The mode field radius of a step index fiber is measured as \(4.5 \mu \mathrm{m}\) at free-space wavelength \(\lambda=1.30 \mu \mathrm{m}\). If the cutoff wavelength is specified as \(\lambda_{c}=1.20 \mu \mathrm{m}\), find the expected mode field radius at \(\lambda=1.55 \mu \mathrm{m}\).

A step index fiber is a type of optical fiber characterized by a core with a uniform refractive index enclosed by a cladding with a lower refractive index. The 'step' refers to the sudden change in refractive index at the interface between the core and the cladding, which creates a sharp boundary that guides light within the fiber. This configuration ensures total internal reflection of light beams, allowing them to travel great distances with minimal loss.

Such fibers are commonly used in telecommunications and networking because they can transmit data over long distances without significant signal attenuation. The simplicity of their design also contributes to their popularity in various optical applications.

The term 'wavelength' refers to the distance between two consecutive points that are in phase on a wave, such as two crests or troughs. Represented by the Greek letter \( \lambda \), wavelengths are a fundamental concept in the study of optics and photonics. The wavelength of light determines its color when in the visible spectrum and also influences its behavior in an optical fiber.

In the context of a step index fiber, different wavelengths of light will have varying mode field diameters—a measure of the distribution of light across the fiber core—which directly affects the fiber's performance and its capacity to carry information.

The cutoff wavelength, \( \lambda_c \), in fiber optics, is a critical wavelength above which a single-mode fiber starts to operate as a multimode fiber. This means that at wavelengths longer than the cutoff, the fiber can support more than one propagation mode—several paths through which light waves can travel.

The cutoff wavelength is an important specification for fiber design because it determines the operating range of the fiber. For a single-mode operation, the operating wavelength must be longer than the cutoff wavelength. Fibers operating close to their cutoff wavelength tend to have a higher dispersion, which can degrade signal quality, hence the importance of carefully considering \( \lambda_c \) in fiber-optic communication systems.

In the realm of step index fibers, the scaling exponent, denoted as \( \alpha \), is a critical parameter that describes how the mode field radius (the effective width of the light beam in the fiber) scales with wavelength. The exponent is derived from the ratio of the logarithms of mode field radius at a reference wavelength to the radius at the cutoff wavelength, and the corresponding wavelengths.

This exponent is essential in determining how the mode field radius changes as the wavelength of light passing through the fiber changes. It plays a key role when calculating expected fiber characteristics, like the mode field radius at different wavelengths, which is necessary in designing fiber-optic systems with optimum performance. By controlling the scaling exponent, fiber manufacturers can tailor fibers for specific applications that require distinct light propagation characteristics.