## Abstract

This paper presents dynamical diffraction properties of a femtosecond pulse in a sinusoidal volume holographic grating (VHG). By the modified coupled-wave equations of Kogelnik, we show that the diffraction of a femtosecond pulse on the VHG gives rise to periodical energy oscillation and pulse splitting. In the initial stage of diffraction, one diffracted pulse and one transmitted pulse emerge, and energy of the transmitted pulse periodically transfers to the diffracted pulse and vice versa. In the latter stage, both the diffracted and transmitted pulses split into two spatially separated pulses. One pair of transmitted and diffracted pulses propagates in the same direction and forms the output diffracted dual pulses of the VHG, and the other pair of pulses forms the output transmitted dual pulses. The pulse interval between each pair of dual pulses is in linearly proportional to the refractive index modulation and grating thickness. By the interference effect and group velocity difference we give explanations on the periodical energy oscillation and pulse splitting respectively.

© 2014 Optical Society of America

## 1. Introduction

Ultrashort dual pulse sequences are very useful in areas of pump-probe measurement, coherent control of quantum states, femtosecond micromachining and femtosecond pulse measurement [1–8]. The common method to generate ultrashort dual pulses is by autocorrelation structure. A semi-reflective mirror is generally used as a beam splitter, which invariably introduces material dispersion and absorption [9]. Based on diffraction of different combinations of transmitted and reflective Damman gratings, Changhe Zhou’s group proposed several schemes to realize ultrashort dual- and multi-pulses [10–15]. Recently, Mantsyzov’s group has analyzed and observed that ultrashort dual pulses can be acquired by diffraction of a femtosecond laser pulse from linear photonic crystals at the Laue geometrical scheme [16–18]. In former papers [19, 20], we found that by properly choosing refractive index modulation of the transmitted VHG, an incident femtosecond pulse can be split into diffracted dual pulses, and the two pulses have the same peak intensity and temporal duration. Moreover, emergence of the diffracted dual pulses is periodic with respect to the refractive index modulation. Based on Bragg selectivity and overmodulation effect of the VHG, we gave an explanation on the periodic evolution and deduced a mathematical expression to depict the evolution periodicity. However, the underlying reason of emergence of diffracted dual pulses in the VHG did not touch on in our former papers. To our knowledge, no other paper which studied on diffraction of ultrashort pulse from single layer VHG had found diffracted dual pulses and pulse splitting, let alone give explanation on pulse splitting.

Diffraction of ultrashort femtosecond pulse by VHG has been studied for many years. Theoretical and experimental studies were first done by Y. Ding and A. M. Weiner. They studied the diffraction in spectral domain and demonstrated that the diffraction bandwidth can be controlled by period and thickness of the VHG [21]. Later, X. Yan *et al.* [22, 23] systematically studied the influence of grating parameters on the temporal and spectral diffraction intensity of transmitted VHG. Wang *et al.* [24] further studied the ultrashort pulse shaping properties of VHGs recorded in anisotropic media. Hernández-Garay *et al.* [25] deduced the bandwidth of the VHG from the diffraction efficiency equation, then studied femtosecond spectral pulse shaping by four different bandwidth VHGs recorded in photopolymerizable glasses. With the development of PTR glass, Siiman *et al.* [26] experimentally studied the diffraction of ultrashort pulse by transmitted VHG recorded on PTR glass. Except for diffraction by single layer VHG, diffraction of ultrashort pulse by two- or multi-layer VHG are also discussed. A. Yan *et al.* [27, 28] studied the diffraction spectral intensity as a function of grating parameters in transmitted and reflective multi-layer VHGs. X. Yan *et al.* [29] generated temporal femtosecond dual pulses in transmitted two-layer VHG, and demonstrated that the pulse interval can be modulated by the buffer layer thickness. They explained that the generation of dual pulses was due to coherent superposition of two diffracted pulses coming from two grating layers, and the variable pulse interval was due to the phase shift provided by the buffer layer.

In this paper, by building the temporal-spatial dynamic images of the diffracted and transmitted pulses, we further study the formation reason of diffracted dual pulses in a transmitted VHG. It is found that in the initial stage of diffraction, the incident femtosecond pulse splits into one transmitted pulse and one diffracted pulse, and energy of the transmitted pulse periodically transfers to the diffracted pulse and vice versa. With an increase in the propagation depth in the VHG, periodical energy oscillation disappears and both the diffracted and transmitted pulses split into two pulses propagating in two different directions. At the output plane of the VHG, transmitted dual pulses and diffracted dual pulses are emitted.

## 2. Theoretical model

The transmitted VHG is a static unslanted phase grating. The grating fringe planes are parallel to y-*z* plane and extend infinitely in y axis. The grating vector **K** is parallel to *x* axis. The refractive index distribution of such a grating can be expressed in one dimension as

_{0}is the background refractive index of the grating material; Δn is the refractive index modulation; grating wavenumber K = 2π/Λ with grating period Λ = λ

_{r}/(2sinθ

_{r}). λ

_{r}and θ

_{r}are Bragg wavelength and angle of the VHG.

A femtosecond pulse incidents on the VHG at Bragg angle θ_{r} to readout. The electric field of the input pulse can potentially be a complicated function of space and time, but for the sake of simplicity we assume that it covers the whole space and it has a Gaussian shape in time domain, as

_{0}central wavelength. Parameter T is related to full width at half maximum (FWHM) Δτ of incident pulse by $T=\Delta \tau /\sqrt{2\mathrm{ln}2}$.

By applying Fourier transform on Eq. (2), the incident pulse is expanded as a series of spectral components, each of which represents amplitude of a plane monochromatic wave,

According to Kogelnik theory [30], an incident wave will couple out a transmitted wave and a diffracted wave inside the grating and the total field is,

_{t}(ω, z) and E

_{d}(ω, z) are complex electric field amplitudes of the transmitted and diffracted waves with respect to the frequency and propagation depth, respectively;

**ρ**and

**σ**are propagation vectors, by virtue of the momentum conservation, connecting to the grating vector by the relation

**σ = ρ-K**.

The total field in the VHG obeys scalar wave equation ${\nabla}^{2}E+{k}^{2}E=0\text{\hspace{0.05em}}\text{\hspace{0.17em}}(k=\omega \text{\hspace{0.05em}}n/c)$ for no charge source in the VHG $\nabla \cdot E=0\text{\hspace{0.05em}}$ [30]. Considering slowly varying envelop approximation and ignoring terms that are quadratic in refractive index change, the modified Kogelnik’s coupled-wave equations describing the diffraction of a femtosecond pulse from VHG are acquired [20]

_{r}’ is the readout angle inside the VHG.

Solving Eqs. (5) and (6), diffracted and transmitted spectral components E_{d} (ω, z) and E_{t} (ω, z) inside the VHG are obtained [20],

_{0}.

Diffraction efficiency spectrum of the VHG is defined as the intensity spectrum of diffraction to that of incident, and the expression is

The temporal diffracted and transmitted fields inside the VHG are given by

Consequently, the temporal diffracted intensity I_{d} (t, z) and transmitted intensity I_{t} (t, z) inside the VHG are expressed as

Equation (11) describes the temporal-spatial dynamic intensity distributions of the diffracted and transmitted pulses in the VHG, which are controlled by the grating parameters and the duration of readout pulse. Here we focus on the diffracted intensity distributions when femtosecond dual pulses emerge.

## 3. Periodical energy oscillation and pulse splitting inside the transmitted VHG

According to Refs [19, 20], when refractive index modulation of the VHG changes in the range of 1.6 × 10^{−4} to 3.05 × 10^{−2}, diffracted dual pulses can be obtained. In the following, Δn = 1.25 × 10^{−2} is chosen to simulate the diffraction. This index modulation can be acquired in BB-640 emulsions [31]. However, it is not easy to acquire large size emulsion with high quality, so we choose photorefractive Fe:LiNbO_{3} crystal as the VHG material. Its saturated refractive index modulation is near this value [32], most important, the photorefractive VHG can be of large size and high quality due to mature crystal growth technique. The background refractive index is n_{0} = 3.134. Other parameters are as follows: 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. Speed of light in vacuum is c = 3 × 10^{8} m/s. The period of the volume grating is Λ = 7.3μm. According to the Bragg condition of VHG Λ = λ_{0}/(2sinθ_{r}), the readout angle inside the VHG satisfies $\mathrm{cos}{\theta}_{r}^{\text{'}}\text{=}\sqrt{\text{1}-{\left(\frac{{\text{\lambda}}_{\text{0}}}{\text{2\Lambda}{\text{n}}_{\text{0}}}\right)}^{\text{2}}}$.

Figures 1(a) and 1(b) show temporal-spatial dynamical evolution of the diffracted and transmitted intensity inside the VHG. On the incident plane, the transmitted intensity is the maximum, whereas the diffracted intensity is the minimum because of no coupling. When the incident pulse propagates further into the VHG, according to the coupled-wave theory of Kogelnik [30], the energy of the transmitted pulse will couple to the diffracted pulse, then the diffracted intensity will attain its maximum and the transmitted intensity will attain its minimum on the output plane. However, if the refractive index modulation or thickness of the VHG is large enough, overmodulation effect will have effect on the diffraction [31, 33]. The transmitted intensity may attain a maximum while the diffracted intensity may be a minimum on the output plane. However, the energy coupling in the VHG is abnormal here. It is seen that in the initial stage of diffraction, the energy of both the transmitted pulse and diffracted pulse are oscillated. In the latter stage of diffraction, both the diffracted and transmitted pulses split into two sub-pulses propagating in two different directions.

Figures 2(a) and 2(b) further show the energy oscillation of both the transmitted and diffracted pulses when the propagation depth is in the range of 0-1.0mm. It is seen that only one transmitted pulse and one diffracted pulse exist in the VHG, and energy of both two pulses are periodically oscillated. When energy of the transmitted pulse attains its maximum, energy of the diffracted pulse will attain its minimum. It seems that energy of the transmitted pulse is periodically transferred to the diffracted pulse and vice versa. This effect is similar to the “Pendellösung” effect. The spatial period of energy oscillation, at which complete energy transfer occurs, is estimated to be 0.12mm on both the transmitted and diffracted pulses from Fig. 2.

The “Pendellösung” effect was first observed by Shull in 1968 in studying diffraction of slow neutron beams by thick perfect crystal [34], then predicted and observed in second-order Bragg scattering of matter waves from a standing light wave acting as a thick grating [35], in atomic wave from standing light waves [36]. The Laue diffraction formalism in X-ray diffraction from perfect crystals provides the theoretical explanation on this effect [37]. The experimental observation of this effect is not trivial since it requires thick and highly modulated optical media. Recently, M. L. Calvo has observed periodic oscillatory behavior of the angular selectivity in VHG recorded on a photopolymerizable glass with high refractive index modulation, it is the first time that this effect is observed for light diffraction in amorphous material [38]. Later, V. A. Bushuev and S. E. Svyakhovskiy predicted and observed the “Pendellösung” effect in diffraction of ultrashort pulse from linear photonic crystal [16–18]. Here we find the periodical energy oscillation with respect to the propagation depth in the VHG. This effect is of special interest for the experimental determination of the unit cell of the crystal structure and the scattering amplitude, for the realization of all-optical switching in a volume grating.

Figures 3(a) and 3(b) show the pulse splitting of the transmitted and diffracted pulses in the VHG when the propagation depth is in the range of 1.5mm to 4mm. It is seen when the propagating depth is larger than 3.0mm, both the transmitted and diffracted pulses split into two independent sub-pulses propagating in two different directions, respectively. The temporal waveforms of the four independent sub-pulses are similar and maintain their shape even in the range of 4mm to 7.8mm. However, when the depth is smaller than 3mm, two sub-pulses in both the diffracted and transmitted pulses are overlapped. The larger the propagation depth, the smaller the overlapped zone gets.

Above discussions show that to observe periodical energy oscillation and pulse splitting in the VHG, the thickness of the VHG is at least 3mm. To further depict the evolution behavior of the diffracted pulse after pulse splitting, the thickness must be larger than 3mm. In our discussion, to fully understand the pulse evolution behavior, the maximum thickness is chosen as 7.8mm.

In the following, we will give explanations on the periodical energy oscillation and pulse splitting in the VHG when a femtosecond pulse is diffracted by a transmitted VHG.

## 4. Explanation on pulse splitting in the latter stage of diffraction

Figures 1 and 3 show that when the depth is larger than 3mm, four sub-pulses, two transmitted sub-pulses E_{t1}, E_{t2} and two diffracted sub-pulses E_{d1}, E_{d2}, exist in the VHG simultaneously. In the following we will demonstrate that one transmitted sub-pulse E_{t2} and one diffracted sub-pulse E_{d2} propagate in the same direction, so does the other pair of sub-pulses. Then we prove that the occurrence of diffracted dual pulse is due to the difference of group velocity of the one pair of two sub-pulses.

Figure 4(a) shows the temporal pulse distributions
in the diffracted direction of the VHG when the refractive index modulation is fixed at
Δn = 1.25 × 10^{−2} and the grating thickness changes in the range
of 3.5 to 7.5mm. It is seen that all diffracted pulses comprise dual pulses and all pulses have
the same peak intensity and temporal duration. As the thickness is small, dual pulses have a
nonzero center which is due to the overlap of two diffracted pulse. As the thickness is larger
than 5.0mm, two diffracted pulses will totally separate from each other. Moreover, waveform of
each diffracted pulse does not change with the increase of the thickness of the VHG. When
studying the transmitted pulse, we will find the same phenomena. Combining Figs. 1, 3 and 4, we can conclude that the transmitted sub-pulse E_{t2} and
diffracted sub-pulse E_{d2} propagate in the same direction and form the output
diffracted dual pulses. Meanwhile, the other pair of transmitted sub-pulse E_{t1} and
diffracted sub-pulse E_{d1} propagates in the same direction and forms the output
transmitted dual pulses in the transmitted direction of the VHG.

Defining pulse interval as the time interval between two peaks of diffracted dual pulses, it is seen that the pulse interval is in linearly proportional to the thickness of the VHG, shown as Fig. 4(b). When the thickness is 6mm, the estimated pulse interval is 250fs.

Figure 5(a) further shows the temporal pulse
distributions in the diffraction direction of the VHG when the grating thickness is fixed at
7.8mm, while the refractive index modulation changes in the range of 6.50 ×
10^{−3} to 3.05 × 10^{−2}. It is seen all diffracted
pulses include two same pulses and the pulse interval is also in linearly proportional to the
refractive index modulation of the VHG, shown as Fig.
5(b). In Fig. 5(b), when the refractive index
modulation is 1.25 × 10^{−2}, the estimated pulse interval is about
325fs.

In summary, the pulse interval is in linearly proportional to the thickness and refractive index modulation of the VHG.

From Figs. 4 and 5 we know that all diffracted pulses have the same shape in case of changing thickness or refractive index modulation of the VHG. It means that after the pulse splitting as what Fig. 3 shows, each diffracted pulse propagates with its own invariable group velocity. Moreover, as the pulse interval is in linearly proportional to the thickness and refractive index modulation, the group velocities of diffracted dual pulses are different in the VHG. To prove our assumption, in the following we will deduce the relation among pulse interval, thickness and the difference of refractive index modulation.

On diffracted output plane of the VHG, transmitted sub-pulse E_{t2} and diffracted sub-pulse E_{d2}, propagate the same distance d. If the group velocity of each pulse is fixed, the pulse interval can be written as

_{t}and υ

_{d}represent the group velocity of two sub-pulses E

_{t2}and E

_{d2}respectively. The existence of pulse interval means that the two group velocities are different, that is, each sub-pulse propagates at its own group velocity.

The group velocity is in inversely proportional to the effective refractive index of the material, that is υ_{i} = c/n_{i}, with i = t, d representing the transmitted and diffracted pulses. After simple deduction, the relation among pulse interval, propagation depth and difference of effective refractive indices is

_{eff}= | n

_{t}-n

_{d}|.

To compare with the estimated value of Fig. 5(b), substituting parameters of thickness d = 7.8mm and pulse interval *Δt _{pi}* = 325fs into Eq. (13), we get

*Δn*= 1.25 × 10

_{eff}^{−2}, which is consistent with the estimated values in Fig. 5(b). If substituting d = 6mm and

*Δn*= 1.25 × 10

_{eff}^{−2}into Eq. (13), the calculated pulse interval is 250fs, which is consistent with the estimated values in Fig. 4(b). It further proves that the assumption, the occurrence of pulse splitting in the diffraction direction is due to the difference of group velocities between two diffracted sub-pulses, is right.

Furthermore, difference of effective refractive indices *Δn _{eff}* = 1.25 × 10

^{−2}equals to the refractive index modulation

*Δn*of the VHG in our simulation, where

*Δn*is the difference of the maximum and background refractive index of the grating. It means that the slower and faster diffracted sub-pulses propagate in different positions of the VHG, the faster (or slower) pulse propagates mainly in the background material layer where the refractive index is n

_{0}and the slower (or faster) pulse propagates in the middle of the grating fringe layer where the refractive index is the maximum n

_{0}+ Δn (or minimum n

_{0}-Δn), therefore, the propagation of the diffracted fields in the VHG is localized.

## 5. Explanation on periodical energy oscillation in the initial stage of diffraction

From former discussions, we know in the latter stage of diffraction, there are four sub-pulses propagating in the VHG. Now we extend this conclusion to initial stage of diffraction to explain the periodical energy oscillation.

According to Eq. (13), the pulse interval is in linearly proportional to the propagation depth. In the initial stage of diffraction, the pulse interval between two diffracted sub-pulses and two transmitted sub-pulses are so small that all four sub-pulses are overlapped and propagate in the similar direction. According to optics, two diffracted sub-pulses will interfere, so will two transmitted sub-pulses. Periodical energy oscillation is the interference fringe, which is a periodic function of the propagation depth in the VHG. It is well known that the phase difference between two neighboring interference maximum is 2π, thus the relation between the phase difference and spatial period of interference fringe is

Substituting the defined parameters of section 3 into Eq. (14), the spatial period of the interference fringe is acquired, d_{s} = 0.12mm.

The spatial period of energy oscillation can also be calculated by diffraction efficiency Eq. (9). The maximum diffracted intensity occurs under Bragg incidence, so the off-Bragg parameter ξ should be zero. The two adjacent diffracted maximums occur when coupling parameters ${\nu}_{1}=(2k+1)\frac{\pi}{2}$and${\nu}_{2}=(2k+3)\frac{\pi}{2}$, with k an integer. Substituting ν_{1} and ν_{2} into $\nu (\omega ,z)=\frac{\omega \text{\hspace{0.17em}}\Delta nz}{2c\mathrm{cos}{\theta}_{r}\text{'}}$, the distance of propagating depth z in the VHG between two adjacent maximums is d_{s} = 0.12 mm.

The energy oscillation period calculated by diffraction efficiency equation equals to that calculated by interference effect and equals to the estimated value in Fig. 2. It further confirms that the periodic energy oscillation can be explained by pulse interference.

Form above discussions, we can give an image of pulse evolution in the initial stage of diffraction in the VHG. When a femtosecond pulse incidents on the VHG, two transmitted sub-pulses and two diffracted sub-pulses will emerge. When the propagation depth in the VHG is small, the two pair sub-pulses are overlapped, thus the interference happens and periodic energy oscillation emerges.

## 6. Conclusion

In this paper, based on the modified coupled-wave equations of Kogelnik, we have discussed dynamic diffraction of a femtosecond pulse in the transmitted VHG. Due to the coupling effect of the VHG, four sub-pulses exist in the VHG simultaneously. In the initial stage of diffraction, as interference occurs on both two diffracted sub-pulses and two transmitted sub-pulses, periodic energy oscillation emerges on both the transmitted and diffracted pulses. In the latter stage of diffraction, both the diffracted and transmitted pulses split into two sub-pulses propagating in two different directions, and one pair of transmitted sub-pulse and diffracted sub-pulse propagate in the same direction and forms the output diffracted dual pulses and the other pair forms the output transmitted dual pulse. As each pair two pulses propagate with different group velocities, after a certain propagation depth, two pulses will separate from each other and pulse splitting will emerge.

## Acknowledgments

This work was financially supported by National Natural Science Foundation of China (Grants No. 11274225, 11174195), Shanghai Natural Science Foundation (13ZR1414800,14ZR1415400) and Innovation Program of Shanghai Municipal (12YZ002).

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