## Abstract

Based on the modified Kogelnik’s coupled-wave theory, time- and frequency-domain diffractions of a femtosecond pulse from transmitted volume holographic gratings (VHGs) are theoretically studied. Results show that when the refractive index modulation of the VHG changes in a certain range, the number of temporal diffracted pulse will evolve from one to two, then to three, and this pulse number evolution is periodic. This particular phenomenon can be explained by diffraction intensity spectrum and the overmodulation effect of refractive index modulation of transmitted VHG. Moreover, we find centers of all temporal diffracted pulses translate along the negative time axis, and the translation is irrelevant to the refractive index modulations. We will use time delay of volume grating to give a reasonable explanation.

©2013 Optical Society of America

## 1. Introduction

Due to the wavelength and angular selectivity of volume holographic grating (VHGs), the recording and diffraction of VHGs by ultrashort pulse are different from those of the continuous waves (CWs). Gaylord *et al.* [1] was the first to record a holography on photorefractive Fe:LiNbO_{3} crystal by a pulsed laser and measure its reconstructed diffraction efficiency by a CW laser. In 1974, Shah *et al.* [2] used the same approach as that of Gaylord to measure and to propose “photorefractive sensitivity”. Chen *et al.*’s experiment [3] showed that the highest photorefractive sensitivity can be obtained at around 30% for Fe-doped LiNbO_{3} crystals by the pulsed laser, which is one order of magnitude higher than that of a CW hologram. In 1983, G. C. Valley [4] systematically discussed the recording and readout of photorefractive VHGs by short and ultrashort pulses.

The Bragg selectivity associated with the volume holography can also be exploited for ultrashort pulse shaping. One attractive feature of this technique is good spectral resolution (~0.1 angstrom), by which shaped waveforms on the order of 100ps is easily obtained. Reviews of various femtosecond pulse shaping techniques and experiments by VHGs up to 1995 can be found in Ref [5]. In 1998, Y. Ding *et al.* [6] studied diffraction of ultrashort pulse by static VHG and proposed that the diffraction bandwidth can be controlled by the period and thickness of the VHG. Later, Yan’s group [7, 8] studied the distributions of diffraction intensity spectrum and instantaneous diffraction intensity with respect to the grating parameters in transmitted VHG. Yi *et al.* [9] compared the instantaneous diffraction and transmitted properties of ultrashort pulse from reflective and transmitted VHGs. Wang *et al.* [10] studied the pulse shaping properties of VHGs in anisotropic media. Yan *et al.* [11, 12] studied the frequency-domain diffraction and pulse shaping property of ultrashort pulses by transmitted and reflective multilayer VHGs. Garay *et al.* [13] realized spectral pulse reshaping of ultrashort laser pulse with volume phase holographic grating recorded in photopolymerizable glasses. T. Brixner and G. Gerber [14] was the first ones to study the polarization pulse shaping where the polarization states of a single laser pulse can reach different linear and elliptical orientations with varying ellipticity. In recent years, with the recording of VHG in photo-thermal-refractive (PTR) glass, pulse shaping by PTR glass is discussed [15]. Siiman *et al.* [16] experimentally studied the diffraction of ultrashort pulse by transmitted VHG recorded on PTR.

In this paper, we propose a new scheme to realize ultrashort pulse shaping by modulating the refractive index modulation of transmitted VHG. In our scheme, only one VHG is needed. Depending on the strength of refractive index modulation of the volume grating, the diffracted pulse may be two or three pulses or may remain a single pulse and the pulse number evolution is periodic. Moreover we find centers of diffracted pulses have translation with respect to the input pulse.

## 2. Theory analysis

Figure 1
is the experimental configuration used for our analysis. Two coherent continuous waves with wavelength λ_{p} symmetrically incident on the surface of a photorefractive material at equal outside angles θ_{p} (the angles inside the crystal are denoted by θ_{p}’) to record an unslanted transmitted volume grating through photorefractive effect. The grating is characterized by a refractive index distribution of the form n = n_{0} + Δncos(kx), where n_{0} is the background refractive index of the material, Δn is the maximum refractive index modulation induced by photorefractive effect and k = 2π/Λ is the grating wave number with the grating period Λ = λ_{p}/(2sinθ_{p}). The grating vector is parallel to x coordinate axis.

After recording, one broadband femtosecond pulse E_{r}(t) incidents on the material at an outside angle θ_{r} (angle inside the crystal is denoted by θ_{r}’) to readout the recorded refractive index grating. θ_{r} is chosen to make the central frequency component of the input pulse satisfy the Bragg condition of the VHG. Assuming that only a transmitted wave ${E}_{t}(\omega ,z)\mathrm{exp}(-i\overrightarrow{\rho}\cdot \overrightarrow{r})$ and a diffracted wave ${E}_{d}(\omega ,z)\mathrm{exp}(-i\overrightarrow{\sigma}\cdot \overrightarrow{r})$ propagate inside the grating and both of them are transverse waves, where E_{t}(ω, z) and E_{d}(ω, z) are the complex electric field amplitudes of the transmitted and diffracted waves with respect to the frequency and propagation distance, respectively; **ρ** and **σ** are propagation vectors, connecting to the grating vector by the relation **σ = ρ-K**, where **K** is the grating vector.

The temporal field distribution of the input femtosecond pulse is given by

Where ${\omega}_{0}=2\pi c/{\lambda}_{0}$ is the central angular frequency of the input pulse with λ_{0}being the corresponding central wavelength in vacuum. Parameter T is related to full width at half maximum (FWHM) Δτ of ${\left|{E}_{r}(t)\right|}^{2}$by $T=\Delta \tau /\sqrt{2\mathrm{ln}2}$.

By applying Fourier transform on Eq. (1), the field distribution in the spectrum domain can be obtained

To describe the propagation and coupling between the transmitted and diffracted waves in the volume grating, we use the slowly varying envelop approximation and ignore terms that are quadratic in the index change. After neglecting the absorption of the material, we get the modified Kogelnik’s coupled-wave equations for the VHG [8, 17]

The coupled-wave Eqs. (3) and (4) are applicable to all frequency components of the input pulse. For the broadband input pulse, it is impossible that all frequency components satisfy the Bragg condition of the VHG simultaneously, so off-Bragg term $-\text{i}\frac{\text{2\pi c}{\text{K}}^{\text{2}}}{\text{4\pi}{\text{n}}_{\text{0}}}\left(\frac{\text{1}}{\text{\omega}}-\frac{\text{1}}{{\text{\omega}}_{\text{0}}}\right){\text{E}}_{\text{d}}\left(\text{\omega ,z}\right)$ is introduced. When the readout spectral component satisfies the Bragg condition of the VHG, that is ω = ω_{0}, this term is zero and Eqs. (3) and (4) reduce to the coupled-wave equations of continuous waves [17].

Solving Eqs. (3) and (4), the diffraction spectrum distribution at the output plane of the grating is obtained

_{0}.

Accordingly, at the output plane of the material, the diffraction intensity spectrum is

According to the principle of reverse-Fourier-transform, the temporal diffraction field is given by

Consequently, the temporal diffraction intensity is expressed as

From Eqs. (5)-(8), we know the diffraction intensity spectrum and temporal diffraction intensity are controlled by the grating parameters and the duration of the readout pulse. Detailed discussions of these parameters on the diffraction can be found in papers [7–9]. Here we focus on the influence of the refractive index modulation of the VHG on the diffraction.

## 3. Temporal diffraction intensity distribution with respect to the refractive index modulation of VHG

In this section, the temporal diffraction intensity distributions of diffracted pulse are investigated. In order to compare the distribution of diffracted pulse with that of the input pulse, according to Eq. (1), the temporal intensity distributions of input pulse with durations Δτ = 100fs and 200fs are plotted in Fig. 2 . It can be seen that the input pulse has only one lobe.

In simulation, assuming the following parameters: period of the volume grating is Λ = 7.3μm and thickness is d = 7.8mm. FWHM of the input pulse is Δτ = 100fs, central frequency is ω_{0} = 4π × 10^{14} rad/s and the corresponding central wavelength is λ_{0} = 1.5μm. Background refractive index of the photorefractive material is n_{0} = 3.134. Speed of the light in vacuum is c = 3 × 10^{8} m/s. According to the Bragg condition of VHG we can get Λ = λ_{0}/(2sinθ_{r}), the cosine function of the readout angle in the crystal is $\mathrm{cos}{\text{\theta}}_{\text{r}}{}^{\text{'}}\text{=}\sqrt{\text{1}-{\left(\frac{{\text{\lambda}}_{\text{0}}}{\text{2\Lambda}{\text{n}}_{\text{0}}}\right)}^{\text{2}}}$。

Figure 3
shows the distributions of temporal diffraction intensity with respect to the refractive index modulation when it changes from 1.0 × 10^{−4} to 3.0 × 10^{−4} with the index-step of 0.2 × 10^{−4}. From Fig. 3(a), it is found when the refractive index modulation Δn is 1.0 × 10^{−4}, only one single diffraction lobe appears, center of which is almost flat. When Δn increases from 1.2 × 10^{−4} to 1.4 × 10^{−4}, central part of the diffraction lobe decreases, whereas the two side lobes are increasing. When Δn increases to 1.6 × 10^{−4}, two side lobes are so large that one diffracted pulse evolves into two pulses with the same pulse durations and peak intensities. If Δn increases further, center of the dual-pulse starts to increase, at last dual-pulse evolves into tri-pulse. If Δn is appropriately chosen, three similar diffracted pulses can be acquired. The acquired femtosecond dual-pulse and tri-pulse are useful in many fields, such as femtosecond laser induced micro-machining and low-loss optical waveguide [18], coherent control of quantum states [19–21]. In measurement of femtosecond pulses, such as Spectral Phase Interferometer for Direct Electrical-Field Reconstruction (SPIDER) and Frequency-resolved optical Gating(FROG), dual-pulse is also quite necessary [22, 23].

Further increasing the refractive index modulation to 2.0 × 10^{−4}, two side lobes of the tri-pulse decreases symmetrically, while the central lobe increases. At last the tri-pulse directly evolves into one lobe pulse. Figure 3(b) shows the evolution of one lobe diffracted pulse. It can be seen that when the refractive index modulation increases, a single diffraction lobe with increasing peak intensity appears. When the refractive index modulation increases to 3.0 × 10^{−4}, the peak intensity of the diffracted pulse decreases again.

If further increasing the refractive index modulation of VHG in three different ranges, respectively from 3.0 × 10^{−4} to 4.0 × 10^{−4}, 5.0 × 10^{−4} to 6.0 × 10^{−4} and 7.0 × 10^{−4} to 8.0 × 10^{−4}, it is found that the waveform evolutions in each range are similar to those of Fig. 3(a). The number of diffracted pulse also evolves from one to two, then to three, at last to one. The refractive index modulations when dual-pulse emerges are 3.6 × 10^{−4}, 5.6 × 10^{−4} and 7.6 × 10^{−4}. When tri-pulse emerges, the refractive index modulations are 3.8 × 10^{−4}, 5.8 × 10^{−4} and 7.8 × 10^{−4} respectively. If the refractive index modulation changes in the ranges of 4.0 × 10^{−4} to 5.0 × 10^{−4}, 6.0 × 10^{−4} to 7.0 × 10^{−4} and 8.0 × 10^{−4} to 9.0 × 10^{−4} respectively, there is only one diffraction lobe, evolutions of which are similar to those of Fig. 3(b).

From former descriptions, we know when the refractive index modulation changes from 1.0 × 10^{−4} to 9.0 × 10^{−4}, the number of diffracted pulse will evolve from one pulse to dual-pulse, then to tri-pulse, and this pulse number evolution is periodic. Taking the emergence of dual-pulse as an investigation, we can get the evolution period of refractive index modulation is about 2.0 × 10^{−4}. If we further increase refractive index modulation from 1.0 × 10^{−3} to 9.0 × 10^{−3} with the index-step of 2.0 × 10^{−3}, it can be found when index modulations are 5.0 × 10^{−3}, 7.0 × 10^{−3} and 9.0 × 10^{−3}, three pairs of dual-pulse with approximate duration and peak intensity emerge, shown as Fig. 4
.

In summary, by modulating the refractive index modulation of VHG, one readout pulse can be shaped into two same pulses or three same pulses or one pulse with different peak intensity and pulse duration. Our proposed pulse shaping scheme needs only one VHG, avoiding the complex optical path adjusting.

The corresponding refractive index modulation can be acquired from several practical materials. In VHG recorded on PTR glass, the reported refractive index modulation can be in the range of 0.8 × 10^{−4} to 10^{−3} [15, 16]. In BB-640 emulsions, the index modulation can reach the order of 10^{−2} [24]. In photopolymers based in polyvinyl alcohol (PVA)/acrylamide, the index modulation can change from 1.76 × 10^{−3} to 7.7 × 10^{−3} [25]. In the most commonly used photorefractive crystal Fe:LiNbO_{3}, the induced refractive index modulation can be as small as 10^{−6}, but the saturation value of which are in the order of 10^{−3} [26].

Moreover, comparing Figs. 3 and 4 with Fig. 2, we find centers of all diffracted pulses translate along the negative time axis and the translations are equal in value, which means the time delay is irrelevant to the refractive index modulations of the VHG. From Figs. 3 and 4, the estimated time delay is about −88fs.

## 4. Explanation on the periodic evolution of diffracted pulse

The periodic evolution of diffracted pulse can be explained by diffraction intensity spectrum and overmodulation effect of refractive index modulation.

It is well known that Kogelnik’s diffraction oscillates with the refractive index modulation in case of transmitted VHG [17]. When the refractive index modulation is increased, a single diffraction lobe with increasing amplitude appears until reaching the optimal refractive index modulation. Then further increase of the refractive index modulation induces an overmodulation effect of the refractive index modulation that corresponds to the appearance of larger side-lobes and this effect shows some kind of periodicity. The influence of overmodulation of the refractive index modulation on the periodic evolution of diffracted pulse is easy to be explained by diffraction intensity spectrum.

Figure 5
shows the normalized diffraction intensity spectrum distributions when the refractive index modulation changes from 1.0 × 10^{−4} to 2.8 × 10^{−4}, where the central frequency component, corresponding to frequency ω_{0} = 4π × 10^{14} rad/s, satisfying the Bragg condition of the VHG, gets the largest intensity. From Fig. 5(a), it is found when the refractive index modulation increases from 1.0 × 10^{−4} to 1.8 × 10^{−4}, the peak intensity corresponding to the central frequency starts to decrease from 1, while the side lobes increase, which means more input intensity diffracts to frequency components that do not satisfy the Bragg condition. From Fig. 5(b), it can be seen when the index modulation increases from 2.0 × 10^{−4} to 2.8 × 10^{−4}, the peak intensity increases from 0 to 1. Further increasing the refractive index modulation, the peak intensity will repeat the evolution from 1 to 0, then 0 to 1, which means that the peak intensity evolution is periodic, so the evolution of diffraction intensity spectrum is periodic too. The periodic diffraction intensity spectrum distribution results in a periodic temporal diffraction intensity distribution, that’s the reason we can see periodic pulse waveform evolution.

The evolution period can be acquired by Eqs. (5) and (7). Equation (7) shows that the temporal diffraction field E_{d}(t, d) can be regarded as a superposition of a series of monochromatic diffraction field with different frequencies and weights. From Eq. (5), it can be seen that the weights E_{d}(ω, d) are modulated by a sine function, so does the amplitude of temporal diffraction field E_{d}(d, t). Therefore the temporal diffraction intensity is modulated by the square of the sine function. As the sine function is related to the refractive index modulation of VHG, the temporal diffraction intensity distribution is also a function of refractive index modulation and changes periodically with it.

In term ${\mathrm{sin}}^{2}\sqrt{\nu {\text{}}^{2}+\xi {\text{}}^{2}}$, $\nu =\frac{\omega \text{\hspace{0.17em}}\Delta nd}{2c\mathrm{cos}{\theta}_{r}\text{'}}$is related to Δn, while ξ is not, so the period is determined by ν, and the period of ν is

Accordingly, the period of refractive index modulation is

Substituting the parameters defined in section 3 into Eq. (10), the evolution period of refractive index modulation is 1.99 × 10^{−4}, which coincides with the value 2.0 × 10^{−4} we get from the numerical simulations in Figs. 3 and 4.

## 5. Time delay of the diffracted pulse with respect to the input pulse

It is well known that a periodic structure exhibits strong group-velocity dispersion and such a dispersive property can be used to control the time delay or pulse front tilt of diffracted pulse.

The group velocity of the diffracted pulse through the VHG can be obtained by differentiating the phase shift per unit length with respect to the angular frequency ω [27]

Where φ is the phase shift of the diffracted pulse through the VHG, which can be calculated from Eq. (5). $l$is the propagation length of the diffracted pulse in the VHG. In our simulation, the grating period is Λ = 7.3μm, accordingly, the readout angle in vacuum is about 6°, and in the crystal is 2°. It is reasonable to assume $l$ approximately equal to the thickness d of the grating.Consequently, time delay of diffracted pulse through the VHG is

Combining Eq. (5) with Eq. (12), the time delay of the diffracted pulse relative to the input pulse is

Equation (13) shows that the time delay of the diffracted pulse is a function of the grating’s thickness and period, readout angle of input pulse, and is irrelevant to the refractive index modulation of the VHG. This can explain the phenomena shown in Figs. 3 and 4, where the time delays of all diffracted pulses are the same when the refractive index modulation changes. Equation (13) also shows that the time delay is negative, which is consistent with the simulation results of Figs. 3 and 4, where all centers of the diffracted pulses are shifted along the negative time axis.

In our condition, the time delay is caused by the thickness, period and readout angle of the VHG. Substituting the parameters defined in section 3 into Eq. (13), the calculated time delay is −87.7fs, which complies with the estimation of −88fs from Figs. 3 and 4.

## 5. Discussions and conclusions

In former sections, periodic waveform evolutions of temporal diffracted pulse with respect to the refractive index modulation of VHG are studied when the input pulse is defined. Here, we discuss the influence of durations and shapes of the input pulse on the diffraction periodicity.

Firstly, no matter how the durations and shapes of input pulse change, the periodic waveform evolution will exist. From section 3 we know that the periodic evolution is due to Kogelnik’s diffraction oscillation with respect to the refractive index modulation. When the index modulation is overmodulated, the diffraction oscillation has periodicity and this periodicity is an inherent property of transmitted VHG, irrelevant to input pulse. Secondly, durations and shapes of the input pulse have no influence on the evolution period (Δn)_{p}. Equation (10) shows that the evolution period is determined by the grating’s period and thickness, the central wavelength of input pulse, background refractive index of the material, and is irrelevant to durations and shapes of input pulse. In summary, whatever the durations and shapes of the input pulse are, the periodic evolution behavior of temporal diffracted pulse will exist and the evolution period will not change. As an example, we change the duration of input pulse to Δτ = 200fs, the distributions of temporal diffraction intensity with respect to six refractive index modulations are shown in Fig. 6
. In simulation other parameters are the same as those former defined. It can be seen that when the refractive index modulation increases, the number of diffracted pulse will evolve periodically and the evolution is in the same manner as those of Fig. 3. Moreover, the estimated evolution period is about 2.0 × 10^{−4} and time delay is about −88fs, these values are consistent with those of Δτ = 100fs.

Moreover, in this paper we take the unslanted VHG as an investigation, now we show that the discussion is also appropriate to study slanted VHG. Our discussions are based on the coupled-wave equations, which are appropriate to unslanted and slanted VHGs. If the grating is slanted, the coupling constant ν and off-Bragg parameter ξ need change to include an oblique parameter determined by the angle between the grating fringe and normal of the material. But those changes do not influence the diffraction oscillation, so the periodic evolution behavior will exist again, but the evolution period will change with the changing of ν.

Section 2 shows that the modified Kogelnik’s coupled-wave equations are acquired when the VHG is in sinusoidal form. If the grating is non-sinusoidal, the coupled-wave equations cannot be used directly. Maybe we can decompose the non-sinusoidal grating into the superposition of sinusoidal gratings with different grating periods, the coupled-wave equations can be used on each grating. Diffraction of femtosecond pulse by the non-sinusoidal volume grating is so complex that it deserves another long paper, here we won’t discuss it.

In conclusion, based on the Kogelnik’s coupled wave equations, the temporal diffraction characteristics of the transmitted volume holographic grating illuminated by an ultrashort pulse are discussed. Results show that the waveform evolution behavior of the diffracted pulse is greatly influenced by the refractive index modulation of the grating, but is irrelevant to the durations and shapes of input pulse. When the refractive index modulations are in the range of 1 × 10^{−4} to 1 × 10^{−3}, the number of the diffracted pulse changes from one pulse to two pulses, then to three pulses, and this pulse number evolution is periodic and the period is 1.99 × 10^{−4}. The periodic evolution is due to the overmodulation effect of refractive index modulation of transmitted VHG.

Comparing to the input pulse, the diffracted pulses translate along the negative time axis. We use group velocity dispersion in periodic structure to explain it. The results show that the time delay is irrelevant to the refractive index modulations and duration of input pulse. In our discussion, the time delay is caused by grating thickness, period and readout angle. The time delay is −87.7fs, coinciding with the data estimated from the numerical simulation.

## Acknowledgments

This work was financially supported by National Natural Science Foundation of China (Grants No. 60908007, 11274225, 11174195), Shanghai Leading Academic Discipline Project (No. S30105) and Shanghai Municipal Education Commission Innovation Project (12YZ002).

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