###### Chinese Optics Letters, Vol. 17, Issue 11, 113201 (2019)

## Generation of femtosecond dual pulses by a transverse standing wave in a volume holographic grating

Xiaoyan Wang, Xiaona Yan^{*}, Kailong Jin, Ye Dai, Zuanming Jin, Xihua Yang, and Guohong Ma

Author Affiliations

- Physics Department of Science College, Shanghai University, Shanghai 200444, China

## Abstract

Based on Kogelnik’s coupled-wave theory, it is found that when a femtosecond pulse is incident on a transmitted volume holographic grating, two transverse standing waves along the grating vector direction will be generated inside the volume holographic grating (VHG). Due to field localization of two standing waves, they have two different velocities along the propagation depth. On the output plane of the VHG, femtosecond dual pulses are generated in both the diffracted and transmitted directions. Results show that the pulse interval is determined by the refractive index modulation and thickness of the grating, while the waveform of the dual pulses is independent of the grating parameters.

Femtosecond dual pulses, due to their short pulse duration and high spectral resolution, have found applications in various fields, such as femtosecond micromachining, ultrafast pump-probe spectroscopy, and coherent control of quantum states^{[1–4]}.

The common method to generate femtosecond dual pulses uses autocorrelators. However, due to material dispersion and absorption, the beam splitter needs to be as thin as 2 μm in some strict conditions, which significantly limits its practical applications. Later, Zhou’s group proposed several schemes to generate femtosecond dual pulses based on diffraction of plane Dammann gratings^{[5–7]}. Volume holographic gratings (VHGs), compared with plane gratings, have the advantages of high diffraction efficiency and strict Bragg selectivity. Based on these properties, our group generated femtosecond dual pulses in a VHG by adjusting refractive index modulation^{[8]}. In this Letter, based on transverse standing-wave distribution in the VHG, we propose a new scheme to generate femtosecond dual pulses.

Using the general solutions of Kogelnik’s coupled-wave equations^{[9]}, we found that when a femtosecond pulse is incident on the VHG, four waves are coupled out and coherently combine into two transverse standing waves along the grating vector direction. Due to field localization, two standing waves will propagate with two different velocities along the propagation depth; thus, with the increase of the depth in the VHG, two standing waves will separate from each other. On the output plane of the VHG, each standing wave is divided into a transmitted pulse and a diffracted pulse. Thus, dual pulses are output in both the transmitted and diffracted directions. It is found that the waveform of each diffracted pulse is independent of the VHG parameters, while the pulse interval between dual pulses depends on the grating thickness and refractive index modulation. Thus, interval modulated femtosecond dual pulses are generated.

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Temporal- and frequency-domain diffractions of a femtosecond and nanosecond pulse by single-layer VHGs^{[10–18]} and multi-layer VHGs^{[19–21]} have been discussed earlier, but none of them touched on transverse standing-wave distribution in the VHG. Moreover, previous work on temporal diffraction of single-layer VHGs^{[10,11]} found only one diffracted pulse, with a waveform that can be modulated by VHG parameters due to Bragg selectivity. However, in this Letter, on conditions when diffracted dual pulses emerge the waveform of each pulse is independent of the VHG parameters. Moreover, Refs. [8,11] acquired diffracted dual pulses, but the overmodulation effect is used to depict its physical origin. This means that the diffraction of a femtosecond pulse by the VHGs needs in-depth study.

The discussed VHG is a static unslanted phase grating whose grating planes are normal to the

Figure 1. Diffraction of a plane wave by a transmitted VHG.

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A temporal femtosecond pulse is incident on the recorded VHG at Bragg angle

By applying a Fourier transform on Eq. (

According to Fourier optics, the incident femtosecond pulse can be assumed as a superposition of plane waves with different frequencies; the complex amplitude of each wave is represented by Eq. (

The total field in the VHG satisfies the scalar wave equation

Assuming that the general solutions of Eq. (^{[9]}:

Combining Eqs. (

It is clearly seen from Eq. (

Figure 2. Wave vector diagram inside the VHG with a plane wave reading out the VHG.

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In the following, based on different combinations of wave fields

In Eq. (

By applying an inverse Fourier transform on the sum of the four fields, the temporal total intensity inside the VHG is

Figure

Figure 3. Temporal total intensity inside the VHG when the propagation depth

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In Eq. (

Now based on Eq. (

Figure 4. Dependence of diffracted dual pulses on VHG parameters: (a) thickness, (b) refractive index modulation, and (c) period. Pulse interval with respect to (d) thickness and (e) refractive index modulation of the VHG.

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Accordingly, intensities of the two standing waves are, respectively,

Comparing Eq. (

Two standing waves propagate along different grating layers; therefore, their propagating velocities are different. According to the group velocity expression, the velocities of the two standing waves along the

Equation (

On the output plane of the VHG, the faster standing wave arrives first. According to momentum conservation, it will break up into two pulses: the transmitted pulse propagating in the transmitted direction, and the diffracted pulse propagating in the diffracted direction. If the VHG is thick enough, the slower standing-wave pulse will arrive at the output plane with a certain time delay, where it will also break up into one transmitted pulse and one diffracted pulse. Thus, both transmission and diffraction will comprise femtosecond dual pulses.

Figures

In the following, we use the group velocity expression to deduce the relation between the pulse interval and VHG parameters.

Equation (

This time delay is the pulse interval between the dual pulses. Equation (

From above discussions, we know that by adjusting thickness and refractive index modulation of the VHG, we can generate femtosecond dual pulses with a variable pulse interval.

Figure

According to former discussions, we know that the dual pulses originate from two separate transverse standing waves, so waveforms of the diffracted dual pulses are determined by standing waves’ waveforms. When the thickness or refractive index modulation of the VHG is large enough, the two transverse standing waves will separate. In this case, each standing wave propagates along specific VHG layers, just as it propagates along a uniform medium, therefore increasing VHG thickness will not affect waveforms of the two standing waves and also have no effect on the waveforms of diffracted dual pulses. When the refractive index modulation changes, according to Eq. (

Figure

Figure 5. Temporal total intensity inside the VHG when the grating periods are 7.3 and 13 μm.

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When the refractive index modulation or thickness of the VHG decreases, according to Eq. (

Figure 6. Temporal diffracted intensity under a smaller

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Figure

Figure 7. Dependence of temporal diffracted intensity on the duration of incident femtosecond pulses.

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Actually, as the femtosecond pulse is incident on the VHG, the dispersion of the material will have an influence on the diffraction. However, we do not consider it here. The reason is that, according to the discussions of Fig.

In conclusion, diffraction-induced pulse splitting in a VHG and its underlying physical mechanisms are discussed. It is found that when a temporal Gaussian femtosecond pulse is incident on the transmitted VHG, four pulses are generated, which further combine into two transverse standing waves inside the VHG. Spatial localization of the two transverse standing waves in the VHG results in a different longitudinal group velocity. On the output plane, both diffraction and transmission include dual pulses with a pulse interval determined by the thickness and refractive index modulation of the VHG, while pulse waveforms are not affected by the VHG parameters.

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