OPTICAL REVIEW Vol. 8, No. 5 (2001) 378-381 Introductlon of the Inlpulse Response Functlon in Common-Path Interferometers With Fourier Plane Filters S. R. KITCHEN*, S. G. HANSON and R. S. HANSEN Optics and Fluid Dynamics Department, Ris~ National Laboratory, DK-4000 Roskilde Denmark (Received Apru 23, 2001 ; Accepted June 6, 2001 ) In the present paper it will be shown how the introduction of a Fourier plane filter can create various types of common-path interferometers for measuring changes in surface tilt or curvature of an object surface. This is obtained by placing a holographic optical element in the Fourier plane of a 4-f optical system. The interferometers are analysed by using the paraxial approximation of the Huygens-Fresnel integral formalism, and the interferometer functions are given by a novel formalism using impulse response functions. Based on this technique, an interferometer for measuring dedicated changes in surface deflection is presented. This interferometer is insensitive to rigid surface rotations and displacements. The interferometer can be embedded in systems based on single point measurement of a time dependent deflection, i.e. vibrometers, as well as in full-field measurements such as electronic speckle interferometers. Key words: vibrometry differential electronic speckle interferometry, common-path interferometry, holographic optical elements, 1. Introduction Speckle shearing interferometry or shearography is a well- known method for measuring the first derivative of deflections for rough surfaces, which combined with electronic speckle interferometry (ESPI) makes it possible to obtain real-time correlation fringes on a monitor. l) A typical approach in shearography is to illuminate an ob- ject with coherent light and observe the scattered light through a Michelson-type interferometer, where one of the mirrors has a slight incline, the size of the incline determining the size of the shear. The introduction of a holographic grating as the shearing element can yield advantages in respect to light effi- ciency, simplicity and cost efficiency.2~5) Holographic optical elements (HOE) can also provide compact and self-aligning systems for other types of interferometers, such as electronic speckle interferometry systems (ESPI systems) or vibrome- ters . 6) We here introduce a system that can facilitate an arbitrary type of common-path interferometer and with the proper de- sign of the HOE can measure a desired type of surface de- formation. The optical setup is a 4-f system with an HOE placed in the Fourier plane as illustrated in Fig. I . It is here assumed-for the sake of simplicity-that the object is illuminated with coherent light in a backscattering mode and that the system without the Fourier filter makes up an imaging configuration. Furthermore, it is assumed that the HOE divides the incident light into two fields with com- parable strength and these fields interfere in the observation plane (p-plane) at the CCD-array. To ensure fully developed speckle it is additionally assumed that the surface roughness is large compared to the wavelength and that the illuminating spot size exceeds any lateral surface roughness scale.7) This paper was originally presented at the 2001 International Conference (2nd Joint OSJ-SPIE Conference) on Optical En*'ineering for Sensing and Nanotechnology, ICOSN 2001 which was held June 6-8, 2001 at the Pacifico-Yokohama Conference Center, Yokohama, Japan. *E-mail: steven.kitchen @ risoe.dk The correlation fringe pattern is viewed by subtracting the speckle pattern recorded with the object in the undeformed state from the speckle pattern after a deformation of the object surface has taken place, which results in the following average squared intensity distribution on the monitor: n(p) = ((1(p) - I/(p))2) o( (Io(p))(11(p)) - Re{ (UO ( P) Uo!* ( p)) (Uj* ( P) Uf ( p)) } , (1) where U (p) and I (p) denote the field strength and inten- sity, respectively, suffix O and I denotes the undiffracted and diffracted fields, respectively. The prime indicates the fields after deformation, the square brackets denote an ensemble av- erage, and Re{ . } denotes the real part. The fields are analysed using the paraxial approximation of the Huygens-Fresnel principle utilising the ABCD-matrix formalism for the system and introducing soft (Gaussian) apertures 8) The Green's function connecting the optical field leaving the object plane (r-plane) with the field in the obser- vation plane without the Fourier plane filter takes the follow- ing form: k2cr (r + p)2 , = [ 2 ' G(r p) 47rf exp co ( 2 ; 2= ) ( , ) a) 2f kcr (2) 1-i A k(T 2f where a is the I /e2 intensity radius for the aperture in the Fourier plane, k is the wave number, and f is the focal length of the lenses. A describes an optional slight defocus of the 4-f system, viz. the offset of the target from the exact image plane. The scattered optical field from the object can be expressed as : U(r, t) = VT~5~f(r, t), (3) where \V (r , t) is the complex reflection coefficient with a ran- 378