Nonlinear optical phenomena

Nonlinear optical phenomena arise from changes in the optical properties of a material in the presence of an intense electric field from electromagnetic radiation. Here we explore two phenomena that not only can be studied conveniently with intense laser beams but are commonly used in the laboratory to modify the output of lasers for specific experiments, such as those described in Section 14.6. In frequency doubling, or second harmonic generation, an intense laser beam is converted to radiation with twice (and in general a multiple) of its initial frequency as it passes through a suitable material. It follows that frequency doubling and tripling of a Nd–YAG laser, which emits radiation at 1064 nm, produce green light at 532 nm and ultraviolet radiation at 355 nm, respectively. We can account for frequency doubling by examining how a substance responds nonlinearly to incident radiation of frequency ω = 2πν. Radiation of a particular frequency arises from oscillations of an electric dipole at that frequency and the incident electric field E induces an electric dipole of magnitude µ, in the substance. At low light intensity, most materials respond linearly, in the sense that µ = αE, where α is the polarizability (see Section 18.2). To allow for nonlinear response by some materials at high light intensity, we can write

µ=αE+ βE²+ . . .

where the coefficient β is the hyperpolarizability of the material. The nonlinear term βE² can be expanded as follows if we suppose that the incident electric field is E₀ cos ωt:

βE²=βE²₀ cos²ωt = βE²₀(1 + cos 2ωt)

Hence, the nonlinear term contributes an induced electric dipole that oscillates at the frequency 2ω and that can act as a source of radiation of that frequency. Common materials that can be used for frequency doubling in laser systems include crystals of potassium dihydrogen phosphate (KH₂PO₄), lithium niobate (LiNbO₃), and β barium borate (β-BaB₂O₄). Another important nonlinear optical phenomenon is the optical Kerr effect, which arises from a change in refractive index of a well-chosen medium, the Kerr medium, when it is exposed to intense laser pulses. Because a beam of light changes direction when it passes from a region of one refractive index to a region with a different refractive index, changes in refractive index result in the self-focusing of an intense laser pulse as it travels through the Kerr medium (Fig. 20.64). The optical Kerr effect is used as a mechanism of mode-locking lasers (Section 14.5). A Kerr medium is included in the cavity and next to it is a small aperture. The procedure makes use of the fact that the gain, the growth in intensity, of a frequency component of the radiation in the cavity is very sensitive to amplification and, once a particular frequency begins to grow, it can quickly dominate. When the power inside the cavity is low, a portion of the photons will be blocked by the aperture, creating a significant loss. A spontaneous fluctuation in intensity—a bunching of photons— may begin to turn on the optical Kerr effect and the changes in the refractive index of the Kerr medium will result in a Kerr lens, which is the self-focusing of the laser beam. The bunch of photons can pass through and travel to the far end of the cavity, amplifying as it goes. The Kerr lens immediately disappears (if the medium is well chosen), but is re-created when the intense pulse returns from the mirror at the far end. In this way, that particular bunch of photons may grow to considerable intensity because it alone is stimulating emission in the cavity. Sapphire is an example of a Kerr medium that facilitates the mode locking of titanium sapphire lasers, resulting in very short laser pulses of duration in the femtosecond range. In addition to being useful laboratory tools, nonlinear optical materials are also finding many applications in the telecommunications industry, which is becoming ever more reliant on optical signals transmitted through optical fibres to carry voice and data. Judicious use of nonlinear phenomena leads to more ways in which the properties of optical signals, and hence the information they carry, can be manipulated.

Fig. 20.64 An illustration of the Kerr effect. An intense laser beam is focused inside a Kerr medium and passes through a small aperture in the laser cavity. This effect may be used to mode-lock a laser, as explained in the text.

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Strong electrolytes

Conductance and conductivity

Superconductors

Induced magnetic moments

Magnetic susceptibility

Light emission by solid-state lasers and light-emitting diodes

Light absorption by molecular solids, metallic conductors, and semiconductors

Insulators and semiconductors

The occupation of orbitals

The formation of bands

Molecular solids and covalent networks

Energetics