SIMULATION AND VISUALIZATION IN THE EDUCATION OF COHERENT OPTICS J. KORNIS, P. PACHER Department of Physics Technical University of Budapest H-1111 Budafoki út 8., Hungary e-mail: kornis@phy.bme.hu, pacher@phy.bme.hu Abstract Coherent optical metrology is a basic course for physics engineering students specialised in optics. Simulation proved to be a useful aid in studying it: the effect of the various parameters and the errors of the components can be separately traced for optimising various optical arrangements. The simulation package developed helps in studying the phenomena covered in the subject: diffuse surfaces can be generated and can be placed into various optical arrangements, displacements and deformations can be applied on them as well. To assist students to visualise and study various optical arrangements a set of 3D models of optical elements has been developed to create a virtual optical laboratory. KEYWORDS: optical measurements, speckle metrology, simulation, virtual optical laboratory Introduction A large fraction of physics engineering students at the Faculty of Natural Sciences of the Budapest University of Technology and Economics choose specialisation in optics. The students specialised in optics have in their curriculum (Optics module) a two semester course "Optical measurements". This subject is closely related to the lectures as "Holographic interferometry", "Optical metrology", "Speckle metrology" and "Computer aided picture processing". In the "Optical measurements" course students have to perform about twenty laboratory measurements related to various fields of optics. An essential part of measurements deals with interferometric methods including classical, holographic and speckle interferometry. Here we will concentrate on speckle measurements: speckle photography and speckle interferometry [1] and will demonstrate some results obtained with a numerical simulation package developed for speckle metrology. The program package can generate diffuse surfaces with optional features. On the generated surfaces displacements and deformations can be applied; the generated objects can be placed into various optical arrangements. The simulation helps the students in the visualisation of the laser speckle and contributes to the understanding of the signal and noise nature of it. The pictures generated by computer simulation can be used as an input for further picture processing purposes, including different filtering methods, etc. Another teaching aid, which serves for visualisation and study of various optical arrangements will be demonstrated as well.
1. Simulation of Speckle Patterns Illumination of a rough surface with coherent light produces a random intensity distribution in front of the surface, called speckle pattern. Because the speckle pattern follows the movement of the scattering surface the speckle can be used for displacement and deformation measurement. Depending on the magnitude of deformation the speckle metrology can be divided into two categories: speckle photography - when the deformations are greater than the speckle size - and speckle interferometry, when the deformations are smaller than the speckle size. The optical setups for speckle metrology are very simple, therefore it is well suited for laboratory measurements for undergraduate students. In speckle photography the object is illuminated with a coherent light source and the image of the object is recorded before and after the deformation or displacement. The optical setup for speckle interferometry is based on the Michelson interferometer. The only difference is, that the mirrors are replaced by diffuse surfaces: one is the investigated object and the other is a reference surface. The detector, typically a CCD camera, records the interferometric speckle patterns generated by reflection of coherent light on the two surfaces before and after the deformation. The subtraction of these two pictures by a computer results in a correlation fringe system and the deformation can be determined by the evaluation of the fringe pattern. The program developed can generate diffuse surfaces with arbitrary roughness and reflectivity. On the generated surfaces (objects) one can apply rigid body motions and deformations or a sequence of these. The objects can be placed into various optical arrangements. For these arrangements one can choose spherical or collimated illuminating beams and detector elements with various resolution and non-linear characteristics. It is possible to create moving sequences as well, to visualise the changes in the pattern. With the numerical simulation package the effect of the various parameters and the errors of the optical components can be separately traced and the results can be used for optimising the various optical arrangements. The so-called objective speckle pattern is produced by the free space propagation of the reflected light from the object on the surface of the detector. The modelling of the objective speckle pattern is performed by Monte Carlo simulation: several points are chosen randomly from the object, and then the speckle pattern on the screen is obtained by coherently adding up the spherical waves originating from these random points. In our investigations, the object consisted of 1024 x 1024 points. As few as 200 randomly selected object points are enough to produce a simulated objective speckle pattern the statistical properties of which (the second-, third- and fourth-order normalised moments of intensity) approximate the theoretical values satisfactorily [2], [3]. The size of the simulated object was 50 mm x 50 mm, the distance between the detector plane and the object was 1 m, and the detector had 512 x 512 pixels. The speckle pattern is shown in Fig. 1.a and the phase map of the speckle pattern on Fig. 1.c. Fig. 1.b shows the longitudinal section of the speckle pattern generated by a 90-degree rotation of the simulated detector. The phase distribution is wrapped into the +90 and -90 interval (white dots correspond to +90, black to -90 phase).
a. b. c. Fig 1. Cross section (a.) and vertical section (b.) of a calculated objective speckle pattern and the phase distribution (c.) in the cross section (white points correspond to +90, black points to -90 phase). To visualise the interaction of holographic fringes and the speckle field a reference wave can be added to the speckle field, which corresponds to a holographic arrangement. A series of intensity and phase maps with gradually increasing intensity of the reference beam have been calculated and the resulting intensity maps are shown in Fig. 2. Fig. 2. Intensity distribution of an objective speckle pattern with increasing reference amplitude The basic setup for measurement and calculation of a subjective speckle pattern, i.e. for modelling speckle interferometry is a Michelson interferometer. The program allows to specify the features of the light source (intensity distribution of the beam, wavelength and coherence length), the objects, the intensity ratio of the two beams after the beamsplitter. The focal length and aperture size of the imaging lens are adjustable as well. It is possible to modify the speckle size (by changing the aperture size of the imaging lens) and the resolution of the detector (pixel size of the CCD camera). Fig. 3.a. shows an example of a simple rotation along the vertical axis. The angle of rotation is 0.02 o, the illuminating wavelength is 632 nm. The fringe pattern is in a very good agreement with the theory. A more complicated fringe pattern can be seen on Fig. 3.b. The correlation fringe pattern is a result of rotation along two axes and four different deformations. Using a two wavelength light source the shape of the object can be measured with interferometric accuracy. An example of the twowavelength shape measurement is shown in Fig. 3.c. The contour interval is 5.28 µm.
a. b. c. Fig. 3. Correlation fringe system due to rotation along the vertical axes (a.) More complex correlation fringe pattern (b.) An example for shape measurement (c.) With the program computer generated amplitude and phase holograms can be simulated as well. On Fig. 4. the hologram of a point source and its reconstructions are shown. Monte Carlo method have been used for the reconstruction of the hologram. The pictures shown differ in the number of randomly selected points used for the reconstruction. As it can be seen on Fig. 4., 200 randomly chosen points of the hologram are enough to obtain a good quality reconstructed picture (Fig. 4.c). a. b. c. Fig. 4. Computer generated hologram of a point source (a.) Reconstruction of the hologram with Monte Carlo simulation, using 50 (b.) and 200 (c.) randomly selected points of the hologram. 2. Virtual Optical Laboratory To assist students in studying various optical arrangements an other program has been developed. The 3D models of more than hundred optical elements has been developed; these elements (Fig. 5.) can be used to set up various optical arrangements in a virtual optical laboratory. With the program the time and cost required for the design of an experiment can be drastically reduced. The calculated results presented in Fig. 3. a-b. are based on a Michelson type interferometer arrangement. The inputs for the simulation were the size of the object, the parameters of the illumination and the characteristics of the CCD camera. Assigning 3D objects to the optical elements (the investigated object, different kinds of optical elements) the optical arrangement belonging to the mathematical description of the interferometer can be built in the virtual space. On Fig. 6. the
computer generated optical arrangement of a Michelson interferometer is shown in the virtual laboratory. Fig. 5. Virtual optical devices on the optical table Fig. 6. Virtual optical arrangement corresponding to the result on Fig. 3.a-b. More complex optical arrangements are shown in Fig. 7-8. The arrangements can be visualised from different spatial directions and the 3D pictures can be printed out.
Fig. 7. Virtual optical arrangement for generating Holographic Optical Elements (HOE) Fig. 8. A part of the previously presented 3D arrangement using different illumination Acknowledgement This work has been supported by the Hungarian National Scientific Research Foundation (OTKA) under the project 4-136/94.
References 1. J. W. Goodman, Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, edited by J. C. Dainty, Springer Verlag, Berlin pp. 9-75 (1975) 2. J. Kornis and N. Bokor, A. Németh A numerical simulation package for speckle metrology Proc. of the Int. Conference on Applied Optical Metrology, SPIE Vol. 3407 pp. 297-302 (1998) 3. J. Kornis and N. Bokor, Simulations In Speckle Metrology, Simulation and Experiment in Laser Metrology, Akademie Verlag, Berlin, pp. 233-237 (1996)