Phase retrieval without prior knowledge via single-shot Fraunhofer diffraction pattern of complex object An-Dong Xiong1 , Xiao-Peng Jin1 , Wen-Kai Yu1 and Qing Zhao1* Fraunhofer diffraction is a well-known phenomenon achieved with most wavelength even without lens. A single-shot intensity measurement of diffraction is generally considered inadequate to reconstruct the original light field, because the lost phase part is indispensable for reverse transformation. Phase retrieval is usually conducted in two means: priori knowledge or multiple different measurements. However, priori knowledge works for certain type of object while multiple measurements are difficult for short wavelength. Here, by introducing non-orthogonal measurement via high density sampling scheme, we demonstrate that one single-shot Fraunhofer diffraction pattern of complex object is sufficient for phase retrieval. Both simulation and experimental results have demonstrated the feasibility of our scheme. Reconstruction of complex object reveals depth information or refraction index; and single-shot measurement can be achieved under most scenario. Their combination will broaden the application field of coherent diffraction imaging. Fraunhofer diffraction, also known as far-field diffraction, problem12-14. These matrix complement methods require 4N- does not necessarily need any extra optical devices except a 4 (N stands for the dimension) generic measurements such as beam source. The diffraction field is the Fourier transform of Gaussian random measurements for complex field15. the original light field. However, for visible light or X-ray Ptychography is another reliable method to achieve image diffraction, the phase part is hardly directly measurable1. with good resolution if the sample can endure scanning16-19. Therefore, phase retrieval from intensity measurement To achieve higher spatial resolution around the size of atom, becomes necessary for reconstructing the original field. In a beams with shorter wavelength like hard X-ray or electrons broad sense, there are two ways to compensate for the lost are widely used these days. But these high energy particles phase part. One way is adding prior knowledge about the will heat the sample or even break the chemical bonds20. object; another way is taking multiple different Multiple measurements (like ptychography or matrix measurements. In 1978, Fienup developed an algorithm complement methods) appear difficult to be implemented for based on alternating projection using nonnegativity as a fragile sample under hard beams. Both priori knowledge and constraint of the recovered image2,3. Reconstruction for multiple measurements have their advantages in different noncrystalline object from the intensity measurement of circumstance if the object can meet certain condition. Fraunhofer diffraction was achieved by Miao et al. in 19994,5. However, sometimes conflicts occur between our needs and With the development of compressive sensing theory, demand of measuring methods. For example, getting 3D sparsity of the image in certain transformation is also information of biological sample ‘alive’ with high resolution incorporated to solve phase retrieval problem6-9. Recently, means complex object and only one beam shot with few method called PhaseMax is raised pursuing the solution optical devices. The most demanding phase retrieval shows maximizing the inner product with an approximation little noise endurance and usually ends up in simulation. vector10,11. These approaches above use prior knowledge to Application of phase retrieval is usually conducted by make up for the lost phase part. On the other hand, if several assuming the object is real, even when the diffraction pattern different measuring matrixes can be cast on the diffraction is non-centrosymmetric (meaning complex object). Here, we setups, matrix completion methods like Phase Lift and demonstrate the feasibility of reconstructing complex object Wirtinger Flow can perform well even for complex without prior information from the intensity measurement of 1 Center for Quantum Technology Research, School of Physics, Beijing Institute of Technology, Beijing 100081, China * Email: [email protected] single-shot Fraunhofer diffraction in lab condition. 11 w1 l 1 and w 2 l 2 , 22 The light field of two-dimentional Fraunhofer diffraction can ll12 sin[ (k1 k 1 )] sin[ ( k 2 k 2 )] be computed by P . For ll12 ()()k1 k 1 k 2 k 2 2 i( x cos cos x cos sin ) F(,)(,) dx dx O x x e 12. Here, 1 2 1 2 convenience, we set l12 l l below. k (cos cos ,cos sin ,sin ) represents the unit vector of emergent light; is wavelength; O(,) x12 x denotes the light field of the illuminated object. Fast Fourier transform (FFT), whose bases are orthogonal and can be quickly computed, is the most widely used discrete Fourier transform. Due to independence of these measurement vectors, if not provided with any prior information, any phase multiplied by the intensity of diffraction pattern for complex original field is reasonable and can be inverse transformed. To constraint the phase part, we need non- Fig.1 | Inner product of measuring vectors. Modulus of inner product for orthogonal measurement vectors, which means higher 1 1 1 1 l kk(0.0), [( , ),( , )], 100 . k is in the center, sampling density in k domain (diffraction plane). In order to 20 20 20 20 distinguish from FFT, we call this scheme high-density inner product with itself is 1, while inner product with Fourier transform (HFT). ll ** k ( n1 * , n 2 * ) ( n 1 or n 2 ) is 0. If the diffraction field is measured completely including the phase, the measurement can be taken as a linear process, HFT contains more information than FFT in intensity multiplying vectorized signal of original light field ox by measurement. Take the following matrix measurement matrix M. Measurement F at different k 0.2515 0.0404i 0.1371 0.3740 i 0.1159 0.1448 i position k of diffraction plane is the inner product of 0.6229 1.1616i 0.0090 0.3990 i 0.0460 0.2487 i 0.1437 0.3831i 0.0295 0.3449 i 0.9242 1.6586 i corresponding row mk and ox . for example, the magnitude of shifted (shifting the zero- Fk m k o x m k m j m j o x P kj F j , where jj 3 2 1 frequency component to the center) FFT 1 5 1 is m j is a set of complete orthogonal basis. 1 2 3 The inner product P of the measurement of k and centrosymmetric, while the 3×3 times HFT k is given by 2.0068 2.8691 2.5062 1.3127 1.4950 2.6203 2.7675 1.7153 0.5123 2.8418 3 1.8934 5.789 2 2.8311 2.3641 1 1.5491 w2 l 2 w 1 l 1 1 2.8992 2.4411 0.8247 1.2582 2.8169 3.1796 2.2561 1.0879 2.0708 1 2i [ x1 ( k 1 k 1 ) x 2 ( k 2 k 2 )] P dx dx e 21 2.6019 1.5598 0.5990 2.7811 4.0811 3.8968 2.4017 1.0879 2.1744 ll12 ww21 2.5966 1 1.6632 3.9892 5 4.2248 2.0843 1 2.5267 2i 2 i 2 i 2 i ()()kkwl1111 () kkw 111 ( kkwl 2222 )() ( kkw 222 ) 2.7206 0.8689 2.3112 4.4093 4.9224 3.5764 1.0408 1.7153 3.1297 e e e e 2.3040 0.5789 2.5272 3.9595 3.7562 1.9958 0.5767 2.6218 3.3089 22il il 12()()k k k k 1.1348 1 2.5836 2.9810 2 0.4123 1.9560 3 2.6749 1 1 2 2 0.5466 2.0701 2.6465 2.0272 0.9139 1.7703 2.7543 2.6218 1.3625 ll12 is not; because the magnitude part of orthogonal integer sin[ (k1 k 1 )]sin[ ( k 2 k 2 )] P , measurement is centrosymmetric while the phase part is not. ll 12()()k k k k 1 1 2 2 The fractional measurement, linear superposition of integer where w1,2 and wl1,2 1,2 represent boundary of the measurement, can reveal phase information via intensity. original field, k1,2 and k1,2 are the horizon and vertical Phase of Fourier transform is usually believed to carry more component of and . Specially, when information; the image of going around the loop to the same starting point. Loop FFT() O2 Omix iFFT(()) FFT O1 is visually more like FFT() O2 constraint for all loops in multi-dimensional case is O2 . But in the case of HFT, we can see the overlap of two equivalent to loop constraint for all 4-point loops (see set of conjugate phase solution of O1 (complex conjugate supplementary information). Without noise, only two in k domain means inversion of complex conjugate in x conjugate solutions will be generated under these loop domain) without any iteration algorithm from the image of constraints. HFT() O2 Omix iHFT(()) HFT O1 , where O2 is an all- HFT() O2 ones matrix. As can be seen from Fig.2b, shape of O1 emerges at high sampling ratio Rr 2 for certain noise 1 level, where is the interval between adjacent sampled rl measurement. Intensity of Fraunhofer diffraction is usually in dramatically uneven distribution, different types of noise have different effect on performance of phase retrieval. Noise in simulation hereinafter is in the form: a' a rnoi randn anoi randn , where a is the magnitude and randn is a Gaussian random matrix generated by Matlab whose variance is 1. To some extent, in order to retrieve the complex field from single-frame noisy Fraunhofer diffraction, the sampling ratio needed should be Fig.2 | Demonstration of different sampling density. a: coefficient ratio much higher than the regular 4× or 8×. between adjacent and secondary adjacent measurement by sampling density. P 11 Where qkk' j1 , k , k' k , j 0, j 1 b: P20 l r l 12 As shown in Fig.1, the inner products exhibit strong locality.