A scheme to generate individually modulated femtosecond pulse string by multilayer volume holographic grating (MVHG) is proposed. Based on Kogelnik’s coupled-wave theory and matrix optics, temporal and spectral expressions of diffracted field are given when a femtosecond pulse is diffracted by a MVHG. It is shown that the number of diffracted sub-pulses in the pulse string equals to the number of grating layers of the MVHG, peak intensity and duration of each diffracted sub-pulse depend on thickness of the corresponding grating layer, whereas pulse interval between adjacent sub-pulses is related to thickness of the corresponding buffer layer. Thus by modulating parameters of the MVHG, individually modulated femtosecond pulse string can be acquired. Based on Bragg selectivity of the volume grating and phase shift provided by the buffer layers, we give an explanation on these phenomena. The result is useful to design MVHG-based devices employed in optical communications, pulse shaping and processing.
© 2014 Optical Society of America
With the development and maturation of ultrashort pulse laser technology, ultrashort laser pulse has been widely used in nonlinear optics, high-field physics, biomedical imaging, condensed-matter physics, telecommunications, and molecular spectroscopy [1–4]. In some applications, we not only need ultrashort pulse, but also wish that the ultrashort pulse have specific waveforms, such as ultrashort pulse string. Pulse string, comprising two or multiple separated sub-pulses with variable intensity distribution and pulse interval ranging from several femtoseconds to several hundred picoseconds, has found its popularity in femtosecond micromachining to reveal the dynamic evolution of material or to acquire high-quality microstructure [5–12].
The common method to generate femtosecond pulse string is by optical delay line structure, in which mirrors and micro-displacement devices are used to transform space delay among different optical paths into time delay among sub-pulses. However, if the number of the needed sub-pulses is large, the number of mirrors and micro-displacement devices increases correspondingly, dispersion and absorption can’t be neglected. A. M. Weiner employed optical 4f-configuration-based pulse shaper to realize pulse string , in which a spatial light modulator is used as the spectral filter in spectral plane. To acquire the desired pulse string structure, the main work is to design appropriate filter function, which is complicated in some time. Recently, based on diffraction of a femtosecond pulse from a single layer volume holographic grating (VHG), we have realized femtosecond pulse string with one, two or three similar sub-pulses by modulating the refractive index modulations of the VHGs . Then by diffraction of a femtosecond pulse from a 2-layer MVHG, we generated femtosecond dual pulses. The double pulses have the same peak intensity and pulse duration, but have variable pulse interval. The pulse interval can be controlled by the buffer layer thickness . In this paper, we extend our former discussions of 2-layer MVHG to multi-layer MVHG structure to generate femtosecond pulse string.
The concept of MVHGs was first introduced by Tanguay and Johnson . MVHG comprises a recently proposed class of novel diffraction structures in which multiple VHG layers are interleaved with optically homogeneous buffer layers. Each grating layers operate in Bragg diffraction regime, and the buffer layers provide phase modulation between adjacent grating layers. The physical separation of the diffraction and modulation processes in MVHG allows the MVHG to exhibit unique diffraction properties that are unavailable with conventional VHG . By using these unique properties, MVHGs have found various promising applications in optical interconnects , wavelength division multiplexers and demultiplexers , optical filters , multiple beam generator  and diffractive optics element .
The analysis of MVHG is based on Kogelnik’s coupled wave theory  and matrix optics. Diffraction of continuous wave and ultrashort pulse by MVHG has been reported by scientists. Yakimovich discussed the diffraction efficiency and angular selectivity properties of multilayer three-dimensional volume holographic grating . Vre and Hesselink systematically studied the diffraction properties of photorefractive stratified volume holographic optical elements . A. Yan was the first to study the pulse shaping and diffraction properties of MVHG under ultrashort pulse illumination [26–28]. G. Zhang et al. deduced the diffracted field expression of reflectance MVHG by recursion formula and studied its application in light group velocity control . E. F. Pen et al. studied the diffraction properties of multilayer nonuniform holographic structures formed in photopolymer materials . In this paper, we study the diffraction properties of a femtosecond pulse from MVHG when grating parameters change independently.
Comparing with single layer VHG, MVHG has more free parameters and will be more efficient for ultrashort pulse manipulation. In this paper, assuming thickness of each buffer layer and grating layer, and number of grating layers are individually controllable, we discuss the dependence of MVHG diffraction properties on these parameters and the realization of pulse string with multiple sub-pulses. Furthermore, we demonstrate that the waveform of each sub-pulse and pulse interval between adjacent two sub-pulses can be modulated by these parameters independently. The physical origin of these phenomena is given.
2. Theoretical model
The readout structure of an unslanted m-layer MVHG by an ultrashort pulse is schematically shown in Fig. 1. R0 is the readout wave, R and S represent the transmitted and diffracted waves. The system of MVHG is composed of m VHG layers seperated by m-1 buffer layers. In general, the VHGs are recorded in photorefractive material by two plane-wave interference and the buffer layers are isotropic materials. Assuming Ti and di are the thickness of the ith volume grating layer and buffer layer respectively, Λ is the grating period. To decrease Fresnel reflection between adjacent two layers, assuming background refractive indices of the VHG layers and buffer layers are the same, represented by n0.
Based on Kogelnik’s coupled-wave theory and matrix optics, by multiplying the transfer matrix of each layer of the MVHG, the expressions of diffracted field can be derived. In the following, we first give the transfer matrix of the VHG layers and buffer layers.
The complex amplitudes of the transmitted field R and diffractive field S of the ith VHG illuminated by a unit plane wave are given by 
Function of the ith grating layer of the MVHG can be represented by the following transfer matrix
The buffer layers are isotropic material and the transfer matrix of the ith buffer layer can be represented as
By multiplying the transfer matrix of each layer, relation between input and output spectral field of a m-layer MVHG can be obtained,
Assuming the readout femtosecond pulse has a temporal Gaussian distribution,
The field distribution in spectral domain can be obtained by applying Fourier transform on Eq. (8),
Substituting incident field matrix into Eq. (7), the diffracted and transmitted spectrum fields can be acquired.
After inverse Fourier transform on the diffracted spectrum field S(Td, ω), the temporal diffracted field is acquired,
Correspondingly, the temporal diffracted intensity distribution is acquired,
From former deductions, we know that the diffracted intensity is controlled by parameters of the MVHG. In the following, we mainly discuss the influences of the number of grating layers, the thicknesses of grating layers and buffer layers on the distributions of diffracted intensity.
3. Generation of femtosecond pulse string by modulating the number of grating layers of the MVHG
In simulation, assuming the following parameters: central wavelength of the readout pulse is λ0 = 1.5μm and the corresponding central angular frequency is ω0 = 4π × 1014 Hz; Background refractive index of the MVHG is n0 = 3.314. All VHGs have the same spatial period Λ = 7.3μm. Duration of the readout pulse is Δτ = 100fs. Speed of light in vacuum is c = 3 × 108m/s.
Figure 2 shows temporal intensity distribution of the incident Gaussian pulse with duration of 100fs.
Figure 3 shows temporal intensity distributions of the diffracted pulse when the number of VHG layers of the MVHG is 1, 2, 3 and 4, respectively. It is seen when thickness of the grating layers and buffer layers are held constant at Tm = 4mm and dm = 8mm, the addition of grating layers increases the number of diffracted sub-pulses. The number of diffracted sub-pulses equals to the number of VHG layers. In the pulse string, the pulse duration and peak intensity of each diffracted sub-pulse are similar, and the pulse interval between peaks of adjacent two sub-pulses holds constant. It means that the pulse shape and pulse interval are independent of the number of grating layers. Comparing to the incident femtosecond pulse, each diffracted sub-pulse is broadened to 120fs. Further increasing the number of grating layers, the number of diffracted sub-pulses increases too, which is equal to the number of the grating layers N all the time. That is, by stacking N grating layers and N-1 buffer layers into a MVHG, we can acquire N diffracted sub-pulses with similar pulse duration, peak intensity and pulse interval.
4. Independently modulated femtosecond pulse string by MVHG
In practical applications, we not only need femtosecond pulse string with N separated sub-pulses, but also expect the peak intensity and duration of each sub-pulse and pulse interval between adjacent two sub-pulses are more customizable. In this section, by independently modulating buffer layer thickness or VHG layer thickness of the MVHG, we demonstrate such kinds of femtosecond pulse string can be acquired.
4.1 Influence of buffer layer thickness on pulse interval of diffracted sub-pulses
Figures 4 show temporal diffracted intensity distribution as a function of buffer layer thickness in a 3-layer MVHG when thicknesses of all grating layers are fixed at 4mm. In Fig. 4(a), it is seen as thickness of the first buffer layer increases from 2mm to 10mm and the second buffer layer is fixed at 8mm, pulse interval between the first and the second sub-pulses increases correspondingly whereas the pulse interval between the second and the third sub-pulses almost holds constant. It means that the pulse interval between the first and the second sub-pulses depends only on the first buffer layer thickness and independent of the second buffer layer thickness. Figure 4(b) shows that the pulse interval between the second and the third sub-pulses is determined by the second buffer layer thickness and independent of the first buffer layer thickness. Figure 5 further demonstrates that above conclusions can be extended to 4-layer MVHG.
From above discussion, a conclusion can be given that the pulse interval between adjacent sub-pulses is determined by thickness of the corresponding buffer layer.
From Figs. 4 and 5 it is further found, as thickness of a specific buffer layer increases with a constant increment, peak positions of sub-pulses on its left side translate along negative time-axis with a fixed value, whereas positions of sub-pulses on its right side keep constant. Moreover, duration and peak intensity of each sub-pulse are almost unchangeable during this translation. It further indicates that the buffer layer thickness affects only on the pulse interval, but not on the pulse waveform. Figure 6 shows that the pulse interval between adjacent two sub-pulses is in linearly proportional to the buffer layer thickness, which provides us an easy way to acquire the needed pulse interval by appropriately choosing buffer layer thickness.
According to Fig. 6, if the buffer layer thickness is small, the pulse interval will be small too and the adjacent two sub-pulses will be overlapped, shown as the nonzero central intensity in Figs. 4 and 5. If the thickness is so small that superposition of two adjacent sub-pulses shows only one peak, on this condition the number of diffracted sub-pulses is smaller than the number of grating layers N. To make the number of sub-pulses equals to that of grating layers, just as what Fig. 3 shows, the pulse interval should be larger than half addition of durations of adjacent two sub-pulses.
4.2 Influence of grating layer thickness on the shape of diffracted sub-pulses
Figure 7 shows temporal diffracted intensity distributions as a function of grating layer thickness when thickness of the buffer layers keeps constant in a 3-layer MVHG. It is seen all diffracted pulses include three sub-pulses, which means that the number of diffracted sub-pulses is independent of thickness of the grating layer. In Fig. 7(a), as thickness of the first grating layer T1 increases from 1mm to 5mm and thicknesses of the second and third grating layers keep constant, duration and peak intensity of the first diffracted sub-pulse increase correspondingly, however, those of the second and third diffracted sub-pulses keep constant, indicating that the shape of the first diffracted sub-pulse depends on thickness of the first grating layer and independent of thickness of other grating layers and buffer layers. Figures 7(b) and 7(c) further demonstrate that shapes of the second and the third diffracted sub-pulses depend on thickness of the second and the third grating layers, respectively.
Figure 7(d) shows when thickness of the first grating layer increases and the third grating layer decreases simultaneously, the peak intensity and duration of the first diffracted sub-pulse increase and the third sub-pulse decreases. During this process, shape of the second sub-pulse keeps constant as the second grating layer thickness does not change. It further proves that shape of the diffracted sub-pulse can be controlled by thickness of the corresponding grating layer.
Moreover, comparing Fig. 7(d) with Figs. 7(a)–7(c), it is concluded that temporal duration of the total diffracted pulse is determined by total thickness of the MVHG. In Fig. 7(d), although thickness of the first and the third grating layers change, the total thickness of the MVHG keeps constant, it is seen that all three diffracted pulses have the same temporal width. In Figs. 7(a)–7(c), the temporal width of the diffracted pulses increases as the total thickness increases.
Figure 8(a) further shows that shape of the fourth sub-pulse changes with thickness of the fourth grating layer when MVHG includes 4 grating layers. Figure 8(b) shows duration of the fourth diffracted sub-pulse as a function of thickness of the fourth grating layer in a 4-layer MVHG. The relation is almost in linear. In comparison with the duration of input pulse beam, durations of the diffracted sub-pulses are broadened from 116fs to 148fs as thickness of the grating layer increases from 1mm to 7mm.
Figures 7 and 8 further show, as thickness of the ith grating layer increases, duration of the ith diffracted sub-pulse broadens. The front edge of this sub-pulse keeps constant, whereas the back edge translates along the negative time-axis. The translation is the same as the broadening value. Correspondingly, positions of all left sub-pulses translate along the negative time-axis with the same displacement, whereas positions of all right sub-pulses do not change. The result is that the broadening does not change the pulse interval between adjacent two sub-pulses. It further proves that the pulse interval is independent of the grating layer thickness.
4.3 Summary of distributions of diffracted sub-pulses with respect to buffer layer thickness and grating layer thickness
According to former discussions, relations between the parameters of the MVHG and the properties of the diffracted pulse string are now clear. The peak intensity and duration of the sub-pulses are mainly determined by thickness of the corresponding grating layer. Pulse interval between adjacent sub-pulses is mainly determined by thickness of the corresponding buffer layers. Relation of diffracted pulse properties and MVHG’s parameters is summarized in Fig. 9.
5. Explanations on individually modulated femtosecond pulse string
In this section, based on properties of Fourier Transform, Bragg selectivity and coupling effect of the volume grating, we give explanations on the generation of individually modulated femtosecond pulse string.
Taking 2-layer and 3-layer MVHGs as examples, we first explain the influence of buffer layer thickness on the diffracted pulse interval.
From Eq. (5), it is known that parameters M222, M121 M221 and M111 are irrelevant to thickness of the buffer layer d. If only considering the influence of buffer layer thickness d on the diffraction, the first term in Eq. (12) has a relative phase shift to the second term and the phase shift is in proportional to buffer layer thickness d. When Eq. (12) 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. Neglecting coefficients and phase shifts relevant to thickness of the grating layers, the temporal diffracted field as a function of buffer layer thickness d is abbreviated as
In Eq. (13), if thickness of the buffer layer is so large that pulse interval between two sub-pulses is larger than the duration of one sub-pulse, two sub-pulses will separate from each other and double pulses will emerge. It is consistent with the diffracted intensity distribution in Fig. 3(a).
As N = 3, the spectral expression of the diffracted field is,
Equation (14) includes four terms, according to former discussions of N = 2, there will have four diffracted sub-pulses in time domain. However, in Figs. 3 and 4, only three sub-pulses emerge when N = 3. Further study shows that the coefficient of the second term is 10−4 times smaller than those of other three terms, thus the second term can be neglected, then Eq. (14) changes to
Applying Fourier Transform on Eq. (15), the diffracted field as a function of buffer layer thickness in time domain is,
This temporal expression includes three sub-pulses and the pulse intervals between adjacent two sub-pulses are proportional to d1 and d2 respectively, which are consistent with the distributions of diffracted intensity in Figs. 3 and 4 and summary diagram Fig. 9.
According to Eq. (16), center of s3(t) is at t = 0. However, centers of the last sub-pulses in Figs. 3 and 4 are not at t = 0. The reason is that the time delay introduced by the grating layer thickness is not considered in Eq. (16) when we discuss the influence of buffer layer thickness on the diffraction. The time delay provided by the grating layer is in linearly proportional to the grating layer thickness , embodied in Figs. 7 and 8, where positions of sub-pulses translate along negative time axis with the increasing of grating layer thickness.
If MVHG includes m grating layers, a general expression to depict the influence of buffer layer thickness on the diffracted field can be given,Equation (17) shows that the buffer layer thickness determines the pulse interval.
Former discussions show that the coefficient of the second term in Eq. (14) is much smaller than that of other terms. By this conclusion, we construct Fig. 10 to explain the influence of buffer layer thickness on diffraction of MVHG.
Figure 10 shows the diffraction of spectral component of the incident femtosecond pulse by a m-layer MVHG. The incident femtosecond pulse can be assumed as a coherent superposition of spectral components with different frequency and weight determined by Eq. (9). One spectral component is equivalent to a plane wave, when it incidents on the first volume grating of the MVHG, 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. After passing through the first buffer layer, according to Eq. (6), diffracted wave will have a relative phase shift to the transmitted wave, and the phase shift is in proportional to the thickness d1 of the first buffer layer. When they incident on the second VHG, the diffracted wave S1 will directly pass through and no coupled wave emerges. The reason is that S1 has a phase-shift supplied by the first buffer layer, which causes it deviate from the diffraction condition of the second grating layer. However, the transmitted wave R1 has no phase shift, and it will couple out diffracted wave S2 and transmitted wave R2. On the second buffer layer, S1 will have the second phase-shift proportional to the second buffer layer thickness d2. Meanwhile, S2 will have a relative phase-shift to R2. The process will go on until the last grating layer.
When MVHG includes m grating layers, number of m diffracted pulses will output in the diffracted direction and each diffracted pulse comes from a specific grating layer of the MVHG. The superposition of these diffracted pulses composes the diffraction of the MVHG. To prove this assumption is right, in the following, taking propagation of S1 as an example, we will re-deduce the diffraction expression Eq. (17).
If the diffracted wave S1 comes from diffraction of the first grating layer and no diffraction occurs on other grating layers, then it will directly propagates through all other buffer layers and grating layers. The phase shift associated with buffer layers is, in time domain, the phase shift transforms into pulse delay and the output diffracted sub-pulse is . By the same way, expressions of other diffracted sub-pulses can be deduced. The total diffracted field is just what Eq. (17) shows. It proves that our assumption is right, that is, each diffracted pulse comes from a specific grating layer of the MVHG.
Because each diffracted pulse comes from a specific grating layer of the MVHG, duration and intensity peak intensity of each diffracted sub-pulse will be determined by the corresponding grating layer thickness. It is consistent with the results shown in Figs. 7–9, where shape of each diffracted sub-pulse is modulated by the corresponding grating layer thickness.
Comparing to the incident pulse, it is seen that durations of diffracted sub-pulses broaden when the grating layer thickness increases. This can be explained by Bragg selectivity of the grating layer. When thickness of the VHG increases, the incident pulse travels longer distance in the grating layer, Bragg selectivity becomes strict and fewer spectral components are coupled out. According to the property of Fourier Transform, in time domain the duration will broaden. Fewer spectral components are coupled out means that the output intensity will decrease. However, according to diffraction efficiency equations, diffracted intensity increases with the increasing of the thickness. The ultimate result is the intensity increases with the increasing of the grating layer thickness.
6. Discussion and conclusion
In former discussions, to explicitly demonstrate the influence of grating layer thickness and buffer layer thickness on the diffraction, we neglect material dispersion. To the VHG, material dispersion decreases the grating bandwidth, thus the diffracted and transmitted pulses will broaden in time domain. However, the broadening value also depends 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 small . Moreover, each pulse in the pulse string is diffracted only once, so smaller pulse broadening does not affect the occurrence of pulse string. However, as the duration of input pulse gets smaller or the thickness of the grating material gets larger, the influence of material dispersion on the diffraction should be considered. About the influence of dispersion effect of material, laser and grating parameters on the diffraction of MVHG structures is so complicated that it needs another long paper to discuss, we omit here.
In conclusion, we present a scheme in this paper to generate individually modulated femtosecond pulse string by using multilayer volume holographic gratings. By properly choosing thickness of the buffer layer and grating layer, the pulse interval will be larger than half addition of duration of adjacent sub-pulses, thus separated femtosecond pulse string can be generated. The number of sub-pulses in the pulse string is equal to the number of grating layers of MVHG; Duration and peak intensity of each diffracted sub-pulses is individually modulated by the corresponding grating layer thickness; Pulse interval between adjacent diffracted sub-pulses is mainly controlled by the corresponding buffer layer thickness. The underlying reason is that the phase shift introduced by the buffer layer makes the diffracted wave deviate from the diffraction condition of the following volume grating layers, thus no further diffraction occurs. The individually modulated femtosecond pulse string is valuable for femtosecond micromachining technology, pulse shaping and processing, optimizing design and novel applications of volume holographic optical elements.
This work was financially supported by National Natural Science Foundation of China (Grants No. 11274225, 11174195), Shanghai Natural Science Foundation (13ZR1414800,14ZR1415400), Innovation Program of Shanghai Municipal (12YZ002) and Key Basic research program of Shanghai Municipal Science and Technology Commission (No. 14JC1402100).
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