## Abstract

Focusing characteristics of hemispherical microlenses formed on the end of single-mode fibers are investigated. The Gaussian beam passing through a hemispherical microlens formed on the end of a single-mode fiber is always aberrated and truncated due to its spherical aberration and aperturing. However, numerical computations show that for the majority of the microlenses whose truncations and aberrations are small, the truncated or aberrated Gaussian beam can be assumed as a Gaussian beam. The shifts of the size and position of minimum 1/*e* spot size and the shift of the maximum intensity position due to the spherical aberration and the finite size of a microlens are also discussed. To analyze these shifts, Fresnel diffraction integrals are used in the intermediate field region to the hemispherical microlens. Minimum 1/*e* spot sizes for hemispherical microlenses are measured and compared with the theoretical values of minimum 1/*e* spot sizes derived with diffraction theory as well as with those derived using the paraxial theory.

© 1986 Optical Society of America

Full Article |

PDF Article
### Equations (10)

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

(1)
$$\mathrm{\Delta}f=-f/(1+{\pi}^{2}{N}_{G}^{2}),$$
(2)
$${N}_{G}={\omega}_{1}^{2}/\mathrm{\lambda}f.$$
(3)
$$\psi (r,z)=\frac{i{A}_{1}\hspace{0.17em}\text{exp}\left(-ikz-\frac{ik}{2z}{r}^{2}\right)}{\mathrm{\lambda}z}\times {{\int}_{0}^{a}\rho {J}_{0}\left(\frac{k\rho}{z}r\right)\hspace{0.17em}\text{exp}\left\{-{\rho}^{2}\left[\frac{1}{{\omega}_{1}^{2}}+\frac{ik}{2f}+\frac{ik}{2z}-ik\frac{\varphi (\rho )}{{\rho}^{2}}\right]\right\}}^{d\rho},$$
(4)
$$\varphi (\rho )={a}_{2}{\rho}^{4},$$
(5)
$$I(r,z)=\mid \psi (r,z){\mid}^{2}.$$
(6)
$$I(o,z)={\left|\frac{{A}_{1}}{\mathrm{\lambda}z}{\int}_{0}^{a}\text{exp}{\left(-{\rho}^{2}\left\{\frac{1}{{w}_{1}^{2}}+\frac{ik}{2}\left[\frac{1}{Z}-\frac{1}{f}-\frac{2\varphi (\rho )}{{\rho}^{2}}\right]\right\}\right)}^{d\rho}\right|}^{2}$$
(7)
$$W(o)=\frac{{\omega}_{1}{[{A}^{2}+{(B/{Z}_{01})}^{2}]}^{1/2}}{\sqrt{AD-CB}},$$
(8)
$$R(o)=\frac{{A}^{2}+{(B/{Z}_{01})}^{2}}{AC+BD/{Z}_{01}^{2}},$$
(9)
$$\varphi (\rho )=-\frac{1}{8}\frac{{n}^{2}(n-1)}{{R}^{3}}{\rho}^{4}.$$
(10)
$${a}_{2}=-\frac{1}{8}\frac{{n}^{2}(n-1)}{{R}^{3}},$$