Abstract

We propose a new type of waveguide optical amplifier. The device consists of collinearly propagating pump and amplified Stokes beams with periodic imaging of the Stokes beam due to the Talbot effect. The application of this device as an Image preamplifier for Mid Wave Infrared (MWIR) remote sensing is discussed and its performance is described. Silicon is the preferred material for this application in MWIR due to its excellent transmission properties, high thermal conductivity, high damage threshold and the mature fabrication technology. In these devices, the Raman amplification process also includes four-wave-mixing between various spatial modes of pump and Stokes signals. This phenomenon is unique to nonlinear interactions in multimode waveguides and places a limit on the maximum achievable gain, beyond which the image begins to distort. Another source of image distortion is the preferential amplification of Stokes modes that have the highest overlap with the pump. These effects introduce a tradeoff between the gain and image quality. We show that a possible solution to this trade-off is to restrict the pump into a single higher order waveguide mode.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Non linear fiber optics (Academic Press San Diego 2001).
  2. K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
    [CrossRef]
  3. R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
    [CrossRef]
  4. H. M. Pask, "The design and operation of solid-state Raman lasers," Prog. Quantum Electron 27, 3-56, (2003).
    [CrossRef]
  5. N. Bloembergen, "Multimode effects in stimulated Raman emission," Phys. Rev. Lett. 13, 720-724 (1964).
    [CrossRef]
  6. P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators I. Oscillators," Appl. Phys. Lett. 6,210-212 (1965).
    [CrossRef]
  7. P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators II. Amplifiers," Appl. Phys. Lett. 6, 2112-2123 (1965).
    [CrossRef]
  8. S. H. Baek and W. B. Roh, "Single-mode Raman fiber laser based on a multimode fiber," Opt. Lett. 29, 153-155 (2004).
    [CrossRef] [PubMed]
  9. L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and Applications," J. Lightwave Technol. 13, 615-627 (1995).
    [CrossRef]
  10. H. J. Baker, J. R. Lee, and D. R. Hall, "Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers," Opt. Express 10, 297-302 (2002).
    [PubMed]
  11. I. T. McKinnie, J. E. Koroshetz, W. S. Pelouch, D. D. Smith, J. R. Unternahrer, and S. W. Henderson, "Self-imaging waveguide Nd:YAG laser with 58% slope efficiency," Conference on Lasers and Electro-Optics (CLEO), CTuP2, (2002).
  12. M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
    [CrossRef]
  13. L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
    [CrossRef]
  14. A. Kier, ed., "Mid infrared semiconductor optoelectronics," Springer series in Optoelectronics (2006).
  15. V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
    [CrossRef]
  16. S. J. Garth and R. A. Sammut, "Theory of stimulated Raman scattering in two-mode optical fibers," J. Opt. Soc. Am. B. 10, 2040-2047 (1983).
    [CrossRef]
  17. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
    [CrossRef]
  18. W. C. Hurlburt, K. L. Vodopyanov, P. S. Kuo, M. M. Fejer, and Y. S. Lee, "Multiphoton absorption and nonlinear refraction of GaAs in the mid infrared," Conference on Lasers and Electro-Optics (CLEO/QELS), QThM3 (2006).
  19. A. Zajac, M. Skorczakowski, J. Swiderski, and P. Nyga, "Electrooptically Q-switched mid-infrared Er:YAG laser for medical applications," Opt. Express 12, 5125-5130 (2004).
    [CrossRef] [PubMed]
  20. A. E. Siegman, "How to (may be) measure laser beam quality," Tutorial OSA annual meeting (1997).
  21. A. Brignon, G. Feugnet, J. P. Huignard, and J. P. Pocholle, "Large-field-of-view, high-gain, compact diode-pumped Nd:YAG amplifier," Opt. Lett. 22, 1421-1423 (1997).
    [CrossRef]
  22. L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
    [CrossRef]

2006 (1)

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

2004 (2)

2003 (1)

H. M. Pask, "The design and operation of solid-state Raman lasers," Prog. Quantum Electron 27, 3-56, (2003).
[CrossRef]

2002 (2)

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

H. J. Baker, J. R. Lee, and D. R. Hall, "Self-imaging and high-beam-quality operation in multi-mode planar waveguide optical amplifiers," Opt. Express 10, 297-302 (2002).
[PubMed]

1998 (4)

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

1997 (1)

1995 (1)

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and Applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

1994 (1)

M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
[CrossRef]

1983 (1)

S. J. Garth and R. A. Sammut, "Theory of stimulated Raman scattering in two-mode optical fibers," J. Opt. Soc. Am. B. 10, 2040-2047 (1983).
[CrossRef]

1965 (2)

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators I. Oscillators," Appl. Phys. Lett. 6,210-212 (1965).
[CrossRef]

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators II. Amplifiers," Appl. Phys. Lett. 6, 2112-2123 (1965).
[CrossRef]

1964 (1)

N. Bloembergen, "Multimode effects in stimulated Raman emission," Phys. Rev. Lett. 13, 720-724 (1964).
[CrossRef]

Austin, W. L.

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

Baek, S. H.

Baker, H. J.

Bloembergen, N.

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators II. Amplifiers," Appl. Phys. Lett. 6, 2112-2123 (1965).
[CrossRef]

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators I. Oscillators," Appl. Phys. Lett. 6,210-212 (1965).
[CrossRef]

N. Bloembergen, "Multimode effects in stimulated Raman emission," Phys. Rev. Lett. 13, 720-724 (1964).
[CrossRef]

Brignon, A.

Calmes, L. K.

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

Claps, R.

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

Coppinger, F.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Cunningham, D. G.

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

Dimitropoulos, D.

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

Duncan, B. D.

M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
[CrossRef]

Feugnet, G.

Garth, S. J.

S. J. Garth and R. A. Sammut, "Theory of stimulated Raman scattering in two-mode optical fibers," J. Opt. Soc. Am. B. 10, 2040-2047 (1983).
[CrossRef]

Hall, D. R.

Huignard, J. P.

Jalali, B.

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Kimura, T.

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

Lallemand, P.

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators I. Oscillators," Appl. Phys. Lett. 6,210-212 (1965).
[CrossRef]

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators II. Amplifiers," Appl. Phys. Lett. 6, 2112-2123 (1965).
[CrossRef]

Lee, J. R.

McManamon, P. F.

M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
[CrossRef]

Murray, J. T.

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

Nishizawa, J.

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

Norwell, M. C.

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

Nyga, P.

Pask, H. M.

H. M. Pask, "The design and operation of solid-state Raman lasers," Prog. Quantum Electron 27, 3-56, (2003).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and Applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Pocholle, J. P.

Powell, R. C.

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

Raddatz, L.

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

Raghunathan, V.

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

Redina, I.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Roh, W. B.

Saito, T.

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

Salisbury, M. S.

M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
[CrossRef]

Sammut, R. A.

S. J. Garth and R. A. Sammut, "Theory of stimulated Raman scattering in two-mode optical fibers," J. Opt. Soc. Am. B. 10, 2040-2047 (1983).
[CrossRef]

Shori, R.

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

Skorczakowski, M.

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and Applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Stafsudd, O. M.

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

Suto, K.

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

Swiderski, J.

White, I. H.

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

Yegnanarayanan, S.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Yoon, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Yoshimoto, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Zajac, A.

Appl. Phys. Lett. (2)

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators I. Oscillators," Appl. Phys. Lett. 6,210-212 (1965).
[CrossRef]

P. Lallemand and N. Bloembergen, "Multimode effects in the gain of Raman amplifiers and oscillators II. Amplifiers," Appl. Phys. Lett. 6, 2112-2123 (1965).
[CrossRef]

Electron. Lett. (1)

R. Claps, D. Dimitropoulos, and B. Jalali, "Stimulated Raman scattering in silicon waveguides," Electron. Lett. 38, 1352-1354 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Norwell, "Influence of restricted mode excitation on bandwidth of multimode fiber links," IEEE Photon. Technol. Lett. 10, 534-536 (1998).
[CrossRef]

J. Lightwave Technol. (1)

L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and Applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

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

S. J. Garth and R. A. Sammut, "Theory of stimulated Raman scattering in two-mode optical fibers," J. Opt. Soc. Am. B. 10, 2040-2047 (1983).
[CrossRef]

Opt. Eng. (1)

M. S. Salisbury, P. F. McManamon, and B. D. Duncan, "Optical-fiber preamplifiers for ladar detection and associated measurement for improving the signal-to-noise ratio," Opt. Eng. 33, 4023-4032 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Optoelectron (1)

K. Suto, T. Kimura, T. Saito, and J. Nishizawa, "Raman amplification in GaP-AlxGa1-xP waveguides for light frequency discrimination," IEE Proc.: Optoelectron 145, 105-108 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

N. Bloembergen, "Multimode effects in stimulated Raman emission," Phys. Rev. Lett. 13, 720-724 (1964).
[CrossRef]

Phys. Status Solidi A (1)

V. Raghunathan, R. Shori, O. M. Stafsudd, B. Jalali, "Nonlinear absorption in silicon and the prospects of mid-infrared Silicon Raman laser," Phys. Status Solidi A 203, R38-R40 (2006).
[CrossRef]

Proc. SPIE (1)

L. K. Calmes, J. T. Murray, W. L. Austin, and R. C. Powell, "Solid state Raman image amplifier," Proc. SPIE 3382, 57-67 (1998).
[CrossRef]

Prog. Quantum Electron (1)

H. M. Pask, "The design and operation of solid-state Raman lasers," Prog. Quantum Electron 27, 3-56, (2003).
[CrossRef]

Top. Quantum Electron (1)

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Redina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J Sel.Top. Quantum Electron 4, 938-947 (1998).
[CrossRef]

Other (5)

W. C. Hurlburt, K. L. Vodopyanov, P. S. Kuo, M. M. Fejer, and Y. S. Lee, "Multiphoton absorption and nonlinear refraction of GaAs in the mid infrared," Conference on Lasers and Electro-Optics (CLEO/QELS), QThM3 (2006).

A. E. Siegman, "How to (may be) measure laser beam quality," Tutorial OSA annual meeting (1997).

A. Kier, ed., "Mid infrared semiconductor optoelectronics," Springer series in Optoelectronics (2006).

G. P. Agrawal, Non linear fiber optics (Academic Press San Diego 2001).

I. T. McKinnie, J. E. Koroshetz, W. S. Pelouch, D. D. Smith, J. R. Unternahrer, and S. W. Henderson, "Self-imaging waveguide Nd:YAG laser with 58% slope efficiency," Conference on Lasers and Electro-Optics (CLEO), CTuP2, (2002).

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 (5)

Fig. 1.
Fig. 1.

Cross-section of multimode silicon waveguides used in this analysis. For the ease of analysis, waveguide width, a is taken to be much larger than the thickness, b.

Fig. 2.
Fig. 2.

Contour profile of the electric field amplitude (X-Z profile) showing the self-imaging Raman amplifier with the evolution of the pump and Stokes along the length of the multimode silicon waveguide. A single pump and Stokes Gaussian beam is launched into the waveguide. Pump power coupled into the waveguide is 1KW peak (1mW average) and Stokes power is 1μW.

Fig. 3.
Fig. 3.

Evolution of small-signal Raman gain along the length of the multimode silicon Raman amplifier. The input pump and Stokes power are taken to be 1KW peak (1mW average) and 1μW respectively.

Fig. 4.
Fig. 4.

The beam quality (M2) of the Stokes beam calculated along the waveguide. The image periodically repeats itself due to the self-imaging property of the multimode waveguide. (a) Stokes beam propagating through a passive waveguide with no pump launched and (b) Stokes beam propagating through an active waveguide Raman amplifier with input pump intensities of 25MW/cm2 and 50MW/cm2. Pump powers are 1KW peak (1mW average) and 2KW peak (2mW average). The M2 parameter is a measure of the image quality at the focal points with an ideal value of unity. The image deteriorates (M2 increases) with increase in pump power due to preferential amplification of fundamental Stokes mode in comparison with higher order modes, and the presence of the phase-sensitive Raman four wave mixing.

Fig. 5.
Fig. 5.

The electric-field amplitude profile of a test input images (left) and the amplified output images (right). These figures describe the cross sectional (X-Y) profile (as opposed to Figures 2 which shows the propagation (X-Z) along the waveguide length). X and Y are in micrometers. The spatial frequency of the test images are (a) 50 lines per mm, (b) 100 lines per mm, and (c) 250 lines per mm. Pump power of 1KW peak (1 mW average) is launched. The Stokes image experiences ~10dB gain over a length of ~5cm (ie. the first focal length).

Tables (1)

Tables Icon

Table 1. List of parameters used throughout this paper and the values of these parameters used in these simulations

Equations (12)

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

ϕ mn = 4 Z ab Sin ( mπx a ) Sin ( nπy b ) , 0 < x < a and 0 < y < b
ψ in = P Z w 2 π exp ( ( ( x a ) 2 ) 2 4 w 2 ) exp ( ( y b 2 ) 2 4 w 2 ) exp ( )
ψ ( z ) = m n A mn ( z ) ϕ mn e j β mn z
L image p 4 n 0 a 2 λ 0
d A S mn dz = α S mn 2 A S mn + kl κ mn kl A P kl 2 A S mn + k m& l n κ′ mn kl A P mn A P kl * A S kl e jΔβz
d A P mn dz = α P mn 2 A P mn ω P ω S kl κ mn kl A P kl 2 A P mn ω P ω S k m& l n κ′ mn kl A S mn A S kl * A P kl e jΔβz
κ mn kl = ω S ε o 0 b 0 a ϕ P kl ϕ P kl * ( χ Raman ( 3 ) ) ϕ S mn ϕ S mn * ∙dxdy
κ′ mn kl = ω S ε o 0 b 0 a ϕ P kl * ϕ P mn ( χ Raman ( 3 ) ) ϕ S mn * ϕ S kl ∙dxdy
Δ β = ( β P mn β P kl ) ( β S mn β S kl )
κ mn kl = κ m 1 M 1 = ω S ε o χ Raman ( 3 ) 0 b 0 a ϕ P M 1 2 ϕ S m 1 2 dxdy = ω S ε o χ Raman ( 3 ) 3 Z P Z S 2 ab
d A S m 1 dz = A S m 1 κ m 1 M 1 A P M 1 2 , for all m < M
A S m 1 ( z ) = A S m 1 ( 0 ) exp ( κ m 1 M 1 0 z A P M 1 ( z′ ) 2 dz′ )

Metrics