# The influence of grating characteristics on moiré fringe multiplication

- Received:
- Revised:

DOI: 10.1007/BF02325171

- Cite this article as:
- Graham, S.M. & Sanford, R.J. Experimental Mechanics (1988) 28: 329. doi:10.1007/BF02325171

- 10 Citations
- 49 Downloads

## Abstract

Coherent optical techniques of fringe multiplication can be used to enhance the sensitivity of moiré measurements. However, for some coherent optical-processing systems, the significance of the resulting interference pattern is influenced by the characteristics of the gratings being processed. Analysis of one commonly used system for fringe multiplication involving two bar and space amplitude gratings reveals that accuracy of displacements and strains calculated from a fringe pattern is very sensitive to the transmission ratios of the gratings being processed. For variations of the transmission ratio of only 10 percent from the ideal case of 50 percent, the interference patterns produced by optical processing no longer correspond to multiplied moiré-fringe patterns.

### List of Symbols

*a*diffraction order of first diffraction in order sequence

*d*_{i}distance along optical axis

*e*_{d}relative displacement error

*e*_{s}percent error in strain

*F*()field amplitude in focal plane of first lens

*f*_{i}focal length of lens

*i**f*grating frequency (

*l*/mm)- ℱ{}{}
Fourier transform

*g*()field amplitude in image plane

*H*()filter-transmission function

*j*\(\sqrt { - 1} \)

*K*()intermediate function used to simplify a large integral equation

*k*wave number (2π/λ)

*n*diffraction order

*P*_{i}designation for plane

*i**p*pitch of diffraction grating (mm)

*p′*effective grating pitch (mm)

*r*ray group number

*s*inverse of

*z*coordinate (1/*z*)*s*_{i}inverse of distance along optical axis (1/

*d*_{i})*T*_{1}()Fourier transform of the grating-transmission function

*t*_{i}()grating-transmission function for grating

*i**U*()general-field amplitude function

*U*_{i}()incident-field amplitude at plane

*i*- Δ
*u*_{max} maximum difference between predicted and actual displacement

- (
*r, t*) Cartesian coordinates in image plane (

*P*_{6})- (
*u, ν*) Cartesian coordinates in filter plane (

*P*_{4})- (
*u*′, ν′) Cartesian coordinates in plane of second lens (

*P*_{5})- (
*x, y*) Cartesian coordinates in plane of second grating (

*P*_{2})- (
*x*_{1}, y_{1}) Cartesian coordinates in plane of first grating (

*P*_{1})- (
*x′, y′*) Cartesian coordinates in plane of first lens (

*P*_{3})*X, Y*variables used in simplification of integral equations

- Δ
*x* fringe spacing

- ε
applied strain

- ∈
predicted strain

- λ
wavelength of light (mm)

- η, ξ
variables used in simplification of integral equations

- θ
_{i}, θ_{n} angles of incidence and diffraction respectively

- ψ()
complex exponential function representing quadratic phase term