1. Introduction

Angular dispersion optical components (prisms, gratings, etc.) spread out the spectrum of a femtosecond laser pulse, inducing what is known as group delay dispersion (GDD). This phenomenon has quite broad applications, allowing for compression, broadening, and reshaping of a femtosecond laser pulse. The propagation of femtosecond pulses that have passed through single or paired angular dispersion components has also been widely studied [1–22]. These studies are generally based on either the plane wave model [1–7] or the Gaussian beam model [9–20].

In the plane wave model, the variance of pulse width and the GDD effect are chiefly studied by acquiring the spectral phase function with a beam tracing method. In this context, the GDD (a temporal effect) is assumed to be induced by the angular dispersion (a spatial effect) [1–7]. This model also omits information on the inherent size of the laser beam, so it is mainly adopted for wide beams. Gaussian beam models, first established by Martinez [9, 10], are based on Fourier analysis of the beam and require the solution of a Kirchhoff-Fresnel integral. The pulse broadening, pulse front tilt, chirp, and frequency shifting of a Gaussian beam can all be analyzed using this approach, under the assumption that the propagation distance is much less than the Rayleigh length (i.e., the beam is well collimated). Particularly, Martinez presented that for a well collimated Gaussian beam, the effect of angular dispersion on pulse duration is resulted from combined effects of spectral-lateral walkoff and group velocity dispersion [9, 10]. Horvath et al. [16] acquired similar expressions for the width of a femtosecond laser pulse that passes through an ideal grating. While this work also provides an explanation in terms of the frequency domain, which is valid only for distances far greater than the Rayleigh length, this model is limited to cases where the waist of the Gaussian beam overlaps the grating. Varju et al. [17, 18] also acquired an expression for the angular dispersion of a Gaussian beam and plane wave, and verified their result by experiments. On the other hand, no further analysis was carried out on how the angular dispersion affects the pulse width. Recently we have extended the Gaussian beam propagation theory of Martinez [9, 10] to arbitrary propagation distances, and described the variation of spectral lateral walkoff and GDD with distance [19, 20]. In our previous work we have also shown how these effects broaden the pulse width, and experimentally verified the relations.

Pulse width broadening after angular dispersion has not yet been studied for spherical waves, an important alternative model. The spherical wave model has been widely adopted, for example, to describe diffraction at the edges of various optical components (such as pinholes and slits) [16, 23, 24]. In addition to extending our previous work, this paper provides the first pulse broadening analysis of a spherical wave model.

The three models mentioned so far (plane wave, spherical wave, and Gaussian beam) are complementary, describing different aspects of light waves. Could there be common mechanisms at work in the three relations describing pulse width evolution after angular dispersion? Most current works on pulse evolution in the plane wave and Gaussian beam models are based on different framework, and therefore are not directly comparable.

In this paper, Fourier analysis of the electric field is used to define a Kirchhoff-Fresnel integral for the pulse width at arbitrary distances. This approach is extended to the plane wave and spherical wave models in addition to Martinez’s previously published analysis of a Gaussian beam. [9, 10] We show that the analytical expressions for pulse width valid for plane waves, spherical waves, and Gaussian beams can be written in a common, generalized form. The pulse width variation is always a function of two mechanisms: the spectral lateral walkoff and the GDD.

2. General formula of pulse width

The propagation of an electromagnetic field in free space is governed by the Helmholtz equation. Both plane waves and spherical waves are particular solutions of this equation. The Gaussian beam is also a particular solution, but only under the slowly varying amplitude (SVA) approximation [23, 24]. These are the three electric fields conventionally adopted in any analysis of optical systems. Following the method of Martinez [9, 10], which is based on Fourier analysis of the electric field and the Kirchhoff-Fresnel integral, we first analyze the propagation of femtosecond laser pulses under the plane wave, spherical wave models after passing through an angular dispersion component.

The amplitude of a plane wave is perfectly uniform over each infinite plane perpendicular to the wave’s direction of propagation, this plane of constant phase is called the wave front [23, 24]. As shown in Fig. 1, the wave fronts of all spectral components overlap prior to reaching the angular dispersion component. At point O, however, the various spectral components separate from one another. After passing through the angular dispersion component, the width of a plane wave laser pulse at some arbitrary distance z can be derived with the method described in Appendix A, which is

where τ₀ is the full width at half maximum (FWHM) of the initial pulse (assumed to be Gaussian shape). The parameter k is the wave number, and β is the angular dispersion parameter. This pulse width expression can also be acquired using well-known principles of geometrical optics [5–8].

In the spherical wave and Gaussian beam models, however, the wave front is a curved surface [23, 24]. This will result in significant differences in the pulse width after dispersion.

 

Fig. 1. A plane wave passes through an angular dispersion device, represented by a vertical line passing through point O. After the dispersion, each spectral component propagates in a different direction. We may freely define the z-axis as the propagation direction of the central frequency component (ω ₀).Some other frequency ω will then propagate along the z_ω direction, where the angle between the two directions is denoted ε.The angle ε′ between their two wave fronts is equivalent to ε. We also define the distances $\overline{\mathrm{PO}}=d$ and $\overline{\mathrm{OQ}}=z$ .

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The wave front of a spherical wave is shown in Fig. 2 [23, 24]. Similarly, with the method described in Appendix A, the width of a femtosecond pulse with a spherical profile after angular dispersion can be derived as

where d is the distance from the source to the dispersion component and α is the angular magnification factor of the angular dispersion component.

 

Fig. 2. A spherical wave passes through an angular dispersion component. The angle between spectral components ω ₀ and ω is ε. The angle between their wave fronts ε′, however, is less than ε. We also have $\overline{\mathrm{PO}}=d$ , $\overline{\mathrm{OQ}}=z$ . d′=d″=d/α ². P represents the source of the spherical wave.

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Unlike the plane wave and spherical wave models, the Gaussian model has a finite beam size. The amplitude of the electric field has a Gaussian distribution over any given surface of constant phase [23, 24]. Without the well collimated assumption, the pulse width of a Gaussian beam after passing through an angular dispersion component at arbitrary distance z is given by [19]

where w ₀ is the waist size and z_R is the Rayleigh length.

In the previous works, Martinez [10] deduced an analytical expression for pulse width of a Gaussian beam after passing through an angular dispersion component under the well collimated assumption, and pointed out that the pulse broadening is due to two effects: the spectral lateral walkoff and the group delay dispersion. Recently we extended Martinez’s work to an arbitrary propagation distance, by defining the normalized parameters of spectral lateral walkoff and GDD, and expressed in a simpler and more general expression [19, 20]

where the parameter U represents the spectral lateral walkoff, and the normalized V represents the standard term of GDD normalized by a factor τ² ₀/4ln 2 (see the Appendix B for detailed definition). This equation is in a mathematical form practically identical to that proposed by Martinez [9, 10]. U characterizes the effect of spectral components spreading out in the plane perpendicular to the propagation direction of the laser beam. Since angular dispersion spreads out the spectrum, it may not be possible to acquire the entire spectrum at a given position. Thus, according to uncertainty theory, the pulse itself has broadened—this is the spectral lateral walkoff. Yet V describes only the pulse broadening induced by temporal delays between the various spectral components due to optical path length differences in the direction of propagation [19, 20]. The expressions for plane wave and spherical wave pulse broadening (Eq. (1) and Eq. (2)) are much simpler than that of a Gaussian beam (Eq. (3)). In the next section, we will show that broadening in all three beam models can be described by the more general expression of Eq. (4). As is clear from Eqs. (1), (2), and (3), the functional form of spectral lateral walkoff and GDD depend heavily on the beam profile. In the following section, we will analyze the amplitude and phase of the electric field from another perspective and deduce the expressions for spectral lateral walkoff and GDD in the three beam models.

3. Spectral lateral walkoff and group delay dispersion

The broadening of a femtosecond laser pulse after passing through an angular dispersion component is a combination of spectral lateral walkoff and GDD [9, 10, 19, 20]. Expressions for these two effects can be acquired by calculating the derivatives of the electric field’s amplitude and phase [16], the detailed method is shown in Appendix B.

The spectral lateral walkoff in the plane wave model, U _P, is simply

The GDD of the plane wave model, V _P, is

While no spectral lateral walkoff occurs in a plane wave, the GDD increases linearly with propagation distance. Any broadening of the pulse after the angular dispersion component is therefore completely induced by GDD. Note that the functions U _P and V _P allow Eq. (1) to be expressed in the general form of Eq. (4).

The spectral lateral walkoff in the spherical wave model U _S, is also zero:

The GDD of the spherical wave model, V _S, is

Thus, the main difference between the spherical wave model and the plane wave model is that the induced GDD is no longer a linear function of distance. For large z, the GDD of the spherical wave approaches the constant value -4ln 2kβ ² d/α ²τ² ₀. Once again, defining U _S and V _S reduces Eq. (2) to the simplified form of Eq. (4).

The Gaussian beam, however, has the following spectral lateral walkoff [19]:

Its GDD is

In this case the pulse width is determined by both spectral lateral walkoff and GDD. Defining the parameters U _G and V _G permits Eq. (3) to be written in the form of Eq. (4) [19, 20].

It has to be emphasized that when a Gaussian beam propagates over distances much less than the Rayleigh length (d, α ² z≪z _R, this condition corresponds to the assumption of well collimated beam), the GDD can be approximated as V_G=-(4ln 2)·kβ ² z/τ² ₀. This is the same result given in Eq. (6) for the plane wave model. At distances far exceeding the Rayleigh length (z≫z _R), on the other hand, the GDD is $V_G\approx -\frac{4\ln 2·\frac{d}{d+{\alpha }^{2}z}k{\beta }^{2}z}{{\tau }_0^2}.$. This is the same as Eq. (8), which was derived for the spherical wave model. Apparently, over short distances, the GDDs of these three models are basically equivalent. Furthermore, at large distances the GDD effects induced by a spherical wave and a Gaussian beam are the same: $V_S=V_G=-\frac{4\ln 2}{{\tau }_0^2}·\frac{1}{{\alpha }^2}k{\beta }^{2}d$. The final GDD is thus completely determined by the optical parameters of the angular dispersion component and the position of the source point (spherical wave) or beam waist (Gaussian beam). For a spherical wave, when d=0 (that is, when the source overlaps the angular dispersion component) the GDD is 0 at all propagation distances. In this case the spectral lateral walkoff of the spherical wave is also 0. Thus, as indicated by Eq. (4), the width of a femtosecond laser pulse is not broadened by the angular dispersion component.

Let us adopt a laser with central wavelength λ ₀=800nm, pulse width τ₀=100 fs (FWHM, Gaussian shape), α=1, β=0.1rad/µm, which is equivalent to a commonly used Bragg grating with the groove spacing 10µm (100 lines/mm), and d=0.5m. In the case of a Gaussian beam, let w ₀=0.5mm and z_R=1m. Recall that there is no spectral lateral walkoff for either the plane wave or the spherical wave (U_P=U_S=0). The spectral lateral walkoff U_G of the Gaussian beam is depicted in Fig. 2 of Ref. [19], and as it turns out, it would turn to be steady finally. The GDDs of the three models are depicted in Fig. 3.

 

Fig. 3. GDD of the plane wave, spherical wave, and Gaussian beam as a function of distance.

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The GDD induced by a plane wave is linear with distance, while the spherical wave and Gaussian beam quickly converge to a reasonably steady GDD.

The resulting pulse broadening curves of the three beam models are plotted in Fig. 4.

 

Fig. 4. Pulse width as a function of distance for the plane wave, spherical wave, and Gaussian beam after passing through an angular dispersion component.

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Under the plane wave model, pulse width increases in proportion to distance after passing through an angular dispersion component. Under the spherical wave and Gaussian beam models, the pulse width converges to a constant value after a short distance.

4. Discussion

Using the method introduced by Martinez [9,10], based on the Kirchhoff-Fresnel integral and Fourier transformation, we have derived formulas describing the pulse broadening of plane waves, spherical waves, and Gaussian beams after passing through an angular dispersion component. By dividing the pulse broadening into two separate effects: spectral lateral walkoff and group dispersion delay (GDD), calculating the derivative of the electric field’s amplitude and phase [16], and defining the normalized parameters of U and V, the form of these expressions can be greatly simplified.

In the case of a plane wave or spherical wave, since the model provides no information on beam size, all spectral components can be detected at every point in space. Thus, the spectral lateral walkoff is zero for both waves and pulse broadening is entirely determined by the GDD. The Gaussian beam has a finite size, however, so the complete spectrum cannot be detected outside a limited region. This leads to the effect named spectral lateral walkoff, which also contributes to the pulse broadening [10].

The GDD of a Gaussian beam is approximately equal to that of a plane wave at distances far less than the Rayleigh length. At distances far exceeding the Rayleigh length, however, it is approximately equal to that of a spherical wave. The reason for this difference is simply that the wave front of the Gaussian beam is similar to a plane close to the waist, and similar to a sphere far from the waist [23]. The GDD is thus actually determined by the changing shape of the wave front.

For both cases of spherical wave and Gaussian beam, the GDD approaches a limiting value at large distances which depends only on the characteristics of the angular dispersion component and the position of source point (spherical wave) or beam waist (Gaussian beam). The wave fronts of the various spectral components actually overlap at the surface of the disperser and at infinity, so no GDD is induced in this space. As depicted in Fig. 2, the equivalent source point (or beam waist) is located at a distance d/α ² in front of the angular dispersion component. In the reverse propagation direction, the wave fronts of different spectral components do not overlap at this position; thus, the limiting value of the GDD is also determined by this specific distance.

The broadening of a plane wave pulse after passing through an angular dispersion component depends entirely on the GDD, which is a linear function of distance. The pulse width of a spherical wave or Gaussian beam increases rapidly at first, then approaches a constant value. This behavior occurs because the angle between different spectral components approaches zero at large distances in these two models [17]. In other words, the wave fronts overlap again at infinity, so no GDD is induced any more. For a Gaussian beam, there is no division of the spectral components at infinity, so the spectral lateral walkoff is also steady.

Although in Fig. 4 the final pulse width of a Gaussian beam is greater than that of a spherical wave, this conclusion does not hold unconditionally. As indicated by Eq. (4), a spectral lateral walkoff does not necessarily result in a broader pulse. (The parameter U also appears in the denominator of the second term, where it reduces the pulse width.) As can be deduced from Eq. (2) and Eq. (3), the condition that the Gaussian pulse’s final width exceeds that of the spherical pulse is $8\ln 2k{\beta }^2{d}^2<{\alpha }^2{\tau }_0^2\left[z_R+\sqrt{z_R^2+4d^2}\right]$ . The final pulse width in both models is entirely determined by the characteristics of the angular dispersion component and the position of the source (or beam waist).

It is interesting that when the source point of a spherical wave overlaps the surface of the angular dispersion component, the spectral lateral walkoff and GDD remain zero regardless of distance. In this case the pulse is not broadened at all. This is because the wave front propagates uniformly in all direction, each spectral component will retain its spherical shape after passing through the angular dispersion component.

At last, it should be noted that the general formula proposed in this paper has been validated by experiments partially by our previous works for Gaussian beam model [19, 20] and by the experiments of Osvay et al. for plane wave model [7]. Further experiment with spherical wave model is under investigation which requires special measurement techniques.

5. Conclusion

We have derived analytical expressions for the width of a femtosecond laser pulse after passing through an angular dispersion component. The expressions are valid at all distances, and for three beam models: the plane wave, the spherical wave, and a Gaussian beam. Furthermore, we have shown that the three expressions can be generalized to variations of a generalized function of two terms: spectral lateral walkoff and group delay dispersion (GDD). There is no spectral lateral walkoff in the plane wave and spherical wave models, as the pulses are treated as having infinite extent. We find that the GDD of a Gaussian beam approximates that of the plane wave at distances far less than the Rayleigh length, and that of the spherical wave at distances far exceeding the Rayleigh length. Finally, the GDD effect of the spherical wave (and thus the Gaussian beam as well) at infinity is a constant completely determined by the physical characteristics of the angular dispersion component and the location of the source point (or beam waist). The width of a plane wave pulse increases in proportion to the propagation distance after passing through an angular dispersion component, while the spherical and Gaussian pulses approach their limiting values fairly quickly. The final pulse width of a Gaussian beam is not always wider than that of a spherical wave with similar parameters. Under the spherical wave model, there is also a unique condition: if the source point overlaps the surface of the angular dispersion component, the pulse width remains constant at all distances.