Research ArticleVol.29,No.5/1 March 2021/Optics Express 7197Check foOptics EXPRESSupdatesHigh-resolution multi-wavelength lenslessdiffraction imaging with adaptive dispersioncorrectionYUANYUAN LIU,1 QINGWEN LIU,1,4 YOU LI,2.3 BINGXINXU,1 JUNYONG ZHANG,2,5 AND ZUYUAN HE1,6State Key Laboratory of Advanced Optical Communication Systems and Networks,Shanghai Jiao TongChinese Academy of Sciences.Shanghai 201800.ChinaCenter of Materials Science and Optoelectronic Engineering.University of Chinese Academy of Sciences,No.19A,Yuquan Road.Beijing 100049.Chinaliuqingwen @sju.edu.cn5zhangjy829@siom.ac.cnAbstract:Multi-wavelength imaging diffraction system is a promising phase imaging technologydue to its advantages of no mechanical movement and low complexity.In a multi-wavelengthfocused system,spectral bandwidth and dispersion correction are critical for high resolutionreconstruction.Here,an optical setup for the multi-wavelength lensless diffraction imagingsystem with adaptive dispersion correction is proposed.Three beams with different wavelengthsare adopted to illuminate the test object,and then the diffraction patterns are recorded by a imagesensor.The chromatic correction is successfully realized by a robust refocusing technique.High-resolution images can be finally retrieved through phase retrieval algorithm.The effectivenessand reliability of our method is demonstrated in numerical simulation and experiments.Theproposed method has the potential to be an alternative technology for quantitative biologicalimaging.@2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement1.IntroductionOptical phase imaging technology is an important tool to explore the microscopic world,andits applications cover many fields,such as biological specimens,material testing and surfaceshape measurement.However,only the intensity of the scattered signal is detected and the phaseinformation is lost in most optical equipment.Therefore,how to recover the phase information bymaking use of the obtained intensity patterns is always an essential issue in optical measurement.Until now,the complete wavefront can be recovered by using typically two different strategies:interferometry [1,2]and coherent diffraction imaging (CDI)[3-5].In interferometry,a referencebeam is added,which makes the system complicated and also very sensitive to environmentalvibration.CDI is a lensless phase retrieval technique that uses multiple diffraction patterns toreconstruct the complex-valued object by iterative phase retrieval [6-8].Compared with theinterferometry,CDI has the advantages of simple system structure,without the reference beam,and capability of reaching the diffraction resolution limitation theoretically.The convergence ofiterative algorithm is the core problem of CDI.However,weak convergence and limited imagingrange problems exist in early phase retrieval algorithms,such as the Gerchberg-Saxton(GS)[9]and Hybrid Input-output(HIO)[10].In order to overcome the slow convergence of traditional CDI,in recent years,many researchershave improved and developed CDI imaging technology by using the concept of increasing dataredundancy based on multiple measurements.Over sampling of diffraction information can#419128https://doi.org/10.1364/E.419128Journal 2021Received 6 Jan 2021:revised 12 Feb 2021:accepted 13 Feb 2021:published 22 Feb2021Research ArticleVol.29,No.5/1 March 2021/Optics Express7198Optics EXPRESSensure the convergence and accuracy of the iterative algorithm in the object reconstruction process.In 2004,Rodenburg et al.proposed a phase-retrieval method based on lighting-probe scanning ofa specimen [11,12].Subsequently,the ptychographical iterative engine(PIE)has been developedrapidly.For example,Maiden et al.proposed the ePlE algorithm which can simultaneouslyreconstruct the illumination light and the complex amplitude distribution of sample [13]:Asingle-exposure PIE imaging setup based on grating splitting [14]was demonstrated by Cheng Liu,and Ref.[15]reported a super-resolution Fourier Ptychographic Microscopy technology(FPM)based on LED array to achieve multi-angle illumination for frequency domain scanning.Differentfrom the above PIE method,Pedrini and Osten proposed the single-beam multiple-intensityreconstruction technology (SBMIR)[16],and the phase information can be recovered by movingthe camera or the sample for axial scanning to obtain the intensity patterns under differentdistances.On this base,Ref.[17]proposed a method of using SLM as adjustable lens to simplifythe system configuration.Zhengjun Liu et al.proposed the APR algorithm to speed up theconvergence speed in the iterative process [18].As mentioned above,the inevitable problem isthat the sample or CCD has to be continuously moved during the measurement process,whichnot only makes the date acquisition time-consuming,but increases the system scanning error.Many methods have been proposed to solve this problem,such as CMI(coherent modulationimaging)by introducing a phase plate with a known phase distribution [19,20].single exposurebased on hole array [21],and multi-wavelength illumination method [22].Among them,themulti-wavelength imaging setup is a fast coherent diffraction imaging method with simple systemstructure and avoiding mechanical scanning.However,in previous multi-wavelength imagingsystems [23,24],the type of illumination is usually a point source.One problem is that thediffraction angle spectrum diverges too fast,which may limit the detection range of the systemand increase the complexity of the reconstruction.Therefore,the parallel light is used as theillumination,which not only can help to enlarge the detection range of system,but simplify thereconstruction process.In a multi-wavelength focused system,wavelength dispersion correctionis the key for high resolution reconstruction.Here,an adaptive dispersion correction algorithm isproposed in our experiment,which can get the accurate wavelength curvature.Therefore,theconvergence speed and the accuracy of image reconstruction can be greatly improved.In this paper,we describe a system for fast lensless phase imaging that requires only a threewavelength source,a CCD sensor and a computer,but is still able to recover the image withrobustness and high-resolution.Spectral bandwidth is an important parameter in multi-wavelengthimaging system.It is found that under certain conditions,the larger spectral bandwidth,the betterreconstruction.Taking into account the recovery resolution and data acquisition time,the sourceswith three wavelengths,i.e.,532nm,633nm,850nm,are employed for recording the diffractivepatterns in experiment.In the process of digital reconstruction,a robust method of refocusingcriterion combined with iterative search function is proposed to correct the wavelength dispersion.After compensatioin of the wavelength curvature,of the wavelength curvature,high resolutionimages can be recovered.We validate that the proposed method outperforms the existing phaseretrieval algorithm,providing superior accuracy and robustness for various application.2.Method2.1.Fresnel diffraction imagingThe schematic of multi-wavelength imaging system is illustrated in Fig.1(a).The sampleis sequentially illuminated by different wavelengths (2,...,m).After the diffractiontransmission over a distance of 20,the corresponding diffraction intensity recorded by the CCDare (k,...,Im).This imaging system setup ensures the stability of the measurement process and reduces systemerrors.In the iterative phase process of multi-wavelength imaging system,the accuracy of thesimulated light transmission function is a basic factor which affects the resolution of recoveredResearch ArticleVol.29,No.5/1 March 2021/Optics Express 7199Optics EXPRESSimage.According to the scalar diffraction theory,the angular spectrum theory is a strict andaccurate description of the Helmholtz equation.Therefore,the angular spectrum transmissionequation is selected to complete the diffraction calculation in the iterative process,which canobtain a reliable and accurate recovery result [25].A general transformation that describes theformation of a diffraction pattern is given by:U(x.y,d)=U(y,0))H())Where is the incident wavelength,d is the propagation distance,H(.)is the transfer function ofFresnel diffraction,U(x,y,d)and fx,fy are the complex-valued distributions in the object andFourier domains,respectively.Obviously,Eq.(1)indicates that diffraction propagation dependson distance and wavelength,which proves that the diffraction patterns at different wavelengthscan recover complex amplitude distribution information in theory.(a)SampleLaserCLCCDPoint SourceCLFig.1.(a)The schematic of multi-wavelength imaging system;CL,collimating lens;(b)the effect of wavelength dispersion.2.2.Dispersion equations for optical materialsIn our system,the illumination source is parallel beam,so a focusing system is necessary,which isrealized by a collimating lens as shown in Fig.1(a).Next,we discuss the influence of wavelengthchromatic aberration on its collimation in a focused system,which can be understood that thesystem error is caused by the conversion of wavelength.Most of the collimating lenses used forcollimating beams are made of glass.Therefore,when the wavelength is changed,the refractiveindex of the glass also varies,The collimating beams of different wavlength after the lens areshown in Fig.1(b).It is necessary to analyze the influence in our system.A dispersion formula isusually utilized to relate the refractive index and the wavelength for optical glass.According theRefs.[26-28],the Sellmeier formula is the most accurate formulas mentioned.The Sellmeier