Abstract
Digital holography has indeed led to numerous advancement of the classical, analog holographic technology that only permits a hologram to be permanently recorded onto a photographic film. In digital holography, a hologram can be captured from a real object. It can also be numerically generated as an array of numbers that can be stored as digital data, processed through computation, and distributed via digital communication links. In general, the primary purpose of holograms is to display 3-D images. Hence, a digital hologram in digital data form will not be of much practical use if it cannot be visually seen as a 3-D image. This is one of the major disadvantages of a digital hologram compared with the optical hologram, which can be readily captured with our eyes. However, a digital hologram can have different applications apart from generating 3-D images. In fact, recent research has shown that a digital hologram can be utilized in protecting sensitive data (a task referred to as cryptography), or in steganography for embedding large amount of additional data. This chapter describes some of the important applications of digital phase-only holograms in 3-D display, holographic cryptography, and steganography.
5.1 Introduction
Digital holography has indeed led to numerous advancement of the classical, analog holographic technology that only permits a hologram to be permanently recorded onto a photographic film. In digital holography, a hologram can be captured from a real object. It can also be numerically generated as an array of numbers that can be stored as digital data, processed through computation, and distributed via digital communication links. In general, the primary purpose of holograms is to display 3-D images. Hence, a digital hologram in digital data form will not be of much practical use if it cannot be visually seen as a 3-D image. This is one of the major disadvantages of a digital hologram compared with the optical hologram, which can be readily captured with our eyes. However, a digital hologram can have different applications apart from generating 3-D images. In fact, recent research has shown that a digital hologram can be utilized in protecting sensitive data (a task referred to as cryptography), or in steganography for embedding large amount of additional data. This chapter describes some of the important applications of digital phase-only holograms in 3-D display, holographic cryptography, and steganography.
5.2 Holographic Projection and Display
Modern computing technologies have enabled digital holograms to be generated with a commodity computer at video rate of over 25 frames per second. One may wonder why a holographic display unit is unavailable on the consumer market. The bottleneck in the popularity of holographic displays is caused by two major problems. First, a hologram is a complex-valued image composed of magnitude and phase components, and existing display devices can only present either the magnitude or the phase component. Second, holographic images comprise high-frequency fringes with the spatial frequency of optical waves. For the visible light spectrum, the wavelength of light is within the range [380 nm, 720 nm]. As such, the spatial period of fringes in a hologram of acceptable quality has to be under 10 μm, which is significantly less than the pixel size (roughly in the range 50–200 μm) of consumer-grade electronic displays (such as computer monitors and mobile phones).
The guiding objective of this book is to describe different methods developed to date for producing phase-only holograms, addressing the first problem mentioned above. For the second problem, so far digital holograms can only be displayed on special high-resolution devices that are commonly referred to as spatial light modulators (SLMs). Current SLMs are relatively small, in the order of a few centimeters, but are expensive. It will take years for them to evolve to a comparable size to televisions or computer monitors and to become a consumer commodity. Nevertheless, an SLM is still an effective tool for advancing the frontier of holographic technology. This section will describe the experimental setup and the important factors in displaying holograms on an SLM.
5.2.1 Spatial Light Modulator
“Spatial light modulator” is a general term for devices that are capable of modulating the amplitude, phase, or polarization of light. A typical SLM consists of a 2-D rectangular array of cells (pixels). The optical properties of light passing through or reflecting from each cell can be individually controlled by applying an external electrical signal. There are two major types of SLMs. The first one is based on translucent liquid crystal (LC) devices that modulates the light passing through them. The second type, SLM with liquid crystal on silicon (LCoS), reflect and modulate the light that impinges on it. The quality of an SLM is evaluated with a set of parameters obtained through a number of measurements. These parameters, listed and described in Table 5.1, are crucial in a holographic display system.
Table 5.1 Major specification of an SLM
For effective holographic display, an SLM should have a large dimension, small pixel pitch, short response time, high light utilization efficiency, and large fill factor.
Liquid crystal on silicon devices are commonly used in modern overhead projectors, and it is a preferable technology compared to LC devices due to its larger fill factor and higher light utilization efficiency. There are numerous LCoS-based SLM products available on the market, with large deviations in their technical specifications. Some of the popular suppliers include, but are not limited to, Hamamatsu Photonics, Holoeye Photonics AG, and the Jasper Display Corp.
A typical LCoS SLM device consists of a layer of liquid crystal material sandwiched between silicon and glass substrates. The silicon substrate is a 2-D rectangular array of pixel electrodes, and an active matrix circuit is embedded to control the electrical field injected to individual pixel electrodes. When the electric field of a particular pixel electrode is adjusted, the birefringence, and hence the phase retardation of the liquid crystal above the electrode, will change accordingly. A simplified structure of the LCoS SLM is shown in Figure 5.1.
One can easily infer that if the electrical field of the active matrix circuit is adjusted to make the phase-shifting of the 2-D array pixels in the SLM similar to that of a phase-only hologram, the light reflecting from it will reconstruct the 3-D image that is represented by the hologram.
Basically there are two modes of optical reconstruction of a hologram by an SLM. The first mode is holographic projection, whereby the diffracted optical waves emitted by the hologram are projected onto a screen to show the reconstructed image. The screen can be a plane that is parallel to the hologram, or a 3-D surface. If a divergent beam illuminates the hologram, the size of the projected image will increase linearly with the distance between the screen and the hologram. On the other hand, if a plane wave is used instead of a divergent beam, the size of the projected image will be similar in size to the SLM, independent of the location of the screen. In both cases, the hologram should be generated in a way that the reconstructed image is in focus on the entire screen. For a planar screen that is parallel to the hologram, the focused plane is set to be at the location of the screen.
In the second mode of optical reconstructed images, the hologram is displayed as a visible 3-D image on the SLM. These two modes of hologram reconstruction will be described in the subsequent part of this section. To enhance the quality of the reconstructed image, the size of a hologram should be large, and the resolution should be high. A large SLM with fine pixel size, therefore, is desirable for displaying a hologram. The size and resolution of an SLM can be expressed by a dimensionless term known as the space–bandwidth product (SBP). The SBP is the product of the display area and the resolution of an SLM. Suppose an SLM device comprises of M columns and N rows of pixels, each being a square cell of side length δd
meters. The size and bandwidth of the SLM are (Mδd×Nδd)
m2, and (δd)−1
for both dimensions, resulting in SBP = (Mδd×Nδd)(δd)−1(δd)−1
= (M×N).
5.2.2 Holographic Projection
A typical optical setup for holographic projection with a phase-only hologram and a phase-only LCoS SLM is shown in Figure 5.2(a). Assuming a planar screen that is parallel to the hologram, a phase-only hologram is generated from a given source image located at a distance z from the hologram plane. It is then displayed on a phase-only SLM. Next, a point laser beam is converted into an optical plane wavefront with a beam expander, and taken to illuminate the SLM through the beam splitter. Each pixel on the SLM, upon impinging by the incoming beam, scatters a wavefront that is modulated by a phase-shift equivalent to that of the hologram pixel it represents. A screen is placed at a distance z from the hologram to show the projected image, which is sometimes referred to as a “real image.” The collection of optical wavefronts of all the pixel electrodes is reflected by the beam splitter, forming a focused optical reconstructed image on the screen. In this configuration, the illumination beam, the axial direction of the hologram, and the projected image are situated along the same optical path. As such, it is referred to as an inline holographic projection system.
Figure 5.2 (a) An inline holographic projection unit. (b) An off-axis holographic projection unit.
A major disadvantage of the setup in Figure 5.2(a) is that the illumination beam and the optical waves emitted by the SLM are attenuated by the beam splitter. This will lead to a dim reconstructed image as a large proportion of the optical beam has been diverted to other locations. An alternate setup to overcome this problem is to adopt the slightly off-axis arrangement, as in Figure 5.2(b). The optical paths of the illumination beam and the holographic waves are separated. In this off-axis arrangement, the reconstructed image is brighter as the expanded laser beam is used to illuminate the hologram directly.
5.2.3 Holographic Display
A holographic projector is similar to a traditional projector; the main difference is that the focusing of the projected image is produced by the computation of the hologram rather than by using an optical lens. For holographic projection, the focusing surface on which the reconstructed image is formed is fixed by the geometry of the screen. As such, a holographic projector is limited to showing a 2-D image when it is projected onto a flat surface, rather than the full functionality of a hologram’s true 3-D image. This limitation can be overcome with a 3-D holographic display, whereby the reconstructed image of a hologram is conveyed as a “virtual image,” rather than a “real image.” In a holographic projector, a real image is projected from the hologram onto a screen, forming a visible image. A virtual image is a 3-D image that is formed behind the hologram. Observing a virtual image is like looking at a 3-D scene through a window, with the viewing zone limited by the coverage of the SLM. As an observer views through the window at different angles, different parts of the 3-D scene will become visible. In addition, when the viewpoint is shifted, the disparity between objects at different depths in the scene will also change, as if the observer is looking at a real physical environment.
5.3 Holographic Encryption
After learning the primary role of phase-only holograms in image projection and display, we shall now have a look at some practical applications of phase-only holograms in cryptography. Cryptography, also known as encryption, is the art of changing numerical data to a form that can be interpreted by someone with the knowledge of how to revert the changes. It is an important technology for protecting private information from illegitimate access. Modern internet technologies, such as cloud and optical networks, are fully capable of storing and delivering big data sets that are highly vulnerable to security threats and malicious attacks. The problem is particularly critical when the service provider is an unknown common third party, whereby data could be eavesdropped upon during the course of distribution. If a communication link is being eavesdropped upon, all the sensitive contents in the data stream will be exposed, which can lead to severe security and financial impacts. To prevent this, cryptography is mandatory for maintaining the confidentiality of sensitive data and preventing it from being deciphered by unauthorized parties. Data that has been encrypted is generally known as ciphertext. Over the years, numerous research attempts have been conducted in the area of cryptography, resulting in different forms of realization, and some of them are rather complicated. The basic concept of cryptography is illustrated and explained with the simple block diagrams in Figure 5.3.
Figure 5.3 Fundamental framework of cryptography.
Cryptography comprises two parts. The first part is known as the encoder, which converts the source data into a ciphertext using a secret encryption key KE. The key can be in many forms, such as an image, a character string, or a mathematical operation on the input data. The goal is to generate a ciphertext that is significantly different from, and cannot be easily reverted back to, the source data. The second part of cryptography, known as the decoder, carries out the decryption process to revert the ciphertext to the source data with a decryption key KD. Most encryption methods are symmetric, meaning that the encryption and decryption keys are identical (i.e., KD= KE). If the encryption key is exposed, the source data can be easily obtained from the ciphertext. Many intruders, commonly known as cryptanalysts, have been attempting to deduce the secret keys of different encryptions through hacking attacks.
A well-known approach to deduce the encryption key in symmetric encryption is the “known plaintext attack” (KPA), with which the secret key (which is the same in both encryption and decryption) is deduced from the relation between a known source data set and its ciphertext. For example, a simple encryption that is performed by changing each value in the data set S=[32,−200,32,−128,−200,32,32,−128]
into a different one, resulting in the ciphertext Se=[−16,112,−16,5,112,−16,−16,5]
. The encryption/decryption key is a mapping between the original values S=[32,−200,32,−128,−200,32,32,−128]
with the modified values [−16,5,112]
. Suppose the source data and the ciphertext are both available, the secret key can be easily derived by comparing each element in S
with its corresponding counterpart in the ciphertext Se
.
Optical encryption is a variant of classical encryption, mainly applied in encrypting images. Instead of encrypting the source image directly, optical encryption methods are based on modifying the optical waves of a source image along its path of propagation. This approach is more secure than classical encryption, as the optical waves of a source image can be very different from the image itself. Hence even if the source image and its ciphertext are both available, it will be improbable to deduce the modifications that have been made to the optical waves emitted by the source image. Holographic encryption is a kind of optical encryption, whereby the hologram of the source data is encrypted. In this chapter we focus on holographic encryption of phase-only holograms. A hologram is basically used to represent pictorial information; to be explained later, it can also be employed to store and encrypt digital data.
5.3.1 Optical Cryptography
Most of the traditional encryption methods developed to date are designed to operate on the raw data directly. As mentioned before, such a kind of encryption can be cracked using KPA. In the past few decades, rapid advancements in optics have led to the emergence of optical encryption, adopting the theory of optics. Being different from data encryption in general, optical encryption is primarily used to encrypt optical images such as digital photographs. Despite the restricted scope of application, image encryption is important nowadays as there is a huge amount of image data transactions in contemporary social networks, wireless channels, and video broadcasting media. In optical encryption, the image to be encrypted is illuminated with a light source and the optical waves emitted from it are modified (based on an encryption key) at certain point(s) in the course of propagation toward a target destination. Grossly speaking, it is the light waves of the image, rather than the image itself, that are encrypted. Holographic encryption is a kind of optical encryption in which the wavefront selected for encryption is a hologram of the source image. A hologram is a 2-D complex image representing a 3-D image. Commonly, a hologram image comprises high-frequency fringe patterns that offer little clue on the image they represents. It is thus improbable to deduce the encryption key based on the ciphertext and the source image of a hologram.
5.3.2 Double Random Phase Optical Encryption
Effective optical/holographic encryption techniques developed to date are mostly based on the double random phase encoding (DRPE) framework. The method was first developed and reported by Philippe Refregier and Bahram Javidi in 1995 [1]. It is mainly used for encrypting optical images with which the light wave is encrypted in the spatial domain, and subsequently in the frequency space. The schematic of DRPE encryption is shown in Figure 5.4(a).
The source image I(m,n)
to be encrypted is illuminated with an on-axis coherent plane wave. The optical wave emitted by the source image is passed through a random phase mask M1(m,n)=eiφ1(m,n)
( φ1(m,n)
is a random quantity within the range [0,2π)
), which is taken to be the first encryption key. The light wave after passing through the random phase mask, given by IN(m,n)=I(m,n)eiφ1(m,n)
, is Fourier transformed to its frequency spectrum IN˜(ωm,ωn)
with lens L1. A random phase mask, M2(ωm,ωn)=eiφ2(ωm,ωn)
, taken as the second encryption key, is placed at the Fourier plane containing the spectrum of IN(m,n)
. Finally, the noise-added spectrum, denoted by IN˜(ωm,ωn)eiφ2(ωm,ωn)
, is inverse transformed into the spatial domain with lens L2, resulting in a DRPE ciphertext C(m,n)
. Denoting the forward and inverse Fourier transform operations as ℱ
and ℱ−1
, respectively, the ciphertext is given by
(5.1) The decoding process is simply the reversal of the encryption, as shown in Figure 5.4(b). The ciphertext is illuminated with a coherent plane wave, and the optical wave emitted from it is converted into the frequency spectrum with lens L1. A phase mask M2†(ωm,ωn),
which is the conjugate of M2(ωm,ωn)
, is placed on the Fourier plane. Upon exit from the phase mask, the light wave is inverse Fourier transformed to give a decrypted image ID(m,n)
on the image plane. Mathematically:
(5.2) With the use of a digital CCD camera, the magnitude of the decrypted image, given by I(m,n)2
, is recorded.
The DRPE has two major disadvantages. First, the optical setups of the encoder and the decoder are quite complicated, as part of the process has to be conducted in Fourier space. Second, according to the evaluation on the methods in [2–9], the DRPE is vulnerable to different kinds of attacks. In particular, if the encoder is available, the encryption keys can be deduced with the chosen plaintext attack (CPA) and/or the KPA. In these two kinds of attacks, the source images and the ciphertexts are both available, and the encryption key can be revealed by analyzing the relation between them. According to [9], the random phase mask M2(ωm,ωn)
could be easily deduced from the ciphertext of a Dirac impulse (i.e., a single dot). There are different variations on the implementation of the DRPE method to increase its resistance to attacks, such as, but not limited to, phase reservation and compression [10], photon counting [11], space multiplexing of multiple images [12], fractional Fourier transform [13,14], randomized lens-phase function [15], and Arnold transform [16]. Apart from image encryption, DRPE has been extended to encrypt digital data by embedding it in a QR code [17,18].
5.3.3 Single Random Phase Holographic Encryption
Inspired by DRPE, a low-complexity holographic encryption method known as single random phase encryption (SRPE) was proposed in [19]. The concept of SRPE is shown in Figure 5.5(a). A 3-D source image is input to a stochastic hologram generator. In the generator, the source image is first modified in a random manner that retains its visual quality, and then placed in a larger global image G(m,n)
. In the figure, the source image is scaled in size and added with a text message “ABC” that is not overlapping with the image. The modified image is then converted into a phase-only hologram HP(m,n)
. By adding a random phase mask M(m,n)
(the encryption key), the phase-only hologram is converted into a SRPE hologram HE(m,n)=
HP(m,n)M(m,n)
(ciphertext). As is explained in more detail later, the process is equivalent to a one-time pad encryption generally considered to be an “unbreakable” cipher method.
Figure 5.5 (a) Concept of the single random phase encryption (SRPE) method. (b) Implementation of the SRPE encoder. (c) Implementation of the SRPE decoder.
Implementation of the SRPE encoder and decoder is shown in Figures 5.5(b) and 5.5(c), respectively. In the encoder, the source image is embedded into a global image and converted into a phase-only hologram HP(m,n)
. The latter is them overlaid with a random phase mask M(m,n),
resulting in an SRPE hologram HE(m,n)
. In the decoder, the encrypted hologram HE(m,n)
is overlaid with a phase mask M†(m,n)
, which is the complex conjugate of M(m,n)
, thus recovering the phase-only hologram HP(m,n)
of the global image. When HP(m,n)
is illuminated with a coherent beam, the global image and the modified source image that is embedded in it will be reconstructed to form a visible image. Figures 5.5(b) and 5.5(c) show that the complexity of the SRPE method is substantially lower than the DRPE. However, since the ciphertext is representing the global image rather than the source image, it will be improbable to apply KPA or CPA to deduce the encryption key.
The SRPE method is simple in structure and has the following advantages:
1. The source image is modified in an unknown and random manner, so that encryption of the actual image is unknown. The absence of knowledge on what is represented by the phase-only hologram and its encrypted data makes it difficult to deduce the encryption key by plaintext attacks.
2. For the same input source image, a vastly different phase-only hologram results in each round of encryption. Hence, it is not possible to deduce the encryption key by analyzing the correlation (or difference) between a large set of similar source images and their corresponding ciphertexts.
3. After adding the random phase mask, which is taken as the encryption key, it is even more difficult to deduce the encryption key through analyzing the relation between the input image and the SRPE hologram.
4. Upon decryption, the perceptual quality of the original source image is chiefly preserved favorably even though a certain part of it has been distorted.
Following is an explanation of the detail formulation of the SRPE method, which can be divided into three stages.
Stage 1: Stochastic Modification of the Source Image
The source image is modified randomly in a manner that is unknown to the person encrypting the image. Assume the encryption device, whether it is implemented with software or hardware, is a black box so that the details within it are concealed from the operator. To avoid causing too much change to the source image, the random modification is limited to a moderate amount of geometrical transformation (such as scaling and translation). After modifying the source image I(m,n)
, it will be pasted into an arbitrary position in a larger global image G(m,n)
. The rest of the areas in the global image that are not occupied by the source image will be filled with randomly generated content P(m,n)
. Mathematically, the global image is given by
(5.3) where T[]
represents some form of geometrical transform. Equation (5.3) constitutes a stochastic modification of the source image, so that the global image is substantially different from the original source image. The global images generated in different runs of the encryption process will also deviate from one another. This mechanism ensures that neither the cryptanalyst nor the one who performs the encryption will have knowledge of what is being encrypted. Without information on the source data, it will be improbable, if not impossible, to draw any relationship between the ciphertext and the source image. By the same argument, the stochastic modification will also prevent the encryption key from being exposed through large-scale CPA, as the images being utilized to probe the encryption engine will also change in an unknown manner prior to their encryption.
Stage 2: Converting the Global Image to a Phase-Only Hologram
The global image is first converted to a digital Fresnel hologram by convolving the global image with free space impulse response of the light propagation h(m,n;z0)
as
(5.4) The free space impulse response of light propagation is given by h(m,n;z0)=iλz0exp{i2πλ(mδd)2+(nδd)2+z02}
, where λ
is the wavelength of light, and δd
is the pixel size of the hologram. As the hologram is generated from the global image, it is referred to as the global hologram.
The complex-valued hologram H(m,n)
is then converted into a global phase-only hologram HP(m,n)=exp[iθ(m,n)]
with either uni-directional or bi-directional error diffusion.
Stage 3: Encrypting the Phase-Only Hologram with a Random Phase Mask
In this final stage, a random phase mask M(m,n)=exp[iφ(m,n)]
(where φ(m,n)
is a random value in the range [0,2π)) is overlaid onto the global phase-only hologram, resulting in an encrypted hologram
(5.5) In the decoder, as shown in Figure 5.5(c), the global phase-only hologram is recovered by overlaying the conjugate of the random phase mask M(m,n)
onto HE(m,n)
:
(5.6) The random phase mask M(m,n)
can be taken as the secret key for encrypting and decrypting the global image, which is different from the source image I(m,n)
. Hence, given I(m,n)
and HE(m,n)
, it is not possible to deduce the secret key M(m,n)
. After decryption, the global image can be reconstructed from the hologram through optical or numerical reconstruction. From the reconstructed global image, the source image, which is separated from the randomly added contents, will appear with good visual quality.
Numerical simulation of the SRPE method is conducted with the source code in Section 5.5.1. The source image “Fish_1” in Figure 5.6(a) is embedded into the larger canvas “Canvas” comprising of a QR code and a text message “Single random phase encryption,” resulting in a global image as shown in Figure 5.6(b). A phase-only hologram HP(m,n)
is generated from the global image with uni-directional error diffusion and infused with random phase noise M(m,n)
to give an SRPE hologram HE(m,n)
, as shown in Figures 5.6(c). The SRPE hologram is decoded by being overlaid with the conjugate of the random phase mask M†(m,n)
, and the reconstructed image is shown in Figure 5.6(d). It can be seen that apart from some mild noise contamination, the reconstructed image is similar to the original one.
Figure 5.6 (a) Source image “Fish1.” (b) Global image. (c) SRPE hologram of the global image. (d) Reconstructed image of the decoded SRPE hologram on the focused plane.
5.3.4 Enhanced Single Random Phase Holographic Encryption
Despite the effectiveness of SRPE, the source image is modified after encryption. Although the modification does not affect the visual quality, it may not be acceptable in some applications. In view of this, enhanced single random phase encryption (ESRPE) was proposed in [20]. In comparison with SPRE, the ESRPE has similar advantages, but will not lead to any alteration to the source image.
The ESRPE method comprises four stages, as shown in Figure 5.7 and outlined here.
In the first stage, the source image I(m,n)
to be encrypted is converted into an off-axis complex hologram based on angles of incidence θm
(along the horizontal direction) and θn
(along the vertical direction). If θm
= 0 and θn=0
, the complex hologram is equivalent to an on-axis or inline hologram. The source image can be a planar image or a 3-D image. When the off-axis hologram of the source image is illuminated with a plane wave, the source image will be reconstructed on the focused plane oriented at angles θm
and θn
along the horizontal and vertical directions, respectively, from the normal of the hologram plane. The hologram being generated from the source image is known as the source hologram.
In the second stage, one or more secondary images Ik(m,n)|1≤k≤K
(where K≥1
is the total number of secondary images) are randomly generated or selected from a pool of images that is unknown to the operator, and are converted into an off-axis complex hologram. For simplicity of illustration, only one secondary image I1(m,n)
is shown in Figure 5.7. Each off-axis hologram is generated based on a reference beam at an angle of incidence θk;m
(along the horizontal direction) and θk;n
(along the vertical direction). When the off-axis hologram of the secondary image is illuminated with a plane wave, the secondary image will be reconstructed on the focused plane oriented at angles θk;m
and θk;n
along the horizontal and vertical directions, respectively, from the normal hologram plane. The hologram being generated from a secondary image is known as the secondary hologram. The same applies if there is more than one secondary image, in which case there will be a set of secondary images with corresponding secondary holograms.
In the third stage, the source hologram and the secondary hologram(s) are summed up into a global hologram HG(m,n)
and converted into a global phase-only hologram HP(m,n)
with existing techniques such as, but not limited to, error diffusion [21,22]. The global phase-only hologram is very different from the source hologram and any of the secondary hologram(s). As the secondary image is randomly generated or selected from unknown sources, even for identical source images, a totally different global phase-only hologram will be generated in each run of the encryption process.
In the fourth stage, a random phase mask M(m,n)=eiφ(m,n)
is added to the global phase-only hologram HP(m,n)
, resulting in an ESRPE hologram HE(m,n)
. The latter is a ciphertext hologram that bears little correlation to the source image or the secondary image(s).
The mixing of the holograms of the source image and one or more secondary images ensures no prior knowledge of what is being encrypted, so that there is no direct relation between the ESRPE hologram and the source image. It also increases the difficulty of deducing the encryption key through large-scale CPA (e.g., encrypting a large number of source images, each with a small amount of incremental change, to locate the affected regions of the ciphertext hologram).
In the decoder, the ESRPE hologram is overlaid with the conjugate of the random phase mask (encryption key) and illuminated with a coherent beam. The source and the secondary images will be reconstructed at the focused plane along orientations HE(m,n)
and (θk;m,θk;n)
, respectively. If the angular separation between (θm,θn)
and the orientation(s) of the secondary holograms (θk;m,θk;n)
is large enough, the images will be formed at non-overlapping areas on the focused plane, and the source image can be observed visually at its original position without any geometrical change. The specific formulation of each stage of the proposed method is described in the following sub-sections.
Stage 1: Converting the Source Image into an Off-Axis Hologram
For simplicity of explanation, the source is assumed to be a 2-D planar image I(m,n)
, which is parallel to and at an axial distance z0
from the hologram. The source image is first converted into an off-axis complex Fresnel hologram H(m,n)
; R(m,n)
is an inclined plane wave with an angle of incidence (θm,θn)
.
(5.7) where R(m,n)
is an inclined plane wave with an angle of incidence (θm,θn)
.
Stage 2: Converting Secondary Images into an Off-Axis Hologram
The secondary images are assumed to be 2-D planar images Ik(m,n)|1≤k≤K
, which are parallel to and at an axial distance zk
from the hologram. However, more secondary images can be added by the same principle. Similar to stage 1, an off-axis hologram Hk(m,n)
is generated for the kth secondary image with the angle of incidence of the plane wave Rk(m,n)
at angles of incidence (θk;m,θk;n)
.
Stage 3: Generating a Global Hologram
In this stage, the source and the secondary holograms are integrated into a new hologram HG(m,n)
:
(5.8) Subsequently, HG(m,n)
is converted into a global phase-only hologram HP(m,n)
with the error diffusion method. In HP(m,n)
, the magnitude of all the pixels is set to a constant value.
Stage 4: Converting the Global Hologram into a ESRPE Hologram
Finally, the global hologram is multiplied with a random phase mask M(m,n)=eiφ(m,n)
, resulting in an ESRPE hologram given by
(5.9) The ESRPE hologram is taken as the encrypted (ciphertext) hologram, with the phase mask eiφ(m,n)
as the private encryption key. When the ciphertext hologram is overlaid with the conjugate of the phase mask, and illuminated with a coherent beam, the global image, comprising the source image and the secondary image(s), will be reconstructed. The decryption process can be conducted through optical or numerical reconstruction as
(5.10)