Abstract
Freespace optical communication systems exploit the properties of light beams to transfer information through a freespace link. Indeed such systems provide an exciting alternative for communication. Here we introduce information transfer through freespace using a laser beam having its phase encoded with multiple orthogonal aberration modes. We use Zernike polynomials, which form a complete basis set, to represent the aberration modes. The user information is converted to coefficients of the Zernike modes which are summed digitally to obtain the resultant phase profile. A single phase modulating device then reads the resultant phase to shape the wavefront of the beam to be transmitted. The receiving station estimates the coefficients of all modes in the beam from a single measurement of a wavefront sensor, to retrieve the user information. We demonstrate data transfer using multiple modes, each with multiple strengths, and external perturbation compensation using the completeness property of the modes.
Introduction
The past couple of decades have seen rapid developments in freespace optical communication (FSO) systems as an alternative mode of communication that can have advantages in terms of speed and security^{1,2}. Such developments are for both outdoor and indoor applications^{3,4,5,6} of light beams. Light beams with helical wavefronts also known as vortex beams or orbital angular momentum (OAM) beams have been proposed for freespace information transfer, where the information is encoded as OAM states of light^{7,8,9,10}. An OAM mode with l helical phase fronts represented by the complex amplitude \(\exp (il\theta )\), where θ is the azimuthal angle and l is an integer called topological charge, carries OAM per photon equal to lℏ (ref. ^{7}). Therefore, the amount of information to be encoded is decided by the number of l values associated with the light beam. Multiplexing of OAM modes with different l values thus facilitates enhancing the data transfer rate through freespace up to tera bits per second or more^{11,12}, as demonstrated over a propagation distance of ~1 m. In most of the OAMbased FSO systems, first a fixed number of OAM modes are generated and the user data is encoded into each OAM mode by modulating the power (comprising “on” and “off” periods) of the respective mode. Different OAM modes encoded with the user data are then multiplexed often using beam splitters. It is also possible to generate a combination of OAM modes, each with a userdefined power, using a singlephase modulating device^{13}, such as a highdensity phaseonly liquid crystal spatial light modulator (LCSLM)^{14}. However, the process involves implementing a computationally intensive iterative algorithm to calculate the phase hologram, thereby making the modulation rate slow, if user data is to be encoded by periodically modifying the phase holograms. It is to be noted that different OAM modes suffer different divergences on propagation^{15,16}. It is also well known that in an FSO there exists the possibility of eavesdropping by capturing the radiated power^{17}, since it is always easier to measure modulation in power in a beam than any other modulation of the beam. Therefore, an OAM modebased FSO system, relying on modulating the power of various OAM modes for encoding, to some extent compromises on its security. However, a major concern for any OAMbased freespace communication system remains the susceptibility of the mode to atmospheric turbulence^{18}. Indeed, it was shown that air turbulence may lead to vortex instability, with a certain OAM mode after propagation breaking into other OAM modes^{19}.
Therefore, a significant amount of efforts in recent times have been aimed at mitigating the effect of air turbulence in the OAMbased FSO systems. Some such efforts are use of modal diversity of different optical modes toward turbulence^{20}, use of auto focusing Airy beams^{21,22}, use of convolutional neural network for adaptive demodulation of the signal at the receiving station^{23}, use of adaptive optics system to compensate for distorted OAM beams^{24}, and exploring the shape invariance property of OAM Bessel beams^{25}. Unfortunately, most of the turbulence compensation schemes lead to an increase in the processing time at the receiving station^{26}, thereby imposing an upper limit on real time decoding.
In this paper, we propose an orthogonal aberration modebased FSO system. We use Zernike polynomials^{27}, which form a complete set of orthogonal basis functions of two variables, to represent the aberration modes of a single light beam. Multiple Zernike modes whose amplitudes (i.e., coefficients) are encoded with user information are summed digitally before transmitting the resulting phase profile using a dynamic hologram or a nonholographic phase modulating device, such as a deformable mirror. As the user data is encoded in different modulation cycles by modifying only the phase of the beam, the net power in the beam remains unaltered. The receiving station uses a wavefront sensor to measure the amplitudes of all the Zernike modes constituting the wavefront simultaneously before decoding the user information. The proposed mechanism provides flexibility over the number and types of modes used, and enables compensation of air turbulence utilizing the completeness property of the Zernike modes, without requiring any optical correction of the beam. Moreover, the use of multiple amplitudes of each Zernike mode transmitted enhances the information capacity per modulation cycle for a given number of modes in the beam. In this work, we have demonstrated the proposed freespace information transfer mechanism in a laboratory setup comprising computergenerated holographybased transmission and receiving stations.
Results
Generation of beams using computergenerated holography
Computergenerated holography involves computation of the interference pattern between a plane reference wavefront and a userdefined object beam wavefront. The computed interference pattern describes the transmittance of a hologram, which is fabricated as a phase plate or implemented using a spatial light modulator. The object beam wavefront can be reconstructed in one of the diffracted beams when a plane wave is incident on the hologram. Let us consider that the userdefined beam to be generated has the complex amplitude U(x, y) = e^{iΦ(x, y)}, where Φ(x, y) is the phase profile of the beam and (x, y) are the coordinates of the hologram plane. The transmittance function of the hologram can be defined as^{28}
Here, Real(U) represents real part of the function U. t(x, y) can define the amplitude transmittance of the hologram in which case the hologram is termed as binary amplitude hologram. A hologram can also be fabricated with π × t(x, y) as the phase delay introduced by the hologram. Such a hologram is termed as binary phase hologram. Thus, the construction of the hologram does not involve any complex algorithms.
Binary hologram can also be constructed considering a single reference wavefront and a multiple (say, k number of) object beam wavefronts. If Φ_{j}(x, y) is the phase in the hologram plane of the jth beam then \(U(x,y)=\sum _{j = 1}^{j = k}{e}^{i{\Phi }_{j}(x,y)}\). The transmittance function of the binary hologram can still be defined by Eq. (1). The hologram to generate a single userdefined beam can be termed as singlex hologram, while the hologram to generate multiple userdefined beams can be termed as multiplex hologram.
In the case of the binary phase or amplitude hologram the plot of t against Φ, at certain location (x, y), is a square wave^{29}. Since the Fourier transform of a square wave contains all the odd harmonics, the binary hologram when illuminated by a plane wave results in odd diffraction orders, such as ±1, ±3, ±5, and so on. In addition, the binary amplitude hologram also results in the undiffracted 0 order. In general the nth order diffracted beam, where n is an integer, carries a phase equal to nΦ and has relative power \(\frac{1}{{n}^{2}}\) with respect to the +1 order beam^{29}. Therefore, out of all the diffracted beams, the +1 order beam carries the userdefined phase Φ and has the highest power. Even though the higherorder beams also carry an integral multiple of the userdefined phase profile, yet for most applications including in the present work the +1 order beam is preferred as it has more power than the higher orders. In order to separate the +1 order beam from the other diffracted beams including the 0 order beam, Φ should also include a wavefront tilt, τ(x, y) = τ_{x}x + τ_{y}y, where (τ_{x}, τ_{y}) are wavefront slopes with respect to (x, y) axes. Thus, Φ comprises ϕ representing phase difference relative to a plane perpendicular to the beam propagation direction and the tilt τ(x, y) such that Φ(x, y) = ϕ(x, y) + τ_{x}x + τ_{y}y. To construct the multiplex hologram, we thus need k number of (ϕ, τ_{x}, τ_{y}) sets. If the light diffracted from the hologram is focused by a lens, the focal spots corresponding to the ±1, ±3, ±5, ⋯ are located relative to the zero order at τ_{c} × ( ±τ, ±3τ, ±5τ ⋯ ), where τ_{c} is a constant. Thus, one can use an iris diaphragm to isolate the +1 order from the other orders and recollimate the isolated +1 order to recover the userdefined phase Φ. Figure 1 illustrates the generation of a userdefined wavefront using a binary amplitude hologram. The diffraction efficiency of the +1 order beam, using Fourier series analysis, in the case of binary amplitude hologram is found to be \(\frac{100}{{\pi }^{2}}\)% and in the case of binary phase hologram is found to be \(\frac{400}{{\pi }^{2}}\)%. The diffraction efficiency in the +1 order beam can be increased to 100% by constructing a phase hologram with t(x, y) = Mod(Φ(x, y), 2π), where the function Mod(Φ(x, y), 2π) returns the remainder after division of Φ by 2π. Such a hologram can be termed as blazed grating hologram.
Figure 2 shows examples of (a) a binary singlex phase hologram, (b) a binary multiplex phase hologram, (c) a binary multiplex blazed grating hologram, and the representative focal spot patterns (d → f), resulting from binary and blazed grating holograms. The phase profile of a single light beam can however also be defined directly or nonholographically by using a device such as a deformable mirror, in which case there is no diffraction and the userdefined phase is carried by the undiffracted zeroorder beam. Here, the deformable mirror takes a shape proportional to the desired phase profile ϕ(x, y) and a plane wave is incident on the mirror. The reflected beam from the deformable mirror then carries the userdefined phase profile. One can also use a phaseonly LCSLM to directly shape the wavefront in the reflected beam.
Zernike polynomials as orthogonal aberration modes
Zernike polynomials provide a complete set of orthogonal basis functions defined over a unit circle. In this work, we use Zernike polynomial Z_{j}(r, θ) as described by Noll^{27}, where r and θ are the radial and azimuthal coordinates, with r varying from 0 to 1. Although OAM modes do not form a complete set, there exists other modes, such as Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes^{30}, which also form complete sets of orthogonal functions. However, these are complex functions while the Zernike polynomials are real 2D functions. Therefore, owing to the unique properties of the Zernike modes^{31} defined over a circular area, they are found suitable in most of the optical systems to represent the phase profile of a light beam, which is also a real 2D function, without much bothering about the amplitude profile of the beam. In fact, a few of the Zernike polynomials represent classical aberrations balanced by lowerorder aberrations. For instance, (Z_{5}, Z_{6}) represent primary astigmatism, (Z_{7}, Z_{8}) represent primary coma balanced by tilt, and Z_{11} represents primary spherical aberration balanced by defocus. More details on Zernike modes is available in the Supplementary Note 1. The expressions of the Zernike polynomials can be converted to Cartesian system using \(x=r\cos \theta\) and \(y=r\sin \theta\). The desired phase profile of a beam can be obtained by simply summing the Zernike modes, such as \({\phi }(x,y)=\sum _{j}a_{j}Z_{j}(x,y)\), where a_{j} is the coefficient as well as root mean square (RMS) amplitude or simply amplitude (referred to as ϕ_{RMS}) of the Zernike mode Z_{j} in radian. A beam carrying such a phase profile can thus be generated using computergenerated holography technique or nonholographically.
Multiplexing of Zernike modes to encode user information
In this paper, we use one or more number of Zernike modes and multiple ϕ_{RMS} of each Zernike mode to describe the phase profile of a laser beam. Use of multiple ϕ_{RMS} of each Zernike mode is possible, since a_{j} in principle can be any finite real number. However, when we consider the presence of more than one Zernike mode in the beam then we should take into account the intermodal cross talk, while the modes are measured at the receiving station. We nevertheless can choose combinations of Zernike modes which have the minimal intermodal cross talk with respect to one another^{32}. If we consider n number of Zernike modes, each having m different ϕ_{RMS} values, then there will be m^{n} number of unique linear combinations of n Zernike modes. Thus, the user information can be encoded using these m^{n} unique wavefronts of a laser beam to be transmitted through free space in a sequential manner in equal number of modulation cycles of the beam. To be noted that the net power in the beam between different modulation cycles remains unaltered as only the phase profile, and not the amplitude, of the beam gets modified. The user data which can be text, numbers, image, etc. is first converted to basem^{n} number system (named as basem^{n} encoding scheme). Figure 3 depicts an example of the encoding scheme using two Zernike modes each with three values of ϕ_{RMS}. Thus, there are 3^{2} = 9 number of unique wavefronts each of which can be mapped to the digits of the user data in base9 number system using a lookup table, such as Table 1. One may of course use a different lookup table involving same or different Zernike mode and ϕ_{RMS} combinations. More examples of such lookup tables for other encoding schemes are provided in the Supplementary Note 2. It is important to note that here the multiplexing of the Zernike modes is done digitally and the scheme uses only one phase modulating device to generate a beam carrying the resultant phase profile. Therefore, the transmission station has the flexibility over the number of the aberration modes and the number of ϕ_{RMS} values for each aberration mode without any modification of the setup or without requiring any complex hologram computation. It is worth mentioning here that the optimal number of modes and the optimal number of ϕ_{RMS} values is in fact a tradeoff between the data transmission rate and the bit error rate (BER). The number of aberration modes possible is primarily decided by the pixel resolution of the phase modulating device or the maximum spatial frequency content in case of deformable mirror. While the number of ϕ_{RMS} values for each aberration mode is decided by a number of factors, such as pixel resolution of the phase modulator, length of the freespace link and amount of cross talk among the aberration modes. Therefore, practical limit on the maximum number of modes and ϕ_{RMS} levels feasible is decided by the spatial frequency content provided by the phase modulating device, the effect of diffraction due to long distance propagation and the intermodal cross talk. In our proposed scheme, we do not attempt to achieve the optimal information content in the transmitted beam. However, with the use of multiple strengths of each mode for a fixed number of aberration modes to encode the user information, our scheme significantly enhances the information carrying capability of the wavefront relative to the cases where only a fixed strength of each mode is possible.
Decoding of user information with a typeK wavefront sensor
The laser beam with the user information encoded wavefront travels the free space and is then incident on a modal wavefront sensor at the receiving station. The modal wavefront sensor should be able to measure simultaneously the ϕ_{RMS} of all the Zernike modes present over a range of amplitudes. However, the conventional modal wavefront sensors^{33,34} can measure only small aberration amplitudes and suffer from intermodal cross talk when multiple aberrations are to be measured. If large aberration amplitudes are to be measured with the same accuracy then the sensing operation is carried out in a loop, thereby lowering the sensing frame rate^{35,36}. In our work, we use a recently proposed typeK modal wavefront sensor^{37} to measure the multiple Zernike modes (referred to as sensor modes) present in the incident wavefront. The typeK sensor estimates the aberrations from the intensity data captured in a single camera image. It has a large range of linear response that can be more the ten times the linear response range of a conventional modal wavefront sensor and ϕ_{RMS} of two or more sensor modes can be measured, with reduced intermodal cross talk. The outputs of the typeK sensor (denoted as S_{K}) representing ϕ_{RMS} in radian of the sensor modes undergo a thresholding process to take into account any minor deviation between the ϕ_{RMS} used during encoding and ϕ_{RMS} as estimated. For instance, if the ϕ_{RMS} of a Zernike mode during encoding is −0.5 radian, 0, or 0.5 radian, the estimated ϕ_{RMS} is compared with some intermediate value between −0.5 and 0, or 0 and 0.5, say −0.25 or 0.25. If the estimated ϕ_{RMS} is 0.4 radian, it becomes 0.5 radian after thresholding. The thresholded ϕ_{RMS} of all the sensor modes are then converted to a digit of the corresponding number system, using the appropriate lookup table. Figure 4 presents the schematic of the decoding process for the encoded wavefront shown in Fig. 3. Therefore, in our proposed scheme, the sensor modes employed in the wavefront sensor in the receiving station are same as the aberration modes used for encoding in the transmission station. Although use of Zernike modes in wavefront sensors is very common, our scheme facilitates employing the same Zernike modes as information carrier as well.
The typeK modal wavefront sensor comprises primarily a multiplex hologram called the typeK hologram and a camera. The hologram generates a set of +1 order beams for each sensor mode for a given incident wavefront. The hologram effectively adds or subtracts specific amount of the sensor mode to the phase profiles of these +1 order beams. The central intensities of the set of +1 order focal spots are then used to calculate the typeK sensor output S_{K}. Figure 4 inset shows a representative typeK hologram and the resulting +1 order focal spots pattern. The focal spots pattern comprises 30 focal spots, arranged in concentric circles, for three sensor modes with ten focal spots to estimate each sensor mode. There is one more focal spot at the center that corresponds to a +1 order beam without any addition or subtraction of aberration by the hologram. More details of the typeK sensor is available in the Supplementary Note 3.
It is to be noted that for correct decoding of the user information, the receiving station should have two key information, one is the number and types of aberration modes used for encoding (required to design the typeK hologram) and the other is the lookup table (required to connect the sensor outputs to the appropriate basen digit). As the beam propagates through the free space, it undergoes certain amount of divergence, depending on the distance traveled and to some extent on the types of the orthogonal aberration modes used. Divergence thus suffered by the Zernike modes can effect the orthogonality among the modes, which may result in enhanced cross talk between various modes. Therefore, if such cross talks are to be minimized and the same size of the typeK hologram is to be used for all the distances traveled by the beam, the incident beam is first to be demagnified. Instead one may also image the beam aperture in the transmission station to the receiving station. Besides the typeK sensor output may require to be scaled (normalized) before the same is used for decoding the user data. A demonstration using numerical simulation to show how diffraction affects the light beam carrying Zernike modes after it propagates long distances can be found in Supplementary Note 4. It can be seen that in spite of the diffraction suffered, after the demagnification of the beam aperture, various modes can be detected without any noticeable intermodal cross talk.
Experimental implementation
We develop a proofofprinciple setup to demonstrate the working principle of the proposed information transfer. A schematic of the experimental arrangement comprising the transmission station based on a ferroelectric LCSLM (LCSLM_{1}) acting as the binary hologram and a receiving station based on a typeK hologram written on a nematic LCSLM (LCSLM_{2}), separated by a distance of 2.4 m, is seen in Fig. 5(a). Figure 5(b–d) shows some experimental focal spot patterns resulting from the typeK hologram. Instead of using holographic means the transmission station can use a direct (i.e., nonholographic) wavefront shaping device, such as a deformable mirror. A deformable membrane mirror may provide wavefront modulation at frame rates upto tens of KHz. Besides the typeK hologram instead of being implemented using an LCSLM can also be fabricated as a phase plate. The decoding speed can be enhanced significantly by using a set of point photo detectors to record the central intensities of the focal spots instead of the camera.
External perturbations compensation
We first assess the consistency of accurate decoding of user data over a period of time. We encode the wavefront in the transmission station using three Zernike modes, Z_{5}, Z_{6}, and Z_{11} and estimate the strengths of the same three modes in the receiving station using the typeK sensor. We notice that owing to the movement and externally caused perturbations in the information carrying beam, the values of ϕ_{RMS} at the receiving station fluctuates to some extent over time even though there is no change in the aberration strengths from the transmission station. To be noted that for indoor applications the effect of turbulence will be insignificant, however, for outdoor applications the same will play a very important role. Therefore, to test the effectiveness of our turbulence compensation scheme in the case of outdoor applications, we use a table fan to create instability in the air in between the two stations so as to enhance the fluctuations in S_{K}. The table fan creates a circulating turbulent air with an average speed between 4 and 5 m/s, which introduces random aberrations in the transmitted beam. Introduction of such aberrations leads to the degradation of Strehl ratio from the ideal (i.e., aberration free) value of 1. We estimate the effect of the turbulence for our setup created by the table fan in terms of a Strehl ratio of ~0.3. Figure 5(e) shows such an induced case of sensor output variation over different measurement frames when a fixed combination of ϕ_{RMS} = (0, 0.5, 0) of (Z_{5}, Z_{6}, Z_{11}) is transmitted. Figure 5(g) shows the binary pattern display timing of the LCSLM_{1} at the rate of 1440 Hz. However due to limited frame rate of the camera, we display binary holograms at the rate 360 Hz only with an interval of t_{w}, as seen in the right panel of the same figure. In order to make S_{K} steady against the perturbations, we modify the transmission scheme of the encoded wavefronts, such that each information carrying wavefront is preceded by a reference plane wavefront as depicted in the right panel Fig. 5(g). The reference wavefront when incident on the typeK hologram results in a reference focal spot pattern whose central focal spot location gives a measure of beam movement. The same central focal spot can also be used to get a measure of the intensity fluctuations in the transmitted beam. Since the Zernike modes form a complete set of basis functions, any arbitrary change in the reference phase profile introduced by the perturbation can be expressed as a linear combination of a subset of the Zernike modes^{31}. Therefore, primary effect of perturbation other than the beam movement can be estimated from a measure of the typeK sensor outputs corresponding to the reference wavefront. Since typeK sensor has a linear response range upto several radian of ϕ_{RMS} for each sensor mode, corrected S_{K} for each information frame can be obtained, after beam movement compensation, as S_{K} (information frame) – S_{K} (preceding reference frame). Thus, the maximum strength of the perturbation that can be compensated by the scheme is decided by the linear range provided by the typeK sensor, which can be set as per the requirement subject to the spatial resolution available in the typeK hologram^{37}. More details about the perturbation compensation scheme is available in the Supplementary Note 5. On application of the perturbation compensation, the typeK sensor in the receiving station provides a much more consistent measure of the sensor modes, as indicated by the plot in Fig. 5f. The compensation scheme to be effective the nature of the external perturbation should not vary between the reference wavefront and the following information wavefront. In the case of rapid variation in perturbation, the wavefront modulation should take place at a rate at least twice that of the perturbation variation rate.
Therefore, the perturbation compensation scheme addresses the issue of atmospheric turbulence using the same wavefront sensor, which is used for the decoding of the transmitted beam. Moreover the compensation does not involve any optical correction of the beam as is done in an adaptive optics system. However, to be noted that subtraction of the effect of turbulence as proposed in our scheme will not work if the light beam is encoded using modes, which do not form a complete basis set of real orthogonal functions. Hence, such a perturbation compensation scheme can not be adapted to OAM, LG, or HG modesbased FSO systems. We show in Supplementary Note 5 how our scheme utilizes the completeness property of the Zernike modes and why such a scheme will not work with OAM modes.
Discussion
We choose the three Zernike mode combinations (Z_{5}, Z_{6}, Z_{11}) to encode the user data. In our first experiment, we transmit a gray scale image of size 65 × 50 pixels with each pixel as an 8 bit integer. The image pixels are encoded using base3, 5, 9, 25, and 27 encoding schemes. At the receiving station, the wavefronts are decoded both without perturbation compensation and with perturbation compensation. It can be seen in the Supplementary Note 2 that base3 and 5 encoding schemes use a single Zernike mode, while base9 and 25 encoding schemes use two Zernike modes. However due to use of multiple strengths (i.e., amplitudes) for each mode, the information content of m number of wavefronts in the case of base5 scheme has increased by \(\frac{{5}^{m}1}{{3}^{m}1}\times 100\)% relative to base3 scheme, and the same for base25 has increased by \(\frac{2{5}^{m}1}{{9}^{m}1}\times 100\)% relative to base9 scheme. In our base25 encoding scheme, the maximum number of unique information (i.e., the largest integer value) carried by the beam over eight modulation cycles is estimated to be 25^{8} − 1 = 1.5259 × 10^{11}. Instead if we consider OAM modebased systems using same number (i.e., two) of modes as the base25 scheme, with different l values, over eight modulation cycles the maximum number is estimated to be 4^{8} − 1 = 65535. Details of this comparison of information content in the beam using fixed number of modes can be found in Supplementary Note 6. The effective rate of data transfer in our proposed scheme is decided by both the encoding scheme and minimum interval (i.e., time period of one modulation cycle) at which the userdefined wavefront can be shaped by the phase modulating device. In our experiment with a modulation frequency of 360 Hz and data transfer using 180 wavefronts per second with perturbation compensation on, the data transmission speed is kept at 4.5 and 4.86 Kbits/s for base25 and base27 encoding scheme, respectively.
Figure 6(a) shows the original image transmitted, and Fig. 6(b–f) show the detected images with perturbation compensation when the data is transmitted using different encoding schemes. It is noticed that in the case of base3 and base9 encoding schemes the image is detected without a single wrong pixel. Although in the other cases there is an error of maximum one to four pixels, even though there is still no major observable defects in the detected images. In order to further increase the external perturbation, we use the table fan to introduce turbulence in the medium. The directly detected image transmitted as base25 numbers which travel through the turbulent air is seen in Fig. 6(g) showing several wrong pixels. However on application of perturbation compensation, the number of wrong pixels reduces drastically as seen in Fig. 6(h).
The BERs of image transmissions as presented in Fig. 6 are shown in Table 2. It is observed that BER which is 0.0498 in the case of Fig. 6(g) has improved to 0.0047 in the case of Fig. 6(h).
We then perform yet another experiment to transmit a set of 8 bit integers and alphabets in a similar manner as the above experiment. The results presented in Table 3 confirm that when the user data at the receiving station is decoded with perturbation compensation, the effect of perturbation significantly gets eliminated. Accuracy of the transmission increases when the data is encoded as base3 and base5 numbers.
The relatively small number of wrong detections of user data in some of the schemes is attributed to the low modulation frequency of 360 Hz, owing to the limitation of the camera used in the present setup. Even a threefold increase in the modulation frequency should make the perturbation compensation much more effective so as to eliminate wrong detection of user data altogether in all the schemes. Under extreme cases of turbulence, use of an encoding scheme with a smaller base such as base3 and choosing larger separation between Φ_{RMS} values, such as 0, ±1 instead of 0, ±0.5, will further improve the perturbation compensation. The use of smaller base in combination with blazed grating implementation of the typeK hologram will also improve the signal to noise ratio in the receiving station, in the case of significant drop in signal levels in a long distance freespace link. Further the base27 scheme due to the use of three different Zernike modes simultaneously can be adapted to error correcting schemes, such as triplemodular redundancy code^{38}. The three modes carrying copies of the same information which pass through the same atmospheric link, however, may get affected differently by turbulence. Therefore triplemodular redundancy code may work efficiently to reduce the BER although at the cost of the transmission speed.
To conclude, we have demonstrated freespace information transfer using a laser beam whose wavefront is encoded with a linear combination of Zernike modes, which form a complete set of orthogonal basis. User information is converted to amplitudes of one or more Zernike modes using multiple amplitude values for each Zernike mode. Multiplexing of modes is done digitally without involving any computationally intensive step and a singlephase modulating device is used to transmit the data, as sequentially transmitted wavefronts of a single beam without affecting the net power in the beam. Use of a complete set of orthogonal functions and multiple amplitudes of a single mode, to encode the wavefront, opens the door toward reaching optimum limit of information content in the beam. The receiving station uses a typeK modal wavefront sensor that provides amplitudes of all the Zernike modes present in the incident wavefront from just one measurement using a camera. The use of the typeK sensor enables measurement of both small, as well as large amplitudes of each Zernike mode with the same accuracy. We have also demonstrated a scheme to compensate for external perturbations, such as turbulence in the air, without requiring any additional optical arrangement, so as to make the information transfer steady. Therefore, in addition to indoor applications where the effect of turbulence is limited, our scheme is expected to be ideal for outdoor applications as well when turbulence is present. The proposed information transfer scheme has a built in security feature against eavesdropping since the receiving station must a priori know the number, types, and amplitudes of the Zernike modes used and the lookup table mapping the Zernike mode amplitudes with the user data. In our scheme, we do not use intensity or power associated with a given mode as a degree of freedom to increase the information content as can be done in the OAM modebased system. In lieu of the intensity, our scheme uses the amplitude of the Zernike mode as a degree of freedom, which unlike intensity of an OAM mode is less susceptible to external factors due to our compensation scheme and also more secured against eavesdropping. Our experimental arrangement however can be considered as a proofofprinciple setup only and the same was not optimized in the terms of speed, accuracy, information content in the wavefront, and the distance traveled in free space. There is nevertheless an equivalent scope in the proposed scheme, similar to the OAM modebased systems, to further enhance the information content of the beam by incorporating polarization and wavelength division multiplexing over and above the Zernike mode division multiplexing.
Methods
Construction of binary hologram in the transmission station
Binary hologram to generate the +1 order beam carrying userdefined phase profile ϕ(x, y) with (x, y) = −1 → +1 is constructed over 480 × 480 pixels using τ_{x} = 165π and τ_{y} = −41π. An appropriate linear combination of Zernike modes Z_{j}(x, y) constitutes ϕ(x, y) and t(x, y) is computed using \(U(x,y)={e}^{i[\phi (x,y)+{\tau }_{x}x+{\tau }_{y}y]}\) in Eq. (1). t(x, y) is multiplied by a unit circle centered at (x, y) = (0, 0) and is then converted to a binary image to be displayed on the ferroelectric LCSLM (ForthDD, SXGAR3), which has a pixel pitch = 13.62 μm. To be noted that the +1 order beam from the LCSLM has some residual aberrations which may be corrected holographically in order to get a near ideal +1 order beam.
Incident laser beam
The beam incident on the binary hologram on the ferroelectric LCSLM is derived from a He–Ne Laser (wavelength = 632.8 nm, output power = 10 mW). Experiments are performed at ≈50% of the laser power. The +1 order beam carrying the user information has a diameter of ≈6.5 mm.
Construction of the typeK hologram
The typeK multiplex binary holograms are constructed over 750 × 750 pixels and are displayed as binary images on a nematic LCSLM (HoloEye LCR1080) that has a pixel pitch = 8.1 μm. t(x, y) is computed using k sets of (ϕ(x, y), τ_{x}, τ_{y}), where k = 11 for base3 and 5 encoding schemes, k = 21 for base9 and 25 encoding schemes, and k = 31 for base27 encoding scheme. The +1 order beam at the center of the typeK focal spot pattern has τ_{x} = 153.8π and τ_{y} = 100.9π. For the base3 encoding scheme, the typeK hologram generates five pairs of bias beams with a_{v} = (−1, −0.5, 0, 0.5, 1) radian to detect a single sensor mode. These beams are directed to form a circle using relative tilt with respect to the central beam (defined as Δτ = \(\sqrt{\Delta {\tau }_{x}^{2}+\Delta {\tau }_{y}^{2}}\)) as 21π. For the base5 encoding scheme, the typeK hologram generates five pairs of bias beams with a_{v} = (−1.5, −0.75, 0, 0.75, 1.5) radian to detect a single sensor mode. These beams are also directed to form a circle using Δτ = 21π. TypeK holograms for base9 and base25 encoding schemes are designed for two sensor modes using a_{v} = (−1, −0.5, 0, 0.5, 1) and a_{v} = (−1.5, −0.75, 0, 0.75, 1.5) for each sensor mode, respectively. For both encoding schemes bias beams are directed to form two concentric circles using Δτ = 14π and 24.5π. TypeK hologram for base27 encoding scheme is designed for three sensor modes using a_{v} = (−1, −0.5, 0, 0.5, 1) for each sensor mode and the bias beams are directed to form three concentric circles using Δτ = 11.5π, 21π, and 30π. The bias beams for each typeK hologram use b = 0.7 radian.
Camera used in the typeK wavefront sensor
The focal spot pattern resulting from the typeK hologram is recorded by a CMOS camera (Basler, A504K) having pixel resolution = 1280 × 1024, pixel pitch = 12 μm, bit depth = 8 bit, and a full frame rate = 500 fps.
Data availability
All relevant data are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors wish to acknowledge the financial support from the Department of Electronics and Information Technology (DeitY), India, vide its letter no. 12(4)/2011PDD.
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B.R.B. with input from S.K. conceived the experiment. S.K. developed the experimental setup and performed the experiment. Manuscript was prepared with contributions from both the authors.
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Konwar, S., Boruah, B.R. Leveraging the orthogonality of Zernike modes for robust freespace optical communication. Commun Phys 3, 203 (2020). https://doi.org/10.1038/s42005020004681
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