Waveguide (optics)

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An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber and rectangular waveguides.

Optical waveguides are used as components in integrated optical circuits or as the transmission medium in local and long haul optical communication systems.

Optical waveguides can be classified according to their geometry (planar, strip, or fiber waveguides), mode structure (single-mode, multi-mode), refractive index distribution (step or gradient index) and material (glass, polymer, semiconductor).

[edit] Dielectric slab waveguide

A dielectric slab waveguide consists of three dielectric layers with different refractive indices.

Practical rectangular-geometry optical waveguides are most easily understood as variants of the simple dielectric slab waveguide ^[1], also called planar waveguide^[1]. The slab waveguide consists of three layers of materials with different dielectric constants, extending infinitely in the directions parallel to their interfaces.

Light may be confined in the middle layer by total internal reflection. This occurs only if the dielectric index of the middle layer is larger than that of the surrounding layers. In practice slab waveguides are not infinite in the direction parallel to the interface, but if the typical size of the interfaces is much much larger than the depth of the layer, the slab waveguide model will be an excellent approximation. It should be noted that guided modes of a slab waveguide can not be excited by light incident from the top or bottom interfaces. Light must be injected with a lens from the side into the middle layer. Alternatively a coupling element may be used to couple light into the waveguide, such as a grating coupler or prism coupler.

One model of guided modes is that of a planewave reflected back and forth between the two interfaces of the middle layer, at an angle of incidence between the propagation direction of the light and the normal, or perpendicular direction, to the material interface is greater than the critical angle. The critical angle depends on the index of refraction of the materials, which may vary depending on the wavelength of the light. Such propagation will result in a guided mode only at a discrete set of angles where the reflected planewave does not destructively interfere with itself.

This structure confines electromagnetic waves only in one direction, and therefore it has little practical application. Structures that may be approximated as slab waveguides do, however, sometimes occur as incidental structures in other devices.

[edit] Two-dimensional waveguides

[edit] Strip waveguides

A strip waveguide is basically a strip of the guiding layer confined between cladding layers. The simplest case is a rectangular waveguide, which is formed when the guiding layer of the slab waveguide is restricted in both transverse directions rather than just one. Rectangular waveguides are used in integrated optical circuits, and in laser diodes. They are commonly used as the basis of such optical components as Mach-Zehnder interferometers and wavelength division multiplexers. The cavities of laser diodes are frequently constructed as rectangular optical waveguides. Optical waveguides with rectangular geometry are produced by a variety of means, usually by a planar process.

The field distribution in rectangular waveguide cannot be solved analytically, however approximate solution methods, such as Marcatili's method, are known.

[edit] Rib waveguides

A rib waveguide is a waveguide in which the guiding layer basically consist of the slab with a strip (or several strips) superimposed onto it. Rib waveguides also provide confinement of the wave in two dimensions.

[edit] Optical fiber

A diagram which illustrates the propagation of light through a multi-mode optical fiber.

Optical fiber is typically a circular cross-section dielectric waveguide consisting of a dielectric material surrounded by another dielectric material with a lower refractive index. Optical fibers are most commonly made from silica glass, however other glass materials are used for certain applications and plastic optical fiber can be used for short-distance applications.

[edit] Nonlinear effects

Optical waveguides are ideal for nonlinear interactions because they provide strong beam confinement over long propagation distances. Waveguides are characterized by regions of high refractive index bounded by regions of lower index and can be categorized as planar, channel and fiber guides (Fig. 1). Progress in this area of nonlinear guided-wave optics has concentrated on harmonic generation and parametric amplification, which utilize the large χ(2) obtained in noncentrosymmetric waveguides. However, during the past few years there has been a great deal of interest in all-optical signal processing, which has stimulated work on third-order processes in integrated optics [1], but for fibers the propagation losses are so low that the small third-order nonlinearities in glasses could be compensated for by the long interaction length.

It is the interaction geometry that differentiates guided-wave from bulk nonlinear optics. To obtain high intensities, it is usually necessary to focus laser beams. However, in bulk media, the tighter the focus, the shorter the distance over which it can be maintained. Since the efficiency of a nonlinear interaction depends at least linearly, and in many cases quadratically, on the interaction distance, clearly there is a trade-off in the efficiency between high intensity and interaction length. In a waveguide, however, the beam is confined in one (planar waveguide) or two (fiber or channel waveguide) dimensions to values of the order of the wavelength of light, for distances determined exclusively by the propagation losses. These are typically millimeters to centimeters in integrated-optics waveguides and meters to kilometers in fibers. Thus it is the beam confinement over long distances afforded by waveguides that makes them attractive for nonlinear-optics devices. As the field has progressed, the material properties needed to make efficient devices with well-defined operating parameters have been identified.

[edit] Waveguides characteristics

In planar waveguides light is confined in one dimension but can diffract in the usual way in the plane of the film. For both fiber and channel waveguides the light is confined in both cross-sectional dimensions.

The size of the confinement region is of the order of the wavelength of light. This results in the highest intensities possible for a given input power since the effective beam area is minimized in a waveguide. For example, 1-W power in a 1 μm x 1 μm waveguide produces an intensity of 100 MW/cm2. As is shown in Fig. 2, the guided-wave field is maximum in the region of high index and decays with distance into the media of lower index. Thus, in principle, the nonlinearity can be in either the core (film) or the bounding media.

[edit] Parametric nonlinearities

Parametric nonlinearities are optical ones with an instantaneous response based on the χ(2) or χ(3) nonlinearities of a medium. They give rise to effects such as Frequency Doubling, Sum and Difference Frequency Generation, Parametric Amplification and Oscillation, and Four-wave Mixing. Usually, phase matching is a condition for achieving a high efficiency in such processes. This occurs only in a limited bandwidth. However, by manipulation of the parameters which influence the phase matching, it is possible to shift the wavelength range where the nonlinear interaction is strong.

Parametric processes are usually polarization-dependent: the nonlinearity itself is polarization-dependent, and at least in cases with a χ(2) nonlinearity also the phase matching, because such media exhibit birefringence. The Kerr Effect raises the refractive index by an amount which is proportional to the square-intensity, leading to effects such as Self-phase Modulation and Four-wave Mixing. There are also parametric nonlinearities arising from χ(3). Spontaneous and stimulated Raman scattering is the interaction of light with “optical phonons” and Spontaneous and stimulated Brillouin scattering is the interaction of light with “acoustic phonons” and typically involves counterpropagating waves.

[edit] Optical Kerr Effect

The Optical Kerr Effect is the phenomenon in which the refractive index of the medium changes when the electron orbit is deformed by the strong electric field [2]. The refractive index under the Kerr Effect is expressed as where n₀ denotes and n₂ denote linear refractive index and Kerr coefficient, respectively. The Kerr Coefficient in the silica glass fiber is typically given by n2 = 3.18 × 10-20 m2/V. Some interesting and important nonlinear effects in optical waveguides utilizing Kerr effect are: Self-phase modulation, Optical Solitons and Modulational Instabilities.

A. Self-phase modulation (SPM)

Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the Optical Kerr Effect. This variation in refractive index will produce a phase shift in the pulse, leading to a change of the pulse's frequency spectrum.

Then it is seen from (2) that the angular frequency decreases at the preceding edge and increases at the trailing edge of the pulse. This is also schematically illustrated in Fig. 3 (botton curve). This phenomenon is called Self-phase Modulation, which causes the frequency chirping to the optical pulse. In the time domain, the envelope of the pulse is not changed, however in any real medium the effects of dispersion will simultaneously act on the pulse. In regions of normal dispersion, the "redder" portions of the pulse have a higher velocity than the "blue" portions, and thus the front of the pulse moves faster than the back, broadening the pulse in time. In regions of anomalous dispersion, the opposite is true, and the pulse is compressed temporally and becomes shorter. This effect can be exploited to some degree (until it digs holes into the spectrum) to produce ultrashort pulse compression. Self-phase modulation is an important effect in optical systems that use short, intense pulses of light, such as lasers and optical fibre communications systems

B. Optical Solitons

In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed. In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. Dispersion and non-linearity can interact to produce permanent and localized wave forms. This two effects can be visualized in Fig. 4. [4]. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, there is also the non-linear Kerr effect: the refractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the dispersion effect, and the pulse's shape won't change over time: a soliton. Many exactly solvable models have soliton solutions, including the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation. The soliton solutions are typically obtained by means of the inverse scattering transform and owe their stability to the integrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research. Much experimentation has been done using solitons in fiber optics applications. Solitons' inherent stability make long-distance transmission possible without the use of repeaters, and could potentially double transmission capacity as well.[5].

[edit] Application

Here, we consider a nonlinear slot-waveguide coupler as shown in Fig. 5[6]. All the waveguide parameters and the operating wavelength are: the refractive index of high-index region and cladding are taken as nH = 3.48 (Si) and nC = 1.46 (SiO2), respectively; the waveguide height is h=250 nm; and wH=200 nm. wS is the width of the slot region and nS is the refractive index of the slot region.

The distance between two cores is d, and the operating wavelength is assumed to be 1.55 μm. We assume that the material embedded in the slot region has Kerr-type nonlinearity, and its intensity-dependent refractive index is given by nS = nL(1 + n2|E|2/Z0)1/2, where nL = 1.41545 is the linear part of the refractive index nS, n2 [m2/W] is the nonlinear coefficient, E is the electric field distributions, and Z0 is the free-space impedance. This coupler acts as a polarization-independent coupler in linear regime [7].

Fig. 6(a) shows the normalized output powers in the incident slot waveguide of the coupler as a function of propagation distance for n2P=0.0012 μm2.

Here, P is the optical power. For quasi-TE modes, the output is switched to the incident waveguide because of phase mismatch induced by nonlinearity. However, for quasi-TM modes, there are almost no changes in transmission characteristics compared with the linear case (black dots inFig. 6(b). Fig. 7 shows the normalized optical powers in the bar and cross ports of the coupler as a function of n2P at the propagation distance of 200 μm (coupling length in the linear case). Here, bar and cross ports are the outputs of the waveguides for light incidence and nonincidence, respectively. Intensity-dependent switching characteristics for quasi-TE modes can be clearly seen. Surprisingly, for quasi-TM modes, there are almost no changes in the transmission characteristics, even when high-intensity light is launched. This is quite unusual, because in the conventional nonlinear directional coupler, switching powers for both polarizations are similar, although they are polarization dependent.

This strong polarization dependence can be understood from the guided-mode field distributions of quasi-TE and quasi-TM modes, as shown in Fig. 7(b) and Fig. 7(c), respectively. In these figures, the dominant electric fields (the x and y components of electric fields, Ex and Ey, for quasi-TE and quasi-TM modes, respectively) are shown. We can see that the light is strongly confined in the slot region for quasi-TE modes, whereas for quasi-TM modes, the fields are spread into the cladding region.

Therefore, even if the guided mode with the same optical power is launched, a change in the propagation characteristics of quasi-TE modes caused by the nonlinearity is significantly different from that of quasi-TM modes, resulting in strong polarization-dependent switching characteristics.

[edit] See also

• ARROW waveguide
• Cutoff wavelength
• Dielectric constant
• Digital planar holography
• Electromagnetic radiation
• Erbium-doped waveguide amplifier
• Equilibrium mode distribution
• Leaky mode
• List of telecommunications transmission terms
• Transmission medium
• Waveguide
• Waveguide (electromagnetism)
• Photonic crystal fiber
• Photonic crystal
• Prism coupler

[edit] External links

• AdvR - Rubidium doped waveguides in potassium titanyl phosphate (KTP)

[edit] Notes

1. ^  Ramo, Simon, John R. Whinnery, and Theodore van Duzer, Fields and Waves in Communications Electronics, 2 ed., John Wiley and Sons, New York, 1984.

[edit] References of nonlinear effects

[1] Stegeman, George I. and Seaton, Colin T., "Nonlinear Integrated Optics." Journal of Applied Physics. 15 December 1985, Vol. 58, 12.

[2] Hellwarth, R. W., Third-order optical susceptibilities of liquids and solids. New York : Pergamon Press, 1977. pp. 1–68. Vol. 5. ASIN: B0007AKO7C.

[3] , Self-phase modulation. Wikipedia, the free Encyclopedia. [Online] [Cited: April 26, 2009.] http://en.wikipedia.org/wiki/Self-phase_modulation.

[4] Soliton (optics). Wikipedia, the free Encyclopedia. [Online] [Cited: April 26, 2009.] http://en.wikipedia.org/wiki/Soliton_(optics).

[5] Photons advance on two fronts. EETimes.com. [Online] EE Times. [Cited: April 26, 2009.] http://www.eetimes.com/showArticle.jhtml?articleID=172302644.

[6] Fujisawa, Takeshi and Koshiba, Masanori., "All-optical logic gates based on." J. Opt. Soc. Am. B. 2006, Vol. 23, 6.

[7] —. "Polarization-independent optical directional coupler based on slot waveguides." OPTICS LETTERS. 2006, Vol. 31, 1.