## Abstract

A theoretical model of far-field interference from a sphere has been established, and its applications have been investigated. When two coherent parallel laser beams shine on a smooth sphere surface from opposite directions, the reflected lights form interference fringes at far field. The fringes have hyperbolic shapes and are not uniformly distributed. This paper derives a method for calculating the path-length difference between two parallel reflected lights, analyzes the interference field, and discusses reasons that cause the fringe density variations. A formula for calculating the highest orders of interference fringes is also provided. A method for using a spectrometer, CCD camera, and computer to measure the sphere diameter is demonstrated. The results agree with those from an Abbe comparator. The theory and methods are also suitable for measuring diameters of smooth cylinders.

© 2012 Optical Society of America

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### Equations (13)

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(1)
$$\mathrm{\Delta}OA{A}^{\prime}\cong \mathrm{\Delta}OB{B}^{\prime},\phantom{\rule[-0.0ex]{1em}{0.0ex}}\angle 1=\angle 2=\angle 3=\angle 4=45\xb0,\phantom{\rule{0ex}{0ex}}\angle 5=\angle 6=45\xb0+\alpha ,\phantom{\rule{0ex}{0ex}}\angle 7=\angle 8=45\xb0-\alpha ,{B}^{\prime}F\perp {A}^{\prime}{P}^{\prime},\phantom{\rule[-0.0ex]{1em}{0.0ex}}\angle {A}^{\prime}{B}^{\prime}F=\alpha .$$
(2)
$$\delta =n(D{A}^{\prime}+EB+{A}^{\prime}F),\phantom{\rule{0ex}{0ex}}\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\text{in the air},\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}n\approx 1.$$
(3)
$$\delta =D{A}^{\prime}+EB+{A}^{\prime}F$$
(4)
$$\delta =2\sqrt{2}r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha =2\sqrt{2}r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\theta}{2}.$$
(5)
$$\mathrm{\Delta}\delta =2\sqrt{2}r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{{\theta}_{2}}{2}-2\sqrt{2}r\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{{\theta}_{1}}{2}=N\lambda .$$
(6)
$$d=2r=\frac{N\lambda}{\sqrt{2}(\mathrm{sin}\text{\hspace{0.17em}}\frac{{\theta}_{2}}{2}-\mathrm{sin}\text{\hspace{0.17em}}\frac{{\theta}_{1}}{2})}.$$
(7)
$${\theta}_{0}=215\xb0{31}^{\prime}{0}^{\prime \prime}-\phantom{\rule{0ex}{0ex}}195\xb0{13}^{\prime}{30}^{\prime \prime}=20.2917\xb0\mathrm{.}$$
(8)
$${\beta}_{s}=\mathrm{arctan}\text{\hspace{0.17em}}\frac{S{F}^{\prime}}{f}=\mathrm{arctan}\text{\hspace{0.17em}}\frac{(1373-593)\times 0.00348}{48.5}\phantom{\rule{0ex}{0ex}}=3.2033\xb0,$$
(9)
$${\beta}_{L}=\mathrm{arctan}\text{\hspace{0.17em}}\frac{{F}^{\prime}L}{f}=\mathrm{arctan}\text{\hspace{0.17em}}\frac{(1864-1373)\times 0.00348}{48.5}=2.0177\xb0,$$
(10)
$${\theta}_{1}={\theta}_{0}-{\beta}_{S}=17.0884\xb0,$$
(11)
$${\theta}_{2}={\theta}_{0}+{\beta}_{L}=22.3094\xb0\mathrm{.}$$
(12)
$$N={N}_{S{F}^{\prime}}+{N}_{{F}^{\prime}L}=70.$$
(13)
$$d=2r=0.698\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}.$$