Abstract

We describe the theory of imaging by degenerate four-wave mixing (DFWM) using a standard diffraction theory of imaging by coherent light. We demonstrate that, even with the phase-conjugating geometry, no aberration correction can be achieved by DFWM imaging. We demonstrate the coherent nature of DFWM image formation using spatially modulated signals generated in flame OH in the phase-conjugating geometry. The intensity distribution in the Fourier plane of a telecentric lens system is shown to be the spatial Fourier transform of the object distribution characteristic of coherent imaging. The brightness of the DFWM signals exceeds that of similar laser-induced fluorescence signals that can be discriminated by restricting the aperture of the imaging system while still allowing a spatial resolution of approximately 70 µm. DFWM imaging with the forward-folded boxcars geometry is demonstrated and used in a simple referencing scheme to compensate for structure on the images imposed by nonuniformity of the laser beams employed. Images formed in NO are used to illustrate that structure on a scale of less than 100 µm arising from beam inhomogeneity can be removed by this referencing technique.

© 1997 Optical Society of America

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References

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  1. G. Kychakoff, R. D. Howe, R. K. Hanson, J. McDaniel, “Quantitative visualization of combustion species in a plane,” Appl. Opt. 21, 3225–3227 (1982).
    [CrossRef] [PubMed]
  2. P. Ewart, P. Snowdon, I. Magnusson, “Two-dimensional phase-conjugate imaging of atomic distributions in flames by degenerate four-wave mixing,” Opt. Lett. 14, 563–565 (1989).
    [CrossRef] [PubMed]
  3. K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
    [CrossRef]
  4. G. W. Stroke, An Introduction to Coherent Optics and Holography, (Academic, New York, 1966).
  5. R. A. Fisher, Optical Phase Conjugation, (Academic, New York, 1983).
  6. P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).
  7. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).
  8. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
    [CrossRef]
  9. J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
    [CrossRef]
  10. P. Ewart, S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
    [CrossRef] [PubMed]
  11. D. J. Rakestraw, R. L. Farrow, T. Dreier, “Two-dimensional imaging of OH in flames by degenerate four-wave mixing,” Opt. Lett. 15, 709–711 (1990).
    [CrossRef] [PubMed]
  12. P. Ewart, M. Kaczmarek, “Two-dimensional mapping of temperature in a flame by degenerate four-wave mixing in OH,” Appl. Opt. 30, 3996–3999 (1991).
    [CrossRef] [PubMed]
  13. W. J. Smith, Modern Optical Engineering: the Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990).
  14. B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
    [CrossRef]

1994 (1)

K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
[CrossRef]

1992 (1)

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

1986 (1)

1982 (1)

1966 (1)

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Alden, M.

K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
[CrossRef]

Astill, A. G.

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
[CrossRef]

Dreier, T.

Ewart, P.

P. Ewart, M. Kaczmarek, “Two-dimensional mapping of temperature in a flame by degenerate four-wave mixing in OH,” Appl. Opt. 30, 3996–3999 (1991).
[CrossRef] [PubMed]

P. Ewart, P. Snowdon, I. Magnusson, “Two-dimensional phase-conjugate imaging of atomic distributions in flames by degenerate four-wave mixing,” Opt. Lett. 14, 563–565 (1989).
[CrossRef] [PubMed]

P. Ewart, S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
[CrossRef] [PubMed]

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

Farrow, R. L.

Fisher, R. A.

R. A. Fisher, Optical Phase Conjugation, (Academic, New York, 1983).

Fritzon, R.

K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
[CrossRef]

Goodman, J. W.

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Greenhalgh, D. A.

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

Hanson, R. K.

Howe, R. D.

Huntley, W. H.

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Jackson, D. W.

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Jefferies, I. P.

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

Kaczmarek, M.

Koch, U. A.

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

Kychakoff, G.

Lehmann, M.

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Magnusson, I.

Mann, B. A.

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

McDaniel, J.

Nyholm, K.

K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
[CrossRef]

O’Leary, S. V.

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

P. Ewart, S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
[CrossRef] [PubMed]

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
[CrossRef]

Rakestraw, D. J.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
[CrossRef]

Smith, P. G. R.

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

Smith, W. J.

W. J. Smith, Modern Optical Engineering: the Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990).

Snowdon, P.

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography, (Academic, New York, 1966).

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

Yates, A. J.

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

Appl. Opt. (2)

Appl. Phys. B (2)

B. A. Mann, S. V. O’Leary, A. G. Astill, D. A. Greenhalgh, “Degenerate 4-wave mixing in nitrogen dioxide: application to combustion diagnostics,” Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

K. Nyholm, R. Fritzon, M. Alden, “Single-pulse two-dimensional temperature imaging in flames by degenerate four-wave mixing and polarization spectroscopy,” Appl. Phys. B 59, 37–43 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. Goodman, W. H. Huntley, D. W. Jackson, M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966).
[CrossRef]

Opt. Lett. (3)

Other (6)

G. W. Stroke, An Introduction to Coherent Optics and Holography, (Academic, New York, 1966).

R. A. Fisher, Optical Phase Conjugation, (Academic, New York, 1983).

P. Ewart, I. P. Jefferies, U. A. Koch, P. G. R. Smith, A. J. Yates, “Novel combustion diagnostic techniques using degenerate four wave mixing,” in Laser Spectroscopy: XIth International Conference, Hot Springs, Virginia, 1993, AIP Conf. Proc. 290, L. Bloomfield, T. Gallagher, D. Larson, eds. (American Institute of Physics, New York, 1994).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, U.K., 1980).

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE Press, Bellingham, Wash., 1989).
[CrossRef]

W. J. Smith, Modern Optical Engineering: the Design of Optical Systems, 2nd ed. (McGraw-Hill, New York, 1990).

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Figures (8)

Fig. 1
Fig. 1

Optical arrangement for imaging by DFWM. The probe beam is considered to be introduced in plane z = 0 and propagates as a plane wave toward the object in plane z = u.

Fig. 2
Fig. 2

Telecentric lens system. The aperture in the Fourier plane governs the range of angles over which light can be collected and contribute to the image.

Fig. 3
Fig. 3

Experimental arrangement for imaging of OH distributions in the methane–air flame. The grating imposes a gridlike structure onto one of the pump beams and also onto light produced by DFWM in the flame. The intensified camera, ICCD, can be placed in the image plane of the telecentric lens system, as shown, or in the Fourier plane at the position of the aperture.

Fig. 4
Fig. 4

(a) Image formed by DFWM of OH in the flame with one pump modulated by a 200-µm grating. The image brightness is reduced on the right side of the flame by the effects of absorption. (b) Intensity distribution recorded by placing the camera in the Fourier plane. The diffraction orders from the grating structure in the object are clearly visible. Note the reduction in intensity of the even orders of diffraction. (c) Image formed when the aperture in the Fourier plane is reduced to 0.5 mm, blocking diffraction orders from the spatial frequency determined by the 200-µm grating. (d) Unmodulated DFWM image of OH distribution in the Bunsen-type flame showing the inner cone of unburnt gas and the local increase in OH density at the outer edge of the flame owing to diffusion of air. Note that the OH signal is weaker on the right-hand side of the image because of absorption.

Fig. 5
Fig. 5

(a) Arrangement of pump beams E 1 and E 2 and probe E 3 in the FFB geometry for DFWM. (b) Sheetlike pumps and circular probe used for imaging in the FFB geometry. The image is formed by light generated in the direction of the signal.

Fig. 6
Fig. 6

Experimental arrangement for simultaneous generation of two images by DFWM in the FFB geometry. Images generated by NO in the two cells are shown in Fig. 7. For image referencing one cell is replaced by the medium to be imaged. A flow of nitrogen seeded with NO was used to generate the image shown in Fig. 8.

Fig. 7
Fig. 7

(a), (b) Simultaneous single-shot DFWM images of NO in two separate cells. (c) Ratio of the two images in (a) and (b). (d), (e) Averages of six shots. (f) ratio of averages of six shots in (d) and (e).

Fig. 8
Fig. 8

(a) Raw DFWM image of a small section at the center of a NO-seeded flow. The direction of the flow is indicated by the arrow. The width of the image is approximately 2 mm. Note the inhomogeneity of the image caused by hot spots in the incident laser beams. (b) Image produced by referencing the image of (a) by a simultaneously recorded image in a cell of NO. Much of the laser-imposed structure on a scale of 100 µm was removed, as shown in (c). The upper red curve in (c) is an intensity profile along a horizontal strip in (a), and the lower blue curve is the corresponding profile in referenced image (b). The intensity scale is in units of relative standard deviation from the mean intensity in raw image (a).

Equations (11)

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Eξ, η=-ikdxdy×expiΦP1E0x, yexpikr2πr,
Ex, y=-ikdξdη expiΦP2E*×ξ, ηξ, ηexpiks2πs,
Ex, y=-k2dxdydξdηE0*x, y×expiΦP2-ΦP1ξ, ηexpiks-r4π2rs.
Ep, q=-ikdxdyEx, yexpikt2πt×exp-ikx2+y22fAx, y.
r=u1+x-ξ2+y-η2u21/2,
ru1+x-ξ2+y-η22u2+O1u4,
r-s=x2-x22u+y2-y22u+x-xξ+y-yηu+O1u3,
Ep, q=K  dxdydξdηdxdyE0*x, y×expiΔΦξ, ηAx, y×expik-x2-y22u+x-xξ+y-yηu-xp+yqv.
ΔΦ=ΦP2-ΦP1.
Pa-Pa¯/σa+3,
Pb-Pb¯/σaPb¯/Pa¯.

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