| Phys 198 | April 28, 1997 |
Electronic speckle pattern interferometry is a method of speckle correlation interferometry that is growing in popularity because it simplifies the process of obtaining specklegrams. Instead of holographic plates and photographic darkroom processing, a standard CCD camera is used to acquire the images, requiring only that the camera have the resolution to detect speckles (about 5 microns in diameter), instead of the submicron resolution of holographic film plates. Also, the display can be a standard TV monitor, since only the fringes have to be resolved in the display. The basic idea is to take a picture of the speckled surface under study in its initial state, save this picture in a frame grabber or in computer memory, deform the surface, and take another picture. The fringe pattern describing the deformation that has occurred between the initial and final states is obtained by subtracting the second picture from the stored initial picture. To avoid negative values, the absolute value of the difference between the two images is displayed on the monitor screen. Pictures can be taken at the standard TV picture rate, which is 30 frames per second.
To develop a rationale for this technique, consider the intensity I = I(x,y) of an image of the speckled surface. Let the initial intensity distribution be called Ibefore and the final intensity distribution Iafter. Set up a camera to detect an object wave IO (the image) coming from the object surface and a reference beam IR which is added to the object wave. The detected wave will contain information about the phase difference f between the two waves, due to the varying path differences for different points in the object beam. After the surface is deformed, of course, this phase difference will be changed.
Assume the camera voltage output is linearly proportional to the intensity of the light incident on the camera sensor, and subtract the camera output signals for each of the two states by using a computer
To avoid negative values, rectify this signal and get an output proportional to the absolute value of Vsubt
This signal is fed to a display system and controls the brightness of the display image.
Sign Problem (Carrier technique)
The sign ambiguity inherent in all systems which combine two images a before and an after still has to be accounted for. To eliminate this sign ambiguity, a carrier must be introduced. It is very common to use a piezo-electric translation device (PZT) to introduce a controlled phase into each image. Earlier methods of carrier imposition (e.g., tilting the object or changing the angle of one of the beams) created a known but changing phase pattern across the entire image. Here a constant phase is imposed across the entire image, but a series of pictures is used with each having a known and different phase change. Usually there are four such images with a phase change of a = 90 degrees between each of them. If this is done, and some trigonometric mathematics is performed on combinations of the four images, with I1 being the reference image specified as having no carrier phase change, I2 having 90 degrees introduced with respect to I1, I3 having 180 with respect to I1, and I4 having 270 with respect to I1, it is found that the resulting relationship is given by
where f(x,y) is the phase change at each point (x,y) between the reference and object beams. The same math operations are performed on a second set of four images taken after a surface deformation occurs. Two phase maps are thus obtained, and it is known which map comes before and which comes after the deformation. Simple point-by-point subtraction using a computer gives the desired phase shift due to deformation without any ambiguity as to sign. Taking four images each of the before and after states not only allows removal of the sign ambiguity in the phase change Df due to the deformation occurring between the before and after states but also allows noise reduction.
Phase Unwrapping
The problem still remains of determining the correct magnitude of the phase because the trigonometric functions are multi-valued; although angles have unique trig function values, numerical values of trig functions only give angles modulo 2p.
The most widely used solution to this problem is called phase unwrapping. Assuming that the Nyquist criterion has been met (at least 2 data samples within highest frequency component), it can be assumed that no two successive data points will be 2p radians apart in phase. If two successive points are in fact separated by that great a change in phase, then 2p must be added or subtracted from the second data value, depending on the sign of the original change.
Quantitative Results
The analysis of the fringe pattern follows the method described for Speckle Correlation Interferometry, discussed in the lecture on Speckle Interferometry given April 21, 1997.
Send Mail or Comments: matthysd@vms.csd.mu.edu
 |
 |
Visitors to this page :