Abstract

A nonresonant background suppression technique using coherent cancellation through phase mismatching is discussed and applied for a noncollinear beam configuration. A cuvette structure consisting of a glass, a sample, and a glass layer is regarded. Phase mismatching is shown to be a useful method to suppress nonresonant contributions from cuvette glass walls as well as those originating from the sample. A numerical calculation reveals a limit for the background suppression which can be achieved with this technique. Measurements using ethanol as a sample show the possibility to compensate the nonresonant background originating from the cuvette walls and to effectively suppress the nonresonant contribution in the spectrum of the sample by a factor of 10–50, yielding Lorentzian bands biased by a constant background. Direct measurement of depolarization ratios without interfering nonresonant background is demonstrated for ethanol and shows that this technique can readily be combined with the polarization sensitive CARS technique.

© 1989 Optical Society of America

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References

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  1. W. Kiefer, D. A. Long, Eds., Non-Linear Raman Spectroscopy And Its Chemical Applications, NATO Advanced Study Institutes Series (D. Reidel, Holland, 1982).
    [CrossRef]
  2. S. A. J. Druet, J.-P. E. Taran, “CARS Spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
    [CrossRef]
  3. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  4. Y. Prior, “A Complete Expression for the Third-Order Susceptibility (χ(3))—Perturbative and Diagrammatic Approaches,” IEEE J. Quantum Electron. QE-20, 37 (1984).
    [CrossRef]
  5. E. Yarkoni, Y. Prior, “Exact and Approximate Expressions for (χ(3)) in CARS Diagnostics Measurements,” IEEE J. Quantum Electron. QE-20, 43 (1984).
    [CrossRef]
  6. W. M. Tolles, J. W. Nibler, J. R. McDonald, A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc. 31, 253 (1977).
    [CrossRef]
  7. E. K. Gustafson, R. L. Byer, “Coherent Anti-Stokes Raman Scattering from Small Volumes,” Opt. Lett. 9, 220 (1984).
    [CrossRef] [PubMed]
  8. J. W. Nibler, “Coherent Anti-Stokes Raman Scattering of Gases,” in Non-Linear Raman Spectroscopy and Its Chemical Applications, NATO Advanced Study Institutes Series, W. Kiefer, D. A. Long, Eds. (D. Reidel, Holland, 1982), pp. 261–280.
    [CrossRef]
  9. Estimated from unpublished measurements of the 883-cm−1 vibration of ethanol in water.
  10. G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981).
  11. F. M. Kamga, M. G. Sceats, “Pulse-Sequenced Coherent Anti-Stokes Raman Scattering Spectroscopy: a Method for Suppression of the Nonresonant Background,” Opt. Lett. 5, 126 (1980).
    [CrossRef] [PubMed]
  12. R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
    [CrossRef]
  13. H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
    [CrossRef]
  14. S. A. Akhmanov, N. I. Koroteev, “Spectroscopy of Light Scattering and Nonlinear Optics. Nonlinear-Optical Methods of Active Spectroscopy of Raman and Rayleigh Scattering,” Sov. Phys. Usp. 20, 899 (1977).
    [CrossRef]
  15. Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
    [CrossRef]
  16. T. A. H. M. Scholten, G. W. Lucassen, F. F. M. De Mul, J. Greve, “Compensating Pulse-to-Pulse Fluctuations and Increasing Spectral Reproducibility of Phase-Matched CARS Measurements,” Appl. Opt. 27, 3225 (1988).
    [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  18. S. Guha, J. Falk, “The Effects of Focusing in the Three-Frequency Parametric Upconverter,” J. Appl. Phys. 51, 50 (1980).
    [CrossRef]

1988 (1)

1984 (3)

E. K. Gustafson, R. L. Byer, “Coherent Anti-Stokes Raman Scattering from Small Volumes,” Opt. Lett. 9, 220 (1984).
[CrossRef] [PubMed]

Y. Prior, “A Complete Expression for the Third-Order Susceptibility (χ(3))—Perturbative and Diagrammatic Approaches,” IEEE J. Quantum Electron. QE-20, 37 (1984).
[CrossRef]

E. Yarkoni, Y. Prior, “Exact and Approximate Expressions for (χ(3)) in CARS Diagnostics Measurements,” IEEE J. Quantum Electron. QE-20, 43 (1984).
[CrossRef]

1981 (1)

S. A. J. Druet, J.-P. E. Taran, “CARS Spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

1980 (3)

F. M. Kamga, M. G. Sceats, “Pulse-Sequenced Coherent Anti-Stokes Raman Scattering Spectroscopy: a Method for Suppression of the Nonresonant Background,” Opt. Lett. 5, 126 (1980).
[CrossRef] [PubMed]

S. Guha, J. Falk, “The Effects of Focusing in the Three-Frequency Parametric Upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
[CrossRef]

1977 (2)

S. A. Akhmanov, N. I. Koroteev, “Spectroscopy of Light Scattering and Nonlinear Optics. Nonlinear-Optical Methods of Active Spectroscopy of Raman and Rayleigh Scattering,” Sov. Phys. Usp. 20, 899 (1977).
[CrossRef]

W. M. Tolles, J. W. Nibler, J. R. McDonald, A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc. 31, 253 (1977).
[CrossRef]

1976 (2)

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, N. I. Koroteev, “Spectroscopy of Light Scattering and Nonlinear Optics. Nonlinear-Optical Methods of Active Spectroscopy of Raman and Rayleigh Scattering,” Sov. Phys. Usp. 20, 899 (1977).
[CrossRef]

Bloembergen, N.

H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
[CrossRef]

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Byer, R. L.

De Mul, F. F. M.

Druet, S. A. J.

S. A. J. Druet, J.-P. E. Taran, “CARS Spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Eesley, G. L.

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981).

Falk, J.

S. Guha, J. Falk, “The Effects of Focusing in the Three-Frequency Parametric Upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

Fitzgibbon, R.

Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Greve, J.

Guha, S.

S. Guha, J. Falk, “The Effects of Focusing in the Three-Frequency Parametric Upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

Gustafson, E. K.

Harvey, A. B.

Kamga, F. M.

Koroteev, N. I.

S. A. Akhmanov, N. I. Koroteev, “Spectroscopy of Light Scattering and Nonlinear Optics. Nonlinear-Optical Methods of Active Spectroscopy of Raman and Rayleigh Scattering,” Sov. Phys. Usp. 20, 899 (1977).
[CrossRef]

Kramer, S. D.

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

Lax, B.

Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
[CrossRef]

Lotem, H.

H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
[CrossRef]

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

Lucassen, G. W.

Lynch, R. T.

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
[CrossRef]

McDonald, J. R.

Nibler, J. W.

W. M. Tolles, J. W. Nibler, J. R. McDonald, A. B. Harvey, “A Review of the Theory and Applications of Coherent Anti-Stokes Raman Spectroscopy (CARS),” Appl. Spectrosc. 31, 253 (1977).
[CrossRef]

J. W. Nibler, “Coherent Anti-Stokes Raman Scattering of Gases,” in Non-Linear Raman Spectroscopy and Its Chemical Applications, NATO Advanced Study Institutes Series, W. Kiefer, D. A. Long, Eds. (D. Reidel, Holland, 1982), pp. 261–280.
[CrossRef]

Prior, Y.

E. Yarkoni, Y. Prior, “Exact and Approximate Expressions for (χ(3)) in CARS Diagnostics Measurements,” IEEE J. Quantum Electron. QE-20, 43 (1984).
[CrossRef]

Y. Prior, “A Complete Expression for the Third-Order Susceptibility (χ(3))—Perturbative and Diagrammatic Approaches,” IEEE J. Quantum Electron. QE-20, 37 (1984).
[CrossRef]

Sceats, M. G.

Scholten, T. A. H. M.

Taran, J.-P. E.

S. A. J. Druet, J.-P. E. Taran, “CARS Spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Tolles, W. M.

Yacoby, Y.

Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
[CrossRef]

Yarkoni, E.

E. Yarkoni, Y. Prior, “Exact and Approximate Expressions for (χ(3)) in CARS Diagnostics Measurements,” IEEE J. Quantum Electron. QE-20, 43 (1984).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

IEEE J. Quantum Electron (2)

Y. Prior, “A Complete Expression for the Third-Order Susceptibility (χ(3))—Perturbative and Diagrammatic Approaches,” IEEE J. Quantum Electron. QE-20, 37 (1984).
[CrossRef]

E. Yarkoni, Y. Prior, “Exact and Approximate Expressions for (χ(3)) in CARS Diagnostics Measurements,” IEEE J. Quantum Electron. QE-20, 43 (1984).
[CrossRef]

J. Appl. Phys. (2)

Y. Yacoby, R. Fitzgibbon, B. Lax, “Coherent Cancellation of Background in Four-Wave Mixing Spectroscopy,” J. Appl. Phys. 51, 3072 (1980).
[CrossRef]

S. Guha, J. Falk, “The Effects of Focusing in the Three-Frequency Parametric Upconverter,” J. Appl. Phys. 51, 50 (1980).
[CrossRef]

Opt. Commun. (1)

R. T. Lynch, H. Lotem, S. D. Kramer, N. Bloembergen, “Interference Effects in Third-Order Light Mixing Spectroscopy,” Opt. Commun. 18, 109 (1976).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

H. Lotem, R. T. Lynch, N. Bloembergen, “Interference Between Raman Resonances in Four-Wave Difference Mixing,” Phys. Rev. A 14, 1748 (1976).
[CrossRef]

Prog. Quantum Electron. (1)

S. A. J. Druet, J.-P. E. Taran, “CARS Spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Sov. Phys. Usp. (1)

S. A. Akhmanov, N. I. Koroteev, “Spectroscopy of Light Scattering and Nonlinear Optics. Nonlinear-Optical Methods of Active Spectroscopy of Raman and Rayleigh Scattering,” Sov. Phys. Usp. 20, 899 (1977).
[CrossRef]

Other (6)

W. Kiefer, D. A. Long, Eds., Non-Linear Raman Spectroscopy And Its Chemical Applications, NATO Advanced Study Institutes Series (D. Reidel, Holland, 1982).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

J. W. Nibler, “Coherent Anti-Stokes Raman Scattering of Gases,” in Non-Linear Raman Spectroscopy and Its Chemical Applications, NATO Advanced Study Institutes Series, W. Kiefer, D. A. Long, Eds. (D. Reidel, Holland, 1982), pp. 261–280.
[CrossRef]

Estimated from unpublished measurements of the 883-cm−1 vibration of ethanol in water.

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981).

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Figures (14)

Fig. 1
Fig. 1

Planar degenerate four-wave phase match configuration. k 1 and k 2 are the wavevectors of the pump and Stokes beams, respectively; k 3 config is determined by the configuration ( k 1 , k 2 , θ ), while k3,phys is the length of the wavevector at the CARS frequency. θ is the beam crossing angle, ϕ is the signal collection angle. The angular direction ϕ of k 3 phys may deviate from the direction of k 3 config.

Fig. 2
Fig. 2

(a) Symmetrical three-layer configuration with layer 1 identical to layer 3. The configuration is polarized by plane waves which propagate parallel to the z axis. (b) Plot of the nonresonant intensity contribution originating from the triple layer, and the vibration resonant contribution originating from the sample as a function of the z position for A = 0.

Fig. 3
Fig. 3

Maximum value of F (Fmax) for which the condition A = 0 given by Eq. (13) can be fulfilled. Fmax is shown as a function of the ratio (Δk)1/(Δk)2 for different thickness ratios t/s. In the region to the left of the dashed curve (Δk)2 × s = π and Lcoh = s. On the righthand side of the dashed curve (Δk)2 × s is such that Lcohs. Thus in all cases the reduction in resonant signal is less than a factor of 2.5.

Fig. 4
Fig. 4

Calculated nonresonant CARS intensity for a degenerate triple layer of BK7 glass as a function of the signal collection angle and with the beam crossing angle as a parameter, (a) Configuration and definitions used for the calculation, (b)–(f) Calculation parameters: wavenumber shift 1200 cm−1 (relative to 532 nm); wavevector values are those for BK7 glass; waist diameters of the pump and Stokes beams, 160 μm; intensity calculated at a distance of 50 cm from the sample; coherence length at θ = 0°; Lcoh = 763 μm. (b) Total thickness (s + 2 ⋅ t) 200 μm; (c) total thickness 800 μm; (d) enlargement of (c) showing the double-lobed curves; (e) total thickness 2000 μm; (f) enlargement of (e).

Fig. 5
Fig. 5

Calculated trajectories of the peak CARS signal collection angle ϕc as a function of the sample thickness and with the crossing angle θ as a parameter. Calculation parameters are the same as those in Fig. 4. Positions of double lobes with equal lobe intensity are indicated by a dashed line.

Fig. 6
Fig. 6

Total nonresonant CARS signal vs layer thickness as calculated from an integration of the curves in Fig. 4 over the signal collection angle ϕ: (a) curves for beam crossing angles θ smaller than the matching angle; (b) curves for θ larger than the matching angle.

Fig. 7
Fig. 7

Measured nonresonant CARS intensity at 1000 cm−1 of a glass/ethanol/glass triple layer as a function of the collection angle and with the crossing angle as a parameter: (a),(b) 0.4/0.5/0.4-mm configuration; (c),(d) 0.15/0.25/0.15-mm configuration. Angles are externally measured values.

Fig. 8
Fig. 8

Qualitative plot of the nonresonant intensity originating from the triple layer and from the single sample layer as a function of the crossing angle; definition of the net and gross suppression ratios.

Fig. 9
Fig. 9

Calculated net suppression ratio in the first minimum at the larger angle side of the matching angle, expressed in units of S. The net suppression is shown as a function of the sample thickness s of a glass/ethanol/glass triple layer with F ≈ 1.25.

Fig. 10
Fig. 10

Phase mismatch such that A = F × [ e i χ N R , glass e 1 e 1 e 2 ] .. Compensation of the nonresonant signal of the 0.4-mm cuvette walls. (a) Normal CARS spectrum of ethanol (without glass interference); (b) CARS spectrum of a 0.4/0.08/0.4-mm triple layer with the crossing angle equal to the matching angle; (c) CARS spectrum of the same triple layer with the crossing angle such that A = F × [ e i χ N R , glass e 1 e 1 e 2 ] . The spectra are plotted on an arbitrary intensity scale.

Fig. 11
Fig. 11

Phase mismatch such that A = 0. Compensation of the nonresonant signal of cuvette walls and sample medium. The triple layer is a 0.15/0.25/0.15-mm glass/ethanol/glass configuration. The linearly polarized pump beam (which defines the e 1 = e x direction) is oriented under a 45° angle with respect to the linearly polarized Stokes beam: (a) signal analyzer parallel to the e 1 direction thus the e x χ R e x e x e x contribution; (b) analyzer under a 45° angle with the e 1 direction thus the e x χ R e x e x e x + e y χ R e x e x e y contribution; (c) analyzer perpendicular to the e 1 direction thus the e y χ R e x e x e y contribution.

Fig. 12
Fig. 12

CARS spectra of the 883-cm−1 vibration of ethanol at different collection angles in a double-lobed minimum. The triple layer is a 0.4/0.5/0.4-mm configuration. The beam crossing angle is the same for all three spectra (1.90° at 883 cm−1): (a) positive dispersive band shape at a collection angle (1.27°) smaller than the angle for which the background is minimal; (b) Lorentzian band shape at the minimum position (1.52°); (c) negative dispersive band shape at a collection angle (1.77°) larger than the angle at the minimum.

Fig. 13
Fig. 13

Calculated, normalized spectral line profiles of the 883-cm−1 vibration of a 0.4/0.5/0.4-mm configuration using Eqs. (6) and (10) and F = 1.25 , B / Γ = 1.51 × [ e i χ N R , glass e 1 e 1 e 2 ] , n1=1.56, n2=1.36. (a) Positive dispersive spectral contour at a crossing angle (1.55°) smaller than the angle for which the background is zero; (b) Lorentzian contour at the angle for zero background (1.70°); (c) negative dispersive contour at a crossing angle (1.85°) larger than the angle for zero background.

Fig. 14
Fig. 14

Measured gross suppression ratio S of a layer of water as a function of layer thickness. The matching angle at which a minimum occurs in the background is ~1.6° for an 8-mm layer thickness and ~1.9° for a 1.5-mm layer thickness.

Tables (1)

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Table I Depolarization Ratios and Other Line Parameters of Ethanol as Determined from Fig. 11

Equations (32)

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χ ( 3 ) = χ N R ( 3 ) + χ R ( 3 ) .
I CARS [ χ e 1 e 2 e 3 ] [ χ e 1 e 2 e 3 ] * ,
I CARS = 9 16 ω 3 2 ε 0 4 c 4 n 1 2 n 2 n 3  | χ e 1 e 1 e 2  | 2 I 1 2 I 2 s 2 sinc 2 ( Δ k s /2),
Δ k = | k 3  | phys | k 3  | config .
| k 3  | phys = n 3 ω 3 / c .
| k 3  | config = [ 4 | k 1  | 2 + | k 2  | 2 4 | k 1 | | k 2  | cos  ( θ ) ] 1 / 2 ,
Δ k = Δ k 0 [ | k 1 | | k 2 | / ( 2 | k 1 | | k 2 | ) ] θ 2 ,
L coh = π / Δ k ,
e i χ R sample e 1 e 1 e 2 = B / ( δ i Γ ) ,
I CARS s 2 sinc 2 [ ( Δ k ) 2 s / 2 ] × [ A 2 + 2 A ( B Γ ) ( f f 2   +   1 ) + ( B Γ ) 2 ( 1 f 2   +   1 ) ]  ,
A = e i χ N R , glass e 1 e 1 e 2 { F + 2 sinc [ ( Δ k ) 1 t / 2 ] sinc [ ( Δ k ) 2 s / 2 ] t s ×  cos  [ ( Δ k ) 1 t + ( Δ k ) 2 s 2 ] } ,
( Δ k ) 1 t + ( Δ k ) 2 s = π  .
2 sinc  [ ( Δ k ) 1   t / 2 ] sinc  [ ( Δ k ) 2   s / 2 ] t s  cos  [ ( Δ k ) 1 t + ( Δ k ) 2 s 2 ] = F .
E 1 = e 1 E 0 1  exp [ i   ( k 1 z ω 1 t ) ( x 2 + y 2 ) / w 1 2 ]  ,
E 2 = e 2 E 0 2  exp( i { k 2 [ z cos ( θ ) + x sin( θ ) ] ω 2 t } { [ x cos( θ ) z sin( θ ) ] 2 + y 2 } / w 2 2 ) ,
sin ( ϕ c ) = | k 2 | / | k 3  | config ×  sin ( θ ) .
θ = 1.74 + σ 5.95 × 10 4 ϕ = 1.88 + σ 2.90 × 10 4   with  σ the wavenumber shift   ( c m 1 ) .
V ( G 2 E E 2 G ) d V = S [ G ( E / n ) E ( G / n ) ] d S ,
2 E 3 i ( n / c ) 2 ( 2 E 3 i / t 2 ) = μ 0 ( 2 P 3 i ( 3 ) / t 2 ) ,
2 E 3 i + k 3 2 E 3 i = μ 0 ω 3 2 P 3 i ( 3 ) ,
2 G = k 3 2 G ,
μ 0 ω 3 2 v P 3 i ( 3 ) * G d V .
G = exp  ( i k 3 | r r | ) / | r r ' | ,
S [ G ( E 3 i / n ) E 3 i ( G / n ) ] d S 4 π E 3 i ( r ) .
E 3 i ( r ) = μ 0 ω 3 2 / ( 4 π ) × V   P 3 i ( 3 ) * G d V
E 3 i ( z ) z P 3 i ( 3 ) ( z )  exp { [ i k 3 ( z z ) ] } d z .
P 3 i ( 3 ) ( z ) = 3 8 e i χ ( 3 ) E 1 2 E 2 *  ,
l p q = l q p = 2 n p n q / ( n p + n q ) ,
E 3 i e i χ layer 1 ( 3 ) e 1 e 1 e 2  exp  [ i ( φ 1 2 ) ] ×  sin  ( φ 1 2 ) ( Δ k ) 1 + l 12 2 e i χ layer 2 ( 3 ) e 1 e 1 e 2  exp  [ i ( φ 1 + φ 2 2 ) ] ×  sin  ( φ 2 2 ) ( Δ k ) 2 + l 12 2 l 23 2 e i χ layer 3 ( 3 ) e 1 e 1 e 2 ×  exp  [ i ( φ 1 + φ 2 + φ 3 2 ) ] ×  sin  ( φ 3 2 ) ( Δ k ) 3 ,
e i χ R , sample e 1 e 1 e 2 = B / ( δ i Γ ) ,
I CARS i [  sin  ( φ 2 / 2 ) ( Δ k ) 2 ] 2 × [ A 2 + 2 A ( B Γ ) f f 2   +   1 + ( B Γ ) 2 1 f 2   +   1 ]  ,
A = e i χ N R , glass e 1 e 1 e 2 × [ F + 2  sin  ( φ 1 / 2 ) ( Δ k ) 1 ( Δ k ) 2  sin ( φ 2 / 2 ) ]  cos  [ ( φ 1 + φ 2 ) / 2 ]  ,

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