1 / 2 ]
I = x 0 ( W / 2 ) x 0 + ( W / 2 ) d x cos [ 2 ω c x A B x + z A B z 0 ( x 2 + y 2 + z 0 2 ) 1 / 2 ] x 2 + y 2 + z 0 2 = 1 ( y 2 + z 0 2 ) 1 / 2 × u 1 u 2 d u cos { 2 ω c [ x A B u + z A B z 0 ( y 2 + z 0 2 ) 1 / 2 ( 1 u 2 ) 1 / 2 ] } ( 1 u 2 ) 1 / 2 ,
u 2,1 = x 0 ± ( W / 2 ) [ ( x 0 ± W z ) 2 + y 2 + z 0 2 ] 1 / 2 .
u 0 = x 0 / [ ( x 0 2 + y 0 2 + z 0 2 ) 1 / 2 ]
I c { sin [ 2 ω c x A B ( u 2 + Z ) ] sin [ 2 ω c x A B ( u 1 + Z ) ] } 2 ω x A B ( y 2 + z 0 2 ) 1 / 2 ( 1 u b 2 ) 1 / 2 = c { cos [ 2 ω c x A B ( u a + Z ) ] sin [ ω c x A B ( u 2 u 1 ) ] } ω x A B ( y 2 + z 0 2 ) 1 / 2 ( 1 u b 2 ) 1 / 2 ,
u a = ( u 1 + u 2 ) / 2 , Z = z A B z 0 x A B ( y 2 + z 0 2 ) 1 / 2 ( 1 u b 2 ) 1 / 2 ,
I = { sin [ ω c x A B ( u 2 u 1 ) ] [ ω c x A B ( u 2 u 1 ) ] } ( u 2 u 1 ) × cos [ 2 ω c x A B ( u a + Z ) ] ( y 2 + z 0 2 ) 1 / 2 ( 1 u b 2 ) 1 / 2 .
u 1 u 2 ( 1 u b 2 ) 1 / 2 u 1 ( 1 u 1 2 ) 1 / 2 u 2 ( 1 u 2 2 ) 1 / 2 = W ( y 2 + z 0 2 ) 1 / 2 ,
T ( r , ω ) = 1 2 π ( ρ 0 in ) 2 | r 0 in r ( r c ) | 2 L W y 0 2 + z 0 2 × { sin [ ω c x A B ( u 2 u 1 ) ] ω c x A B ( u 2 u 1 ) } cos [ 2 ω c x A B ( u a + Z ) ] .
k Δ x ˆ ω 0 c 2 x A B f 1 ,
Δ z Δ z c = f 1 2 ,
W Δ z W Δ z c ,
sin [ ω c x A B ( u 2 u 1 ) ] ω c x A B ( u 2 u 1 ) = sin ( ω c x A B W f 1 2 Δ z ) ω c x A B W f 1 2 Δ z = sin ( k Δ W 2 f 1 Δ z ' ) k Δ W 2 f 1 Δ z ' .
Δ z 0 = 2 π f 1 k Δ W = λ Δ W f 1 ,

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