Utilizing multiplexed volume holography, a single optical element, enabling to shape structural variants of non-diffracting Airy wavefronts, from one-dimensional Airy mode to unconventional Airy modes such as vortex Airy and quad Airy modes, has been experimentally realized. Here, beam shaped angularly multiplexed volume holographic gratings (AMVHGs) are recorded in PQ: PMMA photopolymer, where five different spatial wavefronts of Airy beams have been sequentially recorded, for simultaneous reconstruction of different Airy modes, by a conventional Gaussian beam. Spatial and spectral mode selective properties of AMVHGs are demonstrated by narrow-band as well as by broadband light source. In addition, through wavelength degeneracy property, the maximum sensitivity wavelength of blue (488 nm) is used for recording in PQ: PMMA, but the AMVHGs are operated at a broad wavelength band of interest, all the way to longer wavelength in near infrared (850 nm). The K-sphere representation is used to explain the spectral properties of AMVHGs.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The research field of structured light is growing very fast and new light patterns are coming very often . These structured light patterns are desirable in vast variety of applications such as microscopy, imaging, communications and material processing . In last two decades, techniques to produce structured light have evolved rapidly and many efforts have been made in making optical elements which can precisely control spatial properties of light. With the availability of high resolution spatial light modulating devices, alteration, modulation, and tailoring of spatial properties of light field is straightforward [3,4]. However, despite being most commonly used method, high cost, pixelated structures, unwanted diffraction orders, wavelength dependent aberrations, and phase shifts are major issues related to such spatial light modulating devices. Furthermore, in designing the compact optical systems, simple optical elements and procedures are always desired and advantageous.
Gaussian beams are well-known fundamental transverse modes, commonly obtained from most of the laser cavities. On the other-hand unconventional beam shapes are light fields that are the solutions of paraxial wave equations described by special mathematical functions. The best examples include, Airy, Inc-Gaussian, Mathieu, parabolic, and Pearcey beams. The preservation of beam structures during propagation up to longer distance is often referred to as phenomenon of non-diffraction. Airy beam is an important member among non-diffracting beams, and has been attracted much interest due to their excellent property of self-acceleration, self-healing, and diffraction free region along its curved path. The prediction of Airy wavepackets was first made by the Berry and Balazs for the solution of the Schrodinger equation for free particles . A decade ago first experimental demonstration of the optical Airy beam was done by Silvologlou et al . The self-bending characteristics of Airy beams during propagation has changed the perception of propagation properties of optical beams, and many of its variants have been proposed and experimentally demonstrated [7–12]. Airy beams have been used in wide range of applications, such as optical routing , light sheet microscopy , plasma channels , light bullets , acoustics  and particle clearing and optical manipulation of particles [18,19]. Airy beams have also been studied in the context of the matter waves , spin waves , water waves  and temporal and spatiotemporal pulses .
Unconventional beam shapes, like Airy beams, can be obtained from intracavity or extracavity laser methods based on the phenomena of interference, diffraction, and polarization [3,24–26]. Many of the extracavity methods uses special diffractive optical elements designed by computer generated holograms and electron beam writing. In principle, by the above described techniques arbitrary light fields can be generated, but in practice the quality of optical field is limited by the number of phase or amplitude steps created by electron beam writing. In general, due to technological constrains design and implement of optical elements for multiplexing structures from the electron beam writing is difficult. Furthermore, the generated modes from such optical elements are wavelength specific and mode purity is limited. We believe that volume holography can solve many of the existing problems and it is promising candidate to realize optical elements to simultaneously generate structured light patterns. Multiple diffraction orders are the most common characteristics of thin gratings. On the contrary, single dominant diffraction order is defining feature of volume holograms. In addition, multiplexing capability offers great advantage to make volume holographic techniques unique. Volumetric diffraction, offered by thick holographic gratings, shows significantly great performance over thin gratings, and the inherent special features, including Bragg wavelength degeneracy, high angular selectivity, and multiplexing ability together with excellent wavefront reconstruction ability, make volume holography distinctive among other techniques [27–29]. Apart from the use of volume holograms for data storage application, volume hologram has been utilized in advanced applications, such as hyperspectral imaging , and multidimensional imaging . Volume holographic gratings have also been used in astronomical instruments for high resolving power characteristics, resulting from high spatial frequencies recording capabilities . Special optical elements based on volume holographically recorded gratings have smooth and uniform refractive index variations inside the medium. The diffraction behavior of volume gratings can be explained through Born approximation . This feature of volume holograms results in less scattering and high mode purity as compared to surface etched gratings obtained from the electron beam writing. The success of the volume holographic gratings highly depends on the function of the photosensitive materials used in their fabrication. A variety of materials have been tried and tested in the past. The advancement in the photopolymer fabrication technology has also eased the fabrication of volume holographic substrates .
A variety of methods to generate non-diffraction beams of Airy family have been experimentally demonstrated. However, wavelength tunability and simultaneous generation of various Airy modes from a single optical element have not been dealt so far. In this study, our purpose is to apply volume holographic multiplexing for simultaneous wavefront generation of the Airy family, and to study spatial and spectral properties of the Airy beams. Here, we utilize two specific features of volume holography: single diffraction order under Bragg diffraction condition, and multiplexing ability to achieve simultaneous generation of five different Airy modes (i.e. one dimensional Airy beam, Airy beams, dual Airy beam, vortex Airy beam, and quad Airy beams) in the diffraction fan out. Our experimental setup for recording angularly multiplexed volume holographic gratings (AMVHGs) is configured such that all different Airy beams can be simultaneously reconstructed using a conventional Gaussian incident light. To best of our knowledge there are no optical elements that can provide the multiple Airy beams from the single optical components. The volume hologram, consisting of AMVHGs, presented here performs significant tasks since it can act as a spatial mode shaper, a spectral filter, and it can also be used as Fourier filter for imaging. Because of Bragg degeneracy, another advantage is that to obtain beam shapes neither a high quality Gaussian beam nor a specific wavelength or polarization selection is required. Spectral properties of the AMVHGs is studied in detail. In addition, we experimentally demonstrate different mode generation of Airy family from a variety of light sources, including both narrow-band and broad-band fashion, from blue (λ = 450nm) to near infrared (λ = 850 nm).
Paraxial wave equations can have variety of solutions described by the special functions [1–7]. The formation of Airy beams can be explained by catastrophic theory . The spatial intensity distribution of Airy beams may have many structural variants. Here, we discuss some of the simplest cases.
2.1 Airy beams
The physical origin and interesting propagation behavior of Airy beams is attributed to the cubic phase profile. The formation of the Airy beams can be considered as the result of interference of multiple waves that have special phase relationship with each other, such that the interference pattern follows parabolic propagation path. The expressions for the finite energy accelerating Airy beams are obtained from the normalized paraxial wave equation ,Eq. (1) is given by
The Fourier transform of Eq. (2) gives
2.2 One dimensional Airy beams
One dimensional Airy beams have cubic phase modulation only in single direction. These beams have been used in variety of applications including microscopy. At z = 0, a simple expression for the one dimensional Airy beam can be written as,
2.3 Vortex Airy beams
The light beams with phase singularity and screw dislocations are known as optical vortices . Optical vortices are characterized by zero intensity, where, the phase is undefined. The transverse intensity distribution of optical vortices looks like a donut shape with a dark core at center of the beam. Vortex beams have helical shaped wavefronts with azimuthal phase dependence of the form exp(ilφ), which creates phase singularity at the center of the beam. The topological charge (l) is an integer that decides the handedness of rotation of the spiral phase around the vortex core. The spiral phase distribution provides the property of orbital angular momentum to the beam. Optical vortices can be embedded into various kind of beams and the vortices generally follow the propagation dynamics of envelope beams. A vortex Airy beam can be obtained by providing spiral phase to the conventional Airy beams [33,34]. Various experimental and theoretical studies on the vortex Airy beams have been performed. Here, we used a spiral phase with the unit topological charge. The expression of optical field for finite energy vortex Airy beam can be expressed as
2.4. Dual Airy beams
Dual Airy beam is composed of two 2D Airy beams accelerating in opposite direction. The asymmetric transverse intensity distribution of Airy beam is responsible for their acceleration. It has been shown that Dual Airy beams are associated with the symmetric transverse patterns and better self-healing property than single Airy beams . Dual Airy beams can be obtained from the superposition of two cubic phase profiles of the conventional Airy beams. The symmetrical dual Airy beam shapes may be particularly useful in the optical trapping , microscopy experiments .
The expression for the dual airy beams can be expressed as
The main lobe of the two Airy beams coincides but side lobes are situated in symmetrically opposite direction.
2.5. Quad Airy beams
These beams are called quad airy beams because they appear as the combination of the four Airy beams at the four corners of a square. The formation and the propagation of these beams can be explained by geometrical optics and diffraction catastrophe theory . It is theoretically and experimentally shown that each part of the beam component as well as the whole beam possess the Airy beam like properties. As comparison to the single two dimensional Airy beams, quad Airy beams are axially symmetric. To experimentally generate quad Airy beams using a phase only spatial light modulator (SLM), a (3/2) phase mask can be used. The quad Airy beam does not need any Fourier transform of the diffracted light coming from the cubic phase mask. One of the main properties of the quad Airy beam is that they have even number of the main lobes. Airy beams have been generated using the ¼ quadrant phase only patterns . It is shown that, by considering geometrical model and centro-symmetrical phase distribution of the whole quadrant a special Airy beam with four main lobes can be produced. Here, we are following the method given in reference  for creating quad airy beam. The phase function to create the phase mask for quad airy beam can be written as
Simulation results of intensity profiles of various Airy beam shapes, are shown in Fig. 1. The nondiffracting regions and the self-bending of the Airy family beams depend on beam parameters s and ξ. By adjusting these beam parameters, parabolic path and nondiffracting regions can be altered. Due to the requirement of the infinite extent of these beams experimental realization of ideal Airy beams is not possible. Practically these beams can be obtained by modulating the Gaussian beam using cubic phase mask and optical Fourier transform.
3. Experimental setup
Angularly multiplexed volume holographic gratings (AMVHGs) creates three dimensional optical structures in the form of refractive index modulation within photosensitive materials. AMVHGs offer highly selective properties in spatial-spectral fashion due to Bragg degeneracy as comparison to conventional thin gratings [27,28]. We have used PQ: PMMA photopolymer for recording of beam shaped multiplexed volume holographic gratings, because it provides many advantages, such as low refractive index modulation, high sensitivity, high stability, and high multiplexing capability to record higher spatial frequencies, with extremely low shrinkage which makes them suitable for making optical elements [27,28]. In addition, PQ: PMMA provides high angular and wavelength selectivity . The thickness of AMVHGs in PQ: PMMA (~1.8 mm) in our experiment was thick enough to avoid the cross-talk between different modes. Photo sensitivity of PQ: PMMA is high at wavelength (λ = 488 nm). We recorded all the multiplexed gratings using this wavelength; however, AMVHGs can be operated at a broad wavelength band of interest because of Bragg wavelength degeneracy.
Recording parameters such as the threshold fluence (energy/area) to initiate photo-chemical reaction and the exposure time greatly influence the diffraction efficiency. By controlling these two parameters together with the photopolymer fabrication parameters, high diffraction efficiencies can be obtained. In the previous study, a detailed investigation has been performed for the fabrication and recording process of multiplexed holograms in PQ: PMMA photopolymer . Here, we are following the same procedure to perform our experiments. The photopolymer PQ: PMMA substrate is formed by the methyl methacrylate (MMA), 2,2- azo-bis-isobutyrolnitrile) (AIBN), and phenanthrenquinone (PQ) in the weight ratio of 100:0.5:0.7. PQ and the MMA chemicals are purchased from Aldrich chemicals whereas, AIBN was purchased from the Showa company. The steps involved in the preparation of the PQ: PMMA is mentioned in reference . Minimum fluence (energy/area) required to initiate the photochemical reaction for our substrate is around 500 mJ/cm2. However, the optimized parameters for the enhanced diffraction efficiency suggest that the higher fluence and low exposure time provides the better efficiencies. In our case, we chose fluence for all the beams above 900 mJ/cm2 and the exposure time of 15 seconds. The beam diameter for reference and signal beam is 10 mm. Figure 2 illustrates an interferometric setup for recording AMVHGs. The recording setup is designed with a phase only LCSLM in its signal arm to generate desired modes. To record all the multiplexed grating, we used a Gaussian beam from an Ar+ ion laser emitting light at λ = 488nm. The direct laser beam is split into two beams with equal intensity using a beam splitter BS1.
The reflected beam from SLM is phase modulated and optically Fourier transformed through a lens (FT lens) except for the quad Airy beam which does not require Fourier transform. Interference patterns between desired Airy mode and Gaussian beam was recorded in PQ: PMMA. The incidence angle of the signal beam, was varied by a mirror (M1) mounted onto the rotation stage to provide a different angle to the signal beam while sequentially recording AMVHGs. The advantage of recording angularly multiplexed holograms in this way is that during hologram reconstruction a single Gaussian beam is sufficient for simultaneous generation of all the recorded multiplexed Airy family beams. In order to keep the beam overlapping in the same pupil during angular multiplexing an optical relay lens system  is used as shown in Fig. 2.
4. Experimental results
A schematic diagram of reconstruction process for simultaneous generation of five different Airy beams from AMVHGs is shown in Fig. 3. The photograph of experimentally fabricated PQ: PMMA based volume hologram with AMVHGs is shown in Fig. 3(a). The diameter of the volume hologram is 20 mm and thickness is 1.8 mm. A Gaussian incident beam at wavelength λ = 488nm obtained from Ar+ ion laser is used for the reconstruction. Experimental results for intensity profiles of five different Airy beams, namely one-dimensional, Airy, vortex Airy, dual airy, and quad Airy, are shown in Fig. 3(b). The diffraction beam fan-out of five airy beams from the AMVHGs separated by a diffraction angle around 10 degrees (∆Φ = 10°). It is important to notice that an incident Gaussian beam simultaneously produces five different kinds of spatial Airy modes using a volume hologram, which consists of AMVHGs and behaves as a fixed optical element such as a mirror or lens in a conventional imaging system. The quality of all generated modes solely depends on the quality of the signal beam during recording and hence, a very high quality beam shapes can be obtained in our approach. Figure 2(b) and Fig. 3(b) shows the signal airy beam obtained from SLM matches with the reconstructed beam obtained from the AMVHGs. The number of side lobes which corresponds to higher spatial frequencies are recorded and reconstructed well. The small difference which appears in these results is due to the difference of the recording and reconstruction plane.
The experimental results in Fig. 3 clearly show the advantage of AMVHGs method in beam shaping for high dimensional spatial mode multiplexing. To best of our knowledge this is the first attempt for simultaneous generation of variety of Airy modes from the single optical element by single input beam. This photopolymer based volume hologram is single, fixed optical elements, and it is compact, robust, and straightforward way to produce high quality beam shapes.
The Bragg wavelength degeneracy of volume holograms allows use of AMVHGs for multi-wavelength operations . Figure 4 shows reconstructed Airy family beams under different operation wavelengths. Airy beams in blue (λ = 450nm) and green color (λ = 515nm) were obtained from an Ar + laser by tuning cavity of laser, while reconstructed beams in red color (λ = 633nm) were obtained from a He-Ne laser. To further investigate spectral property of AMVHGs, we utilized a pulsed laser operating in infrared regions of spectrum with three longer wavelengths λ = 680 nm, λ = 780 nm, and λ = 850 nm.
Experimental results for the pulsed light operation of different wavelengths are shown in Fig. 5. From this figures it is clear that change of operation wavelengths or light sources does not affect mode quality, which is one of the major limitations in many of beam shaping techniques. Moreover, a white light one-dimensional Airy beam, obtained from incoherent white light LED source (LIUCWHA, Thorlabs), has been used to show the broadband operation. Spectra of incoherent broadband light emitted from the LED source is shown in Fig. 6(a). The red, blue and green wavelength bands selected from the grating are shown in Fig. 6(b). AMVHGs in such incoherent condition acts as a spectral filter with spatial mode selection properties. Schematic diagram of reconstruction setup is shown in Fig. 7(a), and intensity distribution of white light one-dimensional Airy beam is shown in Fig. 7(b). Experimental results show that beam shapes remains intact even for the incoherent white light source. At present in our experiment Airy beam shapes preserve their broadband property as long as different spectral components are overlapped. By using additional prism or a grating to compensate the dispersion a white light beam shape can be obtained up to longer distances .
The spectral components obtained from single slit diffraction from white light LED source is shown in Fig. 7(c). The red, blue, and green beam shaped spectral components of one dimensional white light Airy beam obtained from AMVHGs are shown in Fig. 7(d)-7(f).
Experimental results clearly demonstrate the Bragg-wavelength degeneracy of AMVHGs can be applied to form a spectral-spatial mode shaper. In addition, it is important to mention that the single optical element, consisting of AMVHGs, is capable of performing narrow as well as the broad band operation without change in the spatial mode properties of the beam. Present results in Figs. 6 and 7 show solid evidence of distinct properties of volume holography based beam shaping for both incoherent spatial-spectral mode shaping and spectral filtering. Our findings suggest that AMVHGs can be used for narrowband as well as the broadband beam shaping for coherent as well as incoherent light sources.
The diffraction efficiency of volume holographic gratings is customarily calculated using Kogelnik’s couple wave theory [27,28]. The peak diffraction efficiency depends on the angle of incidence and wavelength of the probe beam. The angular selectivity curve for the grating is obtained by measuring diffracted power as a function of rotation angle of incident reconstructed beam. Figure 8 shows both theoretical and experimental angular selectivity curve obtained using the reference beam used to record holograms. Simulated data is obtained using Kogelniks couple wave theory , full-width-of-half-maximum (FWHM) of angular selectivity curve is ~0.04°. Results in Fig. 8 show the high angular selective property AMVHGs. The angular and wavelength degeneracy of our holograms can be tuned by changing the thickness of the grating.
The Bragg wavelength degeneracy property responsible for the multiple wavelength operation of AMVHGs can be explained using K-sphere [7,8,39]. A schematic diagram of K-sphere is shown in Fig. 9, where, different illuminating wavelengths correspond to different radii. Light with longer wavelength produces smaller radius on K-sphere and vice versa. Under Bragg-matched condition Bragg diffraction occurs for arbitrary combination of Bragg-matched wavelengths and angles. Change in the illumination wavelength results in the corresponding change in the diffraction angle. The diffraction angle for the different wavelengths can be obtained by , where Λ is the period of refractive index modulation, L is the thickness of volume hologram, λ is wavelength of reconstruction light is a diffraction angle corresponding to a Bragg-matched wavelength. Therefore, under a white light reconstruction in the wide angle illumination of AMVHGs, Bragg matched condition is simultaneously satisfied for the all the spectral components.
It is important to note that the difference between the SLM based beam shaping and beam shaped volume holograms lie in terms of their diffraction properties and regime. Due to Bragg wavelength degeneracy and angular selectivity volume phase gratings inherently capable of multi-spectral operation that can be advantageous in situations where, conventional beam shaping methods are difficult. The spectral properties of the generated beam can easily be controlled by the rotation of the holograms. Due to the different orientation of the grating vectors of each grating the diffraction efficiency of all the beams will not be maximum simultaneously when they are reconstructed using wavelength other than the recording wavelength. However, maximum diffraction efficiency for particular beam can be obtained by fine tuning of the angle of incidence of the readout beam. The M/#  can reach ~2 for current multiplexing capability. We are using LCOS SLM in our volume hologram recording setup to generated beam shapes. It has a limitation in terms of power handling capacity. To get higher diffraction efficiency we need a higher fluence and smaller exposure time. It was earlier experimentally demonstrated that the M/# of multiplexed gratings can go up to ~3 [35,36]. One of the possible solution is to use a digital micro mirror device for generating the beam shapes that have higher power handling capacity then the LCOS SLM. The single diffraction order of volume phase gratings can help in making achromatic optical element by avoiding unnecessary complications from the unwanted diffraction orders and the multiple spectral components at the same time. Flexibility of operational wavelengths of AMVHGs may increases their functionality in integration in microscopic systems. Due to its simplicity and compactness these gratings can directly be used in the illumination path of all the wide-field imaging methods like, bright-field, phase contrast, structured illumination and light sheet microscope. These gratings can also provide the way for high dimensional spatial mode multiplexing . These AMVHGs can also be used as spatial spectral feedback device in the laser cavities to directly obtain the spatial modes [41–44].
In conclusion, we have experimentally demonstrated simultaneous generation of five different kinds of Airy beams from a single optical element. Angular multiplexing property of volume holography was used to record five multiplexed gratings in (PQ: PMMA). Both coherent and incoherent light sources, including a CW laser, white light LED, and a pulsed laser, were used to show spectral property of AMVHGs, at a broad wavelength band of interest, from shorter visible color (λ = 450 nm) all the way to longer wavelength in near infrared (λ = 850 nm). The AMVHGs are naturally capable of narrow band as well as the broadband beam shaping operations. Present methods provide many advantages over conventional and previously described methods of beam shaping. The use of AMVHGs in broad spectrum range from visible to infra-red may be useful in wide range of applications. The spectral properties discussed here may also find important uses in the incoherent interferometry, wavelength tunable laser beam shaping, and optical coherence tomography. Present results open up the possibility of high dimensional multiplexing of spatial modes using volume holography for a variety of applications.
Ministry of Science and Technology Taiwan (105-2628-E-002-008-MY3, 106-2221-E-002 −157 -MY3).
The authors gratefully acknowledge support from National Taiwan University and Jasper Display Corporation (NTU-107L7728 NTU-107L7807, and JD4704).
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