Rigid body motion measurements with Fourier lensless holography

A recently proposed1 holographic and moiré technique for in-plane motion measurements is generalized to certain cases of rigid body motion measurements. In this case, a three-recording-plates set, in a trihedral arrangement attached to an object, registers the interference between spherical and plañe coherent wave fronts. The in­ terference fringes are plañe sections of a paraboloid revolution family with a common axis parallel to the plañe wave propagation direction; it also contains the point source originating the spherical wave. If, between two such exposures, the arrangement is translated or rotated in the space, the two overlapping interference patterns registered will produce low frequency moiré patterns. If the movement is small (less than the innermost Fresnel ring), the moiré pattern consists of families of equally spaced straight lines, the spacing and orientation of which are related to the magnitude and orientation of the displacement or rotation. So, as in the case of inplane motions of the photographic píate, these moiré fringes can be used to determine the magnitude and direction of the displacement or the angles of rotation. In addition, each family of moiré fringes gives rise, in the Fourier plañe, to two bright spots separated by a distance proportional to the magnitude of the displacement or rotation. Also, a slanted image of the pupil limiting the plañe wave, reconstructed in the Fourier plañe, is njodulated by Young’s fringes, whose interfringe is also reciprocally proportional to the magnitude of the movement. First, let us consider the intensity moiré pattern produced by the overlapping of the two interference patterns in each of the three recording plates of the trihedral arrangement. In general, the transmittance of the developed plates will be

A recently proposed1 holographic and moiré technique for in-plane motion measurements is generalized to certain cases of rigid body motion measurements.
In this case, a three-recording-plates set, in a trihedral arrangement attached to an object, registers the interference between spherical and plañe coherent wave fronts.The in terference fringes are plañe sections of a paraboloid revolution family with a common axis parallel to the plañe wave propagation direction; it also contains the point source originating the spherical wave.If, between two such exposures, the arrangement is translated or rotated in the space, the two overlapping interference patterns registered will produce low frequency moiré patterns.If the movement is small (less than the innermost Fresnel ring), the moiré pattern consists of families of equally spaced straight lines, the spacing and orientation of which are related to the m agnitude and orientation of the displacem ent or rotation.So, as in the case of inplane motions of the photographic píate, these moiré fringes can be used to determ ine the magnitude and direction of the displacement or the angles of rotation.In addition, each family of moiré fringes gives rise, in the Fourier plañe, to two bright spots separated by a distance proportional to the magnitude of the displacem ent or rotation.Also, a slanted image of the pupil limiting the plañe wave, reconstructed in the Fourier plañe, is njodulated by Young's fringes, whose interfringe is also reciprocally proportional to the magnitude of the movement.
First, let us consider the intensity moiré pattern produced by the overlapping of the two interference patterns in each of the three recording plates of the trihedral arrangement.In general, the transm ittance of the developed plates will be where Ci and are related to the amplitudes of the spherical and plañe waves, k is the wave number vector, and r and r' are position vectors before and after the movement of the trihe dral.ro is a constant position vector from the origin of the coordínales system to the reference point source, as shown in Fig. 1.The position vectors r and r ' are related by ( 2 ) where A is the rotation matrix and b is the translation vector, both characterizing the trihedral movement.
Assuming |r | « |r « |, the transm ittance f given by (1) becomes The last equation consists of a high frequency family of curves modulated by a low frequency one represented by the first cosine.T he moiré fringes condition of maxima will be given by Rewriting Eq. (4), it becomes (5) where x¡ are the components of r, and the coefficients Bj are Bj = i) -(h -b)hj, x'j being the versors th a t characterize the trihedral orientation after rotation.
T hat is, in each of the three planes, the moiré fringe pattern consists of a family of parallel Unes, as in the case of in-plane motions,1 th a t are the intersections between the three re cording plates and a family of equally spaced planes given by Eq. ( 5).
A straightforward calculation shows th a t the spacing A¡¡ and the slope m,-; of the moiré fringes in the (¿,;)-plane are ( 6) (7) W hen the developed plates are separately Fourier transformed in the conventional way,2 the moiré fringes act as a low frequency linear grating producing two bright spots symmetrically located to the zero order, the separation d tJ between them being related to the spacing A ¿y as where A is the reconstruction wavelength and D is the distance between the hologram and the Fourier plañe, when the former is illum inated with a spherical converging beam.
Returning to (1), it may be w ritten as follows: Fourier transforming Eq. ( 8) and taking into account only one diffracted order, the amplitude distribution in the Fourier plañe will be where C and a are constants, and F [ | denotes the Fourier transform.As F | J operates over a bidimensional transparency, all the position vectors are two-coordinate dependent.
In the case of an infinitesimal rotation, Eq. ( 9) becomes in which a is a constant phase delay, * denotes the convolution product, and v is related to the translation b and the rotation with being the rotation's Euler angles, as where + indicates transposition.
As shown in Eq. ( 10), the reconstructed pupil in the Fourier plañe will be m odulated by Young's fringes, the spacing of which is given by the reciprocal of the modulus of v.
R eturning to either Eq.(6) or Eq. ( 7) we can conclude th at in certain cases, it is possible by this m ethod to study 3-D motions.If we deal with only a rigid body translation or a very small rotation, the movement can be determined.In other cases, additional information is needed for determining motion parameters.
T he measurements of A,; can be made, provided th a t the two overlapping interference patterns registered produce at least two moiré fringes on the holographic píate.Therefore, this condition establishes the smallest displacement th at can be measured by this technique, so limiting the sensitivity.On where Eq. ( 2) has been used, I denotes the identity matrix, n is an integer number, and h is a known position vector given by the other hand, the accuracy mainly depends on the precisión of th e A¡j measurements.
Figure 1 shows the experimental setup, and Fig. 2 shows experimental results for the case of a small rotation about an axis th a t lies in the direction of light propagation.

Fig. 1 .Fig. 2 .
Fig.1.Scheme of the experimental setup: H i, H2, and H3 are three recording plates in a trihedral arrangement, r denotes a point position vector on one of the plates, r' indica tes the position vector of the same point after the trihedral motion, and ro is the position vector of the reference source So-P is an empty pupil which diffracts an approximately plañe wave.