Speckles appear as noisy steady light patterns caused by random phase disturbance of coherently propagating light. Their suppression and use in optical metrology make its simulation and evaluation necessary. This contribution addresses the issue of numerical propagation of speckle distributions

Kelly, Damien P.

Meinecke, Thomas

Sinzinger, Stefan

Digitale Bibliothek Thüringen

  Thomas Meinecke, Damien P. Kelly, Stefan Sinzinger: Numerical models for speckle fields Zuerst erschienen in: DGaO-Proceedings. - Erlangen-Nürnberg: Dt. Gesellschaft für angewandte Optik, ISSN 1614-8436. - Bd. 116.2015, B6, insg. 2 S.  URN:  urn:nbn:de:0287-2015-B006-3 URL: http://www.dgao-proceedings.de/download/116/116_b6.pdf  [Download: 14.01.2016]    DGaO Proceedings 2015 – http://www.dgao-proceedings.de – ISSN: 1614-8436 – urn:nbn:de:0287-2015-XXXX-Ysubmitted: XX.XXX.2015       published: XX.XXX.2015Numerical Models for Speckle FieldsThomas Meinecke*, Damien P. Kelly**, Stefan Sinzinger**Fachgebiet Technische Optik, Institut für Micro- und Nanotechnologien, Technische Universität Ilmenau** Fachgebiet Optik-Design, Institut für Micro- und Nanotechnologien, Technische Universität Ilmenaumailto:thomas.meinecke@tu-ilmenau.deSpeckles appear as noisy steady light patterns caused by random phase disturbance of coherently propagating light. Their suppression and use in optical metrology make its simulation and evaluation necessary. This contribution addresses the issue of numerical propagation of speckle distributions.1 IntroductionWhen we work on a practical setup in our lab with coherent light, grainy steady patterns, called speckles, appear. They are caused by light stray-ing either on rough surfaces or within scattering volumes or can be generated synthetically by spa-tial light modulators. They are noisy, random light distributions and can be evaluated statistically. Therefore they are applied as test distributions in metrology, i.e. shearing interferometry, measure-ments of vibrations and rotations, visualization of aberrations and phase retrieval. If a speckle field is captured by a camera one gets an intensity image. Because of its noise, i.e. random characteristics statistical evaluation can be done as follows: either in a histogram where the counts for each intensity value are plotted and can be fitted to a special probability density function PDF, or by calculation of the autocorrelation function, which contains the information of the correlation length, i.e. the aver-age speckle size ASS. The digital generation of complex amplitude U(N,M) of speckle distributions is performed by adding two sets of random, Gaussian distributed numbers; one set represents the real and the other the imaginary part, the in-tensity is then calculated by the multiplication of its complex amplitude times its complex conjugate [1].2 Evaluation of SpecklesIn this contribution the evidence of speckle distri-bution is given by the answer of two questions:1. Does the intensity distribution obey the nega-tive exponential PDF?2. Does the average speckle size follow the rela-tionship of ?Both answers should be positive for the generation of fully developed speckle patterns and after its processing by numerical propagation [2].The statistical behavior is analyzed by making up ahistogram of intensity followed by the fitting of the curves by gamma as well as negative exponential PDF. The gamma PDF depends on 2 parameters of  and , whereas the negative exponential PDF is described by the parameter of μ. If 	 and 	, i.e. μ=1, the gamma PDF reduces to negative exponential function what is a typical property of speckles. The autocorrelation function delivers the correla-tion length, the distance between lateral spatial locations where the intensity values are still corre-lated statistically with each other. This corre-sponds to the average speckle size of the ob-served distribution and is evaluated by measuring the distance between the central global maximum and the first local minimum. Meanwhile this aver-age speckle size follows the relationship of ASS	 with  as the wavelength, z the propa-gation distance and 2W the aperture width for squared, but ASS=1, for circular apertures having a radius of R. Both criteria, i.e. the negative exponential PDF and the ASS are used for speckle validation [1].The propagation of optical wavefields is represent-ed by the Kirchhoff diffraction integral. This can be simplified to a Fresnel transform for the paraxial regime which corresponds to a Fourier transform of a multiplication of the complex amplitude distri-bution U(x’,y’) in object plane times a parabolic phase propagator. This is called direct method (DM). The Kirchhoff diffraction integral can be seen as a convolution of U(x’,y’) by the impulse re-sponse of free space which is generally performed in spectral domain by multiplication of its Fourier transforms. Then the spectral result is transformed back to the spatial domain by inverse Fourier transform. If this impulse response is approximated parabolically we call this spectral method (SM). The distributions and the following propagation can be only represented numerically by sampling. This results in 2 consequences: I) replicas appear due to propagation at a lateral replica period RP’	’with ’ as pixel size at object plane, II) there is a different pixel size  in the diffraction plane for each method. For DM the pixel size scales with the distance by 	’) with N as an integer number of samples while for SM  remains const., i.e. 	’ [2]. This behavior concerns both lateral coordinates of x,y.a  Proceedings 2015  - http:// .dgao-proceedings.de  - ISSN: 1614-8436  - urn:nbn:de:0287-2015-B006-3 submitted: 20.Jul.2015   -   published: 17.Aug.2015DGaO Proceedings 2015 – http://www.dgao-proceedings.de – ISSN: 1614-8436 – urn:nbn:de:0287-2015-XXXX-Ysubmitted: XX.XXX.2015       published: XX.XXX.20153 ObservationsThe observation of the numerical propagation is realized in 3 steps as follows:I) the numerical evaluation of the propagation integrals using DM and SM respectively, with-out considering the sampling and pixel size,II) the adaptation of the spatial extent 	 to the replica period RP’	’. This works only for a unique distance. For further distances RP’increases. Holding 	’, the spatial extent  should be enlarged by zero padding: =RP’	.III) the expansion of the spatial extent  to the diffraction area [3]. For a given distance z and aperture size of 2W,  is enlarged to 	’		. The new replica period RP can be achieved by a new (smaller) pixel size of 	’+2W). The original distribu-tion has to be interpolated followed by an ap-propriate zero padding to the diffraction area.The numerical algorithms of DM and SM were regarded versus the propagation distance. First, the intensity statistics were evaluated concerning the negative exponential PDF, where one wishesto have the parameters =1 and 	 as men-tioned above. Second, the measured correlation length is compared to the relationship of average speckle size ASS.Fig. 1 Observation: step 2 (violet), diffraction area (or-ange), step 3 (green).When just the DM and SM are programmed (step I), the calculated distributions are not comparable to each other, due to the different pixel sizes men-tioned above. The statistics of the intensity distri-bution follows the expectation of the negative ex-ponential PDF, but the reproduction of ASS fails. The measured size corresponds to one pixel rep-resenting the behavior of the pixel size for both DM(scaled with distance) and SM (remains const.) [s.Fig. 1].The adaptation of the spatial extent to the replica period results in intensity statistics as expected and reproduces the functional behavior of the speckle size [s. Fig. 2]. It turned out that the ASSwas only few pixels, so that an exact confirmation could not be reached. Here, the DM and SM yield-ed identical intensity distributions.Considering the diffraction phenomenon under the conditions of step III) delivers once again identical distributions independent of SM or DM. However, the speckle intensities don’t follow the negative exponential PDF anymore. But it should not be surprising because of the intensity decay at the boundaries due to diffraction. The expectation ofthe ASS could be confirmed similar to step II).Fig. 2 Intensity statistics (left) and average speckle size (right) vs. propagation distance.4 ConclusionsFor the numerical propagation of speckle fields obeying the sampling criterion 	’ is inevita-ble. In a strict sense, this criterion is only given for one propagation distance. If one wished to propa-gate to further axial distances or to expand the lateral region, an appropriate zero padding as well as sampling (interpolation of pixel size) are availa-ble methods to adapt this new situation to the sampling criterion. Then, the result is independent on the applied method (DM, SM).References[1] J. W. Goodman: „Speckle Phenomena in Optics: Theory and Applications“, B. Roberts & Company (publ.) (2007), ISBN 0-9747077-9-1[2] D. Li, D. P. Kelly, J. T. Sheridan „Three-dimensional static speckle fields: part I. Theory and numerical investigation“ in J. Opt. Soc. Am. A, Vol. 28, No. 9:1896-1903 (2011)[3] T. Kreis: „Handbook of holographic interferometry: optical and digital methods “, WILEY-VCH, Wein-heim (2005), ISBN: 3-527-40546-1DGaO Proceedings 2015  - http://www.dgao-proceedings.de  - ISSN: 1614-8436  - urn:nbn:de:0287-2015-B006-3 submitted: 20.Jul.2015   -   published: 17.Aug.2015

Numerical models for speckle fields

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Numerical models for speckle fields

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