Space Bandwidth Product Analysis of Digital Holography and Image Reconstruction Process Lokesh Reddy B School of Electronics Engineering VIT University, Chennai Campus Chennai, India lokeshreddy.b2015@vit.ac.in Praveen Phinehas M Anith Nelleri School of Electronics Engineering School of Electronics Engineering VIT University, Chennai Campus VIT University, Chennai Campus Chennai, India Chennai, India Abstract—Space Bandwidth Product (SBP) is a process by which the imaging capacity of an optical is measured. The space bandwidth product analysis for inline and off-axis digital holography process is studied. This paper deals the analysis of two essential parameters i.e. image resolution and field of view of both systems. By this demonstration the maximum spatial frequency that captured by the CCD depends on object size and pixel pitch of CCD camera when distance between object plane to hologram plane is kept constant. Keywords—Digital Holography; Space bandwidth product; Inline holography; Off-axis holography. I. INTRODUCTION The two important steps in digital holographic processes are optical recording and numerical reconstruction of digital holograms. This process can be analyzed by means of Space Bandwidth Product (SBP) of the imaging system. In 1996 Adolf W Lohmann presented study of space bandwidth product in digital holography. Lohmann investigated three different holographic setups with respect to Space Bandwidth Product (SBP) in recording systems [1-2]. For an accurate retrieval of object information from a digital hologram, the sampling theorem needs to be addressed. A Signal must be sampled at least twice the maximum spatial frequency of the signals as defined by WhittakerShannon sampling theorem [3-4]. In other words two pixels of the CCD must sample at one fringe period. The Space bandwidth product addresses about sampling criteria of optically recoding holograms using CCD and reconstructed imaging performance using numerical reconstruction So, The SBP of digital holographic system must be larger than the object. Let us consider object u(x, y) is a band limited. This object has the lateral dimension Xx and Xy, and the spatial frequency is limited in Ȗx and Ȗy. The space bandwidth product is defined by SBP=XxXyȖxȖy [5]. In this present paper, the simulation of In-line Fresnel and off-axis Fresnel holographic recording systems and its reconstruction was carried out and analyzed with respect to SBP. c 978-1-5090-5913-3/17/$31.00 2017 IEEE II. METHODOLOGY Consider the light propagation from object plane to hologram plane or CCD Plane and it is given by [6] (1) is the complex amplitude in the Where hologram-plane, is the complex amplitude in the object-plane, k=2ʌ/Ȝ is the wave number, Ȝ is the wavelength and D is the distance between object and hologram. The CCD sensor at hologram plane records the interference pattern between object and reference wave. The interference pattern obtained in inline recording system can be described as [7] (2) Where and correspond to the amplitude of reference wave and object wave and is spatial frequency of the object respectively. Therefore SBP of an inline system in CCD plane [8] is given by (3) Where is lateral size of the object, ߣ is wavelength and D is the distance between object to CCD camera. The above equation shows the condition for recording of an in-line digital holography system. Now consider an off-axis hologram recording system. The interference pattern occurs by incident reference wave and object wave interfering at an angle ș. The interference pattern obtained in off-axis recording system can be described as. (4) The SBP of off-axis system can be described as [9] (5) The above equation shows that off-axis system provides a SBP four times greater than the in-line holographic system by using similar object. Similarly to obtain the SBP of image plane in the both system arrangements are given by [10-11] 194 (6) (7) Where N is the number of pixels, is lateral size of is pixel pitch of the CCD camera. the object and TABLE I. SIMULATION PARAMETERS FOR DIGITAL INLINE SYSTEM SIMULATION OBJECT NUMBER PIXELS DISTANCE (D ) RECONSTRUCTED PIXELS 1 196 25MM 196 2 576 25MM 536 3 1521 25MM 1489 4 1936 25MM 1931 5 2916 25MM 2912 6 4096 25MM 4092 Fig.1 Diffraction field of object beam In Fresnel holography [12], the object wave diffracts with an angle (Į) is defined as Ƚሺ ሻൌɉ Ǥ ɉ ǡ Ǥ ሾͳ͵ǦͳͶሿǤ The complete SBP simulation can be done in three steps: Step1: Object function is generated in MATLAB. In each simulation case, the object pixels are varied for the both systems to analyze the SBP. We chose the square object as sample for analysis. Step2: The hologram is produced by computing the Fresnel propagation from the object plane to CCD plane or hologram plane by interfering object wave and reference wave. This field in the hologram plane or CCD plane is called holographic recording. Find the object pixels in the object plane. Step3: We want to retrieve the object field in the image plane from the hologram plane. Thus we need to perform back propagation from the hologram plane. This step is called holographic reconstruction. Find the total reconstructed pixels in the image plane or reconstruction plane. TABLE II. SIMULATION PARAMETERS FOR DIGITAL OFF-AXIS SYSTEM SIMULATION OBJECT NUMBER PIXELS DISTANCE (D ) RECONSTRUCTED PIXELS 1 196 240MM 192 2 576 240MM 510 3 1521 240MM 1434 4 1936 240MM 1849 5 2916 240MM 2812 6 4096 240MM 3970 The simulation parameters for inline recording system are shown in Table I and similar range of reconstructed pixels for digital off-axis recording system as shown in Table II. III. NUMERICAL SIMULATION AND RESULT ANALYSIS Numerical simulation was done to demonstrate the space bandwidth product. The simulation parameters are set as follows-CCD sensor size of 256X256 with of pixel pitch , Ȝ=532nm, the recording distance for DIn-line=25mm and Doffaxis=240mm. 2(a) 2017 International Conference on Nextgen Electronic Technologies 2(b) 195 2(c) 2(d) Fig.2 Simulation results for digital inline system-2(a) Reconstruction result for simulation No.1 2(b) Reconstruction result for simulation No.2. 2(c) Reconstruction result for simulation No.4. 2(d) Reconstruction result for simulation No.6. 3(c) 3(d) Fig.3 Simulation results for digital off-axis system- 3(a) Reconstruction result for simulation No.1 3(b) Reconstruction result for simulation No.2. 3(c) Reconstruction result for simulation No.4. 3(d) Reconstruction result for simulation No.6. Fig. 2(a) shows the result for simulation No.1 for inline system as shown in Table I and has a good reconstruction result. The pixels of the object are increased in each simulation case and consequently the reconstructed pixels are reduced. The reconstruction result for simulation No.2 looks same as that of the original object. The reconstruction results in simulation No.4&6 are not good agreement with the original image and also there seems to be some noise/unwanted pattern at the edges of the background. Thus in comparison to the previous reconstruction outputs the latter result has a low resolution. This is due to the incorrect sampling. In off-axis case, the reconstruction pixels obtained are similar to object pixels at distance of 240mm. From Fig.3 (a) it is clear that the reconstruction result for simulation No.1 of off-axis system is similar to that of inline system and in all other cases, off-axis has lower resolution reconstructed image compared to inline system. The above simulation results for both systems shows that the reconstructed image in inline system gives good resolution when compared with that of the off-axis system. This is because the field of view in off-axis hologram reconstructing scheme is one fourth of inline system. Fig.4 Inline system - SBP vs Object size (mm) Fig.5 Off-axis system - SBP vs Object size (mm) 3(a) 196 3(b) Fig.4 and Fig.5 shows that comparison of inline and off-axis system simulated using Eq. (6) and Eq. (7). In inline system, limitations are not applied for effective field of view when compared to that of offaxis system so more information can be detected with higher resolution. 2017 International Conference on Nextgen Electronic Technologies From the numerical experiment, we verified that the maximum spatial frequency that captured by the CCD depends on object size and pixel pitch of CCD camera. In all simulated cases the off-axis system has lower resolution image reconstruction than the inline system. ĐŬŶŽǁůĞĚŐŵĞŶƚ The authors acknowledge the financial support from the Defence Research and Development Organization (DRDO), Govt. of India under the grant No. ERIP/ER/1403162/M/01/1573. ZĞĨĞƌĞŶĐĞƐ [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Measurement Science and Technology 13, R85-R101 (2002). D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE 7072, 707215 (2008). L. Xu, J. Miao and A. Asundi, “Properties of digital holography based on in-line configuration,” Opt. Eng. Vol. 39, no. 12, pp. 3214–3219, 2000. R. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill, 1986), pp. 69–97. A. W. Lohmann, “The space-bandwidth product, applied to spatial filtering and to holography,” IBM Research Paper, RJ-438 (1967). 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