Abstract

Laser Rayleigh imaging has been applied in a number of flow and flame studies to measure concentration or temperature distributions. Rayleigh cross sections are dependent on the index of refraction of the scattering medium. The same index of refraction changes that provide contrast in Rayleigh images can also deflect the illuminating laser sheet. By applying a ray-tracing algorithm to the detected image, it is possible to correct for some of these beam-steering effects and thereby improve the accuracy of the measured field. Additionally, the quantification of the degree of beam steering through the flow provides information on the degradation of spatial resolution in the measurement. Application of the technique in a well-studied laboratory flame is presented, along with analysis of the effects of image noise and spatial resolution on the effectiveness of the algorithm.

© 2005 Optical Society of America

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References

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  1. G. Kychakoff, R. Howe, R. K. Hanson, J. McDaniel, “Quantitative visualization of combustion species in a plane,” Appl. Opt. 21, 3225–3227 (1982).
    [CrossRef] [PubMed]
  2. G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
    [CrossRef] [PubMed]
  3. R. K. Hanson, “Combustion diagnostics: planar imaging techniques,” Proc. Combust. Inst. 21, 1677–1691 (1986).
    [CrossRef]
  4. M. C. Escoda, M. B. Long, “Rayleigh scattering measurements of the gas concentration field in turbulent jets,” AIAA J. 21, 81–84 (1983).
    [CrossRef]
  5. D. R. Dowling, P. E. Dimotakis, “Similarity of the concentration field of gas-phase turbulent jets,” J. Fluid Mech. 218, 109–141 (1990).
    [CrossRef]
  6. R. W. Dibble, R. E. Hollenbach, “Laser Rayleigh thermometry in turbulent flames,” Proc. Combust. Inst. 18, 1489–1499 (1981).
    [CrossRef]
  7. D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
    [CrossRef]
  8. V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
    [CrossRef]
  9. Y.-C. Chen, M. S. Mansour, “Topology of turbulent premixed flame fronts resolved by simultaneous planar imaging of LIPF of OH radical and Rayleigh scattering,” Exp. Fluids. 26, 277–287 (1999).
    [CrossRef]
  10. J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
    [CrossRef]
  11. R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
    [CrossRef]
  12. P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
    [CrossRef]
  13. W. Merzkirch, Flow visualization (Academic, 1974).
  14. P. A. Kalt, M. B. Long, “OMA—image processing for Mac OS X,” www.oma-x.org (accessed May 2005; includes the ray-tracing algorithms used here).
  15. R. S. Barlow, ed., “International workshop on measurement and computation of turbulent nonpremixed flames,” www.ca.sandia.gov/TNF/ (accessed May 2005).
  16. R. S. Barlow, J. H. Frank, “Effects of turbulence on species mass fractions in methane/air jet flames,” Proc. Combust. Inst. 27, 1087–1095 (1998).
    [CrossRef]
  17. A. N. Karpetis, R. S. Barlow, “Measurements of scalar dissipation in a turbulent piloted methane/air jet flame,” Proc. Combust. Inst. 29, 1929–1936 (2002).
    [CrossRef]
  18. J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
    [CrossRef]

2002 (3)

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

A. N. Karpetis, R. S. Barlow, “Measurements of scalar dissipation in a turbulent piloted methane/air jet flame,” Proc. Combust. Inst. 29, 1929–1936 (2002).
[CrossRef]

J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
[CrossRef]

2001 (2)

R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
[CrossRef]

P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
[CrossRef]

1999 (1)

Y.-C. Chen, M. S. Mansour, “Topology of turbulent premixed flame fronts resolved by simultaneous planar imaging of LIPF of OH radical and Rayleigh scattering,” Exp. Fluids. 26, 277–287 (1999).
[CrossRef]

1998 (2)

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

R. S. Barlow, J. H. Frank, “Effects of turbulence on species mass fractions in methane/air jet flames,” Proc. Combust. Inst. 27, 1087–1095 (1998).
[CrossRef]

1990 (1)

D. R. Dowling, P. E. Dimotakis, “Similarity of the concentration field of gas-phase turbulent jets,” J. Fluid Mech. 218, 109–141 (1990).
[CrossRef]

1986 (2)

R. K. Hanson, “Combustion diagnostics: planar imaging techniques,” Proc. Combust. Inst. 21, 1677–1691 (1986).
[CrossRef]

D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
[CrossRef]

1984 (1)

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

1983 (1)

M. C. Escoda, M. B. Long, “Rayleigh scattering measurements of the gas concentration field in turbulent jets,” AIAA J. 21, 81–84 (1983).
[CrossRef]

1982 (1)

1981 (1)

R. W. Dibble, R. E. Hollenbach, “Laser Rayleigh thermometry in turbulent flames,” Proc. Combust. Inst. 18, 1489–1499 (1981).
[CrossRef]

Barlow, R. S.

A. N. Karpetis, R. S. Barlow, “Measurements of scalar dissipation in a turbulent piloted methane/air jet flame,” Proc. Combust. Inst. 29, 1929–1936 (2002).
[CrossRef]

R. S. Barlow, J. H. Frank, “Effects of turbulence on species mass fractions in methane/air jet flames,” Proc. Combust. Inst. 27, 1087–1095 (1998).
[CrossRef]

Bergmann, V.

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

Catrakis, H. J.

P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
[CrossRef]

Chen, Y.-C.

Y.-C. Chen, M. S. Mansour, “Topology of turbulent premixed flame fronts resolved by simultaneous planar imaging of LIPF of OH radical and Rayleigh scattering,” Exp. Fluids. 26, 277–287 (1999).
[CrossRef]

Dibble, R. W.

R. W. Dibble, R. E. Hollenbach, “Laser Rayleigh thermometry in turbulent flames,” Proc. Combust. Inst. 18, 1489–1499 (1981).
[CrossRef]

Dimotakis, P. E.

P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
[CrossRef]

D. R. Dowling, P. E. Dimotakis, “Similarity of the concentration field of gas-phase turbulent jets,” J. Fluid Mech. 218, 109–141 (1990).
[CrossRef]

Dowling, D. R.

D. R. Dowling, P. E. Dimotakis, “Similarity of the concentration field of gas-phase turbulent jets,” J. Fluid Mech. 218, 109–141 (1990).
[CrossRef]

Drake, M.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

Escoda, M. C.

M. C. Escoda, M. B. Long, “Rayleigh scattering measurements of the gas concentration field in turbulent jets,” AIAA J. 21, 81–84 (1983).
[CrossRef]

Fielding, J.

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

Forkey, J. N.

R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
[CrossRef]

Fourguette, D. C.

P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
[CrossRef]

D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
[CrossRef]

Frank, J. H.

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
[CrossRef]

R. S. Barlow, J. H. Frank, “Effects of turbulence on species mass fractions in methane/air jet flames,” Proc. Combust. Inst. 27, 1087–1095 (1998).
[CrossRef]

Hanson, R. K.

R. K. Hanson, “Combustion diagnostics: planar imaging techniques,” Proc. Combust. Inst. 21, 1677–1691 (1986).
[CrossRef]

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

G. Kychakoff, R. Howe, R. K. Hanson, J. McDaniel, “Quantitative visualization of combustion species in a plane,” Appl. Opt. 21, 3225–3227 (1982).
[CrossRef] [PubMed]

Hollenbach, R. E.

R. W. Dibble, R. E. Hollenbach, “Laser Rayleigh thermometry in turbulent flames,” Proc. Combust. Inst. 18, 1489–1499 (1981).
[CrossRef]

Howe, R.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

G. Kychakoff, R. Howe, R. K. Hanson, J. McDaniel, “Quantitative visualization of combustion species in a plane,” Appl. Opt. 21, 3225–3227 (1982).
[CrossRef] [PubMed]

Kaiser, S. A.

J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
[CrossRef]

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

Karpetis, A. N.

A. N. Karpetis, R. S. Barlow, “Measurements of scalar dissipation in a turbulent piloted methane/air jet flame,” Proc. Combust. Inst. 29, 1929–1936 (2002).
[CrossRef]

Kychakoff, G.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

G. Kychakoff, R. Howe, R. K. Hanson, J. McDaniel, “Quantitative visualization of combustion species in a plane,” Appl. Opt. 21, 3225–3227 (1982).
[CrossRef] [PubMed]

Lapp, M.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

Lempert, W. R.

R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
[CrossRef]

Long, M. B.

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
[CrossRef]

D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
[CrossRef]

M. C. Escoda, M. B. Long, “Rayleigh scattering measurements of the gas concentration field in turbulent jets,” AIAA J. 21, 81–84 (1983).
[CrossRef]

Mansour, M. S.

Y.-C. Chen, M. S. Mansour, “Topology of turbulent premixed flame fronts resolved by simultaneous planar imaging of LIPF of OH radical and Rayleigh scattering,” Exp. Fluids. 26, 277–287 (1999).
[CrossRef]

McDaniel, J.

Meier, W.

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

Merzkirch, W.

W. Merzkirch, Flow visualization (Academic, 1974).

Miles, R. B.

R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
[CrossRef]

Penney, C.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

Pitz, R.

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

Stricker, W.

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

Wolff, D.

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

Zurn, R. M.

D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
[CrossRef]

AIAA J. (1)

M. C. Escoda, M. B. Long, “Rayleigh scattering measurements of the gas concentration field in turbulent jets,” AIAA J. 21, 81–84 (1983).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

V. Bergmann, W. Meier, D. Wolff, W. Stricker, “Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4/H2/N2 jet diffusion flame,” Appl. Phys. B 66, 489–502 (1998).
[CrossRef]

Combust. Sci. Tech. (1)

D. C. Fourguette, R. M. Zurn, M. B. Long, “Two-dimensional Rayleigh thermometry in a turbulent nonpremixed methane hydrogen flame,” Combust. Sci. Tech. 44, 307–317 (1986).
[CrossRef]

Exp. Fluids. (1)

Y.-C. Chen, M. S. Mansour, “Topology of turbulent premixed flame fronts resolved by simultaneous planar imaging of LIPF of OH radical and Rayleigh scattering,” Exp. Fluids. 26, 277–287 (1999).
[CrossRef]

J. Fluid Mech. (2)

D. R. Dowling, P. E. Dimotakis, “Similarity of the concentration field of gas-phase turbulent jets,” J. Fluid Mech. 218, 109–141 (1990).
[CrossRef]

P. E. Dimotakis, H. J. Catrakis, D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 433, 105–134 (2001).
[CrossRef]

Meas. Sci. Technol. (1)

R. B. Miles, W. R. Lempert, J. N. Forkey, “Laser Rayleigh scattering,” Meas. Sci. Technol. 12, R33–R51 (2001).
[CrossRef]

Proc. Combust. Inst. (6)

J. Fielding, J. H. Frank, S. A. Kaiser, M. B. Long, “Polarized/depolarized Rayleigh scattering for determining fuel concentrations in flames,” Proc. Combust. Inst. 29, 2703–2709 (2002).
[CrossRef]

R. W. Dibble, R. E. Hollenbach, “Laser Rayleigh thermometry in turbulent flames,” Proc. Combust. Inst. 18, 1489–1499 (1981).
[CrossRef]

R. K. Hanson, “Combustion diagnostics: planar imaging techniques,” Proc. Combust. Inst. 21, 1677–1691 (1986).
[CrossRef]

R. S. Barlow, J. H. Frank, “Effects of turbulence on species mass fractions in methane/air jet flames,” Proc. Combust. Inst. 27, 1087–1095 (1998).
[CrossRef]

A. N. Karpetis, R. S. Barlow, “Measurements of scalar dissipation in a turbulent piloted methane/air jet flame,” Proc. Combust. Inst. 29, 1929–1936 (2002).
[CrossRef]

J. H. Frank, S. A. Kaiser, M. B. Long, “Reaction-rate, mixture-fraction, and temperature imaging in turbulent methane/air jet flames,” Proc. Combust. Inst. 29, 2687–2694 (2002).
[CrossRef]

Science (1)

G. Kychakoff, R. Howe, R. K. Hanson, M. Drake, R. Pitz, M. Lapp, C. Penney, “Visualization of turbulent flame fronts with planar laser-induced fluorescence,” Science 224, 382–384 (1984).
[CrossRef] [PubMed]

Other (3)

W. Merzkirch, Flow visualization (Academic, 1974).

P. A. Kalt, M. B. Long, “OMA—image processing for Mac OS X,” www.oma-x.org (accessed May 2005; includes the ray-tracing algorithms used here).

R. S. Barlow, ed., “International workshop on measurement and computation of turbulent nonpremixed flames,” www.ca.sandia.gov/TNF/ (accessed May 2005).

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Figures (8)

Fig. 1
Fig. 1

Pixel stencil used in the ray-tracing implementation.

Fig. 2
Fig. 2

Single-shot Rayleigh image from a turbulent methane–air jet flame (Flame D) at 5 jet diameters downstream. (a) original image, (b) corrected image using columnwise ray tracing, (c) calculated sheet-intensity deviation based on the columnwise ray-tracing approach, (d) intensity deviation along the vertical line at the right of the images in (a) and (c). Images are 37 mm × 9.6 mm, corresponding to 5.2 d × 1.3 d.

Fig. 3
Fig. 3

(a) Rayleigh image from a turbulent flame. (b) Rayleigh image corrected for beam steering by using the iterative ray-tracing algorithm. Images are 37 mm × 9.6 mm.

Fig. 4
Fig. 4

Performance of the beam-steering correction for different noise levels in the original image. Solid curve, iterative ray-tracing algorithm; dashed curve, columnwise technique. See text for details on the performance measure.

Fig. 5
Fig. 5

Performance of the beam-steering correction for different pixel spacings. Solid curve, iterative ray-tracing algorithm; dashed curve, columnwise technique.

Fig. 6
Fig. 6

Ray tracing of scattered light from the probe volume to the detector. (a) Extended Rayleigh image with flame, lens, and detector, (b) corresponding intensity distribution, is displayed in an arbitrary logarithmic color scale corresponding to log(rays/pixel). The Spatial coverage of the images is 61 mm × 12 mm.

Fig. 7
Fig. 7

Deviation of rays propagating across a turbulent flame. Two curves are shown. One for the actual data taken at 1 atm; the second is the simulated behavior at 10 atm. To within the noise of the pdf, the curves are the same.

Fig. 8
Fig. 8

(a) Simulated Rayleigh image from a 10 atm flame; (b) image corrected by using the columnwise ray-tracing algorithm; (c) blue, sheet-intensity modulation from the right side of the uncorrected image; red, sheet intensity modulation calculated by the ray-tracing algorithm. Images are 37 mm × 9.6 mm.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

S Ray = K I 0 N V i x i σ i = K I 0 N V σ eff ,
σ = 24 π 3 λ 4 N 2 ( n 2 - 1 n 2 + 2 ) 2 ,
( n 2 - 1 ) ( n 2 + 2 ) 2 3 ( n - 1 )
S Ray = K I 0 V 32 π 3 3 λ 4 ( n - 1 ) 2 N .
n - 1 S Ray N .
n - 1 S Ray
n - 1 S Ray .
n 1 sin θ 1 = n 2 sin θ 2 .
β = π 2 - tan - 1 Δ n x Δ n y , n 2 = n 1 + r ^ · n ,
η r = ( N R - N L ) Orig - ( N R - N L ) Corr ( N R - N L ) Orig .

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