# A geometrical approach to holographic interferometry

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## Abstract

Geometrical considerations are used to obtain quantitative data for the components of the displacement vector. Use is made of a dual-beam illumination and a variant is described, using point light sources and a single point of observation. Contour lines for the displacement vector in the viewing direction and in a perpendicular direction are obtained as the algebraic sum and difference of two interference patterns. Densification of the initial pattern is used to obtain moiré patterns for the out-of-plane and in-plane displacements of flat Surfaces and for the derivatives of these displacements. To determine the complete displacement field of objects of arbitrary shape, one holographic plate is sufficient, using two times two-point light sources.

## Keywords

Light Source Fluid Dynamics Quantitative Data Single Point Displacement Vector## List of Symbols

*a,b,c,e*distances in a circle construction (Appendices A and B)

*d*displacement vector

*d*_{n}, d_{r}..component of the displacement vector in the direction

*n, r*,..*k*fringe-order number

- ℓ
optical-path length; direction of illumination

*n*direction of the sensitivity vector

*p*direction of the displacement vector

*r*direction of observation; radius

*t*direction perpendicular to the r-direction (plane V or V′)

*u*arbitrary direction (x-y plane)

*x,y,z*coordinates

*L*point of illumination

*N*intersection of the n-direction with the circle through L and R

*O*origin of the coordinate system

*P, P′*point of the object surface, before and after displacement

*Q, S*points of the object surface

*R*point of observation

*V, V′*planes tangent to surfaces of revolution

*W, W′*planes in which the displacement components are determined

- α
angle between ℓ- and z-direction

- α′
angle between ℓ- and r-direction (point R is moving)

- α
^{'} angle between ℓ- and r-direction (point R is fixed)

- \(\tilde \alpha , \hat \alpha , \bar \alpha \)
given values for α, in which

*tg*α=0.5-1.0-1.5- β
angle between ℓ- and n-direction

- γ,γ′
angle between r- and z-direction

- δ
change in optical-path length

- ϑ
angle between p- and n-direction

- λ
wavelength of laser light

- ξ
*x/r*- ϕ
angle between p- and z-direction

- ψ
angle between plane W′ and the x-z plane

- ω
angle between n- and z-direction

- Θ
_{1}, Θ_{2} interference patterns giving contour lines for δ, for illumination from L

_{1}and L_{2}- Θ
_{1}^{'}, Θ_{2}^{'} (Θ

_{1}+ Φ), (Θ_{2}+ Φ)- Θ
_{2}^{''} (Θ

_{2}- Φ)- Φ
interference pattern due to a rotation of the reference plane for Θ (Φ=

*Ax+By+C*)- Ψ
interference pattern for the sum and difference of Θ

- Ψ
_{1}, Ψ_{2} (Θ

_{1}+ Θ_{2}), (Θ_{1}- Θ_{2})- Ψ
_{1}^{''}Ψ_{1}^{'} (Θ

_{1}^{'}+ Θ_{2}^{''}), (Θ_{1}^{'}- Θ_{2}^{'})- Ω
_{1}, Ω_{2} interference pattern due to a rotation of the model, for illumination from L

_{1}and L_{2}

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## References

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