Abstract

This paper contains a discussion of the relation between the angular distributions of the coherently emitted radiation and the angular distribution of the incident beams in a degenerate four-wave mixing experiment, when two of the input beams are derived from the same laser source. The objective of the discussion is to obtain beam configurations that permit the use of such spectroscopic techniques when good spatial resolution is required so that the conventional collinear phase matching geometry is inappropriate. Two different configurations are indeed obtained and experimentally demonstrated that have, in the worst direction (longitudinal), spatial resolution of the order of 1 mm, with beam apertures comparable with those used in the standard collinear experiments and without the complicating requirement of two distinct pump beams.

© 1981 Optical Society of America

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References

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  1. A. C. Eckbreth, United Technologies Research Center Report 78-41 (1978); see also, M. Pealat, J. P. Taran, F. Moya, Opt. Laser Technol. 12, 21 (1980).
    [CrossRef]
  2. D. V. Murphy, B. L. Barshall, R. K. Chang, A. C. Eckbreth, Opt. Lett. 4, 167 (1979).
    [CrossRef] [PubMed]
  3. K. A. Marko, L. Rimai, 1980 SAE Congress and Exposition, Detroit (Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, Pa. 15096), paper 800138.
  4. A. C. Eckbreth, Appl. Phys. Lett. 32, 421 (1978).
    [CrossRef]
  5. K. A. Marko, L. Rimai, Opt. Lett. 4, 211 (1979).
    [CrossRef] [PubMed]
  6. L. Rimai, K. A. Marko, L. C. Davis, in Proceedings, Seventh International Conference on Raman Spectroscopy, Ottawa, 1980, W. F. Murphy, Ed., p. 666 (North-Holland, Amsterdam, 1980).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

1979 (2)

1978 (1)

A. C. Eckbreth, Appl. Phys. Lett. 32, 421 (1978).
[CrossRef]

Barshall, B. L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Chang, R. K.

Davis, L. C.

L. Rimai, K. A. Marko, L. C. Davis, in Proceedings, Seventh International Conference on Raman Spectroscopy, Ottawa, 1980, W. F. Murphy, Ed., p. 666 (North-Holland, Amsterdam, 1980).

Eckbreth, A. C.

D. V. Murphy, B. L. Barshall, R. K. Chang, A. C. Eckbreth, Opt. Lett. 4, 167 (1979).
[CrossRef] [PubMed]

A. C. Eckbreth, Appl. Phys. Lett. 32, 421 (1978).
[CrossRef]

A. C. Eckbreth, United Technologies Research Center Report 78-41 (1978); see also, M. Pealat, J. P. Taran, F. Moya, Opt. Laser Technol. 12, 21 (1980).
[CrossRef]

Marko, K. A.

K. A. Marko, L. Rimai, Opt. Lett. 4, 211 (1979).
[CrossRef] [PubMed]

K. A. Marko, L. Rimai, 1980 SAE Congress and Exposition, Detroit (Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, Pa. 15096), paper 800138.

L. Rimai, K. A. Marko, L. C. Davis, in Proceedings, Seventh International Conference on Raman Spectroscopy, Ottawa, 1980, W. F. Murphy, Ed., p. 666 (North-Holland, Amsterdam, 1980).

Murphy, D. V.

Rimai, L.

K. A. Marko, L. Rimai, Opt. Lett. 4, 211 (1979).
[CrossRef] [PubMed]

K. A. Marko, L. Rimai, 1980 SAE Congress and Exposition, Detroit (Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, Pa. 15096), paper 800138.

L. Rimai, K. A. Marko, L. C. Davis, in Proceedings, Seventh International Conference on Raman Spectroscopy, Ottawa, 1980, W. F. Murphy, Ed., p. 666 (North-Holland, Amsterdam, 1980).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

Appl. Phys. Lett. (1)

A. C. Eckbreth, Appl. Phys. Lett. 32, 421 (1978).
[CrossRef]

Opt. Lett. (2)

Other (4)

K. A. Marko, L. Rimai, 1980 SAE Congress and Exposition, Detroit (Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, Pa. 15096), paper 800138.

A. C. Eckbreth, United Technologies Research Center Report 78-41 (1978); see also, M. Pealat, J. P. Taran, F. Moya, Opt. Laser Technol. 12, 21 (1980).
[CrossRef]

L. Rimai, K. A. Marko, L. C. Davis, in Proceedings, Seventh International Conference on Raman Spectroscopy, Ottawa, 1980, W. F. Murphy, Ed., p. 666 (North-Holland, Amsterdam, 1980).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959).

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Figures (8)

Fig. 1
Fig. 1

Intensity distributions in exciting beam and geometry of calculation. r′ describes the volume over which the third-order polarization is nonzero; r gives the point where the radiated coherent Raman field is calculated. O, the origin, is the focus for the incident beam. Δθp is the angular thickness of the annular pump beam for both cases considered; for case 1a, the Stokes beam is assumed rectangular with angular dimensions Δθs × Δϕs. For case 1b it is assumed circular with half-angle θs,max.

Fig. 2
Fig. 2

Dependence of the normalized radiated anti-Stokes power on azimuth θ for numerical conditions given in Table I. (a) Case 1a for three different values of ϕ, where ϕ = 0 corresponds to radiation emitted in a direction diametrically opposed to that of the incident Stokes beam and yields the center of the anti-Stokes spot. Rmax = 10−2 cm, zmax = 0.5 cm. (b) and (c) correspond to the case of Fig. 1(b) with Rmax = 0.1 cm and two different values of zmax (0.5 and 0.1 cm, respectively).

Fig. 3
Fig. 3

Dependence of radiated signal on longitudinal dimension of probed volume. zmax in centimeters, Rmax = 10−4 cm. Dashed line represents the peak intensity (θ = 0.0195, ϕ = 0) for case 1a. Solid line represents the intensity at θ = 0.0275 for case 1b. The values of 1/a (0.0815) and 1/b (0.0516) are indicated on the abscissa.

Fig. 4
Fig. 4

Dependence of radiated intensity on transverse dimensions of probed volume. Rmax in centimeters, zmax = 10−4 cm. Dashed line represents the peak intensity (θ = 0.0195, ϕ = 0) for case 1a. Solid line represents the intensity at θ = 0.0275 for case 1b. Triangles represent the intensity for case 1b at θ = 0.

Fig. 5
Fig. 5

Schematic of the experimental arrangement used to display th distribution of the CARS (and CSRS) radiation. Shown is a typical beam input configuration as seen in cross section on the plane of the input lens. Subsequent boxes indicate the beam cross sections at the points shown. The following notation is used: s, Stokes beam; p, pump beam; a, bxcrs—the anti-Stokes emission from noncollinear phase matching, a, coll—the anti-Stokes emission from collinear phase matching. The inset at lower right shows how the nonplanar wave vectors satisfy the phase matching condition.

Fig. 6
Fig. 6

In both (a) and (b) the photo on the upper left shows the configuration of the input beams in the plane of the input lens. The photo on the upper right shows the distribution of the CARS radiation seen in the plane of the SIT vidicon face when it is displaced from the monochromator focal point. The geometric constructions for each case, which are used to help deduce the angular distributions of the signal, are shown below the photos.

Fig. 7
Fig. 7

This figure contains the photos and diagram for the CSRS process investigated. As in Fig. 6, the upper left photo shows the configuration of the input beams while the photo at the upper right shows the distribution of the CSRS signal. In this case, the pump beam is the center beam in the upper left photo. The geometric construction is indicated in the diagram below the photos.

Fig. 8
Fig. 8

Relative intensity of the CARS signal from the interior of the flame cone of a small propane torch is shown. The torch position is moved along the beam and data recorded every millimeter. The insert shows the propane CARS spectra observed on the OMA, near 3000-cm−1 Raman shift.

Tables (1)

Tables Icon

Table I Values of Parameters Used in Numerical Calculations: (1a) Denotes Parameters Used Only for the Configuration Shown in Fig. 1(a); (1b) for Fig. 1(b); All Others are Common to Both

Equations (31)

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E i ( r , t ) exp [ i ( k a r ω a t ) ] k a 2 r j k l χ i j k l E * s 0 , j E p 0 , k E p 0 , l U , i = x , y ,
ω a = 2 ω p ω s .
E p ( r ) E p 0 exp ( i k z z ) J 0 ( k ρ ) sin ( k z ' Δ θ p / 2 ) k z ' Δ θ p / 2 ,
z ' = r cos θ ,
ρ = r sin θ ,
k z = k p cos θ p ,
k = k p sin θ p .
E s ( r ) E s 0 exp ( i k s z z i k s x ' ) sin ( k s z ' Δ θ s / 2 ) k s z ' Δ θ s / 2 F y ,
F y = sin ( k s y ' Δ ϕ s / 2 ) k s y ' Δ ϕ s / 2 ,
x = r sin θ cos ϕ ,
y = r sin θ sin ϕ ,
k s z = k s cos θ s ,
k s = k s sin θ s ,
U = U z U ,
U z = 2 0 z max d z ' cos ( 2 k z k s z k a z ) z ( sin a z ' a z ' ) 2 sin b z ' b z ' ,
a = k Δ θ p / 2 ,
b = k s Δ θ s / 2 ,
k a z = k a cos θ ,
U = 2 π 0 R max ρ d ρ [ J 0 ( k ρ ) ] 2 J 0 ( A ρ ) ,
A = [ ( k a cos ϕ k s ) 2 + k a 2 sin 2 ϕ ] 1 / 2 ,
k a = k a sin θ .
E s ( r ) E s 0 exp ( i k s z ) 2 J 1 ( k s ρ ξ max ) / k s ρ ξ max ,
ξ max = 2 sin ( θ s , max / 2 ) θ s , max .
U z = 2 0 z max d z ' cos ( 2 k z k s k a z ) z ' ( sin a z ' a z ' ) 2 ,
U = 4 π k s ξ max 0 R max d ρ ' [ J 0 ( k ρ ' ) ] 2 J 1 ( k s ρ ξ max ) J 0 ( k a ρ ' ) .
I ( θ , ϕ ) | χ | 2 I s I p 2 U 2 ,
2 k p cos θ p = k s cos θ s + k a cos θ a ,
k s sin θ s = k a sin θ a .
θ s k s k a k θ p .
2 k p cos θ p = k s + k a cos θ a .
θ a 2 k p k a θ p .

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