## Abstract

The evolution of the frequency chirp of a laser pulse inside a classical pulse compressor is very different for plane waves and Gaussian beams, although after propagating through the last (4th) dispersive element, the two models give the same results. In this paper, we have analyzed the evolution of the frequency chirp of Gaussian pulses and beams using a method which directly obtains the spectral phase acquired by the compressor. We found the spatiotemporal couplings in the phase to be the fundamental reason for the difference in the frequency chirp acquired by a Gaussian beam and a plane wave. When the Gaussian beam propagates, an additional frequency chirp will be introduced if any spatiotemporal couplings (i.e. angular dispersion, spatial chirp or pulse front tilt) are present. However, if there are no couplings present, the chirp of the Gaussian beam is the same as that of a plane wave. When the Gaussian beam is well collimated, the introduced frequency chirp predicted by the plane wave and Gaussian beam models are in closer agreement. This work improves our understanding of pulse compressors and should be helpful for optimizing dispersion compensation schemes in many applications of femtosecond laser pulses.

©2009 Optical Society of America

## 1. Introduction

A classical pulse compressor (commonly including four gratings or prisms, or a pair of gratings and prisms with a double-pass configuration) [1,2] can stretch or compress a femtosecond laser pulse by introducing variable amounts of positive or negative frequency chirp (also referred to as just the “chirp”). Pulse compressors are ubiquitous in ultrafast optics because of their ability to tailor the pulse’s temporal duration, which is essential for making and maintaining intense, short pulses. Important applications include chirped-pulse amplification (CPA) and material-dispersion compensation, the latter of which is necessary for generating ultrashort pulses [3–5]. Since the duration of the pulse after the compressor depends on its group-delay dispersion, it is vital to able to accurately calculate this quantity [6–8].

In some practical applications, the classical 4-dispersive element compressor is not always suitable, and instead, a single angular dispersion element [9–11] or a pair of angular dispersion elements (also just referred to as an “element”) in a single-pass configuration [12–14] are used to for dispersion control. In these cases, spatiotemporal couplings (meaning that there are *x*-*ω*, or equivalently *x*-*t* cross terms in the field) such as angular dispersion, spatial chirp, and pulse front tilt are present in the output pulse, making it even more difficult to calculate the chirp that is added to the pulse by the compressor.

To model compressors, either a plane wave [1,2,9–12], or a Gaussian beam [5–8,15–18]) model is usually used. Martinez first showed that these two models sometimes give very different results. Namely, the propagation dependence of the chirp of a pulse while propagating inside a classical pulse compressor is very different for the two types of beams. Interestingly, however, after propagating through the 4th dispersive element (i.e., once all of the angular dispersion is removed), both models predict the same results [15]. Similarly it has been shown several times, that, when a pulse passes through a single angular disperser, the chirp of a Gaussian beam increases nonlinearly with propagation distance away from the disperser, while that of a plane wave increases linearly [8]. Though compressors are very commonly used, it seems that there is still more to learn about how they affect ultrashort pulses.

Usually, the diffraction integral is used to investigate the propagation dependence of an ultrashort pulse, but this approach is quite complex making it difficult to use to understand how compressors work and why they effect plane waves differently than they do Gaussian beams [15]. Actually, the evolution of the chirp is entirely caused by a spatio-spectral phase *ϕ*(*x*,*z*,*ω*) that is added to the pulse by the compressor (since the chirp is defined as the second order derivative of the pulse’s spectral phase with respect to angular frequency) [19]. Therefore, it is sufficient to determine the chirp’s evolution by calculating this phase change that is acquired by the pulse due to propagation through the compressor.

In this paper, we have derived expressions for the propagation dependence of the frequency chirp of a femtosecond Gaussian laser pulse after passing through each element of a classical four-element compressor. These expressions were derived by calculating the phase change acquired by the Gaussian beam after propagating through each element of the compressor. These calculations reveal the physical mechanism by which chirp is acquired by an ultrashort pulse in a pulse compressor. As the effects of misalignment of the dispersive elements are very complex [20–22], we consider only perfectly aligned compressors in this paper.

## 2. Phase acquired by a Gaussian pulse in a compressor

A steadily propagating electromagnetic field in free space is governed by the scalar approximation of the Helmholtz equation. The plane wave is the simplest solution to this equation, and the Gaussian beam is also a special solution which is obtained if the slowly varying amplitude (SVA) approximation is made. The Gaussian beam is a very good model for realistic laser beams, so it is frequently adopted for modeling optics experiments. The phase of a Gaussian beam propagating in free space is given by [23]:

*r*= (

*x*

^{2}+

*y*

^{2})

^{1/2}is the distance from the z axis,

*R*(

*z*) =

*z*+

*z*

_{R}^{2}/

*z*is the radius of curvature of the wave front,

*z*=

_{R}*kw*

_{o}^{2}/2 is the Rayleigh range of the Gaussian beam, and

*w*is the beam waist size. Equation (1) describes the phase shift of a Gaussian beam at the point (r, z) relative to the original point (0, 0). In this equation, the first term is the geometrical phase shift which is the same as the phase of a plane wave. The second term represents a phase shift relative to the radial position. This is due to the finite beam size of the Gaussian beam. No such term exists in the expression for plane waves because they are infinite in space. The third term is the Gouy phase shift which is relative to the geometrical phase shift, and is also unique to a Gaussian beam [23].

_{0}As shown in Fig. 1 , in order to study the chirp evolution of a femtosecond Gaussian laser pulse propagating through a pulse compressor, we first need to define all of the system parameters. For a single angular dispersion element, the deflection angle of a femtosecond laser pulse after passing through the element is described by [15,16]:

*ω*is the frequency. For the four elements in a perfectly aligned pulse compressor, the parameters must obey the following equations [15]:

*x*-

*z*plane and the Gaussian beam still follows the propagation rules of free space in the y direction. In this case, we omit the information for the y axis in order to simplify our calculation. When an arbitrary spectral component

*ω*of a femtosecond Gaussian laser pulse passes through the first element and arrives at the point (

*x*,

_{1ω}*z*), the waist position is at a distance

_{1ω}*d*/

*α*+

^{2}*z*relative to the apex of the element for the opposite propagation direction for which the equivalent Rayleigh range is

_{1ω}*z*=

_{R1}*z*/

_{R}*α*, where

^{2}*d*is the distance between the beam waist and the first dispersive element [18]. Thus the phase expression for the Gaussian beam can be written as:

In the same way, when an arbitrary spectral component *ω* passes through the second, third and fourth element and arrives at the locations (*x _{2ω}*,

*z*), (

_{2ω}*x*,

_{3ω}*z*) and (

_{3ω}*x*,

_{4ω}*z*), the corresponding waist position is equivalent to

_{4ω}*d*+

*α*+

^{2}z_{1ω}*z*,

_{2ω}*d*/

*α*+

^{2}*z*+

_{1ω}*z*/

_{2ω}*α*+

^{2}*z*and

_{3ω,}*d*+

*α*+

^{2}z_{1ω}*z*+

_{2ω}*α*+

^{2}z_{3ω}*z*with respect to the apex of the first element for the opposite propagation direction, and the corresponding Rayleigh ranges are of

_{4ω}*z*=

_{R2}*z*,

_{R}*z*

_{R}_{3}=

*z*/

_{R}*α*, and

^{2}*z*=

_{R4}*z*. Then the corresponding phase expressions for the Gaussian beam are:

_{R}*ω*.

## 3. Chirp evolution of a Gaussian pulse in a compressor

As shown in Fig. 1, after passing through the first angular dispersion element, each spectral component of the femtosecond laser pulse is separated in space, and propagates a different distance. The reference spectral component propagates a distance *z*
_{1}, while an arbitrary spectral component *ω* propagates a distance *z*
_{1}
* _{ω}*, which can be related to

*z*

_{1}using the angle

*θ*

_{1}. The relevant formulae are [17]:

And the corresponding first and second order derivatives are:

*β*=

*dθ*

_{1}/

*dω*is the angular dispersion introduced by the element. The second order derivative of the expression for${\varphi}_{1}$with respect to spectral frequency

*ω*(which describes the introduced chirp of the Gaussian beam passing through the first element) can then be obtained as follows:

Comparing the above expression with the phase function ${\varphi}_{1}$ (Eq. (6)), we can see that the first term is the chirp introduced by the geometrical phase shift, which is negative. The second term is the chirp introduced by the radially dependent phase term which has an *x*-*ω* coupling, and this is positive. The chirp introduced by the Guoy phase shift is usually far less than the first two terms, so for simplicity we have left it out of Eq. (16) and will neglect it for the rest of our discussion.

After the pulse has passed through the second element (which is anti-parallel with the first element and separated by a distance *L*
_{1}), the angular dispersion is totally eliminated but different spectral component of the pulse are still separated transversely in space, or spatial chirp, and also the *x*- *ω* coupling term in the phase (wave front tilt dispersion) are present [24–28]. In this interval, the relevant distances are:

*ξ*=

*αβL*is the spatial chirp present in the pulse before entering the second element. Then the corresponding first and second derivatives of

_{1}*z*

_{2}

*and*

_{ω}*x*

_{2}

*with respect to*

_{ω}*ω*can be obtained as follows:

And thus the chirp of the pulse after passing through the second element is:

Comparing Eq. (23) with the phase function ${\varphi}_{2}$ (Eq. (7)), we can see that the first term is the chirp introduced by the geometrical phase shift, which is negative. As the second element removes angular dispersion, no extra chirp will be introduced by the geometrical phase shift after passing through the second element. The second term represents the frequency chirp introduced by the radially dependent phase when the pulse travels the distance *z*
_{2}, which is positive. Here we can see that though the angular dispersion has been eliminated after passing through the second element, *x*-*ω* couplings, namely, spatial chirp and wave-front-curvature dispersion are still present, and these combined with a changing beam spot size introduce additional frequency chirp as the pulse propagates through this region of the compressor.

After the pulse has passed through the third element, angular dispersion is present again. As shown in Fig. 1, the distance for any spectral component between the third and the fourth elements is *L*
_{3}cos*θ*
_{3} [2], and thus the relevant formulae are:

Then the first and second derivatives of *z*
_{3}
* _{ω}* and

*x*

_{3}

*with respect to frequency*

_{ω}*ω*can be obtained as follows:

*dθ*

_{3}/

*dω*= -

*β*. Also, the chirp of the femtosecond laser pulse after passing through the third element is:

Comparing the above expression with the phase function ${\varphi}_{3}$ in Eq. (8), we see that the first term is the chirp from the geometrical phase term acquired from propagating between the first and the second elements, which is negative. The second term is the chirp from the geometrical phase shift due to propagation between the third and the fourth elements, which is also negative. Finally, the third term is the chirp introduced by the radially dependent phase shift due to propagation through the same distance, which is positive. If the propagation distance *z*
_{3} = *L*
_{3}, i.e. when the pulse arrives at the entrance of the fourth element, the chirp from the last term becomes zero.

Provided that *z*
_{3} = *L*
_{3}, and the angular dispersion from the first and third elements are equal and opposite, after the pulse has passed through the fourth element, all of the spectral components overlap transversely in space (i.e. no spatiotemporal couplings are present), and both the angular dispersion and all other spatiotemporal couplings have been removed, giving:

So, after passing through the fourth element, neither the geometrical phase shift nor the radially dependent phase shifts introduce any chirp and then we get [15]:

In this formula we can see that the final frequency chirp of the Gaussian beam is the same as that of a plane wave after passing through the pulse compressor. This final expression depends only on the geometrical phase shift and has no dependence on the radially dependent phase shift. From the above analysis of the chirp of the Gaussian beam, we can easily obtain the corresponding chirp of the plane wave by considering only the geometrical phase term. As the term describing the geometrical phase shift of the Gaussian beam is equal to the phase term of the plane wave, the chirp introduced by this term is the corresponding chirp of the plane wave:

From the above analysis, we can see that, for the Gaussian beam, the chirp introduced by the geometrical phase term is negative while that introduced by the radially dependent phase shift is positive at any position after passing through the element; it becomes zero when no *ω*-*x* coupling terms are present in the phase. Also, the radially dependent phase shift in the phase function of the Gaussian beam is the fundamental reason for the difference between the frequency chirp of a Gaussian beam and plane wave when passing through the classical pulse compressor. The frequency chirp evolution of the Gaussian beam and plane wave over the whole propagation process are shown in Fig. 2
, using the parameters *α* = 1, *β* = 0.1 rad/μm, *d* = *z _{R}* =

*L*

_{1}=

*L*

_{2}=

*L*

_{3}= 1 m.

As shown in Fig. 2, though the frequency chirp evolution of Gaussian beams is different from that of plane waves, the final result is the same for both models. In the following section, we provide a detailed analysis of the physical mechanism of this phenomenon.

## 4. Comparison of the Gaussian beam and plane wave models for a pulse compressor

Here we compare the propagation dependent frequency chirps predicted by the two models in order to better understand their differences. For the plane wave model, in each interval of propagation, the following expressions are obtained:

where the*z*’s are the corresponding propagation intervals as shown in Fig. 1. We can see that the frequency chirp of the plane waves only changes in the interval where the angular dispersion is nonzero (e.g. after the pulse has passed through the first and third elements, as shown in Fig. 2). The corresponding chirp evolution of the Gaussian beam in each interval is:

In contrast to the plane wave model, for Gaussian beams, the chirp changes as the pulse propagates even when the angular dispersion is zero. When the angular dispersion is zero but the *x*-*ω* coupling in the phase is nonzero, the frequency chirp of the Gaussian beam still changes along the propagation distance (as shown in Fig. 2, after the pulse has passed through the second element). After the pulse has passed through the first and third elements (where the angular dispersion is not zero), the changes in the frequency chirp consist of two parts; one is introduced by the geometrical phase shift (corresponding to the first term in Eqs. (42) and (44)) and the other is due to the radially dependent phase shift (corresponding to the second term in Eqs. (42) and (44)).

Calculating the difference in the chirp predicted by the two models over corresponding intervals (Eq. (42) is substracted from Eq. (38), and, etc.), we get the following results:

From the above equations we can clearly see that over any propagation interval, the fundamental reason for the difference between frequency chirp of the Gaussian beam and plane wave is due to the radially dependent, or the *x*-*ω* coupling term in the phase of the Gaussian beam.

As shown in Fig. 2, (A) In the interval 0-*L*
_{1}, the chirp of the Gaussian beam increases less rapidly than that of the plane wave because the radially dependent phase shift introduces a positive change in the chirp (see Eq. (46)). (B) In the interval *L*
_{1} to *L*
_{2}, there is no further change of the chirp of the plane wave while the chirp of Gaussian beam continues to change, due to the spatiotemporal couplings in the radially dependent term. However, these changes can be either positive or negative, as determined by the parameters of the beam itself (such as the Rayleigh length *z _{R}*) and the propagation distances (such as d,

*L*

_{1,}

*L*

_{2}) (see Eq. (47)). (C) In the interval

*L*

_{2}to

*L*

_{3}, the chirp of the Gaussian beam increases faster than that of the plane wave, because in this interval, different spectral components of the pulse show a tendency to converge, that is to say, the spatial chirp is becoming smaller, which is the opposite of what happens between 0 to

*L*

_{1}(where the beam is diverging and the spatial chirp is increasing), so the chirp introduced by radially dependent phase shift becomes negative (see Eq. (48)). When the propagation distance

*z*

_{3}=

*L*(at the entrance of the fourth element), the changes in the difference of the chirp between the Gaussian beam and the plane wave in this interval offsets their difference in the intervals 0 to

_{3}*L*

_{1}and

*L*

_{1}to

*L*

_{2}exactly, leading to the final chirp of Gaussian beam and plane wave being the same. Actually, Eqs. (46) ~(48) also reveal this rule:

Thus, for both the Gaussian beam and the plane wave, the final chirps are the same. The fundamental reason for this difference is that the Gaussian beam has an *x*-*ω* coupling (or a spatiotemporal coupling) in its phase which introduces some chirp, and this term is absent in the phase of plane wave.

If the spot size of the Gaussian beam increases, or if the Rayleigh range is longer and the beam is better collimated, the propagation dependence of the chirp predicted by the Gaussian beam is closer to that of the plane wave, as shown in Fig. 3 .

This phenomenon is also shown in Eqs. (46) ~(49) where you can see that if the Gaussian beam is well collimated, that is to say, *d*, *L*
_{1}, *L*
_{2}, *L*
_{3}<< *z*
_{R},(which is called the approximate condition for good collimation [15,16]), $\Delta {\varphi}_{PG}^{\text{'}\text{'}}=0$. Also, we can see that the chirp changes of the Gaussian beam can be either positive (see subfigures b and c) or negative (see figure a) in the interval *L*
_{1} to *L*
_{2}, and depends on the parameters of the Gaussian beam and the propagation distance.

## 5. Discussion

Using the Kirchhoff-Fresnel diffraction integral, we can obtain the complete electric field of the pulse while propagating through compressor which will tell us not only the frequency chirp, but also the spatial chirp, pulse front tilt, spot size, and etc [6–8,15,16]. However, for studying the evolution of the chirp, an analysis of the whole electric field, is not necessary, as the chirp is directly determined by the phase shift of the laser pulse. In this paper, we used a much simpler and more straightforward method, by analyzing the phase acquired by a Gaussian pulse due to propagation through a compressor, and directly calculated the second order derivative of the phase with respect to the spectral frequency *ω*. This gives us the chirp evolution of the pulse when passing through a pulse compressor. We found the *x*-*ω* coupling term in the phase of the Gaussian beam to be the fundamental reason for its chirp evolution inside a pulse compressor being different from that of a plane wave.

When the spot size of the Gaussian beam increases, or when it is well collimated and has a longer Rayleigh range, the chirp acquired by the Gaussian beam is much closer to that of a plane wave, as shown in Fig. 3. The reason for this is that the radially dependent phase term of the Gaussian beam becomes less significant (i.e. the curvature of the wave front increases). If the Rayleigh range is long enough, no radially dependent phase term exists, leading to equivalent chirps for the Gaussian beam and the plane wave models. Although the spot size of the Gaussian beam has no influence on the final chirp after propagation through the 4th dispersive element (assuming a well aligned compressor), it could significantly influence the value of the chirp when the pulse is passing through a single angular dispersion element, a pair of elements in a single-pass structure, or when the pulse compressor is not perfectly aligned. This of course affects the duration of the output laser pulse.

Another phenomenon worth noting is that for Gaussian beams, as long as spatiotemporal couplings exist (i.e. *x*-*ω* cross terms), even though there is no angular dispersion, an additional frequency chirp will be introduced as the pulse propagates. Martinez implied such a phenomenon but provided no explanation [15]. In essence, propagation of an ultrashort pulse in the presence of spatiotemporal couplings is a three-dimensional (*x*, *ω* and *z*) effect, and it causes the spatial terms of the Gaussian to mix with the frequency terms, and phase terms are transferred to the intensity (and vice versa). Namely, the first dispersive element introduces angular dispersion which propagation changes into spatial chirp, wave-front-curvature dispersion (or pulse front tilt, if viewed in the time domain), frequency chirp, and others. And once the angular dispersion vanishes after the second prism, there are still coupling terms remaining—the spatial chirp and the wave-front-tilt dispersion—so propagation again transfers these terms into frequency chirp. Specificially it is the radially dependent phase term in the Gaussian model that allows for this mixing, and if this term vanishes, then the frequency chirp of Gaussian beam will be the same as that of plane wave (such as after the pulse has passed through the fourth element, which also makes this term vanish). The spatial chirp and frequency chirp are commonly regarded as independent parameters; the spatial chirp describes the transverse separation (perpendicular to the propagation direction) of different spectral components while the frequency chirp describes the longitudinal delay (along the propagation direction). Here we again illustrate that this is not the case [26–28], and we have shown that these two quantities can be coupled by propagation.

## 6. Conclusion

In this paper, we studied the chirp evolution of a Gaussian beam when passing through a classical pulse compressor by directly calculating the acquired spatio-spectral phase. Compared with the chirp evolution predicted by the plane wave model, we found that a spatiotemporal coupling or an *x*-*ω* dependent term in the phase of the Gaussian beam is the fundamental reason for the difference in these two models’ predictions for pulse compressors. For a Gaussian beam, the existence of spatiotemporal couplings also introduces an additional frequency chirp even when no angular dispersion exists. If the Gaussian beam is well collimated, the frequency chirp evolution is closer to that of a plane wave after passing through the angular dispersion elements. This work provides a deeper understanding of the physical mechanism of the frequency chirp evolution when a laser pulse passes through a classical pulse compressor. Our analysis will also be helpful for optimization of dispersion compensation schemes in many applications of femtosecond laser pulses.

## Acknowledgments

This work was supported by the National Natural Science Foundation (NSFC) (30900331, 30927001), and Program for Changjiang Scholars and Innovative Research Team in University.

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