## Abstract

Pulse evolution after the acousto-optic deflector (AOD) and its dependence on AOD parameters is not clear, but is important to AOD applications in scanning microscopy. By considering the abnormal Bragg AOD as a combination of a dispersion media and a diffraction grating with Littrow structure, the dispersion characteristics of AOD when scanning Gaussian laser beam of femtosecond pulses were analyzed. Pulse width about AOD parameters was explored theoretically and verified with experiments. It is found that after exiting AOD, the pulse width of laser beam either merely decreases or first decreases and then increases before it finally goes steady with the propagation distance, depending on the interaction between the material dispersion and angular dispersion.

©2007 Optical Society of America

## 1. Introduction

Acousto-optic deflector (AOD) can control light beam (such as intensity, direction and frequency of laser beam) fleetly and efficiently [1–5] and is widely applied in areas involving beam steering such as micro-machining [6] and microscopic imaging [7, 8]. Comparing with mechanical scanning, AOD has significant advantages. For instance, AOD deflects the laser beam by wave diffraction without any physical movements of the scanning component. It is inertia-free and involves less mechanical oscillation interference [5–8]. Furthermore, by electrically setting the active frequency, AOD allows random access to the region of interest (ROI), which confines the scanning to ROI only and increases the signal-to-noise ratio and the frame-capture rate [7, 8]. Femtosecond pulse laser based multiphoton microscopy has been widely applied in biomedical imaging [9, 10], and requires fast and ‘silent’ steering with AOD [11–12]. However, steering femtosecond pulse laser with AOD is frustrated due to two facts: the AOD is highly dispersive, and the femtosecond pulse laser is not monochromatic. Severe dispersion problem results in pulse width lengthening and spot distortion, which diminishes the advantages of ultrashort pulse laser in applications [6, 11–13]. In recent years, technical efforts have been conducted to compensate for the dispersion and to assure AOD scanning femtosecond pulse laser in micro-machining [6], and microscopic imaging [14–20].

To achieve efficient AOD scanner system and dispersion compensation, it’s necessary to study the dispersion characteristics of AOD [13]. Dispersion analyses based on Gaussian beam model for grating were given by Martinez [21, 22] and Horvath *et al*.[23] more than a decade ago. Martinez [22] studied the dispersion characteristics of femtosecond pulse Gaussian beam
propagating through a single ideal grating and deduced the expressions of pulse width and wave front, etc. Horvath *et al*. [23] acquired the trends of pulse shape and pulse width variances by numerical simulation. However, these works are with some assumptions. Martinez assumed the beam propagating distance is far less than Rayleigh length. Horvath’s model requires that the waist of Gaussian beam exactly overlaps the surface of grating, etc. These assumptions are not valid for AOD case. Moreover these analyses are based on ideal grating model without material dispersion. These limitations prevent the application of the Gaussian beam model of grating to the analysis of AOD. As a result, in all the technical efforts
[14–20], the compensation scheme was only evaluate roughly with the plane wave model. There observed a lot of discrepancies between the experiments and expectations. Specifically, some basic issues like the pulse width dependence on the propagation distance and on the AOD parameters are not clear, which are of significant importance to the design of AOD system and compensation scheme. Recently a new general Gaussian beam model for angular disperser was proposed without the assumptions made by Martinez and Horvath *et al*., which provides potential tools to address these issues of AOD [24]. Based on these studies [22–24], in this paper the temporal dispersion property of the AOD is studied by considering the AOD as a combination of a dispersion media and diffraction grating with Littrow structure. The dependence of the pulse width on the AOD parameters and propagation distance is explored theoretically and verified with experiments.

## 2. Theoretical analysis

In scanning femtosecond pulse laser, the AOD experiences both material dispersion and angular dispersion due to the inherent dispersive nature of the acousto-optic crystal material and acousto-optic interaction. Therefore, in analyzing the dispersion property of the AOD, it is reasonable to treat the AOD as a combination of dispersion media with linear group velocity dispersion and a diffraction grating.

As it’s shown in Fig. 1, Gaussian laser beam of femtosecond pulses incidents at an AOD with an angle γ, the first order of diffraction is with deflecting angle θ = θ (γ,ω). The first order Taylor expansion of the deflection angle is given as [21, 22],

where $\alpha =\frac{\partial \theta}{\partial \gamma}$ represents the coefficient at which the deflecting angle varies with the incident angle for each single wavelength of incident light; $\beta =\frac{\partial \theta}{\partial \omega}$ stands for the coefficient the deflecting angle varies with frequency when different spectra components incident at the same angle.

For an AOD, the diffracting efficiency should be high and flat across the whole active bandwidth. Therefore most AODs are designed to work in the abnormal Bragg diffraction mode. Equations governing the angle and frequency are expressed as [1, 3],

where ${f}_{0}=\frac{v}{\lambda}{\left({n}_{i}^{2}-{n}_{d}^{2}\right)}^{\frac{1}{2}}$ is defined as extremum frequency (at this frequency the diffracting angle of AOD is 0), *n _{i}* is the refraction index of incident light, nd is the refraction index of the diffracting light. As for the case of abnormal Bragg diffraction the refraction indexes of the incident and diffracting light are different. In Eqs. (2) and (3),

*f*is the frequency of the ultrasonic wave inside the AOD crystal,

*v*is the travel speed of the ultrasonic wave in AOD media, λ is the light wavelength of in vacuum.

As shown in Fig. 1, light beam refracts at both the incident and exiting surfaces. Substituting Eqs. (2), (3) into the refraction function yield [3],

Therefore, the external scanning characteristics of the abnormal Bragg AOD are the same as that of the normal Bragg AOD. Derivative Eq. (4) with respect to γ, ω respectively, the dispersion parameters of the AOD can be written as,

In actual AOD application, both γ_{0}, θ_{0} are very small [3], hence,

Therefore, referring to diffracting characteristics, abnormal Bragg AOD can be considered as a grating of Littrow structure, with a grating constant Λ = *v*/*f*.

Femtosecond pulse laser is appended with linear chirp by material dispersion of AOD crystal and with angular dispersion by grating characteristics of AOD. Under this consideration, applying Martinez’s method [21, 22], the expression of the complex amplitude of Gaussian beam at arbitrary distance *z* after AOD is acquired as:

$$\mathrm{exp}\left[i\frac{\mathrm{kzq}\left(d\right)}{2q\left(d+{\alpha}^{2}z\right)}\left(\frac{{x}^{2}}{{z}^{2}}+{\beta}^{2}{\omega}^{2}+\frac{2\mathrm{x\beta \omega}}{z}\right)\right]$$

where *k* is the wave number of light in vacuum, *GDD*
_{m} represents the group delay dispersion introduced by AOD crystal material, τ_{0} is the original pulse width, defined as Full width of Half Maximum (FWHM). As angular dispersion occurs only in XZ plane, similar to reference 21, and 22, coordinate y is omitted here.

The propagation distance in AOD as *L*
_{1}+*L*
_{2} is comparatively small to *d* and *z*, and is neglected in the *q* parameter of Gaussian beam. Apply Fourier transform, yield,

Referring to the method in Ref. 21-24, only consider the relevant terms about pulse width in Eq.(8), the general pulse width expression of the femtosecond pulse Gaussian beam depending on the AOD parameters is derived as,

where

Parameters *U*, *V* describe effects of the spectral lateral walk-off and group delay dispersion on the pulse width [21, 22, 24]. Actually *V* is the combination of the group delay dispersion introduced by AOD crystal material, *GDD _{m}*, and that induced by angular dispersion ${\mathrm{GDD}}_{A}=-k{\beta}^{2}z\bullet \frac{\left(d+{\alpha}^{2}z\right)d+{z}_{R}^{2}}{{\left(d+{\alpha}^{2}z\right)}^{2}+{z}_{R}^{2}}\left[25\right].$

*GDD*is usually positive for general material, yet

_{m}*GDD*is usually negative and varies with the distance.

_{A}In previous studies analyzing the pulse width variance of the light beam after dispersion media or angular dispersion components, the infinite plane wave model is commonly used [26, 27]. In these studies, the beam size is assumed to be infinite, which corresponds to *U* = 0, so that the effect of spectral lateral walk-off is neglected. Therefore, the pulse variance equation of infinite plan wave model can be simplified from Eq.(11) as,

where -*kβ*
^{2}
_{z} (recorded as *GDD _{A}*) is the group delay dispersion induced by the angular dispersion in infinite plane wave model. It differentiates from that of Gaussian beam model (

*GDD*) by a coefficient $\eta =\frac{\left(d+{\alpha}^{2}z\right)d+{z}_{R}^{2}}{{\left(d+{\alpha}^{2}z\right)}^{2}+{z}_{R}^{2}}\left[24\right].$ Eq.(12) can be comprehended as that, AOD induces negative GDD as angular component, which compensates the positive GDD induced by material dispersion, therefore during the propagation, the pulse is first compressed to original pulse width and then broadened. However the actual pulse width variance is somewhat different (see the experiment part).

_{A}## 3. Experiment and results

Experiments were conducted by measuring the pulse width across active frequency of the AOD and across the propagation distance to verify the formula derived above. The pulse width of the laser beam was measured by using an autocorrelator based on two-photon absorption (Model: Carpe, APE). The experiment parameters are listed as follows. Femtosecond pulse laser exits from the Ti: Sapphire laser (Model: MAITAI, Spectral physics) with original pulse width 124 fs, at a repetition rate 80 MHz central wavelength 800nm, and a beam waist *w*
_{0} = 0.5 mm. The TeO_{2} AOD (Model: PSGDG-3/Q, SIPAT) works in abnormal Bragg mode with slow shear acoustic wave. *d* = 1.5 m. The overall GDD induced by the material dispersion of all optical components (intensity modulator, etc.) in the system is GDD_{T} ~15000 fs^{2}. The pulse width of the laser beam after the inactive AOD is 362 fs, remains constant at *z* = 1, 2, 3 m, e.g. remains constant along the propagation distance. The same results were observed for the 0*th* order of diffraction beam from the active AOD. As working beam in AOD is the first order of diffraction, in the following experiments all the measurements were applied to the first order diffracting beam if not stated.

The variance of the pulse width after the active AOD with the propagating distance is measured beforehand. Fig. 2 shows the experiments results at the central frequency *f* = 70 MHz for an instance. For contrast, the theoretical values predicted by the Gaussian beam model formula (9) and the infinite plane wave model formula (12) are also plotted.

It is interesting to note the measured pulse width after the AOD is smaller than the pulse width of the laser passing the inactive AOD or that of the 0*th* order diffraction. The pulse width first decreases with the propagation distance, and then remains stable afterwards. The experiment results have coincidence with the Gaussian beam model. The fact that the pulse
width decreases after passing the AOD is contradictory to the expectation without considering the material dispersion, but is reasonable. According to Eqs. (11), negative *GDD _{A}* increases in magnitude, which cancels

*GDD*, decreases the gross GDD, and therefore decreases the pulse width. As

_{m}*z*increases,

*U*and

*GDD*(accordingly

_{A}*V*) approaches a constant with propagation, so the pulse width also approaches a constant when

*z*is adequately big (in Fig. 2, around 2.5 m).

To the infinite plane wave model, as indicated by the blue curve in Fig. 2 and Eq.(12), the pulse width would first decrease in much sharper manner, and minish to the original pulse width (124 fs) and then increase to infinite with the propagation distance. These differences could be clearly explained by compare Eq. (9) and (12). First, both the experimentally measured minimal pulse width and that predicted by Gaussian beam model is longer than that of infinite plane wave model. According Eq. (12), though the GDD effect (characterized by *V*) would turn to 0 in a certain position, the spectral lateral walk-off (characterized by *U*) exists. Yet because the pulse broadening due to *U* is not included in the infinite plane wave model, as far as the value of GDD is 0, pulse width can achieve the original value. The rest differences are due to the angular dispersion difference (coefficient η) between the Gaussian model and the infinite plane wave model. As η < 1, *GDD _{A}* predicted by Gaussian beam model is smaller than that by infinite plane wave model, thus the pulse width decreases in a much milder manner comparing with that of the plane wave mode. As the propagation distance increases, η gradually goes to 0, thus the angular dispersion is to decay and pulse width turns to steady. While in the infinite plane wave model, without the limitation of parameter η, the pulse width would infinitely increase theoretically as the

*GDD*increase with propagation distance

_{A}*z*(see Eq. (12)).

When scans the light beam, AOD works at different frequencies, thus pulse width variations at different frequency points are studied. Fig. 3 plotted these results measured at different active frequencies (maximal and minimal frequency 50, 90 MHz, and central frequency 70 MHz for instances). For contrast, the theoretical results predicted by Gaussian beam model according to Eq. (9) are also depicted. It is clear the measurements are quite consistent with the theoretical predictions.

It is interesting to note that the trend of pulse width variation along distance is different across frequencies. As is shown in Fig. 3, at certain frequency (*f* = 50, and 70 MHz, for instances), the pulse width first decreases then decreases to steady, but at another point (90 MHz), the pulse width first decreases then smoothly increases with the propagation distance, and has a valley point of pulse width at certain position. Pulse width descends more rapidly at high frequency points than at low ones, this is because of the bigger β parameter of high frequency points. In the propagation of laser, *f* = 90 MHz has smaller pulse width than *f* = 50 MHz. This is also by the reason of bigger β value of high frequency points, the effect of GDD, *V*, can be minished to 0 in considerable short distance, and that with the propagation distance, redundant angular dispersion is induced, thus results in the once again increasing of pulse width. While β is comparatively small at low frequency points, therefore the angular dispersion is insufficient to compensate the material dispersion, thus *V* exists, the pulse turns to be wider. Therefore, when using angular dispersion components to compensate for the material dispersion, merely by adjusting compensating distance can not always acquire favorable compensating effect.

## 4. Discussion

In this paper, we considered the AOD in abnormal Bragg diffraction mode as a combination of dispersion media and diffraction grating. By adopting Gaussian beam model, the dispersion
effect of AOD when scans femtosecond pulse laser was studied, and a general expression of pulse width about the AOD parameters was presented. Experiments were carried out for verification. The experimental results of pulse width essentially accords the theoretical curves, thus proved that in analyzing the dispersion characteristics of AOD, Gaussian beam model including material dispersion has significant advantages to plane wave model. Our experiments show the assumption that the abnormal Bragg AOD be considered as a combination of certain material dispersion and a Littrow grating of constant Λ = *v*/*f* is also reasonable.

Comparing with Martinez’s ideal grating model [21, 22], our model considered the impact of material dispersion and precluded the well collimation (*d*«*z _{R}*, α

^{2}

*z*«

*z*, where

_{R}*z*=

_{R}*kw*

_{0}

^{2}/2 is the Rayleigh length of Gaussian beam,

*w*

_{0}is the size of beam waist). This model is more appropriate to actual AOD device. Therefore the expression of pulse width presented here is more general, and could explain the experimental results fairly good. Under the conditions of well collimation and

*GDD*= 0, these conclusions can be readily simplified to Martinez’s model (see Eqs. (21) and (23) in Ref. 22).

_{m}It is interesting to compare the pulse width measurement results of Fig. 3 with Horvath’s numerical calculation based on ideal grating Gaussian beam model (Fig. 7 in Ref.23). First, the pulse width of Gaussian beam will eventually goes steady after exiting AOD or ideal grating, which is a basic feature of Gaussian beam. This is due to the fact that after the angular dispersion components, the angles between the equal-phase-planes of different spectra components would decrease with propagation distance and finally turns to 0 [25]. This effect stabilifies the effects of spectral lateral walk-off and GDD, and thus ensures the pulse width steady after sufficient propagation distance [24]. Second, the inherent material dispersion of AOD comparing to ideal grating results in the significant difference in the variance trends of pulse width. For AOD, the pulse width does not always increase with the propagation distance as that of the ideal grating. In the same propagating distance, when it is far less than Rayleigh length, pulse width at high frequency points (with high dispersion ability) often appears shorter than that at low frequency points. While out of Rayleigh length, pulse width at high frequency points can be either longer or shorter than that at low frequency points. Yet to ideal grating, as shown in Fig. 7 of Ref. 23, the pulse width after ideal grating increases with the propagation distance, and in the same position, the pulse broadening of grating with more angular dispersion is bigger than that of grating with less angular dispersion. So the pulse broadening of ideal grating model without material dispersion is quite straightforward, yet to AOD, the trend of pulse width variance is quite complicated due to the inclusion of material dispersion and the coupling between the GDD and spectral lateral walk-off (as presented in Eqs. (9), (10), and (11)). The GDD is induced by material dispersion and angular dispersion. At different frequency points, the AOD exhibits different angular dispersion. Accordingly, the Gaussian laser beam experiences different amount of GDD and spectral lateral walk-off after propagation a certain distance. This resulted in considerable diversities for both the respective trend and value of pulse width variance. These diversities can not be explained by ideal grating model based on Gaussian beam.

Neither can this trend of pulse width variance be explained by the infinite plane wave model. As predicted by the infinite plane wave model, the pulse width is only dependent on GDD (see Eq. (12)). The negative GDD induced by angular dispersion during propagation can always counteracts the positive *GDD _{m}* induced by material dispersion. Therefore, the pulse width always has the trend of first decrease, declines to the original pulse width, and then increase. Because of the continuous addition of negative GDD by the angular dispersion, the increasing trend of the pulse width would also persist. However this obviously disobeys with the experiment results, as depicted in Fig. 2. The pulse width acquired by experiments is always longer than the original pulse width, and the position where the shortest pulse width takes place is different with the plane wave model, and the pulse width also goes to steady eventually.

The principle analysis presented here would be helpful in practical AOD applications, for instance, in designing the dispersion compensation scheme for AOD scanner for femtosecond pulse laser micromachining and microscopic imaging. Moreover, this analysis could also be extended to other acousto-optic device such as AO modulator, and acousto-optic tunable filter, in applications such as pulse shaping, dispersion managing, and femtosecond pulse laser device etc.

## 5. Conclusion

By considering the abnormal Bragg AOD as a combination of dispersion media with certain material dispersion and diffraction grating, the dispersion characteristics of AOD when scanning femtosecond pulse laser were analyzed based on Gaussian beam model. A general expression of the pulse width about AOD parameters was acquired theoretically and tested by experiments. The experimental results commendably verified the theoretical conclusions. This indicates the model proposed here is more efficient in exploring the dispersion characteristics of an AOD than the plane model and Gaussian model with ideal grating. These results provide insight in understanding the dispersion properties of the acousto-optic device when diffracting Gaussian beam of femtosecond pulses laser, which are closely relevant to the design of acousto-optic device system including AOD and its dispersion compensation scheme for applications involving femtosecond pulse laser beam passing acousto-optic device.

## Acknowledgments

The authors would like to express special thanks to Dr. Xiao Yuan of HUST for helpful discussion. Supported by NSFC grants, 60278017, 30328014, and NIH grant (W.R.C., 5R01DC003918). Correspondence should be addressed to sqzeng@mail.hust.edu.cn.

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