# Dispersion

This is **Section 1.1** of the Laser Optics Resource Guide.

Optical components require close attention to the characteristics of the component’s substrate, especially in laser applications. A wide range of optical glass and crystalline materials are used as substrates for optical components. Understanding the properties of these materials and the defects that can be introduced in the manufacturing process will ensure you choose the correct optics for your application. A number of key substrate characteristics including dispersion, absorption, thermal properties, homogeneity, and subsurface damage will significantly affect the substrate chosen for optical components.

Dispersion is the dependence of light’s phase velocity or phase delay as it transmits through an optical medium on another parameter, such as optical frequency, or wavelength. Several different types of dispersion can occur inside an optic’s substrate: chromatic (*Figure 1*), intermodal, and polarization mode dispersion.^{1}

**Figure 1: **Refractive index of UV Grade fused silica as a function of wavelength

The refractive index is the ratio between the speed of light in vacuum and a light wave’s phase velocity while travelling through a medium, such as air or glass. In pulsed laser applications light is commonly described using frequency because time is generally more critical and the frequency of light is a fixed value, while its wavelength is dependent on the refractive index it is travelling within. Wavelength (λ) is related to angular frequency (ω), refractive index (n), and the speed of light (c) by:

**(1)**$$ \lambda = \frac{2 \pi \, c}{\omega \, n} $$

The refractive index of a material is often described using the Selmeier formula and the material constants B_{1}, B_{2}, B_{3}, C_{1}, C_{2}, and C_{3}:

**(2)**$$ n^2 \! \left( \lambda \right) - 1 = \frac{B_1 \, \lambda ^2}{\lambda ^2 - C_1 } + \frac{B_2 \lambda^2}{\lambda^2 - C_2} + \frac{B_3 \lambda^2}{\lambda ^2 -C_3} $$

Chromatic dispersion is a dependence of light’s phase velocity ν_{p} in a medium on its wavelength, resulting mostly from the interaction of light with electrons of the medium. Chromatic dispersion is described by the Abbe number (*Figure 2*), which corresponds to the reciprocal of the first partial derivative of refractive index with respect to λ, and partial dispersion, which corresponds to the second derivative of refractive index with respect to wavelength.

**Figure 2:** Abbe diagram plot showing the refractive index of common glass types vs their Abbe number. CTE (coefficient of thermal expansion) is defined in the Thermal Properties of Optical Substrates app note.

The Abbe number is given by:

**(3)**$$ V_D = \frac{n_D - 1}{n_F - n_C} $$

n_{D}, n_{F} and n_{C} are the substrate’s refractive indices at the wavelengths of the Fraunhofer D- (589.3nm), F- (486.1nm), and C- (656.3nm) spectral lines. The Abbe number of a material may also be described at any wavelength using the derivative of refractive index with respect to wavelength:

**(4)**$$ V_{\lambda} = -\frac{1}{2} \left(n - 1 \right) \frac{\text{d} n}{\text{d} \lambda} $$

In laser applications, the primary concern is how dispersion will affect the properties of a laser pulse traveling through the medium, which is described by group velocity - the variation of the phase velocity of light in a medium relative to its wavenumber:

**(5)**$$ v_g = \left( \frac{\partial k}{\partial \omega} \right)^{-1} = c \left[ \frac{\partial}{\partial \omega} \left( \omega n \! \left( \omega \right) \right) \right] ^{-1} = \frac{c}{n \! \left( \omega \right) + \omega \frac{\partial n}{\partial \omega}} = \frac{c}{n_g \! \left( \omega \right)} $$

The wavenumber (k) is 2π/λ - this concept is sometimes also referred to a spectral phase. When light of multiple wavelengths travels through a material, it is common for the longer wavelength (low frequencies) to travel slightly faster than the shorter wavelengths due to a frequency (or wavelength) dependence of the group velocity.^{2} This causes a spectral variation of the phase of the wavefront in the same way that light travelling through a prism is broken into its component colors from spectral dispersion of the material. As the group velocity is given as the first derivative of phase velocity with respect to frequency, the group velocity dispersion (GVD) is the derivative of the inverse group velocity with respect to frequency:

**(6)**$$ \text{GVD} = \frac{\partial}{\partial \omega} \left( \frac{1}{v_g} \right) = \frac{\partial}{\partial \omega} \left( \frac{\partial k}{ \partial \omega} \right) = \frac{\partial ^2 k}{\partial \omega ^2} $$

Just as the group velocity is similar to spectral dispersion, in that both correspond to the first derivative of refractive index with wavelength or frequency, the GVD is used similarly to the partial dispersion as they are both second derivatives with respect to wavelength or frequency. Designing optics for low-GVD is similar to designing for good chromatic performance, except the focus is placed on group velocity and GVD rather than the related Abbe number and partial dispersion.

GVD is highly wavelength dependent and has typical units of fs^{2}/mm. For example, the GVD of fused silica is +57 fs^{2}/mm at 589.3nm and −26 fs^{2}/mm at 1500nm. Somewhere between those wavelengths (at about 1.3μm), there is a zero-dispersion wavelength where GVD is zero. *Figure 3* shows the GVD of fused silica vs wavelength. For optical fiber communications, the GVD is typically defined as the derivative with respect to wavelength instead of frequency and is usually specified with units of ps/(nm km).

**Figure 3: **GVD vs wavelength for fused silica with a zero-dispersion wavelength around 1.3μm

As seen in *Figure 3*, there is a significant amount of variation of the GVD with wavelength. This can be extremely problematic for very short laser pulses below approximately 30fs in duration because short pulses have an inherently broad spectrum. Chromatic dispersion may also make refraction angles at optical surfaces frequency-dependent, causing angular dispersion and frequency-dependent path lengths. This significantly impacts broadband systems such as ultrafast laser systems.

Intermodal dispersion is a dependence of the group velocity of light in a waveguide, such as a multimode fiber, on the optical frequency and the propagation mode.^{2} In multimode optical fiber communication systems, this severely limits the achievable data transmission rate, or bit rate. Intermodal dispersion could be prevented by using single-mode fibers or multimode fibers with a parabolic refractive index profile.

Polarization mode dispersion is the dependence of light’s propagation characteristics in a medium on polarization state, which can be relevant in high data rate single-mode fiber systems. All three types of dispersion may cause temporal broadening or compression of ultrashort pulses in free space or optical fibers, potentially causing separate pulses blend together and become unrecognizable (*Figure 4*).

**Figure 4: ** Dispersion can cause laser pulses traveling down fibers to spread until they become unrecognizable

**References**

^{1} Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology, RP Photonics, October 2017, www.rp-photonics.com/encyclopedia.html.

^{2} Ghatak, Ajoy, and K. Thyagarajan. “Optical Waveguides and Fibers.” University of Connecticut, 2000.