## Abstract

The intensity distribution of the beam from a laser operated in the TEM_{00} mode is Gaussian. In some applications it is desirable to have a uniform intensity over a certain region in space. For example, when a Gaussian beam is incident on a smooth surface containing small isolated defects the amount of light scattered by a defect will depend on the position of the defect relative to the center of the beam. In the past, several techniques have been devised to convert a Gaussian intensity profile into a uniform intensity over a specified region in space. In the present work a method of normalization is described that makes direct use of the Gaussian intensity distribution of the TEM_{00} mode. By this method, the amount of light scattered by a defect can be normalized to the value that would be observed if the defect were located at the center of the beam for a defect small in size compared with the 1/*e*^{2} diameter of the Gaussian profile. Experimental data were obtained that verify the theory.

© 1980 Optical Society of America

Full Article |

PDF Article
### Equations (23)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$I(r)={I}_{0}\hspace{0.17em}\text{exp}(-2{r}^{2}/{a}^{2}),$$
(2)
$$\begin{array}{l}{W}_{1}={W}_{0}\hspace{0.17em}\text{exp}(-2{r}_{1}^{2}/{a}^{2})\hfill \\ {W}_{2}={W}_{0}\hspace{0.17em}\text{exp}(-2{r}_{2}^{2}/{a}^{2})\hfill \end{array}\},$$
(3)
$${W}_{1}={W}_{0}\hspace{0.17em}\text{exp}(-2{r}_{1}^{2}/{a}^{2}),$$
(4)
$${W}_{2}={W}_{0}\hspace{0.17em}\text{exp}(-2{r}_{2}^{2}/{a}^{2}),$$
(5)
$${W}_{3}={W}_{0}\hspace{0.17em}\text{exp}(-2{r}_{3}^{2}/{a}^{2}),$$
(6)
$${r}_{2}^{2}-{r}_{1}^{2}=\frac{1}{2}{a}^{2}\hspace{0.17em}\text{ln}({W}_{1}/{W}_{2}),$$
(7)
$${r}_{3}^{2}-{r}_{1}^{2}=\frac{1}{2}{a}^{2}\hspace{0.17em}\text{ln}({W}_{1}/{W}_{3}).$$
(8)
$${r}_{1}\hspace{0.17em}\text{cos}\beta +{r}_{2}\hspace{0.17em}\text{cos}\psi =L,$$
(9)
$${r}_{3}\hspace{0.17em}\text{cos}\alpha +{r}_{1}\hspace{0.17em}\text{sin}\beta =L,$$
(10)
$${r}_{1}\hspace{0.17em}\text{sin}\beta ={r}_{2}\hspace{0.17em}\text{sin}\psi ,$$
(11)
$${r}_{1}\hspace{0.17em}\text{cos}\beta ={r}_{3}\hspace{0.17em}\text{sin}\alpha .$$
(12)
$$2L{r}_{3}\hspace{0.17em}\text{cos}\alpha -A={L}^{2}+{r}_{3}^{2}-{r}_{2}^{2},$$
(13)
$${L}^{2}-A=2L{r}_{3}\hspace{0.17em}\text{sin}\alpha ,$$
(14)
$${r}_{2}^{2}-A={r}_{3}^{2}-B,$$
(15)
$$A=\frac{1}{2}{a}^{2}\hspace{0.17em}\text{ln}({W}_{1}/{W}_{2}),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}B=\frac{1}{2}{a}^{2}\hspace{0.17em}\text{ln}({W}_{1}/{W}_{3}).$$
(16)
$${r}_{3}^{2}=\frac{{({L}^{2}-A)}^{2}+{({L}^{2}+B)}^{2}}{4{L}^{2}}.$$
(17)
$${W}_{0}={W}_{3}\hspace{0.17em}\text{exp}\left[\frac{{({L}^{2}-A)}^{2}+{({L}^{2}+B)}^{2}}{2{a}^{2}{L}^{2}}\right].$$
(18)
$${r}_{1}\hspace{0.17em}\text{cos}\beta ={r}_{3}\hspace{0.17em}\text{sin}\alpha -h,$$
(19)
$${r}_{3}^{2}=\frac{{[{(L+h)}^{2}-A-{h}^{2}]}^{2}+{\left\{B+{L}^{2}+{h}^{2}-\frac{h}{L}[{(L+h)}^{2}-A-{h}^{2}]\right\}}^{2}}{4{L}^{2}},$$
(20)
$${W}_{0}={W}_{3}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{{[{(L+h)}^{2}-A-{h}^{2}]}^{2}+{\left\{B+{L}^{2}+{h}^{2}-\frac{h}{L}[{(L+h)}^{2}-A-{h}^{2}]\right\}}^{2}}{2{a}^{2}{L}^{2}}\right];$$
(21)
$${W}_{N,M}={W}_{0}\hspace{0.17em}\text{exp}[-2({u}_{x}^{2}+{u}_{y}^{2})/{a}^{2}],$$
(22)
$${u}_{x}=ML+L-Nh-{r}_{3}\hspace{0.17em}\text{sin}\alpha ,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{u}_{y}=NL+{r}_{3}\hspace{0.17em}\text{cos}\alpha ,$$
(23)
$$\alpha =\text{arcsin}\hspace{0.17em}\left[\frac{{(L+h)}^{2}-A-{h}^{2}}{2L{r}_{3}}\right].$$