Scanning the Z

The ultra-fast nonlinear optical (NLO) refractive index n₂ causes the index to be modified according to n = n₀ + n₂I, where I is the irradiance and n₀ is the linear refractive index. This, in turn, results in the optical field phase φ being altered on propagation along z according to dφ/dz = kn₂I, where k = 2¼n₀/λ. This NLO phase shift has multiple consequences including temporal and spectral pulse broadening, which can limit the data rates and cross talk between wavelength division multiplex channels in optical fiber.¹

The point here is how to measure n₂ in optical fibers. Each measurement technique has its own advantages. If possible, multiple complementary methods should be used to fully understand the nonlinear response, since multiple effects often occur simultaneously. Glass fibers are usually used at wavelengths well below their absorption cutoff frequency, however, where effects such as two-photon absorption are negligible. Then n₂ becomes the primary nonlinear response.

If a direct measurement of n₂ for a fiber is needed, a spectral measure is often sufficient, captured by monitoring the spectral broadening of a pulse, however, the irradiance within the fiber required by that method is difficult to determine accurately.

Another method that has gained popularity is the Z-scan.² This method measures spatial modulation of phase, however, so it requires bulk samples (e.g. performs, not fibers). The Z-scan has become widely used due primarily to its simplicity and the fact that it is often simple to interpret.

Figure 1. In the Z-scan, we translate an optically polished sample along the axis of a beam of known (typically Gaussian) irradiance distribution while capturing the far-field transmittance through a fixed aperture.

In the Z-scan, a beam of known irradiance distribution (typically Gaussian) is focused to a beam waist at which the confocal parameter is greater than the sample thickness; the latter simplifies modeling (see figure 1). We scan the optically polished sample, typically about 1-mm thick, along the propagation direction z and monitor the transmittance through a fixed aperture T_A in the far field.

Figure 2. In a plot of transmittance versus sample position (here, carbon disulfide), a peak prior to beam waist followed by a valley indicates a positive n₂. In the case of a negative n₂, the two are reversed.

Assuming no nonlinear absorption, the n₂ results in beam defocusing or focusing depending on the sign of n₂ and on the sample position. A positive n₂ produces a decrease in T_A (valley) with the sample positioned just before the beam waist (farther from the detector) and an increase in T_A (peak) for the sample positioned just after the waist (see figure 2). For a negative n₂, the order of the peak and valley is reversed. The magnitude of n₂ is directly proportional to the change in the normalized transmittance between the peak and valley.

This method requires high-irradiance short pulses with I ~ 1 GW/cm² to achieve good signal-to-noise ratio, since n₂ ~ 10^-14 cm²/W. Larger values are often associated with non-ultra-fast nonlinearities such as thermal expansion, electrostriction, or the production of excited states via linear or nonlinear absorption. The first two have slow turn-on times and can be eliminated with the use of picosecond or shorter pulses. We can also eliminate nonlinear refraction caused by excited states with the use of ultra-short pulses.

As powerful as the technique is, a single Z-scan does not specify the specific nonlinear response without other knowledge, however, this is a topic for another note. oe

1. Fiber Optics Handbook, ed. M. Bass, McGraw Hill, New York, NY (2002).

2. M. Sheik-bahae et al., J. of Quant. Elect. 26, p. 760 (1989).