Based on Kogelnik’s coupled-wave theory and matrix optics, generation of femtosecond double pulses by modulating thickness of the buffer layer of two-layer volume holographic grating (TL-VHG) is discussed. Expressions of diffraction field when a femtosecond pulse incidents on the TL-VHG are deduced. Simulation results show when thickness of the buffer layer increases from 6mm to 11mm or even larger, one incident pulse splits into double femtosecond pulses with the same duration and peak intensity, and pulse interval is linearly proportional to the thickness. The reason of these phenomena is due to the interference of diffraction waves reconstructed from two gratings and phase shift resulting from the buffer layer thickness. Time-delay of diffracted double pulses is explained by group time delay of periodic media. It is shown that the slope of the pulse interval with respect to the thickness of buffer layer is 2 times of that of pulse time-delay. Furthermore, we demonstrate it is possible to control the output double pulses’ duration and pulse interval by varying the grating thickness.
©2013 Optical Society of America
Multilayered diffractive optical elements offer powerful opportunity to harness both phase and amplitude modulation for benefits in diffraction efficiency and beam shaping. Stratified holographic optical elements (SHOEs), referred to a structure composed of a stack of thin or thick grating layers interleaved with optically homogeneous buffer layers, are characterized by specific properties determined by an interference of the reconstructed waves from gratings and provide possibility to control the shape of selective response. Thin SHOEs, working in Raman-Nath diffraction regime, possess some similar properties as those of volume holographic gratings (VHGs), such as a strict angular selectivity that differs qualitatively from the selectivity of conventional single-sheet volume holograms  and high diffraction efficiency. For example, thin SHOEs composed of two-layer , three-layer  and five-layer  thin phase gratings have diffraction efficiencies of 67%, >80% and 95%. Thin SHOEs may be used to design waveguide grating filter , waveguide grating coupler , multiple beam generator [7, 8], lidar beam scanner , holographic multiplexing and phase multiplexing .
SHOEs with stacks of multiple volume gratings instead of thin gratings are commonly called stratified volume holographic optical elements (SVHOEs) or multilayer volume holographic gratings (MVHGs), which operate within the Bragg diffraction regime. Analysis of MVHG is based on the coupled-wave theory of Kogelnik  and Matrix optics. Many scientists have studied on the theory of MVHG. Yakimovich showed that the characteristics of angular selectivity have a series of local maxima, width of which is determined by the sum width of the whole system, and the envelope of the maxima coincides with the selectivity contour of a single array . A closed-form expression was derived by Vre and Hesselink to describe the diffraction properties of layered transmission and reflection geometry of MVHG and they also provide the possible application in dynamic multiple-wavelength filter . Yan applied ultrashort pulse as the illumination to the transmission and reflective MVHGs and showed that the spectral distributions of diffracted light depend on MVHG parameters [14, 15]. Zhang et al. proposed a recursion formula for the reflective MVHG and studied its applications on group velocity controlling . With the development of the MVHG theory, applications such as wavelength division multiplexers and de-multiplexers, dynamic multiple wavelength filter have been realized by the MVHG .
In this paper, based on TL-VHG, we discuss femtosecond pulse splitting by modulating the thickness of the buffer layer. The acquired femtosecond double pulses are useful in coherent control of quantum states and femtosecond micromachining etc [18–21]. In femtosecond pulse measurement, such as Spectral Phase Interferometer for Direct Electrical-Field Reconstruction (SPIDER) and Frequency-Resolved Optical Gating(FROG), double pulses are also quite necessary [22, 23].
2. The structure of TL-VHG
Figure 1 is the recording and reconstruction structure of TL-VHG, where a buffer layer is encapsulated between two volume grating layers providing optical contact. The two grating layers are photorefractive material, while the buffer layer is an optically homogeneous material, which provides the phase incursion between neighboring grating layers. Ti (i = 1, 2) and d are thickness of the two grating layers and buffer layer. n0 is the mean refractive index of both the grating and buffer materials. VHGs in two grating layers are recorded simultaneously with two symmetrically incident coherent plane waves s and r through photorefractive effect, while the buffer layer does not record grating for it is not photosensitive material. Then the TL-VHG is readout by a femtosecond pulse u(t), the transmitted and diffracted pulses are represented by R(t) and S(t).
Two VHGs are assumed to extent infinitely in the x-y plane and have the same grating vector and grating period Λ. Parameters of the two gratings are chosen to make Q = 2πTiλ/(n0Λ2)>>1 to guarantee that they are both VHGs, where λ is the readout wavelength.
Two VHGs are phase gratings and the corresponding refractive index distribution is
3. Diffraction model of TL-VHG by an ultrashort pulse
Supposing an ultrashort time-domain Gaussian pulse incidents on the input surface of TL-VHG with angle θr, where ω0 is central frequency of incident pulse. andis the full width at half maximum (FWHM). The Fourier transform of the incident pulse is
It shows that the incident time-domain pulse consists of different frequency components, and weight of each component is represented by Eq. (2).
To the ith VHG in the TL-VHG, there have two waves propagating in it, the total field can be written as
By solving the coupled-wave Eq. (4), the transfer relation of the ith VHG is written in the following way with [Mi] the transfer matrix,14]
The buffer layer can also be represented by a transition matrix [D], the propagation of the transmitted and diffracted waves over the buffer layer satisfies
is a defined detuning parameter in the buffer layer
TL-VHG consists of two grating layers and one buffer layer. In the input plane, the boundary conditions are and, relation of the incident and output complex amplitudes is governed by
Inverse Fourier transform of S(T, ω), the instantaneous diffracted field and diffracted intensity distributions on the output plane are get
From above discussions, we know that the instantaneous diffracted intensity is determined by the thickness of grating layers and buffer layer, period and refractive index modulation of two grating layers. Taking photorefractive LiNbO3 crystal as VHG material, in the next section, we simulate the instantaneous diffracted intensity distribution with respect to the thickness of buffer layer and discuss the conditions when diffracted double pulses generate.
4. Instantaneous diffracted intensity distribution with respect to the thickness of buffer layer and the generation of femtosecond double pulses
In simulation, assuming the background refractive index of the grating layers and buffer layer n0 = 3.314; refractive index modulations of the two VHGs are both n1 = 2 × 10−5; Thickness and period of both gratings are T1 = T2 = 5.5mm and Λ = 7.3 × 10−6m. Central wavelength and the corresponding central angular frequency of the incident pulse is λ0 = 1.5μm and ω0 = 4π × 1014rad/s, FWHM of the incident femtosecond pulse is Δτ = 100fs.
Figure 2 is the intensity distribution of the incident pulse when Δτ = 100fs. It shows that the incident pulse includes only one pulse and the symmetric center of the pulse is t = 0.
Figure 3(a) and 3(b) show instantaneous diffracted intensity distributions of the TL-VHG when thickness of the buffer layer changes from 0mm to 3.5mm. It is found when the thickness d = 0, the diffracted intensity distribution is a flat-top single pulse with duration Δτ = 206fs. Comparing to the incident pulse, duration of the temporal diffracted pulse is broadening, which is due to Bragg selectivity of VHG. When d = 0, the TL-VHG is reduced to a single thick volume grating with thickness T = T1 + T2. The grating bandwidth is determined by 
The bandwidth ΔλP of the incident pulse is determined by
From Figs. 3(a) and 3(b), it is seen when thickness of the buffer layer increases from 0.2mm to 3.5mm, central intensity of all diffracted pulses decreases and gets much smaller with the increasing of the thickness. The front edges of all diffracted pulses are overlapped, meanwhile the back edges translate along the negative time axis. To a specific incremental step of buffer layer thickness in Figs. 3(a) and 3(b), the relative translation displacement of back edges of neighboring diffracted pulses are the same. However, this displacement increase with the incremental step of the buffer layer thickness.
Moreover, all diffracted pulses are central symmetric and the symmetric center is the center of diffracted pulse. It is found that centers of all diffracted pulses are positioned at negative time axis and the translation displacement with respect to the origin of time axis increases with the increasing of buffer layer thickness.
Figure 4(a) and 4(b) show the instantaneous diffracted intensity distributions of the TL-VHG when thickness of the buffer layer changes from 4mm to 11mm. It can be seen when thickness of the buffer layer increases, the central intensity of diffracted pulse decreases further. When thickness increases to 5.5mm, the central intensity is near zero that one incident pulse splits into two diffracted pulses. Further increasing the thickness, all diffracted pulses include two sub-pulses, the first sub-pulses are overlapped while the second sub-pulses translate along the negative time axis with the increasing of buffer layer thickness. The translation displacement is in direct proportional to the thickness of buffer layer. Moreover, all diffracted pulses have the same peak intensity and pulse duration. Duration of one sub-pulse is 96fs, which is smaller than that of incident pulse, but the whole duration of one diffracted double pulses is larger than that of incident pulse. The duration of diffracted double pulses increases with the increasing of the thickness.
Further increasing the thickness of buffer layer from 11mm, it is found that femtosecond double pulses shown as Fig. 4(b) will emerge again. To a specific thickness, defining pulse interval as the separation distance between two peaks of sub-pulses, from former discussions, we know that by modulating the thickness of buffer layer of TL-VHG, one incident pulse can be split into two diffracted pulses with different pulse intervals.
5. Explanation on the emergence of diffracted double pulses and the relationship of pulse interval with the thickness of buffer layer
In this section, based on the coupled-wave theory and diffracted field expressions, an explanation on generation of femtosecond double pulses is given by taking the diffraction of one spectral component in the input pulse as an example.
The incident femtosecond pulse can be assumed as the coherent superposition of spectral components with different frequency and weight determined by Eq. (2). One spectral component is equivalent to a plane wave, when it incidents on the first volume grating of the TL-VHG, according to the coupled-wave theory of Kogelnik, one diffracted wave and one transmitted wave with the same frequency as that of the incident plane wave will emerge on the right output plane of the first grating. When passing through the buffer layer, according to Eqs. (7)–(9), the diffracted wave will have a relative phase shift to the transmitted wave, and the phase shift is in direct proportional to the thickness of buffer layer. When they incident on the second volume grating of the TL-VHG, the diffracted wave will couple with the volume grating, one new diffracted wave and one new transmitted wave will emerge. The new diffracted and transmitted waves will output respectively on the transmission and diffraction directions of the whole TL-VHG. Meanwhile the transmitted wave output from buffer layer will couple out a new diffracted wave and a new transmitted wave in the second grating, which will output respectively on the diffraction and transmission directions of the whole TL-VHG. Thus, on diffraction direction of the output plane, there has one diffracted field and one transmitted field, the spectral expressions of these two fields can be deduced by Eqs. (5)–(11),
Equation (16) shows that in frequency domain, the diffraction of the whole TL-VHG includes two waves. Parameters M221, M111, M222 and M121 are irrelevant to the thickness of the buffer layer d, so the second wave in Eq. (16) has a relative phase shift to the first wave and the phase shift is in proportional to thickness d. When Eq. (16) is transformed into time domain, according to Fourier optics, two sub-pulses will emerge and the phase shift in frequency domain will transform into pulse interval in time domain. As the phase shift is in direct proportional to the thickness of the buffer layer d, so is the pulse interval.
Diffracted intensity is the coherent superposition of these two sub-pulses. When thickness of buffer layer is smaller than a specific value, the pulse interval will be smaller than the FWHM of each sub-pulse. On this condition, these two sub-pulses will overlap and the superposition result is one pulse with central intensity decreasing. The larger the thickness of the buffer layer gets, the larger the central intensity decreases. It is consistent with the evolution of diffracted intensity distributions shown in Figs. 3(a), 3(b) and 4(a). When thickness of the buffer layer is so larger that the pulse interval between two sub-pulses is greater than the FWHM of each sub-pulse, two sub-pulses will have no overlapped zone and will separate with each sub-pulse keeping its own frame. The pulse interval between these two sub-pulses increases with the increasing of the thickness. This result is consistent with the distributions of Fig. 4(b).
Figure 5 shows relation of pulse interval with respect to the thickness of buffer layer, which is fitted by the data acquired from Figs. 3 and 4. It can be seen that the pulse interval is linearly proportional to the thickness of the buffer layer, just as what we discussed above. It proves the explanation on generation of double pulses we given here is reasonable. The line slope of Fig. 5 is βI = 22.5 fs/mm.
6. Time-delay of diffracted pulse with respect to the time axis
It is seen from Figs. 3 and 4 that the center of diffracted pulse translates along the negative time axis. In this section, based on group velocity and group time delay, we give an explanation on the relation of translation displacement with the thickness of buffer layer.
It is well known that a periodic structure exhibits strong group-velocity dispersion and such a dispersive property can control the time delay of diffracted pulse. The group velocity of the diffracted pulse through the TL-VHG can be obtained by differentiating the phase shift per unit length with respect to the angular frequency ω Eqs. (5)–(11). is the propagation length of the diffracted pulse in the TL-VHG. In our simulation, the grating period is Λ = 7.3μm, accordingly, the incident angle in vacuum is about 6°, and in the crystal is 2°. It is reasonable to assume approximately equal to the thickness T of TL-VHG.
Consequently, time-delay of diffracted pulse through the VHG is
In our TL-VHG structure, the time-delay of the diffracted pulse relative to origin of the time axis is shown in Fig. 6.
It is seen from Fig. 6, when the thickness of buffer layer changes from 1mm to 12mm, the time-delay of all diffracted pulse are negative. Meanwhile, with the increasing of the thickness of buffer layer, the absolute value of time-delay increases, this result is consistent with the values shown in Figs. 3 and 4. Moreover, from Fig. 6 it is found that the time-delay is in direct proportional to the thickness of buffer layer. The absolute slope of the time-delay line is |βt | = 11.25 fs/mm, which is half of that of pulse interval line shown in Fig. 5. This indirectly demonstrates that two diffracted sub-pulses are the same and the pulse interval between them comes from the translation, while the translation is due to the changing of the thickness of buffer layer.
7. Discussions and conclusions
From former discussions, we know when thickness of the buffer layer of TL-VHG increases from 6mm, one incident femtosecond pulse will split into two diffracted pulses with the same duration and peak intensity. Moreover, the pulse interval can be modulated by thickness of buffer layer. Here, by changing thickness of volume grating, we show duration of diffracted pulse can be modulated.
Figure 7 shows the instantaneous diffracted intensity distribution of TL-VHG when thickness of buffer layer is fixed at d = 7mm, while the thickness of two gratings change from 1mm to 6mm. It is seen that the diffracted pulses are femtosecond double pulses too. To a specific thickness of grating, the two pulses have the same pulse duration and peak intensity. But to different thickness of grating, the duration and peak intensity of diffracted pulses are different, which increase with the increasing of the thickness of grating layer. Figure 7 also shows that pulse interval between diffracted double pulses changes with the thickness of grating.
Thus, by appropriately choosing the thickness of grating layers and buffer layer, we can get diffracted femtosecond double pulses with desired pulse interval and pulse duration.
The ultrashort pulse has a broad spectrum, thus the influence of material dispersion on the diffraction needs to be discussed. The light dispersion in the TL-VHG occurs at the interface of air-VHG since the refractive index of the grating layers and buffer layer are the same. Therefore, once the incident pulse is dispersed, each spectral component maintains its direction of propagation along the TL-VHG without noticeable change. As a first approximation we can consider a unique background refractive index for grating layers and buffer layer, equal to n0 . In Eq. (1), the refractive index modulation n1 fulfills n1<<n0. Hence n1 can be considered as a constant for the wavelength range of incident pulse. Furthermore, to the VHG, the dispersion of the grating material will decrease the grating bandwidth, thus the diffracted and transmitted pulses will broaden in time domain . However, the pulse broadening resulted from dispersion effect also depends heavily on the duration of input pulse and grating parameters. When duration of input pulse is 100fs and the grating length is not very large, the broadening from dispersion effect is very small . From two above reasons, we can neglect the influence of grating material dispersion on the diffraction.
Furthermore, pulse splitting we discussed here is due to the interference of diffracted pulses, so smaller pulse broadening does not affect the occurrence of pulse splitting. However, as the duration of input pulse gets smaller or the thickness of the grating material gets larger, the influence of dispersion effect on the diffraction should be considered. About the influence of dispersion effect of material, laser and grating parameters on the diffraction of TL-VHG and MVHG structures is so complicated that it needs another long paper to compute, we will discuss it in the future.
In conclusion, generation of femtosecond double pulses by modulating the thickness of buffer layer and grating layer is discussed when a femtosecond pulse incidents on a TL-VHG. Results show when thickness of grating layer is fixed, while the thickness of buffer layer changes, diffracted double pulses with the same pulse duration and peak intensity emerges. The pulse interval of double pulses is in direct proportional to the thickness of buffer layer. While fixing the thickness of buffer layer and changing the thickness of grating layer, diffracted double pulses will emerge again, but the duration are modulated. From the coupled-wave theory of Kogelnik and matrix optics, we give it a reasonable explanation.
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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