 Research
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Optical coherence encryption with structured random light
PhotoniX volume 2, Article number: 6 (2021)
Abstract
Information encryption with optical technologies has become increasingly important due to remarkable multidimensional capabilities of light fields. However, the optical encryption protocols proposed to date have been primarily based on the firstorder field characteristics, which are strongly affected by interference effects and make the systems become quite unstable during light–matter interaction. Here, we introduce an alternative optical encryption protocol whereby the information is encoded into the secondorder spatial coherence distribution of a structured random light beam via a generalized van Cittert–Zernike theorem. We show that the proposed approach has two key advantages over its conventional counterparts. First, the complexity of measuring the spatial coherence distribution of light enhances the encryption protocol security. Second, the relative insensitivity of the secondorder statistical characteristics of light to environmental noise makes the protocol robust against the environmental fluctuations, e.g, the atmospheric turbulence. We carry out experiments to demonstrate the feasibility of the coherencebased encryption method with the aid of a fractional Fourier transform. Our results open up a promising avenue for further research into optical encryption in complex environments.
Introduction
As a fundamental attribute that describes statistical properties of random light fields, optical coherence has played an important role in understanding and tailoring lightmatter interactions [1, 2]. Spatial coherence, in particular, governs a multitude of intriguing optical phenomena and features in a number of applications [3, 4]. Even a straightforward adjustment of a spatial coherence length of light can go a long way to improve the image quality of a microscope [5, 6] and to reduce the turbulence induced field degradation in optical communications [7]. The early theoretical [8, 9] and experimental [10] work on light beams with nonuniform spatial coherence has triggered research into engineering spatial coherence structure of random sources. This initial interest has been further invigorated by the work of Gori and coauthors [11, 12] who have established a general framework for generating random beams with statistically uniform (Schellmodel) [13–19] and nonuniform [20–27] spatial coherence distributions. The spatial coherence structure engineering at the source has been shown to endow optical fields with a number of nontrivial features, such as diffractionfree propagation [9, 24], efficient selfhealing [28, 29], selffocusing [20, 30, 31], selfsteering [32, 33], and selfshaping [34, 35] capabilities as well as periodicity reciprocity [36, 37]. A rich repertoire of propagation scenarios induced by the spatial coherence engineering enables a host of promising applications to photovoltaics [38], diffractive imaging with lowcoherence light [39], optical target tracking [40, 41], and particle trapping [42, 43] among others.
At the same time, information encryption with the aid of optical technologies has been studied extensively in the past decade due to remarkable multidimensional capabilities and ultrafast modulation speed afforded by the light fields [44, 45]. Numerous protocols for optical information encryption have been proposed since the double random phase encoding (DRPE) was developed by Réfrégier and Javidi [46]. Much research on optical information encryption has been carried out to date using the DRPE or DRPE related techniques, including a fractional Fourier domain DRPE [47], lensless DRPE in the Fresnel domain [48], and multidimensional random phase encoding [49], among others. Recent progress in the light field structure engineering [50–52], has highlighted the degrees of freedom of a structured light field as powerful tools for information encoding. For example, the optical encryption protocols based on the phase structure modulation [53] and orbital angular momentum and polarization state mode division multiplexing [54–60] have been developed lately.
To the best of our knowledge, however, the optical encryption protocols proposed thus far have been primarily based on the modulation of the firstorder characteristics of coherent optical fields, such as the phase, amplitude, and polarization of light. The latter are quite sensitive to any interference effects inevitably arising during light propagation and its interaction with the matter [61]. On the other hand, the secondorder correlations of statistical light fields [1] are known to be fairly robust to environmental noise such as the atmospheric turbulence, especially in the low coherent limit [7, 62]. This observation begs a natural question: Can any information be encrypted into the spatial coherence structure of a statistical light field and robustly transmitted through a hostile environment?
In this work, we propose an efficient protocol for the encryption of optical image information into the spatial coherence structure of a random beam with the help of a generalized van Cittert–Zernike theorem involving the encoding system with encryption keys. We argue that the measurement of the spatial coherence structure to decode the information enhances the protocol security because the measurement of a statistical characteristic requires collecting much more data than is the case for acquiring deterministic quantities employed in the traditional optical encryption protocols. Moreover, we demonstrate experimentally that the advanced protocol is extremely robust to the environmental fluctuations, such that the information can be well reconstructed despite the atmospheric turbulence of any strength. Our results pave a way toward robust optical coherence encryption in complex noisy environments.
Method
Principle of coherencebased encryption
We schematically represent an optical encryption protocol based on the spatial coherence structure engineering in Fig. 1. The proposed protocol consists of three stages. At the first stage, a plaintext (i.e., an amplitude image) is encoded into the spatial coherence structure of a random light beam via a generalized van Cittert–Zernike theorem. Next, the spatial coherence structure of the beam, containing a ciphertext, is measured through a generalized Hanbury–Brown Twiss experiment. Finallly, the information encoded into the plaintext is decoded by using the measured spatial coherence structure and the encryption keys involved in the encoding system shown in part (a) to Fig. 1.
Encoding
In Fig. 1a we display the encoding stage of our protocol. The plaintext information, i.e., an image with its (real) transmission function t(v) is first embedded into a completely incoherent light source by transmitting coherent light through a dynamic diffuser, e.g., a rotating groundglass disk (RGGD). The characteristic inhomogeneity scale is much smaller than the beam spot size on its surface [61]. The secondorder correlations of the light field generated by the incoherent source in the spacefrequency domain can be characterized by the crossspectral density as
where v_{1} and v_{2} are two arbitrary position vectors in the source plane, δ(·) is a Dirac delta function and we drop the frequency dependence of W hereafter, for brevity. We then transform the incoherent light field into a partially coherent beam with the help of an optical system with a transfer function H(r,v). The crossspectral density of the partially coherent beam then reads
where r_{1} and r_{2} are two arbitrary position vectors in the output plane of the optical system and the asterisk denotes a complex conjugate.
We call the transform in Eq. (2) a generalized van Cittert–Zernike theorem as the original one describes the evolution of the crossspectral density of light, generated by an extended incoherent source, on free space propagation [1]. The optical imaging system that we employ does not merely consist of simple lenses and stretches of free space. Rather, it is, in general, a complicated system containing the encryption keys. We remark here that our protocol can accommodate any encoding system of the type previously employed in optical encryption protocols with coherent light. The encryption systems characterized by linear and nonlinear transfer functions will give rise to random light beams with statistically uniform (Shellmodel) [15] and nonuniform [27] spatial correlations, respectively. Similar to the case of conventional, coherent encryption protocols, the complexity of the encoding protocol ensures the encoded data security. The attack analysis for different types of encryption protocols was previously studied elsewhere [63, 64] and will not be addressed here.
The difference between the traditional encoding protocols and our coherencebased one is that in our case, given plaintext information is encoded into secondorder field correlations rather than into a deterministic first order characteristic, such as light field amplitude, phase, or polarization. As a consequence, the information about an entire statistical ensemble of a large number of field realizations is required to reconstruct such field correlations, making the information concealed in the ciphertext hard to compromise. Furthermore, a relative insensitivity of the secondorder field correlations to the environmental fluctuations augur well for the robustness of our protocol in noisy environments.
Ciphertext measurement
The spatial coherence of a partially coherent light beam can be measured, typically, with the classic Young’s twopinhole experiment. However, the tiny size of the pinhole opening limits the light efficiency of the measurement and the large sets of pinholeseparation data increases the measurement time. Here, to fully recover a complexvalued spatial coherence structure distribution, specified by the crossspectral density, we apply a recently advanced generalized Hanbury–Brown Twiss technique [40, 41]. To this end, we first introduce a pair of independently controlled coherent reference fields E^{R1}(r) and E^{R2}(r) and combine them with the output random light beam [see Fig. 1b]. The field realizations of two composite random fields can then be written as
where E(r) is a field realization of the partially coherent beam which yields the crossspectral density of the beam as W(r_{1},r_{2})=〈E^{∗}(r_{1})E(r_{2})〉. Hereafter unless specified otherwise, the angle brackets denote ensemble (time) averaging over the field realizations.
The intensityintensity crosscorrelation of the two composite fields at any pair of points can be expressed as
where I^{C1}(r)=E^{C1∗}(r)E^{C1}(r) and I^{C2}(r)=E^{C2∗}(r)E^{C2}(r) denote the random intensities of the two composite fields. On substituting from Eqs. (3) and (4) into Eq. (5) and employing a Gaussian moment theorem [61], we obtain
where I^{U1}(r)=I^{R1}(r)+I(r) and I^{U2}(r)=I^{R2}(r)+I(r). Here I^{R1}(r)=E^{R1}(r)^{2} and I^{R2}(r)=E^{R2}(r)^{2} are the intensities of the two reference fields, I(r)=E(r)^{2} is an intensity of the random beam; prime and double prime denote, respectively, the real and the imaginary parts, and Δϕ=arg[E^{R1}(r_{1})]−arg[E^{R2}(r_{2})] is a phase difference between the two reference fields at points r_{1} and r_{2}, with arg standing for the phase of a complex function. We note that the Gaussian moment theorem is applicable here since the field of a random beam is produced by an RGGD and therefore, it obeys Gaussian statistics [61].
We can infer from Eq. (6) that the information about the real and imaginary parts of the crossspectral density function is contained in the intensityintensity crosscorrelation function G^{C}(r_{1},r_{2}). Hence, by setting the phase difference Δϕ to either 0 or π/2, we can separately extract the real or imaginary part. Further, we notice that the first two terms in Eq. (6) form irrelevant background which can be removed by first evaluating an auxiliary intensityintensity crosscorrelation as
where I^{U1}(r) and I^{U2}(r) are (fluctuating) combined intensities of the reference and random beams. With the help of the Gaussian moment theorem, Eq. (7) can be written as
It follows from Eqs. (6) and (8) that the real and imaginary parts of the crossspectral density can be obtained as
where ΔG(r_{1},r_{2},Δϕ)=G^{C}(r_{1},r_{2})−G^{U}(r_{1},r_{2}) is expressed in terms of the phase delay Δϕ.
It follows from Eqs. (3)–(10) that the crossspectral density function can be fully recovered by evaluating the intensityintensity correlations in Eqs. (5) and (7). In the experiment, the random intensities I^{C}(r) and I^{U}(r) of N realizations are recorded at different time instants and the intensityintensity correlations are then evaluated as [17]
where \(I^{\mathrm {C}}_{n}(\mathbf {r})\) and \(I^{\mathrm {U}}_{n}(\mathbf {r})\) are intensities of the nth field realization. We remark that whenever a partially coherent beam is generated by a Fourier transforming system, such as a thin lens, the time average in Eqs. (5) and (7) can be replaced with the space average over a single instantaneous speckle pattern [40, 65].
Decoding
Once the crossspectral density W(r_{1},r_{2}) has been fully recovered, the plaintext can be decoded by using an inverse transform of Eq. (2) with the correct encryption keys. As is elucidated in Fig. 1c, the decoded paintext is correctly recovered only for the matched spatial coherence and encryption keys.
Example
To demonstrate the feasibility of our encryption method, we present an example of our protocol in which the optical encryption system performs a fractional Fourier transform of a given order, c.f., [47]. The corresponding transfer function then reads
where A=−icscφ_{E}/λf, φ_{E}=p_{E}π/2 and p_{E} is an order of the fractional Fourier transform. The magnitude of p_{E} can be regarded as an encryption key of the protocol. In our experiment, the fractional Fourier transform system can be realized by a thin convex lens of focal length f_{convex}=f/ sinφ_{E}, which is placed midway between the input and output planes of the encoding system. The distances between the convex lens and the input/output planes are equal to f tan(φ_{E}/2). Further, the encryption key falls in the range: p_{E}∈(0,1], with p_{E}=1 corresponding to the ordinary Fourier transform limit.
It follows from Eqs. (2) and (13) that the ciphertext can the be expressed as
where I_{E}(v)=t^{2}(v) denotes the plaintext intensity and Δr=r_{1}−r_{2}. We can infer from Eq. (14) that the crossspectral density W(r_{1},r_{2}) is a twopoint fractional Fourier transform of the image intensity I_{E}(v). Thus, the image can be recovered by an inverse fractional Fourier transform as
Here I_{R}(v) is an intensity recovered from the measured crossspectral density function W_{R}(r_{1},r_{2}) and p_{R} is a decoding key. Next, φ_{R}=p_{R}π/2. Only if does the measured W_{R}(r_{1},r_{2}) match the ciphertext W(r_{1},r_{2}) and p_{R}=p_{E}, the plaintext intensity is correctly recovered: I_{R}(v)=I_{E}(v).
Notice that Eq. (15) can be viewed as a fourdimensional Fourier transform of the function \(W_{\mathrm {R}}(\mathbf {r}_{1},\mathbf {r}_{2}) \exp \left [ \frac {\mathrm {i}\pi }{\lambda f}\left (\text {cot} \varphi _{\mathrm {R}} \mathbf {r}_{1}^{2}  \text {cot} \varphi _{\mathrm {R}} \mathbf {r}_{2}^{2}\right) \right ]\), which can, in turn, be regarded as the crossspectral density of a “effective” chirped random beam. Further, it follows from Eq. (14) that such a beam can be assumed to be generated from an incoherent light source with its intensity being I_{R}(v) with a Fourier transforming imaging system. Thus, per our previous discussion, the crossspectral density of the effective chirped random beam can be recovered by using space instead of time averaging in the generalized Hanbury Brown–Twiss experiment. The corresponding reference fields read
Here the quadratic phase can be introduce by a concave lens. The distance between the lens and the output reference field is d=f/cotφ_{R}−f_{concave}, where f_{concave} is a focal length of the concave lens. On substituting from Eqs. (16) and (17) into Eqs. (3), (4) and (5) and taking a Gaussian average, we obtain the real and imaginary parts of the recovered crossspectral density of the chirped random beam from Eqs. (9) and (10). Next, the plaintext can be recovered by an inverse Fourier transform of the measured crossspectral density. We note in passing that the encryption key p_{R} for decoding the plaintext is now used in the chirped reference fields of Eqs. (16) and (17) that are introduced at the ciphertext interrogation stage.
We exhibit our experimental setup in Fig. 2. We first convert an unpolarized monochromatic, coherent beam of carrier wavelength λ=633 nm, generated by a He–Ne laser, into an xpolarized beam by a linear polarizer (LP) and then split the beam by a beamsplitter (BS) into two beams that go into the top and bottom arms, respectively, as shown in Fig. 2. We employ the top arm in the figure to generate reference light beams with their electric fields given by Eqs. (16) and (17). The purpose of the bottom arm is to encode the image information into the crossspectral density of a random beam as illustrated in Fig. 1a. In the bottom arm, we impinge the xpolarized beam, having been expanded by a beam expander (BE), onto a spatial light modulator (SLM) to which we preload an optical image with a transmission function t(v). We then project the outgoing SLMmodulated coherent light onto a rotating groundglass disk (RGGD) by a 2f imaging system formed by a thin lens L_{1} of focal distance f_{1}=100 mm. In our experiment, the beam spot on the RGGD is much larger than the characteristic inhomogeneity scale of the latter. Thus, the emerged light from the RGGD can be regarded as incoherent and its crossspectral density can be well approximated by Eq. (1). The incoherent light by the secondary source is then transmitted through a fractional Fourier transform system represented by a thin lens L_{2} of focal length f_{2}=250 mm. The random beam with the encoded into its crossspectral density image information is thus generated in the output plane of the fractional Fourier transform system. The distance l between the RGGD and L_{2} (or between L_{2} and the output plane) can be varied to realize a fractional Fourier transform of any given order p_{E}. As a part of the fractional Fourier transform system, a halfwave plate (HWP) with its fast axis making the angle π/8 with the xpolarization direction is inserted to transform the xpolarized random beam into a π/4linearly polarized one.
In the top arm, we first transmit the xpolarized beam through a neutraldensity filter (NDF), an HWP, and a quarterwave plate (QWP). We set the fast axes of the HWP and QWP to make the angle π/8 with respect to and to be parallel to the xpolarization direction, respectively. Therefore, we can generate a righthanded circularly polarized coherent beam immediately past the QWP. We then expand the circularly polarized reference beam by the BE to produce a collimated beam of virtually uniform intensity distribution. Due to a fixed π/2 phase difference between the two orthogonal components of the circularly polarized beam, its x and ypolarized components can be viewed as two coherent reference fields required for the measurement of the crossspectral density of the random probe beam. We can introduce the quadratic phases of the reference fields in Eqs. (16) and (17) by adding a concave lens (CL) of focal distance f_{concave}=100 mm behind the BE. We can adjust the quadratic phase by changing the distance d between the CL and the output plane of the reference beam.
We next combine the reference beam, generated in the top arm, and the random beam, generated in the bottom arm, with a BS to measure the crossspectral density of the composite beam (i.e., the ciphertext), as illustrated in Fig. 1b. We split the x and ycomponents of the composite field by a polarization beamsplitter (PBS) and image them onto CCD_{1} and CCD_{2}, respectively, by a 2f imaging system formed by a couple of thin lens L_{3} of focal distance f_{3}=250 mm. We use a directdigital synthesis (DDS) signal generator as an external trigger to control the two CCDs to simultaneously capture the random intensities of the x and ycomponents of the field.
In our experiment, the message in capital characters ‘SUDA’ is loaded into the SLM and the fractional order of the encryption system is set to be p_{E}=0.6 by setting the distance to l=103 mm. To measure the crossspectral density of the composite beam, we record the intensity distributions I_{x}(r) and I_{y}(r) of the x and ycomponents of the π/4linearly polarized random beam by blocking the reference arm, the intensity distributions \(I^{\mathrm {R}}_{x}(\mathbf {r})\) and \(I^{\mathrm {R}}_{y}(\mathbf {r})\) of the x and ycomponents of the circularly polarized reference beam by blocking the encryption arm, and the intensity distributions \(I^{\mathrm {C}}_{x}(\mathbf {r})\) and \(I^{\mathrm {C}}_{y}(\mathbf {r})\) of the x and ycomponents of the composite field. The real and imaginary parts of the recovered crossspectral density function follow from
where 〈·〉_{s} denotes a spatial average. We can reconstruct the image I_{R}(v) by evaluating an inverse Fourier transform of the measured crossspectral density W_{R}(r_{1},r_{2}). We control the magnitude of the encryption key p_{R} by adjusting the distance d between the CL and the BS. For the correct key, i.e., p_{R}=p_{E}=0.6, the distance is d=178 mm.
Results and discussion
In Figs. 3a and b, we depict our experimental results for the measured crossspectral density of the effective chirped random beam, corresponding to the fractional order p_{R} in the reference arm equal to the fractional order p_{E} at the encoding stage. In Fig. 3d we show the intensity I_{R}(v) of a recovered image obtained with the aid of an inverse Fourier transform of the measured crossspectral density. We also display in Fig. 3c the image illuminating the RGGD, i.e., the plaintext, for comparison. It can be inferred from Fig. 3 that the crossspectral density of a random beam can, indeed, be viewed as an effective carrier of optically encoded information. Furthermore, we find that whenever the encryption and decoding key match well, the decoded image is consistent with the input to the system.
To verify the reliability of our encryption protocol, we vary the distance d between the CL and BS to introduce mismatch between p_{R} and p_{E}. In Fig. 4 we exhibit the experimental results for the measured crossspectral density and the corresponding reconstructed image for p_{R}=0.5 (top) and p_{R}=0.7 (bottom). We can infer from Figs. 4c and f that the original plaintext image is unrecognizable as long as p_{R}≠p_{E}. To visualize the evolution of the reconstructed image with p_{R}, we show in Fig. 5 the experimental results for the reconstructed images with p_{R} varying from 0.5 to 0.72, with the increments of 0.02. We observe that the reconstructed image rapidly blurs as the difference between p_{R} and p_{E} grows. The plaintext image, i.e., ‘SUDA’, can only be recognized for p_{R}=0.6±0.04. The existence of such value range is due to the property of the fractional Fourier transform function used in the encoding system. The function contains only a single encryption key. We remark that the security of the system can be further improved, e.g., by using a more complicated response function and by increasing the number of encryption keys in the encoding system [64].
We can infer from Eq. (14) that not only the structure but also the position of the image can be encoded into the crossspectral density function of a random beam. It follows that our protocol allows for moving image encryption. To illustrate this point, we loaded a string of characters ‘SUDA’, moving along a straight line, into the SLM. In the top two panels of Fig. 6, we show the experimental results for the measured crossspectral density of the beam illuminating the moving string at different locations. In the bottom panels, we display the corresponding reconstructed images. In this experiment, p_{E}=p_{R}=0.6. The experimental results in Fig. 6 indicate that the crossspectral density of the illuminating beam can be successfully employed to carry information about the structure and instantaneous position of a moving image. We notice here the time scale for encryption and decryption is about 0.733 seconds, which is shorter than the time difference (1 second) between two locations of the object in our experiment. Thus, the object can be regarded as the “moving” object. Moreover, we remark that the crosstalk appeared in the recovered images is induced by the imperfect circular polarization of the generated reference beam, which however can be decreased by using the wave plates with higher quality, e.g., the true zeroorder wave plates.
Finally, we demonstrate the robustness of our protocol against fluctuations in a noisy environment, such as the turbulent atmosphere. In our experiment, the fluctuations are simulated by a thermally induced turbulence generated by a hot graphitic plate with an adjustable temperature T. The turbulence strength grows with the temperature, which can be seen from the distortion of the intensity distribution, for example see in [40]. In the experiment, the turbulence is added only in the path of the random beam, i.e., the fluctuations are introduced into the 2f imaging system as shown in Fig. 7a, since the crossspectral density measurement with the reference fields can be performed, in general, separately in a turbulencefree environment. In Fig. 7c—f, we show the experimental results for crossspectral density and the corresponding reconstructed images for noisy environments at different temperatures. In Fig. 7b we exhibit the experimental results with no hot plate. We infer from the figures that our protocol is quite insensitive to the environmental noise. Hence, the encrypted images can be recovered despite the presence of even strong medium fluctuations. Thus, the secondorder optical correlations can be regarded as a robust “lock” for our encryption system that serves to protect against attacks aimed at erasing rather than compromising the encrypted information. The robustness of the crossspectral density distribution of light in our protocol against the environmental fluctuations is tightly linked to efficient selfhealing of lowcoherence light [28, 29, 40].
Conclusions
We have presented a protocol for optical information encryption into the secondorder correlations of random light fields expressed in terms of the crossspectral density of the field. We have verified the feasibility of our protocol by carrying out a proofofprinciple experiment by encoding an optical image (plaintext) into the crossspectral density of a random light beam with the aid of a fractional Fourier transform encoding system and decoding the plaintext through the measurement of the said crossspectral density with the help of the recently introduced generalized Hanbury Brown–Twiss technique [40].
We argue that the vast amount of data required to reconstruct the crossspectral density distribution, compared with a relative paucity of data necessary in the traditional encryption protocols based on measuring firstorder deterministic characteristics of light fields, enhances the security of the encryption protocol. We have demonstrated experimentally that the crossspectral density measurement is extremely robust against environmental fluctuations, e.g., the atmospheric turbulence. Therefore, the optical information can be well reconstructed despite a high level of noise in the ciphertext. This circumstance bestows certain advantages on our protocol over the alternatives in the application scenarios where an attacker aims to deliberately destroy the ciphertext rather than temper with it. Our results indicate that the spatial coherence of a random light beam can serve as a versatile tool in optical security and encryption applications. Finally, we remark that the proposed coherencebased encryption system is performed only in a very short optical path, it can, however, be extended to a longdistance optical image transmission system with the aid of the coherence phase modulation (Liu et al.: Robust farfield imaging by spatial coherence engineering, unpublished).
Availability of data and materials
The experimental data that support the works of this study are available from the corresponding authors on reasonable request.
Declarations
Abbreviations
 DRPE:

Double random phase encoding
 RGGD:

Rotating groundglass disk
 LP:

Linear polarizer
 BS:

Beamsplitter
 BE:

Beam expander
 SLM:

Spatial light modulator
 HWP:

Halfwave plate
 QWP:

Quarterwave plate
 CL:

Concave lens
 PBS:

Polarization beamsplitter
 DDS:

Directdigital synthesis signal generator
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Funding
National Key Research and Development Project of China (2019YFA0705000), National Natural Science Foundation of China (NSFC) (91750201, 11874046, 11974218, and 11904247), Innovation Group of Jinan (2018GXRC010), Local Science and Technology Development Project of the Central Government (YDZX20203700001766), China Postdoctoral Science Foundation (2019M661915), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJB140017), Natural Sciences and Engineering Research Council of Canada (RGPIN201805497).
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Yahong Chen, Fei Wang, and Yangjian Cai conceived the original idea and designed the experiment. Deming Peng, Zhaofeng Huang, and Yonglei Liu conducted the experiment. All authors analyzed their experimental results and contributed to writing and proofreading the manuscript. Yangjian Cai supervised the overall project. The authors read and approved the final manuscript.
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Peng, D., Huang, Z., Liu, Y. et al. Optical coherence encryption with structured random light. PhotoniX 2, 6 (2021). https://doi.org/10.1186/s4307402100027z
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DOI: https://doi.org/10.1186/s4307402100027z