Rotation Measurement Using Speckle Photography with LiNbO 3 Crystal: Theoretical and Experimental Analysis

Photorefractive crystals present varied features charming presence, such as high resolution, and normal handling. Depending on the portability of eras-ing images, photorefractive crystals are convenient for read-write implemen-tations and hence find potential use in speckle photography, speckle interferometry, image processing and holography. A two-beam coupling arrangement using a LiNbO 3 crystal as a recording medium for real-time rotation measurement using the coherent and low-power laser source is presented in this paper. A speckle photography technique is advanced for the measurement of a small rotation of a transmitted glass slide sample. New theoretical analysis is formulated for a general case of a slide rotation. Experimental studies are carried out to verify the outcome of the theoretical predictions and accuracy of measurement. Uncertainty of rotation measurement is studied and quanti-fied. The proposed technique is a simple, attractive and alternative method for fringe analysis. The method promises a high degree of accuracy and in-creased range for rotation measurement in real-time.


Introduction
Measurement of displacement, deformation, rotation or tilt of different objects is a basic requirement in engineering problems. Considerable disturbance can be created by any object motion in the optical setup. It is sometimes necessary to have an estimate of the motion an object has undergone. Speckle photography is the most widely used technique in this regard [1]- [8].
This technique essentially consists of recording incoherent superposition of speckle patterns generated before and after deformation. The fourier transform (FT) analysis of the recorded specklegrams reveals the information of object deformation. There was a great effort to achieve the process work in real time by replacing the conventional photographic material with dynamic photorefractive materials [9] [10] [11] [12], which makes it be possible the observation of double and multiple-exposure speckle fringes in real-time.
In-plane and out-of-plane displacement components of an object motion have been measured accurately using speckle photography [13] [14] [15], or in real time using a photorefractive speckle correlator [13]. The rotational motion (axis of rotation orthogonal to the line of sight) of the glass plate sample leads to a reduction in the contrast of the given Young's fringes.
Speckle photography studies are made using a simple two-beam coupling configuration. In speckle photography, a diffused object illuminated with the transmitted laser beam coming from the glass plate is imaged inside the crystal, and a pump beam is added at this plane. When the glass slide is rotated, the speckle patterns shift consequently. The speckle patterns due to each beam and the pump beam produce index gratings. Thus, the fields from respective points on the object interfere after passage through the crystal and produce Young's fringe patterns.
Speckle photography was first demonstrated by Tiziani et al. [16] for real time displacement, tilt and vibration analysis with a BSO (Bi 12 SiO 20 ) crystal. An Argon Ion laser was used to create the dynamic grating of the speckle pattern inside the crystal and a low-power He-Ne laser read the grating to obtain the fringe pattern. Nasser A. Moustafa et al. [17] presents an optical configuration in speckle photography to extend the applicability of the two-beam coupling configuration using LiNbO 3 as a recording medium for the evaluation of in-plane displacement of a transparent rough object, and measurement uncertainty of in-plane displacement was studied on a wide range of displacements using experimental and new theoretical formulae for displacement measurement.
Mohammad Tiziani et al. [18] present a new optical configuration in speckle shear photography to measure in-plan displacement and strain in real time using BaTiO 3 crystal as a recording medium. Due to starin, the fringes in each pattern are of different width and orientation, resulting in the generation of a moire pattern. The strain is obtained from the width and orientation of the fringes in the moire pattern.
The speckle photography technique is also implemented in a BaTiO 3 crystal The aim of the work is to present a simple speckle technique for analysing the rotation angle by using the recorded speckle with LiNbO 3 as a recording medium. Uncertainty of rotation angle measurements on a wide range of displacement is given. A new mathematical formula has been discovered that confirms with high accuracy the experimental work.

Theoretical Background
Let us consider the transmittance of a plane wave through a transmitted glass slide of refractive index µ and thickness t surrounded by a free space.
where a and φ are the amplitude and phase of the incident wave respectively.
For measuring the rotation angle of the glass plate sample using the theory of refraction of the incident wave. The glass slide is put on a graduated rotatable disc. Spatially coherent light transmitted through the glass slide and illuminated a rough surface, as shown in Figure 1. The obtained speckle is recorded twice, one before the rotation, and one after the rotation of the glass slide. The two images are combined digitally, and the resultant image will contain a pair of identical speckle patterns separated by a distance ξ ∆ which will be calculated experimentally.
The magnitude of the rotation angle θ can be related to the fringe spacing of the observed Young's fringes at the CCD plane, the distances between the laser source and the diffuser, and between the diffuser and the CCD plane.
The amplitude distribution after the first exposure in the observation plane is given as (2) z 1 and z 2 are the distances between the laser source and the diffuser, and between the crystal plane and the observation plane CCD respectively.
Equation (2) can be further written as If the glass plate rotated by an angle θ with the incident wave, the angle of refraction will be 1 θ . The two angles are related by the Snell's law, 1 sin sin In this case, the complex amplitude transmittance of the glass slide will be  [20]. Knowing that, so the complex amplitude transmittance will be During the second exposure, the amplitude distribution ( ) 2 , A x y after the rotation θ of the glass slide can be expressed as Equation (4) In this experiment ξ ξ ∆  , this means that zero ξ ξ ∆ ≈ .
So Equation (5) can be putted in the form: By adding the two Equations (3) and (6) ( ) where new x represent the fringe spacing which given by the new formulae [17]: where M is the magnification of the imaging system.
At maximum intensity ( ) Here m is an integer, it represents straight line fringes parallel to the y-axis.
At m = 1, Equation (13) shows an equation of the rotation angle, and it can be solved as: Figure 2 shows the experimental set-up of speckle recording in a LiNbO 3 crystal.

Experimental Results and Discussions
This crystal is 5 mm × 5 mm × 5 mm in size. The optical configuration is implemented to study the rotation angle of a glass slide sample, which has a thickness 10 3 µm. The glass slide is illuminated with a coherent low power laser beam (λ = 670 nm, and P = 10 mW).
The spatially coherent light transmitted through the glass slide illuminates a rough surface, the laser beam spot at the diffuser plane was kept 2 cm in diameter  Nearly half portion of the transmitted beam illuminates the diffuser. Remaining portion of the laser beam enters the crystal directly. The glass slide sample is putted on a rotatable disc. The pumb beam which given from the same laser source is adjusted at an angle 35˚ (external) with respect to the object beam.
Consequently, there is an interference between the bump beam and the object beam inside the crystal creating the necessary specklegram. If a rotation given to the glass slide, the transmitted beam falls also on the object, and will result in the shift of its image at the crystal plane and this will disturb the two-beam coupling.
At this instant of time in the observation plane, we have two diffracted waves, and these two waves on interference, give rise to Young's fringes at the observa- The distance between the crystal plane and the observation plane z 2 was 45 cm.

Measurement Uncertainties of Method
The uncertainty of θ was estimated by combining the standard uncertainties of the parameters

Conclusion
We have presented a simple speckle technique for analyzing the rotation angle by using the recorded speckle with LiNbO 3 as a recording medium. Uncertainty of rotation angle measurements on a wide range of changing the rotation of the glass palte sample is given. New theoretical formulae were discovered to give high documentation for the experimental work. The uses of laser light of wavelength 0.6700 μm give the optimum combined uncertainty for given accurate measurement of the rotation angle. It can be concluded that the experimental measurements of the rotation angle are in good agreement with the theoretical new formulae. This assures an increase in the efficiency of measurement, in cases where the accuracy of the measurement is of great interest.