OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 18527–18536
« Show journal navigation

Periodical energy oscillation and pulse splitting in sinusoidal volume holographic grating

Xiaona Yan, Lirun Gao, Ye Dai, Xihua Yang, Yuanyuan Chen, and Guohong Ma  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 18527-18536 (2014)
http://dx.doi.org/10.1364/OE.22.018527


View Full Text Article

Acrobat PDF (1446 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

1. Z. Zhao, X. Tong, and C. Lin, “Alignment-dependent ionization probability of molecules in a double-pulse laser field,” Phys. Rev. A 67(4), 043404 (2003). [CrossRef]

8

8. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express 11(1), 68–78 (2003). [CrossRef] [PubMed]

]. 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

9. J. L. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16(1), 39–41 (1991). [CrossRef] [PubMed]

]. 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

10. G. Li, C. Zhou, and E. Dai, “Splitting of femtosecond laser pulses by using a Dammann grating and compensation gratings,” J. Opt. Soc. Am. A 22(4), 767–772 (2005). [CrossRef] [PubMed]

15

15. T. Wu, C. Zhou, J. Zheng, J. Feng, H. Cao, L. Zhu, and W. Jia, “Generation of double femtosecond pulses by using two transmissive gratings,” Appl. Opt. 49(24), 4506–4513 (2010). [CrossRef] [PubMed]

]. 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

16. V. A. Bushuev, B. I. Mantsyzov, and A. A. Skorynin, “Diffraction-induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A 79(5), 053811 (2009). [CrossRef]

18

18. A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 115(1), 56–67 (2012). [CrossRef]

]. In former papers [19

19. Z. Gao, X. Yan, Y. Dai, X. Yang, and G. Ma, “Generation of femtosecond double pulse by adjusting the refractive Index modulation of volume holographic grating,” Appl. Phys. B 112(1), 67–72 (2013). [CrossRef]

, 20

20. X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013). [CrossRef] [PubMed]

], 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

21. Y. Ding, D. D. Nolte, Z. Zheng, A. Kanan, A. M. Weiner, and G. A. Brost, “Bandwidth study of volume holography in photorefractive InP:Fe for femtosecond pulse readout at 1.5 μm,” J. Opt. Soc. Am. B 15(11), 2763–2768 (1998). [CrossRef]

]. Later, X. Yan et al. [22

22. X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.) 115(11-12), 512–516 (2004). [CrossRef]

, 23

23. B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol. 40(7), 906–911 (2008). [CrossRef]

] systematically studied the influence of grating parameters on the temporal and spectral diffraction intensity of transmitted VHG. Wang et al. [24

24. C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A 23(12), 3191–3196 (2006). [CrossRef] [PubMed]

] further studied the ultrashort pulse shaping properties of VHGs recorded in anisotropic media. Hernández-Garay et al. [25

25. M. P. Hernández-Garay, O. Martínez-Matos, J. G. Izquierdo, M. L. Calvo, P. Vaveliuk, P. Cheben, and L. Bañares, “Femtosecond spectral pulse shaping with holographic gratings recorded in photopolymerizable glasses,” Opt. Express 19(2), 1516–1527 (2011). [CrossRef] [PubMed]

] 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

26. L. A. Siiman, J. Lumeau, L. Canioni, and L. B. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 34(17), 2572–2574 (2009). [CrossRef] [PubMed]

] 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

27. A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009). [CrossRef] [PubMed]

, 28

28. A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B 96(1), 71–77 (2009). [CrossRef]

] studied the diffraction spectral intensity as a function of grating parameters in transmitted and reflective multi-layer VHGs. X. Yan et al. [29

29. X. Yan, M. Qian, L. Gao, X. Yang, Y. Dai, X. Yan, and G. Ma, “Pulse splitting by modulating the buffer layer thickness of two-layer volume holographic grating,” Opt. Express 21, 31852–31861 (2013). [CrossRef] [PubMed]

] 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
n=n0+Δncos(Kx),
(1)
where n0 is the background refractive index of the grating material; Δn is the refractive index modulation; grating wavenumber K = 2π/Λ with grating period Λ = λr/(2sinθr). λr and θr are Bragg wavelength and angle of the VHG.

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,

Er(ω)=12πEr(t)exp(iωt)dt=T2πexp[T2(ωω0)24].
(3)

According to Kogelnik theory [30

30. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969). [CrossRef]

], an incident wave will couple out a transmitted wave and a diffracted wave inside the grating and the total field is,
E(ω,z)=Et(ω,z)exp(iρr)+Ed(ω,z)exp(iσr),
(4)
where Et(ω, z) and Ed(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 2E+k2E=0(k=ωn/c) for no charge source in the VHG E=0 [30

30. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969). [CrossRef]

]. 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

20. X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013). [CrossRef] [PubMed]

]
cosθr'dEt(ω,z)dz=Ed(ω,z),
(5)
cosθr'dEd(ω,z)dzi2πcK2n0(1ω1ω0)Ed(ω,z)=Et(ω,z),
(6)
where κ=ωΔn/2cis coupling coefficient; ω is the angular frequency of the spectral component included in the input pulse; c is the speed of light in Vacuum; θr’ is the readout angle inside the VHG.

Solving Eqs. (5) and (6), diffracted and transmitted spectral components Ed (ω, z) and Et (ω, z) inside the VHG are obtained [20

20. X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013). [CrossRef] [PubMed]

],
Ed(ω,z)=iν(ω,z)exp(iξ(ω,z))sinν(ω,z)2+ξ(ω,z)2ν(ω,z)2+ξ(ω,z)2Er(ω),
(7)
Et(ω,z)=exp((ω,z))×(cosν(ω,z)2(ω,z)2-iξ(ω,z)sinν(ω,z)2(ω,z)2ν(ω,z)2(ω,z)2)×Er(ω),
(8)
Where ν(ω,z)=ωΔnz2ccosθr'determines the maximum diffraction efficiency when the readout spectral component satisfies Bragg condition of the VHG. ξ(ω,z)=π2czΛ2n0cosθr'(1ω1ω0) represents the deviation from Bragg condition, which is caused by the spectral components with frequency different from central frequency ω0.

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

η(ω,z)=sin2ν(ω,z)2+ξ(ω,z)21+ξ(ω,z)2ν(ω,z)2.
(9)

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

Ed(t,z)=Ed(ω,z)exp(-iωt)dω.Et(t,z)=Et(ω,z)exp(-iωt)dω.
(10)

Consequently, the temporal diffracted intensity Id (t, z) and transmitted intensity It (t, z) inside the VHG are expressed as

Id(t,z)=|Ed(t,z)|2,It(t,z)=|Et(t,z)|2.
(11)

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

19. Z. Gao, X. Yan, Y. Dai, X. Yang, and G. Ma, “Generation of femtosecond double pulse by adjusting the refractive Index modulation of volume holographic grating,” Appl. Phys. B 112(1), 67–72 (2013). [CrossRef]

, 20

20. X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013). [CrossRef] [PubMed]

], 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

31. C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt. 3(6), 504–513 (2001). [CrossRef]

]. However, it is not easy to acquire large size emulsion with high quality, so we choose photorefractive Fe:LiNbO3 crystal as the VHG material. Its saturated refractive index modulation is near this value [32

32. D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.) 8(2), 85–100 (1975). [CrossRef]

], most important, the photorefractive VHG can be of large size and high quality due to mature crystal growth technique. The background refractive index is n0 = 3.134. Other parameters are as follows: FWHM of the input pulse is Δτ = 100fs, central frequency is ω0 = 4π × 1014 rad/s and the corresponding central wavelength is λ0 = 1.5μm. Speed of light in vacuum is c = 3 × 108 m/s. The period of the volume grating is Λ = 7.3μm. According to the Bragg condition of VHG Λ = λ0/(2sinθr), the readout angle inside the VHG satisfies cosθr'=1(λ0n0)2.

Fig. 1 Temporal-spatial dynamics of the incident femtosecond pulse evolution inside the VHG. Intensities of (a) diffracted pulse Id (t, z) and (b) transmitted pulse It (t, z) with respect to time and thickness of the VHG.
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

30. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969). [CrossRef]

], 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

31. C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt. 3(6), 504–513 (2001). [CrossRef]

, 33

33. S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun. 215(4-6), 263–269 (2003). [CrossRef]

]. 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.

Fig. 2 Periodical energy oscillation occurs in the initial stage of diffraction when the VHG thickness is in the range of 0mm to 1.0mm. The energy periodically transfers from (a) transmitted pulse to (b) diffracted pulse and vice versa.
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

34. C. G. Shull, “Observation of Pendellösung fringe structure in neutron diffraction,” Phys. Rev. Lett. 21(23), 1585–1589 (1968). [CrossRef]

], then predicted and observed in second-order Bragg scattering of matter waves from a standing light wave acting as a thick grating [35

35. S. Dürr, S. Kunze, and G. Rempe, “Pendellösung oscillations in second-order Bragg scattering of atoms from a standing light wave,” Quantum Semiclass. Opt. 8(3), 531–539 (1996). [CrossRef]

], in atomic wave from standing light waves [36

36. C. Keller, J. Schmiedmayer, A. Zeilinger, T. Nonn, S. Durr, and G. Rempe, “Adiabatic following in standing-wave diffraction of atoms,” Appl. Phys. B 69(4), 303–309 (1999). [CrossRef]

]. The Laue diffraction formalism in X-ray diffraction from perfect crystals provides the theoretical explanation on this effect [37

37. B. W. Batterman and H. E. N. D. E. R. S. O. N. Cole, “Dynamical diffraction of X rays by perfect crystals,” Rev. Mod. Phys. 36(3), 681–717 (1964). [CrossRef]

]. 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

38. M. L. Calvo, P. Cheben, O. Martínez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pendellösung effect,” Phys. Rev. Lett. 97(8), 084801 (2006). [CrossRef] [PubMed]

]. Later, V. A. Bushuev and S. E. Svyakhovskiy predicted and observed the “Pendellösung” effect in diffraction of ultrashort pulse from linear photonic crystal [16

16. V. A. Bushuev, B. I. Mantsyzov, and A. A. Skorynin, “Diffraction-induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A 79(5), 053811 (2009). [CrossRef]

18

18. A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 115(1), 56–67 (2012). [CrossRef]

]. 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.

Fig. 3 Pulse splitting in the latter stage of diffraction of (a) transmitted pulse and (b) diffracted pulse in the VHG when the propagating depth is in the range of 1.5mm to 4mm.
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 Et1, Et2 and two diffracted sub-pulses Ed1, Ed2, exist in the VHG simultaneously. In the following we will demonstrate that one transmitted sub-pulse Et2 and one diffracted sub-pulse Ed2 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.

Fig. 4 (a) Temporal diffracted intensity when the refractive index modulation is fixed at 1.25 × 10−2 and the thickness of the VHG changes in the range of 3.5mm to 7mm; (b) Relation between pulse interval of the diffracted dual pulses and the thickness of VHG.
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 Et2 and diffracted sub-pulse Ed2 propagate in the same direction and form the output diffracted dual pulses. Meanwhile, the other pair of transmitted sub-pulse Et1 and diffracted sub-pulse Ed1 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.

Fig. 5 (a) Temporal diffracted intensity when the thickness is fixed at 7.8mm and the refractive index modulation changes in the range of 6.50 × 10−3 to 3.05 × 10−2. (b) Relation between the pulse interval and refractive index modulations of the VHG.
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 Et2 and diffracted sub-pulse Ed2, propagate the same distance d. If the group velocity of each pulse is fixed, the pulse interval can be written as
Δtpi=|dυtdυd|.
(12)
υt and υd represent the group velocity of two sub-pulses Et2 and Ed2 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 υi = c/ni, 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
Δtpi=Δneffdc.
(13)
where Δneff = | nt-nd |.

To compare with the estimated value of Fig. 5(b), substituting parameters of thickness d = 7.8mm and pulse interval Δtpi = 325fs into Eq. (13), we get Δneff = 1.25 × 10−2, which is consistent with the estimated values in Fig. 5(b). If substituting d = 6mm and Δneff = 1.25 × 10−2 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 Δneff = 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 n0 and the slower (or faster) pulse propagates in the middle of the grating fringe layer where the refractive index is the maximum n0 + Δn (or minimum n0-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

Δφ=2π=2πλΔnds.
(14)

Substituting the defined parameters of section 3 into Eq. (14), the spatial period of the interference fringe is acquired, ds = 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 ν1=(2k+1)π2andν2=(2k+3)π2, with k an integer. Substituting ν1 and ν2 into ν(ω,z)=ωΔnz2ccosθr', the distance of propagating depth z in the VHG between two adjacent maximums is ds = 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).

References and links

1.

Z. Zhao, X. Tong, and C. Lin, “Alignment-dependent ionization probability of molecules in a double-pulse laser field,” Phys. Rev. A 67(4), 043404 (2003). [CrossRef]

2.

C. Daniel, J. Full, L. González, C. Lupulescu, J. Manz, A. Merli, S. Vajda, and L. Wöste, “Deciphering the reaction dynamics underlying optimal control laser fields,” Science 299(5606), 536–539 (2003). [CrossRef] [PubMed]

3.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett. 96(5), 057403 (2006). [CrossRef] [PubMed]

4.

T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett. 86(25), 251103 (2005). [CrossRef]

5.

S. Noel, E. Axente, and J. Hermann, “Investigation of plumes produced by material ablation with two time-delayed femtosecond laser pulses,” Appl. Surf. Sci. 255(24), 9738–9741 (2009). [CrossRef]

6.

F. Bourquard, J. Colombier, M. Guillermin, A. Loir, C. Donnet, R. Stoian, and F. Garrelie, “Temporal pulse shaping effects on aluminium and boron ablation plumes generated by ultrashort pulsed laser ablation and analyzed by time- and space-resolved optical spectroscopy,” Appl. Surf. Sci. 258(23), 9374–9378 (2012). [CrossRef]

7.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).

8.

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express 11(1), 68–78 (2003). [CrossRef] [PubMed]

9.

J. L. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett. 16(1), 39–41 (1991). [CrossRef] [PubMed]

10.

G. Li, C. Zhou, and E. Dai, “Splitting of femtosecond laser pulses by using a Dammann grating and compensation gratings,” J. Opt. Soc. Am. A 22(4), 767–772 (2005). [CrossRef] [PubMed]

11.

E. Dai, C. Zhou, and G. Li, “Dammann SHG-FROG for characterization of the ultrashort optical pulses,” Opt. Express 13(16), 6145–6152 (2005). [CrossRef] [PubMed]

12.

J. Zheng, C. Zhou, and E. Dai, “Double-line-density gratings structure for compression and generation of double femtosecond laser pulses,” J. Opt. Soc. Am. B 24(4), 979–984 (2007). [CrossRef]

13.

B. Bai, C. Zhou, E. Dai, and J. Zheng, “Generation of double pulses in-line by using reflective Dammann gratings,” Optik (Stuttg.) 119(2), 74–80 (2008). [CrossRef]

14.

W. Liu, B. Bai, C. Zhou, S. Qu, E. Dai, and G. Li, “Generating femtosecond double pulses using Damman reflection grating,” Acta Phys. Sin. 56, 3292–3298 (2007).

15.

T. Wu, C. Zhou, J. Zheng, J. Feng, H. Cao, L. Zhu, and W. Jia, “Generation of double femtosecond pulses by using two transmissive gratings,” Appl. Opt. 49(24), 4506–4513 (2010). [CrossRef] [PubMed]

16.

V. A. Bushuev, B. I. Mantsyzov, and A. A. Skorynin, “Diffraction-induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A 79(5), 053811 (2009). [CrossRef]

17.

S. E. Svyakhovskiy, V. O. Kompanets, A. I. Maydykovskiv, T. V. Murzina, S. V. Chekalin, A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Observation of the temporal Bragg-diffraction induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A 86(1), 013843 (2012). [CrossRef]

18.

A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys. 115(1), 56–67 (2012). [CrossRef]

19.

Z. Gao, X. Yan, Y. Dai, X. Yang, and G. Ma, “Generation of femtosecond double pulse by adjusting the refractive Index modulation of volume holographic grating,” Appl. Phys. B 112(1), 67–72 (2013). [CrossRef]

20.

X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013). [CrossRef] [PubMed]

21.

Y. Ding, D. D. Nolte, Z. Zheng, A. Kanan, A. M. Weiner, and G. A. Brost, “Bandwidth study of volume holography in photorefractive InP:Fe for femtosecond pulse readout at 1.5 μm,” J. Opt. Soc. Am. B 15(11), 2763–2768 (1998). [CrossRef]

22.

X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.) 115(11-12), 512–516 (2004). [CrossRef]

23.

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol. 40(7), 906–911 (2008). [CrossRef]

24.

C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A 23(12), 3191–3196 (2006). [CrossRef] [PubMed]

25.

M. P. Hernández-Garay, O. Martínez-Matos, J. G. Izquierdo, M. L. Calvo, P. Vaveliuk, P. Cheben, and L. Bañares, “Femtosecond spectral pulse shaping with holographic gratings recorded in photopolymerizable glasses,” Opt. Express 19(2), 1516–1527 (2011). [CrossRef] [PubMed]

26.

L. A. Siiman, J. Lumeau, L. Canioni, and L. B. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett. 34(17), 2572–2574 (2009). [CrossRef] [PubMed]

27.

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009). [CrossRef] [PubMed]

28.

A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B 96(1), 71–77 (2009). [CrossRef]

29.

X. Yan, M. Qian, L. Gao, X. Yang, Y. Dai, X. Yan, and G. Ma, “Pulse splitting by modulating the buffer layer thickness of two-layer volume holographic grating,” Opt. Express 21, 31852–31861 (2013). [CrossRef] [PubMed]

30.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969). [CrossRef]

31.

C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt. 3(6), 504–513 (2001). [CrossRef]

32.

D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.) 8(2), 85–100 (1975). [CrossRef]

33.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun. 215(4-6), 263–269 (2003). [CrossRef]

34.

C. G. Shull, “Observation of Pendellösung fringe structure in neutron diffraction,” Phys. Rev. Lett. 21(23), 1585–1589 (1968). [CrossRef]

35.

S. Dürr, S. Kunze, and G. Rempe, “Pendellösung oscillations in second-order Bragg scattering of atoms from a standing light wave,” Quantum Semiclass. Opt. 8(3), 531–539 (1996). [CrossRef]

36.

C. Keller, J. Schmiedmayer, A. Zeilinger, T. Nonn, S. Durr, and G. Rempe, “Adiabatic following in standing-wave diffraction of atoms,” Appl. Phys. B 69(4), 303–309 (1999). [CrossRef]

37.

B. W. Batterman and H. E. N. D. E. R. S. O. N. Cole, “Dynamical diffraction of X rays by perfect crystals,” Rev. Mod. Phys. 36(3), 681–717 (1964). [CrossRef]

38.

M. L. Calvo, P. Cheben, O. Martínez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pendellösung effect,” Phys. Rev. Lett. 97(8), 084801 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(050.7330) Diffraction and gratings : Volume gratings
(320.5540) Ultrafast optics : Pulse shaping

ToC Category:
Holography

History
Original Manuscript: June 3, 2014
Revised Manuscript: July 8, 2014
Manuscript Accepted: July 16, 2014
Published: July 23, 2014

Citation
Xiaona Yan, Lirun Gao, Ye Dai, Xihua Yang, Yuanyuan Chen, and Guohong Ma, "Periodical energy oscillation and pulse splitting in sinusoidal volume holographic grating," Opt. Express 22, 18527-18536 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18527


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. Z. Zhao, X. Tong, and C. Lin, “Alignment-dependent ionization probability of molecules in a double-pulse laser field,” Phys. Rev. A67(4), 043404 (2003). [CrossRef]
  2. C. Daniel, J. Full, L. González, C. Lupulescu, J. Manz, A. Merli, S. Vajda, and L. Wöste, “Deciphering the reaction dynamics underlying optimal control laser fields,” Science299(5606), 536–539 (2003). [CrossRef] [PubMed]
  3. S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006). [CrossRef] [PubMed]
  4. T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett.86(25), 251103 (2005). [CrossRef]
  5. S. Noel, E. Axente, and J. Hermann, “Investigation of plumes produced by material ablation with two time-delayed femtosecond laser pulses,” Appl. Surf. Sci.255(24), 9738–9741 (2009). [CrossRef]
  6. F. Bourquard, J. Colombier, M. Guillermin, A. Loir, C. Donnet, R. Stoian, and F. Garrelie, “Temporal pulse shaping effects on aluminium and boron ablation plumes generated by ultrashort pulsed laser ablation and analyzed by time- and space-resolved optical spectroscopy,” Appl. Surf. Sci.258(23), 9374–9378 (2012). [CrossRef]
  7. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
  8. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express11(1), 68–78 (2003). [CrossRef] [PubMed]
  9. J. L. Chilla and O. E. Martinez, “Direct determination of the amplitude and the phase of femtosecond light pulses,” Opt. Lett.16(1), 39–41 (1991). [CrossRef] [PubMed]
  10. G. Li, C. Zhou, and E. Dai, “Splitting of femtosecond laser pulses by using a Dammann grating and compensation gratings,” J. Opt. Soc. Am. A22(4), 767–772 (2005). [CrossRef] [PubMed]
  11. E. Dai, C. Zhou, and G. Li, “Dammann SHG-FROG for characterization of the ultrashort optical pulses,” Opt. Express13(16), 6145–6152 (2005). [CrossRef] [PubMed]
  12. J. Zheng, C. Zhou, and E. Dai, “Double-line-density gratings structure for compression and generation of double femtosecond laser pulses,” J. Opt. Soc. Am. B24(4), 979–984 (2007). [CrossRef]
  13. B. Bai, C. Zhou, E. Dai, and J. Zheng, “Generation of double pulses in-line by using reflective Dammann gratings,” Optik (Stuttg.)119(2), 74–80 (2008). [CrossRef]
  14. W. Liu, B. Bai, C. Zhou, S. Qu, E. Dai, and G. Li, “Generating femtosecond double pulses using Damman reflection grating,” Acta Phys. Sin.56, 3292–3298 (2007).
  15. T. Wu, C. Zhou, J. Zheng, J. Feng, H. Cao, L. Zhu, and W. Jia, “Generation of double femtosecond pulses by using two transmissive gratings,” Appl. Opt.49(24), 4506–4513 (2010). [CrossRef] [PubMed]
  16. V. A. Bushuev, B. I. Mantsyzov, and A. A. Skorynin, “Diffraction-induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A79(5), 053811 (2009). [CrossRef]
  17. S. E. Svyakhovskiy, V. O. Kompanets, A. I. Maydykovskiv, T. V. Murzina, S. V. Chekalin, A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Observation of the temporal Bragg-diffraction induced laser pulse splitting in a linear photonic crystal,” Phys. Rev. A86(1), 013843 (2012). [CrossRef]
  18. A. A. Skorynin, V. A. Bushuev, and B. I. Mantsyzov, “Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: diffraction induced splitting, selective compression, and focusing of pulses,” J. Exp. Theor. Phys.115(1), 56–67 (2012). [CrossRef]
  19. Z. Gao, X. Yan, Y. Dai, X. Yang, and G. Ma, “Generation of femtosecond double pulse by adjusting the refractive Index modulation of volume holographic grating,” Appl. Phys. B112(1), 67–72 (2013). [CrossRef]
  20. X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express21(6), 7560–7569 (2013). [CrossRef] [PubMed]
  21. Y. Ding, D. D. Nolte, Z. Zheng, A. Kanan, A. M. Weiner, and G. A. Brost, “Bandwidth study of volume holography in photorefractive InP:Fe for femtosecond pulse readout at 1.5 μm,” J. Opt. Soc. Am. B15(11), 2763–2768 (1998). [CrossRef]
  22. X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.)115(11-12), 512–516 (2004). [CrossRef]
  23. B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008). [CrossRef]
  24. C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A23(12), 3191–3196 (2006). [CrossRef] [PubMed]
  25. M. P. Hernández-Garay, O. Martínez-Matos, J. G. Izquierdo, M. L. Calvo, P. Vaveliuk, P. Cheben, and L. Bañares, “Femtosecond spectral pulse shaping with holographic gratings recorded in photopolymerizable glasses,” Opt. Express19(2), 1516–1527 (2011). [CrossRef] [PubMed]
  26. L. A. Siiman, J. Lumeau, L. Canioni, and L. B. Glebov, “Ultrashort laser pulse diffraction by transmitting volume Bragg gratings in photo-thermo-refractive glass,” Opt. Lett.34(17), 2572–2574 (2009). [CrossRef] [PubMed]
  27. A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A26(1), 135–141 (2009). [CrossRef] [PubMed]
  28. A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009). [CrossRef]
  29. X. Yan, M. Qian, L. Gao, X. Yang, Y. Dai, X. Yan, and G. Ma, “Pulse splitting by modulating the buffer layer thickness of two-layer volume holographic grating,” Opt. Express21, 31852–31861 (2013). [CrossRef] [PubMed]
  30. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J.48(9), 2909–2947 (1969). [CrossRef]
  31. C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt.3(6), 504–513 (2001). [CrossRef]
  32. D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.)8(2), 85–100 (1975). [CrossRef]
  33. S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003). [CrossRef]
  34. C. G. Shull, “Observation of Pendellösung fringe structure in neutron diffraction,” Phys. Rev. Lett.21(23), 1585–1589 (1968). [CrossRef]
  35. S. Dürr, S. Kunze, and G. Rempe, “Pendellösung oscillations in second-order Bragg scattering of atoms from a standing light wave,” Quantum Semiclass. Opt.8(3), 531–539 (1996). [CrossRef]
  36. C. Keller, J. Schmiedmayer, A. Zeilinger, T. Nonn, S. Durr, and G. Rempe, “Adiabatic following in standing-wave diffraction of atoms,” Appl. Phys. B69(4), 303–309 (1999). [CrossRef]
  37. B. W. Batterman and H. E. N. D. E. R. S. O. N. Cole, “Dynamical diffraction of X rays by perfect crystals,” Rev. Mod. Phys.36(3), 681–717 (1964). [CrossRef]
  38. M. L. Calvo, P. Cheben, O. Martínez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental detection of the optical Pendellösung effect,” Phys. Rev. Lett.97(8), 084801 (2006). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4 Fig. 5
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited