References

1

J. A. Krumhansl, Physics Today, March 1991, pp. 33-38.

2

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 15, 1005 (1964).

3

A. W. Snyder, L. Poladian and D. J. Mitchell, ``Directing one beam of light with another'', 7th Optical Fibre Sensor Conference/15th Australian Conference on Optical Fibre Technology (Sydney, December 1990).

4

T. Thwaites, New Scientist, (12 Jan, 1991) No. 1751, p.4. Reports on the above 1990 lecture [3] by A. W Snyder where it is proposed that light can be used to guide and manipulate light for futuristic photonic devices.

5

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).

6

D. J. Mitchell, A. W. Snyder, and L. Poladian, Elect. Lett. 27, 848 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Comm. 85, 59 (1991). .

7

A. W. Snyder, L. Poladian, and D. J. Mitchell, Opt. Lett. 17, 789 (1992).

8

A. W. Snyder, L. Poladian, and D. J. Mitchell, Opt. Lett. 17, 118, 267 (1992).

9

R. De la Fuente, A. Barthelemy, and C. Froehly, Opt. Lett. 16, 783 (1991); M. Shalaby, F. Reynaud, and A. Barthelemy, Opt. Lett. 17, 778 (1992); A. Barthelemy, C. Froehly, S. Maneuf, and F. Reynaud, Opt. Lett. 17, 844 (1992).

10

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).

11

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H.Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991); 69, 2503 (1992).

12

B. Luther-Davies, X. Yang and A. W. Snyder, in ``Integrated Photonics Research'', Vol. 10, OSA Technical Digest Series (OSA, Washington, DC 1992) p. 104.

13

B. Luther-Davies and X. Yang, Opt. Lett. 17, 496; 1755 (1992).

14

A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).

15

When a monochromatic beam is incident on a nonlinear medium, beams of other frequencies (harmonics, sidebands) may be generated. These other components can usually be neglected so that the electric field and the displacement vector have the forms and , respectively [16]. Assuming a local response, is a nonlinear function of . The constitutive relation can then be taken as a definition of . However, this does not uniquely define .

16

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley, New York, 1984); Phys. Lett 20, 378 (1966).

17

N. Bloembergen, Nonlinear Optics (Benjamin, NY, 1965).

18

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Ch's 13 and 32; Table 12-7 and p. 266; Ch's 18, 25, 19 and 28; Table 13-2; Ch's 15 and 22.

19

Four wave-mixing can be ignored when: (a) the nonlinear response is too slow to follow the beating between the components of different wavelengths, (b) the linear dispersion is sufficient to cause phase mismatching, (c) the two components (of different wavelength) are circularly polarized in the opposite sense.

20

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988); M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2011 (1988).

21

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. Lett. 72, 1012 (1994). Presented also at Nonlinear Guided Wave Phenomena, Cambridge (UK, September 1993).

22

M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Lett. 18, 1406 (1993); Opt. Comm. 103, 145 (1993). This gives the temporal analogue of [21]. Its inspiration comes from the spatial domain, using the physics of linear waveguides, although not explicitly stated. See also M. Haelterman and A.P. Sheppard, Opt. Lett. 19, 96 (1994).

23

D. J. Mitchell, and A. W. Snyder, ``Shape and dynamics of classical soliton depend on its polarization", Opt. Lett. submitted Feb. 1994; Y. Silberberg and Y. Barad, Opt. Lett. 20, 246 (1995).

24

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981); A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, Pa., 1985), pp. 11-15; Important Developments in Soliton Theory, A. S. Fokas and V. E. Zakharov, Eds. (Springer-Verlag, Berlin, 1993).

25

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).

26

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 923 (1973).

27

A. W. Snyder and A. Ankiewicz, J. Opt. Soc. Am. 3, 856 (1986).

28

P. D. Maker, R. W. Terhune, and C. M. Savage, Phys. Rev. Lett. 12, 507 (1964). The actual induced may be different, from the effective e.g. it can correspond to a medium with twisted linear birefringence (Section 3, Example 4).

29

L. Poladian, D. J. Mitchell, and A. W. Snyder, ``Self-guided waves, bright and dark", in Technical Digest of Nonlinear Guided Wave Phenomena, OSA (1993), pp 212-215.

30

A. W. Snyder and W. Young, J. Opt. Soc. Am. 68, 297 (1978); A. W. Snyder, IEEE Trans. Micro. Th. and Tech. MTT-17, 1130 (1969); J. Opt. Soc. Am. 70, 405 (1980).

31

N. N. Akhmediev, V. M. Eleonskii, and L. P. Shil'nikov, Sov. Tech. Phys. Lett. 15, 587 (1989). The analytical solution presented for isotropic material is equivalent, but in a simpler form, to that reported by M. V. Tratnik and J. E. Sipe [20].

32

A. W. Snyder and D. J. Mitchell, Opt. Lett. 18, 101 (1993).

33

A. W. Snyder, D. J. Mitchell, and M. Haelterman, Opt. Comm. 116, 365 (1995).

34

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 871 (1988); V. V. Afanasev, Yu. S. Kivshar, V. V. Konotop, and V. N. Serkin, Opt. Lett. 14, 805 (1989).

35

P. McIntyre and A. W. Snyder, J. Opt. Soc. Am. 68, 149 (1978).

36

A. W. Snyder, D. J. Mitchell, and B. Luther-Davies, J. Opt. Soc. Am. B 10, 2341 (1993).

37

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).

38

S. Kawakami and S. Nishida, IEEE J. Quantum Electron. QE-10, 879 (1974).

39

A. W. Snyder, V.V. Afanasjev, and Yu. S. Kivshar, ``Bright like, dark solitons", submitted to Opt. Commun. (1995).

40

Once the modes of the induced (linear) waveguide are known, so are the modal dispersion curves. The existence of branches in such curves are known as bifurcations.

41

A. W. Snyder and H. T. Tran, Opt. Comm. 98, 309 (1993).

42

D. Anderson, Phys. Rev. A 27, 3135 (1983); A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, Elect. Lett. 26, 643 (1990); R. A. Sammut and C. Pask, Elect. Lett. 26, 1131 (1990).

43

A. W. Snyder, Proc. IEEE, 61, 6 (1981). See also Ch. 22 of Ref. [18].

44

A. W. Snyder, D. J. Mitchell, and L. Poladian, J. Opt. Soc. Amer. B 8, 1618 (1991).

45

D. J. Mitchell and A. W. Snyder, J. Opt. Soc. Amer. B 10, 1572 (1993). While not explicitly stated, this reference was inspired by the linear perspective along the lines described in the text.

46

A. A. Kolokolov, J. Appl. Mech. Tech. Phys. 11, 426 (1975); Lett. Nuovo Cimento 8, 197 (1973).

47

A. W. Snyder, D. J. Mitchell, L. Poladian, D. Rowland, and Y. Chen, J. Opt. Soc. Amer. B 8, 2102 (1991).

48

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972); S. V. Manakov, Sov. Phys. JETP. 38, 248 (1974).

49

D. J. Mitchell, A. W. Snyder, and Y. Chen, Electron. Lett. 26, 1164 (1990).

50

Propagation of monochromatic beams in a linear dielectric is approximated by the (linear) Schrödinger equation. This can be separated into two equations, one for the amplitude (continuity equation) and the other one, for the phase [16]. The latter is a generalization of the eikonal equation of geometric optics with the refractive index n replaced by , where is the refractive index of that waveguide of which |E| is a mode. Thus, a soliton behaves like a ray, travelling up the gradient of , averaged over the soliton (with weight ). This leads to the result of [6].

51

L. Lerner, D. J. Mitchell, and A. W. Snyder, Opt. Lett. 19, 1302 (1994).

52

J. Satsuma and N. Yajima, Supp. Prog. Theor. Phys. 55, 284 (1974). H. A. Haus and M. N. Islam, IEEE J. Quant. Elect. QE-21, 1172 (1985).

53

S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theory and Tech. MTT-23, 123 (1975).

54

A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).

55

H. T. Tran, R. A. Sammut, and W. Samir, Opt. Lett. 19, 945 (1991); Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).

56

Even at saturation, all known nonlinear materials obey the weak guidance condition. Suppose they did not saturate, the characteristic beam half width would then have to be less than the wavelength of light before weak guidance was violated. For example, the classical fundamental soliton in a Kerr medium induces a waveguide height as seen from Section 4, example 1.

57

This means that the spatial wavelength of gratings must obey or .

58

See, e.g. Ref. [16], Chap. 2.

59

A. W. Snyder, D. J. Mitchell, and Y. Chen, Opt. Lett. 19, 524 (1994).

60

Yu.N. Karamzin and A.P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

61

R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993).

62

A.V. Buryak and Yu.S. Kivshar, Opt. Lett. 19, 1612 (1994).

63

L. Torner, C.R. Menyuk, and G.I. Stegeman, Opt. Lett. 19, 1615 (1994).

64

M.J. Werner and P.D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993).

65

E. D. Belokolos, A. I. Bobenko, V. Z. Enolsky, A.R. Its, and V. B. Matveev, ``Algebro-Geometrical Approach to Nonlinear Evolution Equations'' (Springer Verlag, Berlin, 1992).

66

The linear equivalent for temporal problems necessitates a time varying refractive index. If a monochromatic beam is passed through a medium with an oscillating n, side bands with different frequencies may be generated. This model describes phenomena such as harmonic generation and Raman scattering.

67

F. N. Sears, M. W. Zemansky and H. D. Young, University Physics, Addison-Wesley (1982) (6Õth edition) (pp. 721 and 789). Standard high school text.

68

A. W. Snyder, D. J. Mitchell and A. Buryak, ``Qualitative theory of bright solitons - the soliton sketch'', submitted to J. Opt. Soc. Am. B (1995).

A.W.S. was invited to present this topic at the 1995 Gordon Conference (USA) on Nonlinear Optics and Lasers. The paper was invited for the proposed book ``Research Trends in Physics: Nonlinear and Quantum Optics'', N. Bloembergen (Editor-in-Chief), but is published here instead, by invitation.