Abstract

The process of high-gain image amplification in photorefractive crystals is investigated. We show that in the regime of large energy transfer, the spatially nonuniform depletion of the pump wave results in the introduction of distortions into the amplified output image. This distortion mechanism and its dependence on the characteristics of the image to be amplified are described theoretically through numerical solutions of the coupled-wave equations for image-bearing beams. The numerical results are then checked experimentally in photorefractive BaTiO3.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Fainman, C. C. Guest, S. H. Lee, “Optical digital logic operation by two-beam coupling in photorefractive materials,” Appl. Opt. 25, 1598–1603 (1986).
    [CrossRef]
  2. A. Chiou, P. Yeh, J. Hong, “Optical interconnection using photorefractive dynamic holograms,” Appl. Opt. 27, 2093–2096 (1988).
    [CrossRef] [PubMed]
  3. I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.
  4. L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
    [CrossRef]
  5. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
    [CrossRef]
  6. J. Ma, L. Liu, S. Wu, Z. Wang, L. Xu, B. Shu, “Multibeam coupling in photorefractive SBN:Ce,” Opt. Lett. 13, 1020–1022 (1988).
    [CrossRef] [PubMed]
  7. M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [CrossRef]
  8. F. Vachss, L. Hesselink, “Selective enhancement of spatial harmonics of a photorefractive grating,” J. Opt. Soc. Am. B 5, 1814–1821 (1988).
    [CrossRef]
  9. F. Vachss, “Non-linear holographic response in photorefractive materials,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).
  10. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  11. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  12. C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  13. M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
    [CrossRef]
  14. P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  15. F. Vachss, L. Hesselink, “Nonlinear photorefractive response at high modulation depths,” J. Opt. Soc. Am. A 5, 690–701 (1988).
    [CrossRef]
  16. D. A. Temple, C. Warde, “High-order anisotropic diffraction in photorefractive crystals,” J. Opt. Soc. Am. B 5, 1800–1805 (1988).
    [CrossRef]
  17. L. M. Connors, “Optical mixing in photorefractive media with applications to real-time holography and phase conjugation,” Ph.D. dissertation (Queen Elizabeth College, University of London, London, 1984).
  18. F. Vachss, P. Yeh, “Image degradation and preservation in photorefractive amplifiers,” in Digest of 1988 Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988).

1988 (5)

1987 (1)

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

1986 (1)

1985 (2)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1982 (1)

L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
[CrossRef]

1979 (3)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Beckwith, P.

I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.

Bender, C. M.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Chiou, A.

Christian, W.

I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
[CrossRef]

L. M. Connors, “Optical mixing in photorefractive media with applications to real-time holography and phase conjugation,” Ph.D. dissertation (Queen Elizabeth College, University of London, London, 1984).

Cory, W. K.

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

Ewbank, M.

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

Fainman, Y.

Feinberg, J.

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

Fiddy, M. A.

L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Guest, C. C.

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
[CrossRef]

Hesselink, L.

Hong, J.

Huignard, J. P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Khoshnevisan, M.

I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Lee, S. H.

Liu, L.

Ma, J.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

McMichael, I.

I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Neurgaonkar, R. R.

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Orszag, S. A.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Rajbenbach, H.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Shu, B.

Solymar, L.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Temple, D. A.

Vachss, F.

F. Vachss, L. Hesselink, “Nonlinear photorefractive response at high modulation depths,” J. Opt. Soc. Am. A 5, 690–701 (1988).
[CrossRef]

F. Vachss, L. Hesselink, “Selective enhancement of spatial harmonics of a photorefractive grating,” J. Opt. Soc. Am. B 5, 1814–1821 (1988).
[CrossRef]

F. Vachss, P. Yeh, “Image degradation and preservation in photorefractive amplifiers,” in Digest of 1988 Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988).

F. Vachss, “Non-linear holographic response in photorefractive materials,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

Wang, Z.

Warde, C.

Wu, S.

Xu, L.

Yeh, P.

A. Chiou, P. Yeh, J. Hong, “Optical interconnection using photorefractive dynamic holograms,” Appl. Opt. 27, 2093–2096 (1988).
[CrossRef] [PubMed]

F. Vachss, P. Yeh, “Image degradation and preservation in photorefractive amplifiers,” in Digest of 1988 Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988).

Young, L.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

L. M. Connors, T. J. Hall, M. A. Fiddy, “On coupled-wave theory of two-beam self-diffraction,” Appl. Phys. B 28, 31–35 (1982).
[CrossRef]

Ferroelectrics (2)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, II: beam coupling,” Ferroelectrics 22, 961–964 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals, I: steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

J. Appl. Phys. (3)

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

M. Ewbank, R. R. Neurgaonkar, W. K. Cory, J. Feinberg, “Photorefractive properties of strontium barium niobate,” J. Appl. Phys. 62, 374–380 (1987).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect: a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Other (5)

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

F. Vachss, “Non-linear holographic response in photorefractive materials,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1988).

I. McMichael, M. Khoshnevisan, P. Beckwith, W. Christian, “Non-linear ranging imager,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1988), pp. 218–219.

L. M. Connors, “Optical mixing in photorefractive media with applications to real-time holography and phase conjugation,” Ph.D. dissertation (Queen Elizabeth College, University of London, London, 1984).

F. Vachss, P. Yeh, “Image degradation and preservation in photorefractive amplifiers,” in Digest of 1988 Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Image amplification by two-wave mixing showing the resultant nonuniform depletion of the pump.

Fig. 2
Fig. 2

Coupling geometry for the interaction of two complex wave fronts within a photorefractive material of thickness L. The pump wave is shown at the top, and the image at the bottom.

Fig. 3
Fig. 3

Pump-depletion geometry showing the various segments of the image wave interacting with a given pump ray. Shown are the pump source point (x1, 0) and the image source point (x2, 0) for the observation point (x, z). Since the image intensity grows as exp(Γ2z), the pump ray at (x, z) is dependent on image values averaged over a transverse interval of width roughly equal to x0 ≡ (tan θ1 + tan θ2)/Γ2.

Fig. 4
Fig. 4

(a) Numerical solutions of the coupled Eqs. (11a) and (11b) showing the intensity plotted vertically against the transverse (x) coordinate on the horizontal axis. The image and pump intensity profiles (in solid and dashed curves, respectively) are plotted at the input plane (z = 0) 5 mm and 1 cm deep into the photorefractive medium. (b) Comparison of numerical solutions for the transverse image-intensity profile (solid curves) with the approximate result of expression (21) (dotted curves). Intensity versus transverse coordinate profiles are plotted for propagation depths of 1.0 and 1.5 cm at the top and bottom, respectively, using the same input image as in (a). Note that the agreement improves as the depletion of the pump increases for the longer coupling distance.

Fig. 5
Fig. 5

Numerical solutions of Eqs. (11a) and (11b) for the transverse image (solid) and pump (dashed) intensity profiles for a low-contrast input image. Shown are the profiles at the input plane and 1 cm deep into the material. Note the reduced degree of distortion compared with the high-contrast result.

Fig. 6
Fig. 6

Experimental setup. An Ar+ laser source is expanded, collimated, and split into two paths. One path is passed through a transparency and imaged into a BaTiO3 crystal, while the other is sent directly into the crystal and acts as the pump beam. After amplification, the image beam is imaged onto a linear photodetector array to obtain the transverse intensity profile.

Fig. 7
Fig. 7

Experimental results showing transverse intensity profiles of the image beam taken from the linear detector array. Results are shown with the pump beam blocked so no amplification occurs (left) and with the pump beam unblocked so image amplification and the associated distortion occurs (right). At the bottom, numerical solutions of the coupled equations corresponding to these experimental conditions are provided for comparison.

Fig. 8
Fig. 8

Numerical results showing the dependence of image gain and distortion on the choice of the coupling function f(m). In all cases the image (solid curves) and the pump (dashed curves) intensity profiles are plotted versus transverse position at a coupling depth of 1 cm within the medium. The three cases shown use f(m) = [1 −(1 − m2)1/2]/m as obtained in the case of negligible space-charge saturation by Moharam et al. in the top plot, f(m) = [1 − exp(am)]/2a as obtained in the case of strong saturation by Refregier et al. in the center plot, and the small modulation depth result of Kukhtarev et al., f(m) = m/2, shown at the bottom.

Equations (30)

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

E ( r ) = R ( x , y , z ) exp ( i k 1 r ) + S ( x , y , z ) exp ( i k 2 r )
2 E ( r ) + ω 2 c 2 [ + Δ ( r ) ] E ( r ) = 0 ,
I ( r ) | E ( r ) | 2 = | R ( r ) | 2 + | S ( r ) | 2 + R ( r ) S * ( r ) exp [ ( k 1 k 2 ) r ] + c . c . I 0 ( r ) { 1 + m ( r ) cos [ k G r + ϕ 0 ( r ) ] } ,
I 0 ( r ) | R ( r ) | 2 + | S ( r ) | 2 , m ( r ) 2 | R ( r ) S ( r ) | / I 0 ( r ) , k G k 1 k 2 , ϕ 0 ( r ) arg [ R ( r ) S * ( r ) ] .
Δ ( r ) = 2 r eff E 1 ( m ) exp [ i ( k G r + ϕ 0 ) ] + c . c . ,
Δ ( r , η ) = n = 0 Δ n ( r , η ) δ n .
Δ 0 ( r , η ) = 2 r eff E 1 [ m ( η ) ] exp { i [ k G r + ϕ 0 ( η ) ] } + c . c .
2 R + 2 i k 1 R + ( ω / c ) 2 r eff E 1 [ m ( r ) ] exp [ i ϕ 0 ( r ) ] S = 0 ,
2 S + 2 i k 2 S + ( ω / c ) 2 r eff E 1 * [ m ( r ) ] exp [ i ϕ 0 ( r ) ] R = 0 ,
( 2 i k 0 cos θ i ) 1 2 R + ( cos θ i ) 1 k ̂ 1 R + Γ 2 f [ m ( r ) ] | S | | R | R = 0 ,
( 2 i k 0 cos θ i ) 1 2 S + ( cos θ i ) 1 k ̂ 2 S Γ * 2 f [ m ( r ) ] | R | | S | S = 0 ,
Γ i k 0 r eff Ȇ 1 / cos θ i ,
r 1 , 2 O [ ( 1 / d ) 2 + ( Γ / 2 ) 2 ( Γ / 2 ) ( 4 π / λ ) ] ,
( cos θ i ) 1 k ̂ 1 I R + Γ 0 f [ m ( r ) ] ( I S I R ) 1 / 2 = 0 ,
( cos θ i ) 1 k ̂ 2 I S Γ 0 f [ m ( r ) ] ( I S I R ) 1 / 2 = 0 ,
f ( m ) = [ 1 ( 1 m 2 ) 1 / 2 ] / m ,
f ( m ) [ 1 exp ( a m ) ] / 2 a ,
f [ m ( r ) ] ( I S I R ) 1 / 2 = I S I R / ( I S + I R ) I S ,
I S ( x , z ) = I S ( x z tan θ 2 , 0 ) exp ( Γ 0 z cos θ i / cos θ 2 ) = I S ( x 2 , 0 ) exp ( Γ 2 z )
I R ( x , z ) = I R ( x + z tan θ 1 , 0 ) Γ 0 ( cos θ i / cos θ 1 ) 0 z I S [ x + ( z z ) tan θ 1 , z ] d z = I R ( x 1 , 0 ) Γ 1 0 z I S [ x + z tan θ 1 z ( tan θ 1 + tan θ 2 ) , 0 ] exp ( Γ 2 z ) d z = I R ( x 1 , 0 ) exp ( Γ 2 z ) cos θ 2 / cos θ 1 0 x 1 x 2 I S ( x + x 2 , 0 ) exp ( x / x 0 ) d x / x 0 ,
I S ( x , z ) < I R ( x , z ) for z < z 0
I R ( x , z ) < I S ( x , z ) for z > z 0 .
f ( m ) ( I S I R ) 1 / 2 = { I S for I S < I R I R for I S > I R .
I R ( x , z > z 0 ) = I R [ x + ( z z 0 ) tan θ 1 , z 0 ] exp [ Γ 1 ( z 0 z ) ]
I S ( x , z > z 0 ) = I S [ x ( z z 0 ) tan θ 2 , z 0 ] + Γ 2 z 0 z I R [ x ( z z ) tan θ 2 , z ] d z .
I R ( x , z > z 0 ) = exp [ Γ 1 ( z 0 z ) ] { I R ( x 1 , 0 ) q exp ( Γ 2 z 0 ) × 0 z 0 ( tan θ 1 + tan θ 2 ) I S [ x + x 1 z 0 ( tan θ 1 + tan θ 2 ) , 0 ] × exp ( x / x 0 ) d x / x 0 } ,
I S ( x , z > z 0 ) = I S ( x 2 , 0 ) exp ( Γ 2 z 0 ) + Γ 2 z 0 z I R [ x 2 + z ( tan θ 1 + tan θ 2 ) , 0 ] exp [ Γ 1 ( z 0 z ) ] d z Γ 2 q exp ( Γ 2 z 0 ) z 0 z d z exp [ Γ 1 ( z 0 z ) ] 0 z 0 ( tan θ 1 + tan θ 2 ) d x / x 0 I R [ x + x 2 + ( z z 0 ) ( tan θ 1 + tan θ 2 ) , 0 ] × exp ( x / x 0 ) = I S ( x 2 , 0 ) exp ( Γ 2 z 0 ) + q 1 0 ( z z 0 ) ( tan θ 1 + tan θ 2 ) I R [ x 2 + z 0 ( tan θ 1 + tan θ 2 ) + y , 0 ] exp ( y / y 0 ) d y / y 0 exp ( Γ 2 z 0 ) 0 ( z z 0 ) ( tan θ 1 + tan θ 2 ) d y / y 0 exp ( y / y 0 0 z 0 ( tan θ 1 + tan θ 2 ) d x / x 0 ) exp ( x / x 0 ) I R ( x 2 + x + y , 0 ) ,
I S ( x , z > z 0 ) I R ( 0 ) + exp ( Γ 2 z 0 ) { I S ( x 2 , 0 ) 0 d u 0 d υ exp [ ( u + υ ) ] I S [ x 2 + ( u + υ ) x 0 , 0 ] } = I R ( 0 ) + exp ( Γ 2 z 0 ) [ I S ( x 2 , 0 ) 0 d t / x 0 ( t / x 0 ) exp ( t / x 0 ) I S ( x 2 + t , 0 ) ] = I R ( 0 ) exp ( Γ 2 z 0 ) [ 0 d t / x 0 ( 1 + t / x 0 ) exp ( t / x 0 ) d I S d x ( x 2 + t , 0 ) ] ,
I R ( x , z + d z ) = I R ( x + d z tan θ 1 , z ) d z g ( x + d z tan θ 1 , z )
I S ( x , z + d z ) = I S ( x d z tan θ 2 , z ) + d z g ( x d z tan θ 2 , z ) ,

Metrics