# Angle Multiplexing Optical Image Encryption in the Fresnel Transform Domain Using Phase-Only Computer-Generated Hologram

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## Abstract

**:**

## 1. Introduction

_{0},y

_{0}) and (x,y), respectively, as being perpendicular to the optical axis. Therefore, the distance z

_{0}between the two planes is fixed. A coherent light source (plane wave) of wavelength λ illuminates the POF h(x

_{0},y

_{0}) and the amplitude of the diffraction field approximates the target image. In determining the diffraction field g(x,y), the FrT is used, and can be expressed as Equation (1) [13]:

## 2. Preliminaries

_{0},y

_{0}) and (x,y) denote the original coordinates of the input POF and the output image reconstruction planes, respectively; z

_{0}is the distance between the two origins at the two planes; $\left({x}_{0}^{\prime},\text{}{y}_{0}^{\prime}\right)$ and (x′, y′) denote the coordinates of the rotated POF and image reconstruction planes, respectively. The rotation angles of the tilted POF and image reconstruction planes are denoted as ($\varphi $

_{x}, $\varphi $

_{y}) and (θ

_{x}, θ

_{y}), respectively.

**A**and

**A′**at the rotated POF and reconstruction planes, respectively. Equation (3) shows how to determine this distance.

## 3. Proposed Method

_{n}, n = 1, …, N}, and their corresponding rotation angles, ${\mathbf{\Theta}}_{n}=\{{\theta}_{{x}_{n}},\text{}{\theta}_{{y}_{n}}\}$ and ${\mathbf{\Phi}}_{n}=\left\{{\varphi}_{{x}_{n}},\text{}{\varphi}_{{y}_{n}}\right\}$, then ${\theta}_{{x}_{n}}$, ${\theta}_{{y}_{n}}$, ${\varphi}_{{x}_{n}}$, and ${\varphi}_{{y}_{n}}$ denote the angles of the rotation about the ${x}^{\prime},\text{}{y}^{\prime},\text{}{x}_{0}$, and ${y}_{0}$ axes, respectively, for the nth image at the reconstruction plane. The POF, ${\Psi}_{n}\left({x}_{0},\text{}{y}_{0}\right)$, for each image can be retrieved by using the modified GSA (MGSA) [29] with the given parameters, including the distance z

_{0}, wavelength λ, and rotation angles {${\theta}_{{x}_{n}},{\theta}_{{y}_{n}},{\varphi}_{{x}_{n}},{\varphi}_{{y}_{n}}$} in the TFrTs. For multiplexing purposes, each POF is multiplied by a phase term, in which the corresponding spatial translation $({u}_{n},{v}_{n})$ is then defined. Next, all the POFs are summed and denoted as ${\Psi}_{T}\left({x}_{0},\text{}{y}_{0}\right)$, which is the POCGH, and can be displayed in a phase-only device. To correctly reconstruct a specific target image, g

_{n}, the corresponding angle parameters {${\theta}_{{x}_{n}},{\theta}_{{y}_{n}},{\varphi}_{{x}_{n}},{\varphi}_{{y}_{n}}$} should be provided in the TFrT. Various parameter combinations can be adopted in the MGSA for multiplexing and encryption purposes.

#### 3.1. Angle Multiplexing at the Image Reconstruction Plane

_{n}can only be reconstructed at its predefined position, $({u}_{n},{v}_{n}),$ when the reconstruction plane has the corresponding rotation angle ${\mathbf{\Theta}}_{n}$. Otherwise, no images can be reconstructed, and only the noise-like pattern is shown at the plane. However, the origin of the rotation angle for each image is different. That is, the translated images are based on the reconstruction planes whose origins are different from the original ones found in the optical axis.

_{n}in this optical architecture:

_{n}at the plane with a rotation angle ${\mathbf{\Theta}}_{n}$, Equation (7) can be expressed in the form shown in Equation (8),

#### 3.2. Angle Multiplexing at the POCGH Plane

_{n}can only be reconstructed at its predefined position when the POCGH plane corresponds with the rotation angle ${\Phi}_{n}$. Otherwise, no images can be reconstructed, and only the noise-like pattern results can be detected on the plane. Note that the distance ${\widehat{r}}_{2,n}$ from the rotated POCGH to the reconstruction plane can be determined by using Equation (14) shown below:

_{n}at the plane with the rotation angle ${\Phi}_{n}$, Equation (15) can be expressed by the form shown in Equation (16),

#### 3.3. Double Angle Multiplexing at the Both Planes

_{n}can only be reconstructed at its predefined position when both the POCGH and reconstruction planes have the corresponding rotation angles, ${\Phi}_{n}$ and ${\mathbf{\Theta}}_{n}$, respectively. Otherwise, no images can be reconstructed, and only the noise-like patterns are shown at the plane. The distance ${\widehat{r}}_{3,n}$ between the two rotated planes can be determined by using the same equation as Equation (6). That is, ${\widehat{r}}_{3,n}={\widehat{r}}_{1,n}$. The light distribution on the rotated plane can be determined based on the TFrT formula shown in Equation (20):

_{n}at the plane with the rotation angle ${\Phi}_{n}$, the rotation angle ${\mathbf{\Theta}}_{n}$ at the POCGH plane is also required. Equation (20) can be expressed in the form shown in Equation (21),

## 4. Results and Discussion

_{0}be 1.3 m, so that the approximations used in deriving the TFrT equations can be satisfied. All three optical architectures shown in Figure 3, Figure 5 and Figure 6 follow identical parameters. To limit the scaling effects caused by the rotation, we select all the rotation angles that are smaller than 15°. Table 1 shows the rotation angles of the POCGHs and reconstruction planes for the eight target images in the three optical architectures. Note that only the rotation angles ${\mathbf{\Theta}}_{n}$ and ${\mathbf{\Phi}}_{n}$ are used in the first and second architectures, respectively, while the both angles are required for the functioning of the third architecture.

_{0}and a given wavelength λ. For the POCGH with the size of 1920 by 1080 and 8 bits resolution, the key spaces for the brute force attack are 497,664,000 and 14,929,920,000 for the single- and double-plane rotation, respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Generalized optical diffraction architecture for a single image reconstruction using rotation manipulation on the POF and reconstruction planes.

**Figure 3.**The first type of angle manipulation for the proposed method, in which only the reconstruction planes are rotated with an angle ${\mathbf{\Theta}}_{n}$.

**Figure 4.**(

**a**) Top view and (

**b**) lateral view of the translated images in the reconstruction plane whose origin is different from the original one.

**Figure 5.**The second type of angle manipulation for the proposed method, in which only the POCGH are rotated with angle ${\Phi}_{n}$.

**Figure 6.**The third type of angle manipulation for the proposed method, in which the POCGH and reconstruction planes are rotated with the angles ${\Phi}_{n}$ and ${\mathbf{\Theta}}_{n}$, respectively.

**Figure 7.**The eight target images used in the computer simulation and their relative locations at the reconstruction plane.

**Figure 8.**The three reconstructed images (and their magnified and filtered versions) (

**a**) “House,” (

**b**) “Indian,” and (

**c**) “Peppers” at the output planes in the first, second, and third types of angle manipulation, respectively.

**Figure 9.**Angle sensitivities represented by CC values of the reconstructed “Peppers” image under the three different architectures: (

**a**) angle shifting at the reconstruction plane when both the POCGH and reconstruction planes are perpendicular to the optical axis; (

**b**) angle shifting at the rotated POCGH plane when the reconstruction plane is fixed; (

**c**) angle shifting at the reconstruction plane when the rotated POCGH is fixed; (

**d**) angle shifting at the POCGH plane when the rotated reconstruction plane is fixed.

${\mathbf{\Phi}}_{\mathit{n}}$ | ${\mathbf{\Theta}}_{\mathit{n}}$ | |||
---|---|---|---|---|

Image # | ${\mathit{\varphi}}_{{\mathit{x}}_{\mathit{n}}}$ | ${\mathit{\varphi}}_{{\mathit{y}}_{\mathit{n}}}$ | ${\mathit{\theta}}_{{\mathit{x}}_{\mathit{n}}}$ | ${\mathbf{\theta}}_{{\mathit{y}}_{\mathit{n}}}$ |

2 | 10° | 12° | 13° | 15° |

3 | 7° | 8° | 10° | 12° |

4 | 13° | 15° | 7° | 8° |

5 | 6° | 3° | 2° | 5° |

6 | −15° | −13° | −5° | −1° |

7 | −5° | −3° | −12° | −10° |

8 | −12° | −10° | −8° | −7° |

9 | −8° | −7° | −15° | −13° |

**Table 2.**CC values of all the reconstructed images for the first type of angle manipulation (rotation on the reconstruction plane).

ANGLE | IMAGE # | |||||||
---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

${\mathbf{\Theta}}_{2}$ | 0.92 | 0.43 | 0.00 | 0.28 | −0.07 | −0.07 | −0.03 | 0.06 |

${\mathbf{\Theta}}_{3}$ | 0.07 | 0.88 | −0.25 | 0.01 | 0.10 | −0.12 | 0.07 | −0.08 |

${\mathbf{\Theta}}_{4}$ | 0.06 | −0.22 | 0.87 | −0.03 | 0.13 | −0.03 | 0.02 | −0.03 |

${\mathbf{\Theta}}_{5}$ | 0.18 | −0.30 | 0.21 | 0.91 | −0.02 | −0.17 | −0.07 | −0.02 |

${\mathbf{\Theta}}_{6}$ | −0.14 | 0.00 | −0.15 | 0.04 | 0.92 | −0.13 | 0.10 | 0.00 |

${\mathbf{\Theta}}_{7}$ | 0.16 | 0.23 | −0.03 | −0.21 | 0.09 | 0.93 | −0.12 | 0.04 |

${\mathbf{\Theta}}_{8}$ | 0.10 | −0.40 | 0.36 | 0.04 | 0.22 | −0.07 | 0.95 | 0.00 |

${\mathbf{\Theta}}_{9}$ | −0.13 | 0.23 | 0.07 | 0.39 | 0.07 | −0.16 | 0.04 | 0.96 |

**Table 3.**CC values of all the reconstructed images for the second type of angle manipulation (rotation on the POCGH plane).

ANGLE | IMAGE # | |||||||
---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

${\mathbf{\Phi}}_{2}$ | 0.86 | −0.16 | 0.37 | 0.30 | −0.09 | −0.12 | −0.16 | 0.12 |

${\mathbf{\Phi}}_{3}$ | 0.07 | 0.81 | 0.33 | 0.40 | −0.12 | −0.07 | −0.17 | 0.01 |

${\mathbf{\Phi}}_{4}$ | 0.14 | −0.28 | 0.96 | 0.28 | −0.05 | −0.12 | −0.12 | 0.08 |

${\mathbf{\Phi}}_{5}$ | −0.03 | 0.28 | 0.28 | 0.90 | −0.11 | 0.02 | −0.17 | 0.00 |

${\mathbf{\Phi}}_{6}$ | 0.04 | −0.19 | −0.01 | 0.03 | 0.85 | 0.22 | 0.16 | 0.21 |

${\mathbf{\Phi}}_{7}$ | −0.12 | 0.23 | 0.24 | 0.25 | −0.11 | 0.94 | −0.16 | 0.16 |

${\mathbf{\Phi}}_{8}$ | −0.07 | −0.06 | 0.23 | 0.05 | 0.03 | 0.28 | 0.95 | 0.21 |

${\mathbf{\Phi}}_{9}$ | −0.14 | 0.26 | 0.24 | 0.12 | −0.05 | 0.35 | −0.08 | 0.96 |

**Table 4.**CC values of the reconstructed images for the third type of angle manipulation (rotations on both the POCGH and reconstruction planes).

ANGLE | IMAGE # | |||||||
---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

${\mathbf{\Phi}}_{2},{\mathbf{\Theta}}_{2}$ | 0.92 | 0.19 | −0.34 | 0.20 | −0.02 | −0.16 | 0.04 | 0.01 |

${\mathbf{\Phi}}_{3},{\mathbf{\Theta}}_{3}$ | −0.20 | 0.96 | 0.09 | −0.09 | −0.03 | 0.03 | 0.01 | 0.14 |

${\mathbf{\Phi}}_{4},{\mathbf{\Theta}}_{4}$ | 0.00 | 0.05 | 0.86 | −0.01 | −0.07 | −0.08 | 0.02 | 0.13 |

${\mathbf{\Phi}}_{5},{\mathbf{\Theta}}_{5}$ | 0.23 | −0.09 | 0.11 | 0.91 | −0.03 | 0.01 | 0.02 | −0.01 |

${\mathbf{\Phi}}_{6},{\mathbf{\Theta}}_{6}$ | −0.20 | −0.26 | 0.21 | −0.31 | 0.85 | 0.18 | −0.04 | −0.02 |

${\mathbf{\Phi}}_{7},{\mathbf{\Theta}}_{7}$ | −0.25 | −0.14 | 0.20 | 0.10 | 0.03 | 0.92 | 0.02 | 0.09 |

${\mathbf{\Phi}}_{8},{\mathbf{\Theta}}_{8}$ | −0.26 | 0.22 | −0.22 | 0.21 | −0.09 | 0.19 | 0.95 | 0.09 |

${\mathbf{\Phi}}_{9},{\mathbf{\Theta}}_{9}$ | −0.27 | 0.24 | −0.13 | 0.23 | 0.09 | −0.14 | 0.04 | 0.91 |

**Table 5.**Computation time (in seconds) for determining the eight POFs, POCGHs, and image decryption in the three different types of angle manipulation in the proposed method.

8 POFs | POCGH | Total/Average Image Decryption | |
---|---|---|---|

Type-1 | 656.38 | 3.86 | 2.38/0.3 |

Type-2 | 614.34 | 3.79 | 2.40/0.3 |

Type-3 | 768.91 | 4.39 | 3.22/0.4 |

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**MDPI and ACS Style**

Chang, H.T.; Wang, Y.-T.; Chen, C.-Y.
Angle Multiplexing Optical Image Encryption in the Fresnel Transform Domain Using Phase-Only Computer-Generated Hologram. *Photonics* **2020**, *7*, 1.
https://doi.org/10.3390/photonics7010001

**AMA Style**

Chang HT, Wang Y-T, Chen C-Y.
Angle Multiplexing Optical Image Encryption in the Fresnel Transform Domain Using Phase-Only Computer-Generated Hologram. *Photonics*. 2020; 7(1):1.
https://doi.org/10.3390/photonics7010001

**Chicago/Turabian Style**

Chang, Hsuan T., Yao-Ting Wang, and Chien-Yu Chen.
2020. "Angle Multiplexing Optical Image Encryption in the Fresnel Transform Domain Using Phase-Only Computer-Generated Hologram" *Photonics* 7, no. 1: 1.
https://doi.org/10.3390/photonics7010001