Convolution Algorithm for Numerical Reconstruction of Digitally_PLP_MOnroy-2010

Convolution Algorithm for Numerical Reconstruction of Digitally_PLP_MOnroy-2010

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
doi: 10.4302/plp.2010.4.10 P HOTONICS LETTERS OF POLAND, VOL. 2 (4), 171-173 (2010) http://www.photonics.pl/PLP © 2010 Photonics Society of Poland 171 Abstract — In this letter, the range of application of the convolution approach to numerical reconstruction of digitally recorded holograms is extended. By numerical manipulation of the digitally recorded hologram, the convolution approach is used for reconstructing digital holograms of objects larger than the recording device. Experimental results are shown to validate the proposed method. Numerical reconstruction of digital recorded holograms, named digital holography (DH), is supported on the same foundations as of optical holography; it can be modelled as a two-diffraction-step technique. In the first step, an object located at a plane z =0 scatters the impinging coherent wave; the scattered optical field interferes with a reference wave on the so called hologram plane placed at a distance z = d . A 2D-discrete detector (CCD or CMOS camera) is placed at this plane, so that a sampled version of the interference pattern I ( x h , y h ) is recorded and transferred to a computer for its further processing. The recorded intensity carries on information about i) the intensities of the reference and scattered waves, and ii) the interference term between the reference and the scattered wave [1]. In the second step, the 3D-information of the object is retrieved by diffracting the conjugated reference wave as it impinges on I ( x h , y h ). This diffraction process, known as hologram reconstruction, can be carried out by evaluating the Fresnel-Kirchhoff diffraction formula [2]:     00 0 exp ,, 2 exp 1 c o s ii hh Ei k r i Ex yz r iks Ixyrxy d x d y s   . (1) In Eq. (1) we have considered that the reference wave has wavelength λ , amplitude E 0 , spatial distribution r * ( x h , y h ) and r 0 wavefront radius; the hologram extends over an area , 1 i and k =2 πλ -1 is the wave number. The   1c o s term is the inclination factor, such that 0 * e-mail:jigarcia@unal.edu.co for small numerical aperture applications. The distance between any point on the hologram to each point on the reconstruction plane is given by: 22 2 hi s zx x y y   . (2) In many practical applications the reference wave is a homogeneous plane wave impinging perpendicularly to the hologram plane, such that it can be represented as a constant field of amplitude E 0 . This condition and small numerical aperture systems will be considered in the text that follows. Different approaches to compute Eq. (1) are found in the literature; Fresnel approach relies on the parabolic approximation of the distance 11 1      xx yy sz zz for the exp( iks ) term and s z in the denominator. These approximations transform Eq. (1) into:   0 exp e x p ,e x p 2 exp i i h h k z i x y iz z i Ix y x y z i d xd y z     . (3)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 3

Convolution Algorithm for Numerical Reconstruction of Digitally_PLP_MOnroy-2010

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online