## Abstract

Spatial distribution of optical phase and amplitude can be measured by using a single-mode optical-fiber interferometric system. Following the first proposal of the measuring principle by the present authors in 1978, a further development of the measuring system is now reported. The major achievement is the use of a novel spatial filter and a set of photodetectors by which the interference fringe position is located. Combining them with analog signal-processing circuitry and with a mechanical feedback system, fully automated recording of the optical phase and amplitude distribution is successfully achieved. The method is used to measure phase-front distribution near a Gaussian beam waist.

© 1981 Optical Society of America

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### Equations (13)

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(1)
$${I}_{p}(u,l,w)={I}_{d}(u,l,w)+{I}_{r}(u,l,w)+2\sqrt{{I}_{d}(u,l,w){I}_{r}(u,l,w)}\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\frac{2\pi \text{u}}{\mathrm{\Lambda}}+\delta \right),$$
(2)
$$\mathrm{\Lambda}=\mathrm{\lambda}l/d$$
(3)
$$\mathrm{\Delta}Q\equiv {Q}_{U}-{Q}_{L},$$
(4)
$${Q}_{L}^{U}={\int}_{-b}^{b}{\int}_{-a}^{a}{I}_{p}(u\pm c,l,w)dwdu=4ab({I}_{d}+{I}_{r})+4\sqrt{{I}_{d}{I}_{r}}\frac{\mathrm{\Lambda}a}{2\pi}\text{sin}\hspace{0.17em}\left(\frac{2\pi}{\mathrm{\Lambda}}b\right)\hspace{0.17em}\text{cos}\hspace{0.17em}\left(\pm \frac{2\pi c}{\mathrm{\Lambda}}+\delta \right).$$
(5)
$$\mathrm{\Delta}Q=4\sqrt{{I}_{d}{I}_{r}}\frac{2a\mathrm{\Lambda}}{\pi}\text{sin}\delta .$$
(6)
$${Q}_{M}K\simeq 4{a}_{M}{b}_{M}{(\sqrt{{I}_{r}}+\sqrt{{I}_{d}})}^{2}.$$
(7)
$${I}_{d}=\frac{1}{2}\{({I}_{d}+{I}_{r})-{[{({I}_{d}+{I}_{r})}^{2}-{(2\sqrt{{I}_{d}{I}_{r}})}^{2}]}^{1/2}\},$$
(8)
$${I}_{d}=\frac{1}{2}\left\{\frac{{Q}_{U}+{Q}_{L}}{8ab}-{\left[2{\left(\frac{{Q}_{U}+{Q}_{L}}{8ab}\right)}^{2}-{\left(\frac{{Q}_{M}}{2{a}_{M}{b}_{M}}\right)}^{2}\right]}^{1/2}\right\}.$$
(9)
$$\text{sin}\delta =\left[\frac{({Q}_{U}-{Q}_{L})\pi {a}_{M}{b}_{M}}{2a\mathrm{\Lambda}{Q}_{M}}\right].$$
(10)
$$I=\frac{1}{2}\left\{{I}_{d}+{I}_{r}+{I}_{n}-{\left[{I}_{d}^{2}+{({I}_{r}+I)}^{2}+2{I}_{r}{I}_{d}-4{I}_{r}{I}_{d}{\left(\frac{\text{sin}2\pi \alpha}{2\pi \alpha}\text{cos}\delta \right)}^{2}\right]}^{1/2}\right\}.$$
(11)
$$W{(Z)}^{2}={W}_{0}^{2}\left(1+\frac{{Z}^{2}}{{Z}_{0}^{2}}\right),$$
(12)
$$R(Z)=Z\left(1+\frac{{Z}_{0}^{2}}{{Z}^{2}}\right),$$
(13)
$${W}^{\prime}(Z)={[W{(Z)}^{2}+{a}^{2}]}^{1/2},$$