## Abstract

We experimentally demonstrate spatial mode multiplexing of optical beams using multiplexed volume holographic gratings (MVGHs) formed in phenanthrenquinone-poly (methyl methacrylate) (PQ-PMMA) photopolymer. Multiple spatial modes of Laguerre-Gaussian (LG) beams are recorded at the same pupil area of a volume hologram resulting in MVHGs, for simultaneous reconstruction of spatial modes. In addition, a helical phase beam, a non-diffracting beam with conical phase profile, and a parabolic non-diffracting beam with cubic phase profile have also been simultaneously recorded and reconstructed from MVHGs. Utilizing Bragg wavelength degeneracy property of volume hologram these multiplexed modes are reconstructed at multiple wavelengths ranging from blue (450nm) to red (635). Due to combined effect of three-dimensional pupil, Bragg wavelength degeneracy, angular selectivity, together with spatial mode properties these, MVHGs can act as spatial mode filter with spectral filtering property. Advantages of volume holography in beam shaping are discussed. Multiple first diffraction orders with desired beam shapes obtained from the single optical element (i.e. a volume hologram with MVHGs) may find important applications in optical communication experiments, and in volume holographic imaging and microscopy. Experimental results show solid evidence that MVGHs in beam shaping provide a simple, compact, single element, and direct way to multiplex spatial modes.

© 2017 Optical Society of America

## 1. Introduction

Optical spatial modes are solution of the wave equation which describes field distributions transverse to the direction of propagation [1]. An exact shape of spatial modes depends on the boundary conditions setup by optical systems, and accordingly various kinds of laser modes can be experimentally obtained. Some well-know and commonly used spatial modes of laser beams includes Gaussian, Hermit-Gaussian(HG), Laguerre-Gaussian (LG), Bessel-Gaussian (BG), and Airy-Gaussian(AG) modes. Light beams with spiral phase fronts are known from decades, and they have been generated and detected by various means and studied in different contexts [1-2]. These beams are also known as optical vortices and can be described by LG or BG modes. Spatial modes of spiral phase beams carry orbital angular momentum (OAM) of *l*ħ per photon where, *l* is azimuthal index or topological charge of beam that indicates the rate of azimuthal phase variation or twist in wavefronts. Due to infinite dimensionality and orthogonality of basis sets, spatial modes carrying orbital angular momentum are considered as a new degree of freedom for light to code and decode classical and quantum information [1–5]. Applications of OAM states in secure information transfer require generation and detection of large number of eigen-states. There are many methods of generation of OAM states; however, most of them cannot produce multiple states simultaneously. Conventionally, wavelength and polarization have been used in multiplexing and demultiplexing techniques to increase data transfer capacity in optical communication. Use of spatial modes together with the other multiplexing techniques can further increase information transfer capacity [3–5].

There exist various methods for wavefront control and laser beam shaping, but only few methods have storage capability. Holography based beam shaping is a classical topic of research in this field, but mostly restricted to thin holograms. It has been used to shape beams of various kinds of particles ranging from photons in different regions of electromagnetic spectrum to fundamental particle beams like electron and neutron [6-7]. Holographic gratings can be classified as thin or thick grating either according to their diffraction regimes or according to angular and wavelength selective property [8]. A thin grating diffracts in the Raman-Nath regime and produces multiple diffraction orders, while a volume hologram is essentially a thick grating and diffracts in the Bragg regime. Bragg diffraction occurs due to significant thickness in grating that cause mutual interference between diffracted fields inside medium and certain features commonly seen in the thin grating, like higher diffraction orders are non-propagating and a single diffraction order is obtained. Due to inherent three-dimensional (3D) pupil and Bragg selectivity property [9], MVHGs have been considered as ideal candidate for advanced applications like, data storage, neural networks optical correlators and optical communications. Capacity to store large amount of data is the strength of volume holograms and that can be utilized for spatial mode multiplexing in OAM states. Recently, volume holography has been used in a variety of microscopic techniques, and multidimensional imaging [9–14]. We believe applications of volume holography in beam shaping has not been explored in detail yet [15].

The combination of computer generated holography (CGH) and spatial light modulating devices is a versatile tool for producing desired optical field with homogenous or inhomogenous spatial distribution of amplitude, phase and polarization [5]. For instance, some pioneer results in generating singular beams are obtained from computer generated holograms [16]. One of the limitations in use of spatial light modulators (SLMs) is that SLMs produce multiple diffraction orders, and a desired order needs to be separated using spatial filtering setup that increases system complexity due to adding additional required optical components. Unlike volume holography, CGH using SLMs does not have multiplexing capability, angular, and wavelength selective property because SLM based CGH behaves as thin gratings, and thus it is generally adapted for fixed wavelengths. Moreover, multiplexing based on thin gratings or SLMs may degrade beam quality of OAM states.

Motivated by the above described facts, the objective of the present work is experimental demonstration of spatial mode multiplexing of different OAM states using MVHGs. First different spatial modes are obtained by a computer generated phase mask projected onto a SLM, and then they are recorded onto a photosensitive thick medium, resulting in a volume phase grating of different shapes. We used PQ-PMMA as the recording medium due to its excellent and well-studied properties [17]. In our experimental demonstration, multiple LG beams with different topological charge are angularly multiplexed to form MVHGs onto same pupil area in a thick (2 mm) PQ-PMMA recording material. In addition, our results demonstrate both recording and simultaneous reconstruction of spatial modes with and without property of orbital angular momentum together with MVHGs, at a broad wavelength band, all the way from blue 450 nm to red 635 nm. To best of our knowledge using multiplexed volume holography through MVHGs, for simultaneous generation of multiple beam shapes from a fixed, single holographic optical element has not been dealt so far.

## 2. Theory

#### 2.1 Laguerre-Gaussian beam

To study spatial mode multiplexing using MVHGs we used three different kinds of scalar beams, a LG, BG and AG. A simple expression of scalar LG beam can be written as [18],

*E*is constant,

_{0}*r*is radius,

*φ*is azimuthal angle,

*z*is the axial distance,

*ω*is the minimum beam radius at

_{0}*z*= 0,

*R(z) = (z*z is the radius of curvature of the wavefront,

^{2}+ z_{0}^{2})/*z*=

_{0}*kω*is the

_{0}^{2}/2*Rayleigh*range,

*k*is the wave number

*k = 2π/λ*,

_{0}*(2p + |l| + 1)η(z)*is the Gouy phase shift with

*η(z) = arctan(z/z*and $\omega \left(z\right)={\omega}_{0}\sqrt{1+{\left(z/{z}_{0}\right)}^{2}}.$ ${L}_{p}^{\left|l\right|}$ Laguerre polynomial with radial index

_{0}),*p*and azimuthal index

*l*. Calculated intensity and phase profile of LG modes are shown top row of Figs. 1 (a) and (b). The size of doughnut shape intensity depends on the topological charge

*l*of beam. In present study

*p*= 0 is used.

#### 2.2 Bessel-Gaussian beam

An ideal Bessel beam cannot be realized due to infinite energy content in it. For all the practical purpose a Bessel-Gauss beam was used. These beams are known for their characteristic feature of nondiffracting and self-healing properties. The general expression of a linearly polarized Bessel-Gaussian beam of order *n* is given by [19]

*Q(z) = exp*{

*-iβ*[

^{2}z/*2k*(

*1 + iz*/

*z*)]},

_{0}*β*is related to transverse wave vector and decides the number of rings in the beam profile. By changing transverse wave vector coordinates of Bessel-Gaussian beam nondiffracting region can be altered. Calculated intensity and phase profile of BG modes are shown in middle row of Figs. 1(a) and 1(b).

#### 2.3 Airy beam

Together with the nondiffracting property Airy beams are known for their distinctive features of accelerating and parabolic path. The airy beam can be formed by cubic phase modulation of the incident Gaussian beam and Fourier transform of the resultant field will generate Airy beam. The angular Fourier spectrum of exponentially truncated finite-energy Airy beams is proportional to the product of a Gaussian beam and cubic phase is given by [20, 21]

*Ai*is Airy function.

*s = x/x*represents a dimensionless transverse coordinate, a positive parameter less than 1,

_{0}*x*is an arbitrary transverse scale, and

_{0}*ξ = z/k x*is a normalized propagation distance. Calculated intensity and phase profile of Airy modes are shown bottom row of Figs. 1(a) and 1(b). The nondiffracting regions of Airy beams depend on beam parameters

^{2}*s*and

*ξ*. By adjusting these parameters, a parabolic path and nondiffracting regions can be altered.

## 3. Experimental setup

The advantage of high wavelength and angular selectivity of volume hologram offers the possibility of recording MVHGs in the same pupil area. Figure 2 shows an interferometric setup to record multiple angularly multiplexed MVHGs corresponding to different spatial modes. In contrast to the conventional angular multiplexing techniques [22], where a hologram is rotated for each exposure to record multiplexed hologram, here we utilize three signal beams coming at recording medium from different angles with respect to signal reference beam. The advantage of this method is that during hologram reconstruction three beams are simultaneously generated from single incident beam without rotating hologram. We followed hologram substrate preparation and the hologram recording parameters used by Luo et. al in [17]. The thickness of hologram is 1.8 mm and the size of hologram pupil is 2 mm in diameter. The beam power was adjusted according to exposure time and energy density requirement to initiate photochemical reaction in PQ-PMMA. To generate signal and reference beam, a Gaussian beam coming from argon ion laser (Ar^{+}) at *λ* = 488 nm was used. Through a beam splitter (BS) input beam was divided into two parts: a reference beam and a signal beam. The power of signal and reference beams was made identical to achieve high contrast interference fringes inside holographic material. A liquid crystal SLM (LCSLM) was used to generate desired spatial mode beam by modulating phase of an incident Gaussian beam by phase grating. A blazed grating pattern was obtained by interference of the desired signal beam with a tilted plane reference wave as shown in Fig. 1(c), and tilted plane wave was designed to separate first diffraction orders from zeroth order beam from LCSLM. To generate three identical signal beam paths in our demonstration, two flip mirrors M_{2} and M_{3} were placed in path of light beam reflected by LCSLM. By inserting flip mirror M2 and M3 in beam path, signal beam passes through can pass through two different optical path and interferes with its reference beam at different corresponding angles, resulting in angular multiplexing. With such experimental arrangement, three angularly multiplexed holograms can be recorded on same pupil in three separate exposures.

## 4. Experimental results

To experimentally demonstrate spatial mode multiplexing of OAM states using MVHGs, we first sequentially recorded three LG modes with different azimuthal index *l* of beam, which have different phase structures and OAM content. After recording at the maximum material sensitive wavelength of 488 nm using Ar^{+} laser, the volume hologram is first reconstructed with the identical laser (Ar^{+} laser at *λ* = 488 nm). A schematic diagram of reconstruction process to obtained spatial modes is shown in Fig. 3, and each diffraction angle for LG_{01-03} modes are found to be θ_{1} = 45°, θ_{2} = 66°, and θ_{3} = 76°. Incident Gaussian beam can be transformed into two three spatial modes simultaneously at three different angles. Experimentally obtained intensity profiles of reconstructed modes are shown in Fig. 4. It is clear from our experimental results that, by MVHGs three spatial LG modes with different phase front properties and angular momentum content can simultaneously reconstructed.

To study the phase properties of the LG modes obtained from MVHGs. Here, we used a self-referencing technique based on lateral shearing interferometry using a plane parallel glass plate (here a microscopic slide is used for this purpose) [23]. It is a simple and straightforward method and can be applied even with microscopic slide. The topological charge of the LG beams is generally shown by the formation of the fork fringes at the point of singularity. The number of bifurcation in the beam depends upon the topological charge of the beam. Figure 5 shows Shearogram of the LG_{01-03} beams. The formation of two oppositely oriented fork with multiple arms confirms the presence phase singularity at the center of the beam for three LG modes.

While reconstruction of hologram we utilized the wavelength degeneracy property of MVHGs, and operated MVHGs at a broad wavelength band of interest, from blue 450 nm all the way to red 635 nm. Figure 6 shows reconstructed beam shapes for three different colors, using blue, green, and red diode lasers. The spatial and temporal coherence property of reconstructed beams obtained from diode lasers is relatively poor as comparison to light obtained from gas lasers. It is evident that the shape of doughnut beams remains intact even if wavelength and coherence property of illuminating light is changed. Here, volume hologram with MVHGs act as spatial mode shapers with spectral filtering properties. By analogy reconstruction of volume hologram for multiple wavelengths can be qualitatively presented by *K*-sphere formulation to describe diffraction phenomena for a volume hologram obtained for plane wave interference [24, 25]. It is shown that during reconstruction of hologram different wavelengths create different radii on the *K*-sphere. Longer wavelength will produce a shorter radius *K*-sphere and vice versa. Under Bragg-matched condition, only radii of *K*-sphere for different wavelength will be changed while the center of all spheres will coincide. As long as resultant wave vector obtained from a probe beam wave vector and the grating vector lies on the *K*-sphere, Bragg diffraction condition will be satisfied and diffraction phenomena will occur. This is possible for arbitrary combination of Bragg-matched wavelengths and angles; hence, MVHGs can be reconstructed using multiple wavelengths. The diffraction angle corresponding to different wavelengths can be calculated using $\text{\Lambda}=\lambda \u20442L\mathrm{sin}({\theta}_{\lambda}\u20442)$, where $\text{\Lambda}$ is the period of refractive index modulation, *L* is the thickness of volume hologram, $\lambda $ is wavelength of reconstruction light ${\theta}_{\lambda}$ is diffraction angle corresponding to particular wavelength.

Experimental results in Fig. 6 shows that doughnut modes of any order can be recorded and reconstructed simultaneously using MVHGs for arbitrary optical wavelengths. The inherent wavelength degeneracy and spatial multiplexing of volume holograms make the proposed method of beam shaping unique among alternative ways of beam generation. The MVHGs will be particularly useful where the change of wavelength without affecting the beam shapes is required, for example in the light sheet microscopy. The proposed method is robust as comparison to other methods of vortex beam generation in particular, special optical elements such as spiral phase plates or computer generated holograms written on glass substrate, which works only for specific wavelength and results in fractional charge vortex beam with even small change in the wavelength.

The volume hologram is considered as a phase grating and the shape of grating is decided by interference pattern recorded onto it. In order to demonstrate potential of simultaneous generation of variety of beam shapes from a single element, three angular MVHGs are recorded on the same holographic pupil corresponding to three different wavefronts: a helical phase wavefront, conical wavefront and cubical wavefront respectively. In this case multiplexed hologram has three different kinds of volume phase gratings simultaneously in a single element. In case of unconventional beam shapes like LG, BG and, AG beams the shape of each wavefront is peculiar hence shape of interference fringes and the shape of grating will also be unique. The shape of phase grating will look like the phase mask shown in Fig. 1(c) but in a volumetric fashion. Figure 7 shows experimental results of reconstructed LG, BG and AG beams probed from MVHGs using a reference Gaussian beam. There is no crosstalk by observation among different spatial modes; for instance, LG01 doughnut mode generated through MVHGs remains intact even if different shaped wavefronts recorded on same holographic pupil.

Figure 8 shows reconstructed beam shapes by illuminating MVHGs using a Gaussian beam with red, green, and blue diode lasers. This results also shows that reference beam used for reconstruction of hologram need not to be exactly same as recording beam in terms of wavelengths and angles due to MVHG Bragg degeneracy. In addition, the reconstructed beam shapes clearly show great potentials to conduct high dimensional spatial mode multiplexing of arbitrary modes using MVHGs.

To obtain angular selectivity curve, we measured the diffractive power by rotating the hologram with respect to incident reconstructed beam and rotation angle scan was carried out for the MVHGs corresponding to three LG modes. Figure 9 shows the angular selectivity curve for three angular multiplexed holograms, and the average full-width-of-half-maximum (FWHM) is ~0.1° that provides solid evidence that MVHGs offer inherent fine angular selectivity.

## 5. Discussion

A natural question arises about the use of MVHGs to produce beam shapes when a light modulating device, such as LCSLM and digital micro mirror devices (DMDs), can dynamically produce any desired beam shapes and what is the significant difference between the two methods. It is important to mention here that there is fundamental difference between these methods of beam generation. First difference is number of diffraction orders obtained in the two methods in general. For thin grating without blazing there is always multiple diffraction orders are obtained whereas single diffraction order is characteristic feature of thick grating. Second difference is multiplexing is difficult in case of LCSLM which act as thin grating, whereas it is one of inherent features of volume holograms. Implementation of MVHGs in beam shaping offer significant advantage over other conventional techniques of beam shaping. For example, a volume hologram with MVHGs is simple, compact, easy to use, single element, and direct way to generate any order of spatial modes. They also produce Bragg-matched first-order diffraction beams which may offer advantage in many applications, and beam shapes at multiples wavelengths can be obtained without affecting the mode properties. They can be adapted as multiplexing as well as demultiplexing devices [5,26–28]. Beam shaped volume holograms can be low cost alternative to replace LCSLMs in applications, where a compact optical element is required for specific beam shapes, for example, fiber optic endoscope, miniature microscopes, and imaging tools, mobile phone microscope, classical and quantum information transfer lab experiments, in academic laboratories for demonstrations etc. Another point of view of looking at the use of volume holograms is that in many experiments stability of the system is an important factor. In such situation once a high quality beam shape is generated then it can be stored as reference wavefronts in the form of volume holographic grating for future use [29, 30]. In addition, it is important to emphasize that, the use of a volume hologram with MVGHs is a fixed, pre-recorded element placed in optical train, but without requiring the recording of a hologram at any stage of the probe or reconstruction process. This is an important difference compared to the aforementioned CGH and holographic alternatives [22, 31]. In our proposed approach, hologram recording takes place only during the construction of the MVHGs; the recorded holograms then take their place as optical elements in the imaging train, just like lenses, mirrors and other fixed elements in a conventional imaging system. Most of previous analyses of volumetric diffraction have been done using plane or spherical waves. Phenomena of volumetric diffraction is complicated and not straightforward. At present there are no theoretical or simulation models to completely describe volume diffraction phenomena from volume phase grating of arbitrary shapes like fork, conical, or cubical shapes. A possible approach to tackle this problem is by analyzing diffraction pattern by localized diffraction grating [32]. By combing different multiplexing technique together, a large number of modes can simultaneously be generated. It is worth mentioning here that, a phase grating can generate as well as detect mode which makes MVHGs suitable for spatial mode detection as well.

## 6. Conclusions

Simultaneous recording and reconstruction of multi-spatial modes of different kinds on MVHGs, formed in PQ-PMMA photopolymer, are experimentally demonstrated. We have observed that MVHGs can act as a spatial mode shaper with spectral filtering properties. Using present method high quality spatial modes can be obtained even using low spatial coherence sources. The shape and phase properties of spatial modes remain intact even if different kinds of spatial modes are recorded onto the same pupil area of MVHGs. Our findings suggest that spatial modes of any beams in any orders can be simultaneously recorded and reconstructed using MVHGs. Above facts show that PQ-PMMA based MVHGs can be a great potential candidate for high dimensional spatial mode multiplexing with all inherent characteristics of three dimensional pupils, angular selectivity, and Bragg degeneracy of wavelength. Present method provides clear advantage of simple, compact and direct method of beam shaping over the other methods of beam shaping. We believe the method presented here may find important applications in the field of microscopy, optical communication, and singular optics.

## Funding

Taiwanese Ministry of Science and Technology (103-2221-E-002-156-MY3, 105-2628-E- 002-008-MY3,106-2221-E-002-157-MY3). National Taiwan University and Jasper Display Corporation (NTU-106M103, NTU-106R7807, and JD4704).

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