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  • Transmission wavefront shearing interferometryfor photoelastic materials

    Sharlotte L. B. Kramer,* Guruswami Ravichandran, and Kaushik BhattacharyaCalifornia Institute of Technology, Division of Engineering and Applied Science,

    1200 East California Boulevard, Pasadena, California 91125, USA

    *Corresponding author: sharlott@caltech.edu

    Received 6 November 2008; revised 16 March 2009; accepted 19 March 2009;posted 20 March 2009 (Doc. ID 103187); published 23 April 2009

    A general analysis and experimental validation of transmission wavefront shearing interferometry forphotoelastic materials are presented. These interferometers applied to optically isotropic materials pro-duce a single interference pattern related to one phase term, but when applied to photoelastic materials,they produce the sum of two different interference patterns with phase terms that are the sum and dif-ference, respectively, of two stress-related phase terms. The two stress-related phase terms may be se-parated using phase shifting and polarization optics. These concepts are experimentally demonstratedusing coherent gradient sensing in full field for a compressed polycarbonate plate with a V-shaped notchwith good agreement with theoretical data. The analysis may be applied to any wavefront shearinginterferometer by modifying parameters describing the wavefront shearing distance. 2009 OpticalSociety of America

    OCIS codes: 120.4880, 120.7000, 160.4760, 260.1440, 260.5430.

    1. Introduction

    Wavefront shearing interferometry is a well-established optical technique for measuring manyoptical, material, and mechanical properties suchas wavefront slope characterization [1], surface de-formation [2], and even fracture of materials [36].Shearing interferometry essentially is the interfer-ence of a coherent wavefront with a copy of itselfsheared or translated by a distance dshear; this tech-nique is self-referencing and hence is insensitive torigid body motion [25]. The general analysis ofthe interference pattern for standard wavefrontshearing interferometers depends only on the wave-front characteristics and the distance dshear. Once theparameters for producing the sheared wavefront andinterfering the two wavefronts are characterized fora particular shearing method, then the analysis maybe detailed for that method. With several methods toproduce the wavefront shearing, the choice of shear-ing interferometer depends on the requirements of

    the application such as measurement sensitivity orcompactness.

    An important consideration for the analysis is howthe wavefront is formed. For techniques that involvetransmission through a material of interest, theshape and optical properties of the material are con-sidered (e.g., a spherical wavefront emanating froman optically isotropic planoconvex lens.) In the caseof a deformed material that is originally planar,thickness and refractive index variations in the ma-terial result in optical path differences that may berelated to stresses. A general analysis of the opticalpath difference in this case was previously completedfor the method of caustics [79]. Though not a wave-front shearing interferometry technique, the methodof caustics, which has been used for large stress gra-dient applications, does consider optical path differ-ences due to a deformed sample, resulting in ashadow spot in the far field. The method of causticsgives only a point measurement, which motivatedthe development of coherent gradient sensing(CGS), capable of measuring full-field stress or dis-placement gradients when used in transmission orin reflection, respectively [3,4]. CGS is a wavefront

    0003-6935/09/132450-11$15.00/0 2009 Optical Society of America

    2450 APPLIED OPTICS / Vol. 48, No. 13 / 1 May 2009

  • lateral shearing interferometer that achieves shear-ing by a pair of amplitude gratings; sensitivity ad-justment is achievable through choice of gratingline density, separation between the gratings, andlight wavelength. Previously, CGS in transmissionhas been used only for optically isotropic materials[3,4,6]. CGS in reflection has been used for opaqueisotropic materials [3,4], for materials with reflectivecoatings [3,4,10], and for composite materials [5,11].No previous studies have considered CGS in trans-mission for optically anisotropic materials.Taking inspiration from the method of caustics ap-

    plied to photoelastic materials, this paper presentswhat is to our knowledge the first general analysisof an initially planar wavefront transmitted througha photoelastic material, in terms of electric field andoptical path difference, for a general wavefrontshearing interferometer; the analysis is then specifi-cally applied to CGS. The analysis may easily bemodified for any wavefront shearing interferometerby changing the experimental parameters related tothe distance dshear.This study demonstrates that the resultant inter-

    ference pattern is no longer a simple function of a sin-gle phase term related to the sum of principalstresses, denoted sum, as in the case of optically iso-tropic materials. Due to the optical anisotropy fromthe stress birefringence, the interference patternsfrom the x and y coordinates of the electric field,Ex and Ey, respectively, are no longer equivalent.Considering the interference patterns along theorthogonal principal axes of the photoelastic speci-men, denoted Iimage1 and I

    image2 , the phase terms of

    these distinct interference patterns, 1 and 2, aresum diff and sum diff , respectively, where diffis related to the difference of principal stresses. Thus,diff obscures the desired phase information, sum,due to the optical anisotropy of the material. diffis zero for an optically isotropic material and there-fore is not an issue for isotropic materials. For a gen-eral incident electric field, wavefront shearinginterferometry for photoelastic materials results inan image that is the superposition of Iimage1 andIimage2 , which is too complicated to analyze by itself.The desired phase sum may be recovered by usingphase shifting and polarization optics. These con-cepts are demonstrated using CGS for a compressedpolycarbonate thin plate with a V-shaped side notchwith good agreement between experimental andtheoretical data.

    2. Experimental Method and Analysis of Full-FieldPhase Data

    A. Experimental Method

    The CGS method starts with an incident plane waveof a collimated laser beam that transmits through atransparent sample or that reflects off an opaquesample. The working principle of CGS to laterallyshear an incident wavefront, shown in Fig. 1 for hor-izontal shear, is the same for both transmission and

    reflection. Tippur et al. [3,4] give a full description ofthe CGS working principle. The main concept of CGSis that the dshear of the interfered wavefronts is due todiffraction through a pair of Ronchi gratings, G1 andG2, each with pitch p, separated by distance

    suchthat the desired wavefronts E0;1 and E1;0 are se-parated by a lateral shearing distance dshear

    inthe x z or y z plane and propagate at the same an-gle relative to the z axis upon leaving grating G2.The diffracted waves transmit through a filteringlens, which separates the corresponding diffractionorders into horizontal diffraction spots at the focalplane of the filtering lens. An aperture at this focalplane selects either the 1 or 1 diffraction order,meaning only the wavefronts E0;1 and E1;0 propa-gate to the image plane. In Subsection 2.B, analysisof the first-order diffraction shows how the interfer-ence pattern may be related to the first x and y deri-vatives of principal stresses based on assumption of asmall dshear.

    B. Analysis

    1. Electric Field Description of theTransmitted Wavefront

    Assuming a coherent plane wave of monochromaticlight propagating along the z axis, the electric fieldof the wavefront at z zo is given by

    Fig. 1. Working principle for horizontal shearing transmissioncoherent gradient sensing.

    1 May 2009 / Vol. 48, No. 13 / APPLIED OPTICS 2451

  • Einx; y; t Exx; y; t Eyx; y; t; 1a

    Exx; y; t Ax expjkzo t x; 1b

    Eyx; y; t Ay expjkzo t y; 1c

    where Ex and Ey are the amplitudes, Ax and Ay areconstants, is the wavelength, k 2= is the wave-number, is the angular frequency, and x and y arearbitrary constant phase terms. If the plane wavepropagates through a transparent material with re-fractive index no and nominal thickness h, then theresulting electric field magnitudes of this perturbedwavefront in the x and y directions after the samplematerial at z are

    Esamplex x; y; t Ax expjkz t x kno 1h kSxx; y; 2a

    Esampley x; y; t Ay expjkz t y kno 1h kSyx; y; 2b

    where Sxx; y and Syx; y are the optical pathdifferences at each point x; y along the x and y di-rections, as further described in Subsection 2.B.2.

    2. Photoelastic Effect in Transparent Materials

    In general, a plane wave transmitted through a ma-terial experiences some change in optical path lengthdue to both variation in refractive index, nx; y,and variation in thickness, hx; y, in the transmit-ting media. Along a given axis a, the optical path dif-ference is expressed as

    Sax; y hnax; y no 1hx; y: 3

    A full explanation of the optical path difference maybe found in [7]. These variations from an initiallyuniform material can be related to stresses in thematerial. First, a transparent material that experi-ences stress-induced birefringence, also known asthe photoelastic effect, has variations in refractiveindex along the three principal optical axes such that

    n1 n1 no A1 B2 3; 4a

    n2 n2 no A2 B1 3; 4b

    n3 n3 no A3 B1 2; 4c

    where i, i f1; 2; 3g, are the principal stresses andA and B are the two absolute photoelastic constantsof the transparent material. These equations areknown as the NeumannMaxwel