The Effect of Dispersion on Ultrashort Pulses

In the time domain, the electric field for a Gaussian pulse with a carrier frequency, ω₀, pulse duration, Δt, and phase, θ(t), can be described by,

where c.c. denotes the complex conjugate. In this expression, A_t is the amplitude of the pulse, ω₀ determines the color of the pulse, Δt determines the minimum pulse duration and consequently the bandwidth of the pulse, and θ(t) determines the temporal relationship among the frequency components contained within the bandwidth of the pulse. θ(t) plays an important role in altering the pulse duration. It is the term that is responsible for pulse broadening in dispersive media and can be thought of as adding a complex width to the Gaussian envelope.

The description of the Gaussian pulse given by (1) is intuitive in the sense that it is fairly straightforward to conceptualize a pulse in the time domain. However, when dealing with pulses traveling through dispersive media, it can be problematic to work in the time domain. For example, in order to determine the duration of a pulse after traveling through some dispersive material, it is necessary to solve a convolution integral1 which in general must be done numerically. However, due to the fact that convolutions become products upon a Fourier transformation2, it is convenient to solve this type of problem in the frequency domain.

Time and frequency along with position and momentum represent a class of variables known as Fourier pairs2. Fourier pairs are quantities that can be interconnected through the Fourier transform. Performing a Fourier transform on equation (1) yields,

(for the sake of brevity, negative frequency components are omitted). The electric field is now expressed as a function of frequency, Δω and Δt are related through the uncertainty relation1

ΔωΔt=4ln(2),

and the spectral phase, φ(ω), describes the relationship between the frequency components of the pulse. In equation (2), ω as well as Δω represent angular frequencies. Angular frequency can be converted to linear frequency, ν (i.e. the observable quantity), by dividing it by 2 π,

ν = ω/2 π

In terms of the linear frequency, the uncertainty principle is given by,

c_B = ΔνΔτ = 2ln(2)/ π

When an input pulse, E_in(ω), passes through a dispersive medium, the phase added by the material is given simply by the product of the input field with the transfer function of the material. The emerging pulse E_out(ω), is given by,

where φ_Mat(ω - ω₀) is the spectral phase added by the material and R(ω) is an amplitude scaling factor which for a linear transparent medium can be approximated by, R(ω) ≈ 11.

It is a common convention to express spectral phase as a Taylor expansion around the carrier frequency of the pulse as shown below,

This approach allows a more straightforward understanding of the effect of material dispersion on properties of the pulse. Taking into account that

φ(ω) = k(ω)L,

where k is the propagation constant, and L is the length of the medium, while also considering that the group velocity is defined as

ν_G = dω/dk,

it is easy to see that first term in (4) adds a constant to the phase. The second term, proportional to 1/ν_G, adds delay to the pulse. Neither of these terms affects the shape of the pulse. The third term, referred to as group delay dispersion (GDD), is proportional to

also known as group velocity dispersion (GVD). It introduces a frequency dependent delay of the different spectral components of the pulse, thus temporally changing it. The GDD and GVD are related through

φ₂(ω) = k₂(ω)L

The fourth term, referred to as Third Order Dispersion (TOD) applies quadratic phase across the pulse. For the purpose of this tutorial, we will truncate the series at the third term, GDD, only making references to higher order terms when necessary. Truncating equation (4) at the third term allows us to rewrite equation (3) for a Gaussian pulse as,

hence phases in the frequency domain are simply additive. This result underscores the advantage of performing these types of calculations in the frequency domain.

To arrive at the new pulse duration, it is necessary to transform the spectral envelope of equation (5) back into the time domain. Performing this Fourier transform, the pulse envelope is given by,

where φ₂ is the sum of the group delay dispersion of the material and the group delay of the pulse. In order to get the new pulse duration, Δt_out, it is necessary to obtain the intensity, I_out(t), by squaring the electric field in equation (6) and then relating I_out(t) to the general form for a Gaussian pulse,

Solving equation (7) for Δt_out,

provides an expression for the pulse duration. Finally, by solving equation (8) for group delay dispersion while replacing the transform limited pulse duration with the spectral bandwidth of the pulse, GDD can be expressed completely in terms of observables (i.e. pulse width and spectrum),

where Δν = cΔλ/λ2. In general, c_B is a function of the pulse profile as shown in Table 1. It should be noted that equation (9) is strictly for Gaussian pulses.

Table 1: Time-bandwidth product C_B for various pulse profiles

Dispersion in materials is defined by the group velocity dispersion. In order to estimate amount of GDD introduced by a material of length L, one has to calculate the wavelength dependent index of refraction, n(λ), typically in the form of a Sellmeier’s type equation, and then calculate second derivative at the wavelength of interest. GVD is related to the second derivative of refractive index with respect to wavelength by

GDD is simply a product of GVD with the length of the material. The dispersive properties of several optical materials are shown in Table 2.

TOD is the frequency dependence on the GVD. The dispersion properties are specified in units of fs3. TOD of several optical materials are shown in the Table below.

Table 2: Material parameters for fused silica, LakL21, SF10 and BK7 glass

By measuring the spectrum and autocorrelation for a Gaussian pulse, equation (9) can be used to determine the amount of GDD. Figure 1 illustrates the results of a numerical simulation of the electric field for three pulses, all containing 100 nanometers of bandwidth, centered around 800 nanometers. The black curve corresponds to a pulse with the GDD set to zero, the red curve corresponds to a pulse with the GDD set to 5 fs2 and the blue curve corresponds to a pulse with the GDD set to -5 fs2. The pulse with the minimum time duration corresponds to the pulse having zero GDD. For the red pulse (positive chirp), the higher frequency components are lagging behind the lower ones and for the blue pulse (negative chirp), the lower frequency components are lagging behind the higher ones.

Figure 2 shows the width of a Gaussian pulse at 800nm before and after propagation through 20 mm of BK7 glass calculated using equation (8) and data from Table 2.

The amount of introduced GDD in this case is about 1000 fs2, and is equivalent to propagating the beam through only a few optical components. It is clear that the effect is not significant for pulses longer than 100 fs. However, a 25 fs pulse broadens by a factor of 4.

1. J. Diels and W. Rudolf, Ultrashort Laser Pulse Phenomena, Second Edition (Massachusetts, Academic Press, 2006).
2. E. Oran Brigham, The Fast Fourier Transform: An Introduction to Its Theory and Application (New Jersey, Prentice Hall, 1973).
3. To learn about dispersion compensation using a prism compressor, see Newport's Application Note 29