## 1 Introduction  
Information security is significant in ensuring the integrity of the science and engineering applications [1, 2, 3]. In comparison with the digital encryption method, optical encryption is associated with more encryption parameters thus more secure [4, 5, 6]. Taking advantage of the parallel nature of light, the optical encryption system can operate in parallel, and its operation speed is much faster than that of the digital system. The early optical encryption model created by [PERSON] and [PERSON] simply consists of the 4/ encryption system known having high encryption strength [7]. Due to the increase of cryptography attacks [8, 9], more advanced optical encryption methods have been further developed, including fractional Fourier transform [10], gyrator transform [11], optical scanning holography [12], ghost imaging [13], phase truncation [14]. Among them, optical superposition either coherent superposition [15] or incoherent superposition [16, 17, 18] has been actively studied. Compared with incoherent superposition, coherent superposition needs more accurate alignment, which is thus able to ensure the sensitivity of the encryption system. More specifically, [PERSON] proposed a simple optical image encryption scheme based on optical coherence superposition without iterative coding [19]. The encryption of this system is based on mathematical operations, and the decryption uses a beam splitter for interference. In contrast, [PERSON] introduced a new scheme using interference-based two-dimensional binary phase mask and encrypting multiple images simultaneously [20]. [PERSON] proposed an asymmetric coherent optical encryption system, which mainly uses equal modulus decomposition (EMD) to decompose an image into a ciphertext and a key [21]. More recently,Chen proposed an optical image cryptosystem based on two beam coherent superposition and unequal modulus decomposition [22]. Different from the EMD approach, the method uses common vector decomposition to complete the encryption process. Among them, the amplitude term is used as ciphertext and the phase term is used as private key. The system generates four different phases and amplitudes, which improves the security of the encryption system. [PERSON] proposed a double image encryption method based on optical interference and logistic mapping [23]. These parameters, such as wavelength, wheelbase, and logistic mapping conditions are used as additional keys to improve the security of the system. [PERSON] proposed an asymmetric scheme for silhouette-free multiple encryption using coherent superposition and the Fresnel transform [24]. Within this approach, the ciphertext and private key in the encryption process are obtained by random mask, and a function is constructed in the encryption process to improve the security. In addition, different images are decrypted using different private keys. Even if one or two of the correct phase masks are obtained, the information of any original image, including its contour, cannot be obtained. [PERSON] proposed a multiple image encryption using Fresnel domain high-dimension chaotic phase coding [25]. In this photoelectric method, the color image is decomposed into R, G and B channels, each channel is individually phase-coded and modulated, and Fresnel transform is performed to generate the corresponding encrypted channel. It is therefore convenient to update, manage and transmit the decryption key. While these methods have proposed more secure encryption, the efficient use of beam power has not been well addressed. The applied beam is often weak thus adversely impacts the detection accuracy of an object through the decryption process, thus limiting its practical use. In our research, a novel method for multiple-image encryption based on coherent beam combining (CBC) using a conjugate Dammann grating (CDG) is proposed.