1. The real variable 2. Scalars and vectors 3. Tensors 4. Matrices 5. Multiple integrals 6. Potential theory 7. Operational methods 8. Physical applications of the operational method 9. Numerical methods 10. Calculus of variations 11. Functions of a complex variable 12. Contour integration and Bromwich's integral 13. Contour integration 14. Fourier's theorem 15. The factorial and related functions 16. Solution of linear differential equations of the second order 17. Asymptotic expansions 18. The equations of potential, waves and heat conduction 19. Waves in one dimension and waves with spherical symmetry 20. Conduction of heat in one and three dimensions 21. Bessel functions 22. Applications of Bessel functions 23. The confluent hypergeometric function 24. Legendre functions and associated functions 25. Elliptic functions Notes Appendix on notation Index.

Methods of Mathematical Physics. By Harold Jeffreys and Bertha Swirles Jeffreys. Pp. viii, 679. 63s. 1946. (Cambridge University Press)

https://dmpeli.math.mcmaster.ca/PaperBank/PRreview.pdf

Self-focusing and transverse instabilities of solitary waves

We report on the experimental observation of modulation instability of partially spatially incoherent light beams in noninstantaneous nonlinear media and show that in such systems patterns can form spontaneously from noise. Incoherent modulation instability occurs above a specific threshold that depends on the coherence properties (correlation distance) of the wave packet and leads to a periodic train of one-dimensional filaments. At a higher value of nonlinearity, the incoherent one-dimensional filaments display a two-dimensional instability and break up into self-ordered arrays of light spots. This discovery of incoherent pattern formation reflects on many other nonlinear systems beyond optics. It implies that patterns can form spontaneously (from noise) in diverse nonlinear many-body systems involving weakly correlated particles, such as atomic gases at (or near) Bose-Einstein condensation temperatures and electrons in semiconductors at the vicinity of the quantum Hall regime.

/pdf/modulation-instability-and-pattern-formation-in-spatially-4wt1nu59sh.pdf

Modulation instability and pattern formation in spatially incoherent light beams.

We provide a comprehensive experimental and theoretical study of incoherently coupled photorefractive spatial-soliton pairs in all three possible realizations: bright–bright, dark–dark, and dark–bright. We also show that when the total intensity of two coupled solitons is much lower than the effective dark irradiance, the coupled soliton pair is reduced to Manakov solitons. In all cases, mutual trapping of both components in the coupled soliton pair is verified by analyzing, experimentally and numerically, the beam evolution after decoupling.

Coupled photorefractive spatial-soliton pairs

We investigate experimentally the interaction of two-dimensional solitary beams in photorefractive strontium barium niobate crystal. We show that simultaneous collision of many solitons can result in complete annihilation of some of them. This effect depends on the relative phases of the solitons and may be useful for application in the formation of multiport waveguide junctions.

https://www.uni-muenster.de/imperia/md/content/physik_ap/denz/publikationen/97_98/97_98_9.pdf

Annihilation of photorefractive solitons

We present experimental results on the propagation of an interference pattern of two He–Ne laser beams of unequal amplitudes through a photorefractive Bi12TiO20 crystal in the presence of drift nonlinearity. The phenomenon that we have observed is the focusing of the fringes as the nonlinearity of the crystal is increased. We show that such a phenomenon can be quantitatively interpreted in the framework of modulation instability theory.

Experimental evidence of modulation instability in a photorefractive Bi(12)TiO(20) crystal.

We propagate two coherent and parallel beams of a He-Ne laser through a Bi(12)TiO(20) photorefractive crystal in the presence of drift nonlinearity. Our experimental results demonstrate that the beams attract or repel each other according to their initial phase difference. They attract each other when they are initially in phase and they repel each other when they are initially out of phase. These experimental results agree with numerical predictions recently published.

Observation of interaction forces between one-dimensional spatial solitons in photorefractive crystals

We show that a collision between two bright spatial solitons acts as an asymmetric optical Y junction and, in specific cases, as an optical switch for the weak beams that they guide. The output energy of the probe beams in each optical channel can be controlled by adjustment of either the relative direction or the phase of the two colliding solitons or by a change in the wavelength of the probe beams. A physical description of such behavior based on the diffraction of the incident probe beams by a diffraction grating formed by the spatial-soliton collision profile is presented.

/pdf/asymmetric-optical-y-junctions-and-switching-of-weak-beams-2f16ey21mh.pdf

Asymmetric optical Y junctions and switching of weak beams by using bright spatial-soliton collisions

We present a fundamental study on the capability of a crossing of two optical waveguides based on dark spatial solitons to act as a controllable optical beam splitter. Our study is based on the fact that the guided beam is diffracted at the waveguide crossing by an effective phase screen formed by the soliton collision profile. We find that when the two dark solitons are immersed into the same finite bright background, the energy of a guided beam can be split into the desired optical channel according to the collision angle. We also found that even the corresponding phase diffractive screen possesses a quite different structure in the bright and dark soliton cases; the physics involved is the same.

Mechanisms of crossing for an X-junction based on dark spatial solitons

thermore, we have seen the 650-nm luminescence to increase quadratically with incident laser power (Fig. 3), indicating a two photon process in contrast to the one photon process leading to the Ge-Si bond breakage. Finally, our data including variation of the polarization orientation seem to support the polarization dependent selectivity of Ge-Si bond breakage predicted by the model of oriented dip o l e ~ . ~ In the near future we will extend the luminescence monitoring to include fibers with varying co-dopants. %lektromagnetisk Institut, Bld. 348, DTH, DK-2800 Lyngby, Denmark M. J. Yuen, App. Opt. 21,136-140 (1982). T. E. Tsai et al., App. Phys. Lett. 61,

G. E. Torres-Cisneros

Papers

Experimental evidence of modulation instability in a photorefractive Bi(12)TiO(20) crystal.

Observation of interaction forces between one-dimensional spatial solitons in photorefractive crystals

Asymmetric optical Y junctions and switching of weak beams by using bright spatial-soliton collisions

Mechanisms of crossing for an X-junction based on dark spatial solitons

Soliton amplification in inhomogeneously broadened fiber amplifiers