1 8 Ju l 2 01 3 Far Field measurement in the focal plane of a lens : a cautionary note

We study theoretically the accuracy of the method based on the Fourier property of lenses that is commonly used for the far field measurement . We consider a simple optical setup in which the far-field intensity pattern of a light beam passing through a Kerr medium is recorded by a CCD camera located in the back focal plane of a thin lens. Using Fresnel diffraction formula and numerical computations, we investigate the influence of a slight longitudinal mispositioning of the CCD camera. Considering a coherent gaussian beam, we show that a tiny error in the position of the CCD camera can produce a narrowing of the transverse pattern instead of the anticipated and wellunderstood broadening. This phenomenon is robust enough to persist for incoherent beams strongly modified by the presence of noise. The existence of this phenomenon has important consequences for the design and the realization of experiments in the field of optical wave turbulence in which equilibrium spectra reached by incoherent waves can only be determined from a careful far-field analysis. In particular, the unexpected narrowing of the far field may be mistaken for the remarkable phenomenon of classical condensation of waves. Finally, we show that the finite-size of optical components used in experiments produces diffraction patterns having wings decaying in a way comparable to the Rayleigh-Jeans distribution reached by incoherent wave systems at thermodynamical equilibrium. References and links 1. A. Porter, “On the diffraction theory of microscopic vision,” Phil. Mag. 11, 154 (1906). 2. J. Goodman, Introduction to Fourier Optics (Roberts & Co Publishers, 2005), third edition ed. 3. P. Elias, D. Grey, and D. Robinson, “Fourier treatment of optical processes,” J. Opt. Soc. Am. 42, 127–132 (1952). 4. P. Elias, “Optics and communication theory,” J. Opt. Soc. Am. 43, 229–232 (1953). 5. P.-M. Duffieux, The Fourier Transform and its Applications to Optics (Wiley, New York, 1983). 6. J. Rhodes, “Analysis and synthesis of optical images,” Am. J. Phys 21, 337 (1953). 7. W. Gambling, D. Payne, H. Matsumura, and R. Dyott, “Determination of core diameter and refractive-index difference of single-mode fibres by observation of the far-field pattern,” IEEE Journal on Microwaves, Optics and Acoustics 1, 13–17 (1976). 8. K. Hotate and T. Okoshi, “Measurement of refractive-index profile and transmission characteristics of a singlemode optical fiber from its exit-radiation pattern,” Appl. Opt. 18, 3265–3271 (1979). 9. P. J. Samson, “Measurement of wavelength dependence of the far-field pattern of single-mode optical fibers,” Opt. Lett. 10, 306–308 (1985). 10. W. Freude and A. Sharma, “Refractive-index profile and modal dispersion prediction for a single-mode optical waveguide from its far-field radiation pattern,” Journal of Lightwave Technology 3, 628–634 (1985). 11. W. Callen, B. Huth, and R. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103 (1967). 12. C. Nascimento, M. Alencar, S. Chavez-Cerda, M. da Silva, M. Meneghetti, and J. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a gaussian beam in a kerr medium,” Journal of Optics A: Pure and Applied Optics 8, 947 (2006). 13. T. Wulle and S. Herminghaus, “Nonlinear optics of bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993). 14. T. Honda, H. Matsumoto, M. Sedlatschek, C. Denz, and T. Tschudi, “Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror,” Optics Communications 133, 293 – 299 (1997). 15. C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998). 16. E. O’Neill, J. Houlihan, J. G. McInerney, and G. Huyet, “Dynamics of traveling waves in the transverse section of a laser,” Phys. Rev. Lett. 94, 143901 (2005). 17. von F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689 – 704 (1934). 18. F. Zernike, “How i discovered phase contrast,” Science 121, 345–349 (1955). 19. L. Lucchetti, S. Suchand, and F. Simoni, “Fine structure in spatial self-phase modulation patterns: at a glance determination of the sign of optical nonlinearity in highly nonlinear films,” J. Opt. A: Pure Appl. Opt. 11, 034002 (2009). 20. M. M. Birky, “Simultaneous recording of near-field and far-field patterns of lasers,” Appl. Opt. 8, 2249–2253 (1969). 21. M. Cross and P. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of modern physics 65, 851 (1993). 22. N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990). 23. F. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Physics Reports 318, 1 – 83 (1999). 24. E. Große Westhoff, R. Herrero, T. Ackemann, and W. Lange, “Self-organized superlattice patterns with two slightly differing wave numbers,” Phys. Rev. E 67, 025203 (2003). 25. C. Denz, M. Schwab, and C. Weilnau, Transverse-Pattern Formation in Photorefractive Optics, Springer Tracts in Modern Physics (Springer, 2003). 26. F. Arecchi, “Optical morphogenesis : pattern formation and competition in nonlinear optics,” Physica D 86, 297–322 (1995). 27. P. Ramazza, E. Pampaloni, S. Residori, and F. Arecchi, “Optical pattern formation in a kerr-like medium with feedback,” Physica D: Nonlinear Phenomena 96, 259 – 271 (1996). 28. P. Ramazza, S. Boccaletti, and F. Arecchi, “Transport induced patterns in an optical system with focussing nonlinearity,” Optics Communications 136, 267 – 272 (1997). 29. G. Agez, P. Glorieux, C. Szwaj, and E. Louvergneaux, “Using noise speckle pattern for the measurements of director reorientational relaxation time and diffusion length of aligned liquid crystals,” Optics Communications 245, 243 – 247 (2005). 30. G. Agez, P. Glorieux, M. Taki, and E. Louvergneaux, “Two-dimensional noise-sustained structures in optics: Theory and experiments,” Phys. Rev. A 74, 043814 (2006). 31. S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave far-field in ordered and disordered nonlocal media,” Opt. Express 20, 27369–27375 (2012). 32. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express 13, 10729–10741 (2005). 33. D. Faccio, A. Averchi, A. Lotti, P. D. Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation fromspontaneous xwave formation in air,” Opt. Express 16, 1565–1570 (2008). 34. W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nature Physics 3, 46–51 (2007). 35. V. Zakharov, V. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I, Berlin (Springer, 1992). 36. A. Newell and R. B., “Wave turbulence,” Annual review of Fluid Mechanics 43, 59–78 (2011). 37. B. Miquel and N. Mordant, “Nonstationary wave turbulence in an elastic plate,” Phys. Rev. Lett. 107, 034501 (2011). 38. A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063 (2007). 39. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau, and S. Rica, “Condensation of classical nonlinear waves,” Phys. Rev. Lett. 95, 263901 (2005). 40. S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear schrdinger equation,” Physica D 57, 96 (1992). 41. S. Pitois, S. Lagrange, H. R. Jauslin, and A. Picozzi, “Velocity locking of incoherent nonlinear wave packets,” Phys. Rev. Lett. 97, 033902 (2006). 42. B. Barviau, S. Randoux, and P. Suret, “Spectral broadening of a multimode continuous-wave optical field propagating in the normal dispersion regime of a fiber,” Opt. Lett. 31, 1696 (2006). 43. B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot, and A. Picozzi, “Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics,” Opt. Express 15, 9063 (2009). 44. P. Suret, S. randoux, H. Jauslin, and A. Picozzi, “Anomalous thermalization of nonlinear wave systems,” Phys. Rev. Lett. 104, 054101 (2010). 45. P. Suret, A. Picozzi, and S. Randoux, “Wave turbulence in integrable systems: nonlinear propagation of incoherent optical waves in single-mode fibers,” Opt. Express 19, 17852–17863 (2011). 46. U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, “Optical wave turbulence and the condensation of light,” J. Opt. Soc. Am. B 26, 2280–2284 (2009). 47. J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, “One-dimensional optical wave turbulence: Experiment and theory,” Physics Reports 514, 121 – 175 (2012). 48. C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nature Physics 8, 470–474 (2012). 49. S. Durbin, S. Arakelian, and Y. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. 6, 411 (1981). 50. F. Dabby, T. Gustafson, J. Whinnery, and Y. Kohanzadeh, “Thermally self-induced phase modulation of laser beams,” Appl. Phys. Lett. 16, 362 (1970). 51. E. Santamato and Y. Shen, “Field-curvature effect on the diffraction ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic film,” Opt. Lett. 9, 564 (1984). 52. L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent gaussian beams passing through self-focusing and self-defocusing media,” Journal of Optics A: Pure and Applied Optics 7, 409 (2005). 53. E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18, 22067–22079 (2010). 54. R. W. Boyd, Nonlinear Optics (Academic Press, 1992). 55. J. Castillo, V. P. Kozich, and A. M. O., “Thermal lensing resulting from oneand two-photon absorption studied with a two-color time-resolved z scan,” Opt. Lett. 19, 171–173 (1994). 56. R. E. Samad and J. Nilson Dias Vieira, “Analytical description of z-scan on-axis intensity based on the huygens– fresnel principle,” J. Opt. Soc. Am. B 15, 2742–2747 (1998). 57. I. C. Khoo, J. Y. Hou, T. H. Liu, P. Y. Yan, R. R. Michael, and G. M. Finn, “Transverse self-phase modulation and bistability in the transmission of a laser beam through a nonlinear thin film,” J. Opt. Soc. Am. B 4, 886–891 (1987). 58. M. Sheik-bahae, A. A. Said, and E. W. V. Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989). 59. M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. V. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” Journal of Quantum Electronics, IEEE 26, 760–769 (1990). 60. L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent gaussian beams passing through self-focusing and self-defocusing media,” Journal of Optics A: Pure and Applied Optics 7, 409 (2005). 61. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proceedings of the IEEE 93, 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”.