laser, speckle, optical, optics, measurement, vibration, vibrometer, vibrometry, angle, angular, rotation, rotational, SNPPYN

Contents.

**     General.
**     Principle of Angular Transduction.
**     Image Processing.
**     References and Background Reading.
 
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**    General.

This is a short note written specifically to clarify the principle of a certain type of angular motion transducer. The method is based upon the illumination of a target with a beam of laser light, and the measurement of rotations of the target is made by processing the speckle image which may be formed from the light scattered back from the target. The method can only be used where there is some degree of optical roughness or irregularity in the surface of the target, sufficient to scatter some of the light rather than simply reflect it specularly, and the target surface must not be so absorptive as to return too little light to allow the formation of an image.

There is more than one way in which laser illumination and its scattered light can be used to transduce angular movements of a target. We are concerned here with only that method where a spot of illumination at the target allows a speckle image to be captured so that physical displacements of the speckle in the image can be interpreted to represent angular movements of the target.

This article results from work performed within the group called "Laser Speckle Associates" in London, UK. To discuss possible developments and applications in this or related areas contact may be made by email to LSA here, or by using the author information at the foot of this page. We look forward to your call.

Hyperlink references to this page are welcome.

**    Principle of Angular Transduction.

Target illumination light from the laser is passed through the front lens of the apparatus towards the target and forms a beam which is approximately parallel at the expected distance of the target. To do this the light from the laser is brought to a focus at a distance equal to the focal length of the lens, and before passing though it. This lens is the same one which receives light scattered back from the target. For some applications the illumination beam can be delivered at a slant from one side so that the light receiving lens then does not need to carry the strong outgoing beam and the beam splitting mirror may be omitted. Here we consider the concentric configuration as shown below.

The diagram shows the essential parts of a very simple implementation of the method. Variants are possible to suit different applications, but the purpose here is to clarify the principle, and for that reason a simple layout is shown.

The arrangement is shown with the lens and image detector set up so that it would focus an image at infinite distance. That means that it is focused like a telescope. For each direction of parallel light entering the lens there is a single point of arrival in the detector plane.

Light scattered from the target arrives at the lens with different total intensities in the different directions. For each direction α the total wavefront of the light is coherently summed at the image with different relative delays from the various parts of the target spot. As α is increased by each interval corresponding to a path difference of one wavelength across the width of the spot the summation produces a statistically independent result. From this fact the angular width (width of correlation) of the speckle can be estimated as λ/d radians where λ is the wavelength and d is the effective width of the spot.

Having the optics focused at infinity is the condition for the image to be the squared modulus of the complex Fourier transform [FT1] (also called Fourier integral [FT2]) of the complex scattering function of position over the target illumination spot.

Knowing the geometry of the optics the angular width of the speckle can be converted to a corresponding speckle size at the image and this size is λ.f/d where f is the focal length of the lens. A suitable detector must be able to resolve detail comparable with the speckle size. For different purposes the choice between over or undersampling this image can be made in various ways.

The angular sensitivity can be expressed in terms of image shift x per unit angular rotation of the target. The scattered light pattern moves through twice the angle of rotation of the target, doubling the angle in the same way as would be the case for specular reflection at a mirror. Thus the sensitivity is 2.x/α = 2.f. So, if the image shift resolution is h then this corresponds to a target angular movement resolution of h/(2.f) radians.

The resolution performance is affected by a number of factors, particularly the strength of the optical signal and the number of pixels in the image. Given sufficient signal strength the use of a large number of pixels permits interpolation of angles to far smaller than that corresponding to either the pitch of the image detector pixels or the speckle dot width.

An instrument using a target illumination spot of red light with a width (approx = diameter) of 1 millimetre produces an angular speckle width of somewhat less than 1 milliradian. If this beam is applied approximately perpendicularly to the curved surface of a cylindrical target with diameter 25 millimetres rotating about its axis and the image is captured with 256 pixels then the process noise induced by speckle randomness is in the region of 25 microradians RMS per spot transit. With the 1 millimetre spot width and the 79 millimetres circumference there are 79 spot transits per revolution. This produces uniform spectral noise power across the first 79 harmonics of the target rotation rate (revs per unit time used as a freqeuncy) and the level of this noise is 25 microradains RMS in each harmonic. Above the 79th harmonic the intensity of this noise density decreases at 6dB per octave.

For small angular movements of the target in which the area illuminated changes little this multiplicative type of process noise appears only as an uncertainty in the calibration constant of the instrument and the instrument is locally linear. Variation of the calibration coefficient (angular sensitivity) due to this effect for the spot, target and pixel geometry described above is in the region of 0.04% RMS. For such measurements the amount of additive noise in the result becomes the dominant concern and issues of optical signal strength and bandwidth are the main factors determining this.

Notice that variation of the spot size affects inversely the size of the speckle dots, but this does not affect calibration. The calibration constant of such an instrument is dependent upon the optical geometry which affects displacements of the speckle pattern but neither upon the granularity of that pattern nor the distance to the target.

The diagram shows targets at two different distances. Because of the choice of focal arrangements the light scattered at any particular angle arrives at the same image point, regardless of such change of distance. The limit to increased target distance is eventually the aperture of the lens.

Similarly, lateral shift of the target, because it has no effect upon the angle of scatter, leaves the light arrival point unchanged in the image plane. The only effect of such a lateral shift is that part of the spot illumination area is replaced by a new area of scattering surface of the target, and this has the effect of destroying correlation of the image approximately in proportion to the fraction of spot area so replaced. It is a source of uncorrelated noise in the image signal.

**    Image Processing.

The image must be processed by an algorithm suitable for the detection of flow in the image. This type of instrument needs a rather high speed of computation to extract the movement result from the image data. The burden of this task is roughly proportional to the product of the image sampling rate and the number of pixels in the image. For single dimensional measurement a single row image detector may be used, though parallel rows may also be used for this purpose. With a square array it is possible to process for angular movements simultaneously about two axes.

**    References and Background Reading.

[ABo95]	A.G.Booth "Interpolative Estimation of Pattern Motion using Bilinear Functions."
[FT1]	Forrest Hoffman. "An Introduction to Fourier Theory." University of Tennessee, Knoxville
[FT2]	William A. Cooper. "The Analysis of Observations with applications in atmospheric science." National Center for Atmospheric Research