## Abstract

A new design is presented to achieve a hybrid micro-diffractive-refractive lens with wide field of view (WFOV) of 80° integrated on backside of InGaAs / InP photodetector for free space optical interconnections. It has an apparent advantage of athermalization of optical system which working in large variation of ambient temperature ranging from -20 °C to 70 °C. The changing of focal length is only 0.504 μm in the ambient temperature range with the hybrid microlens, which opto-thermal expansion coefficient matches with thermal expansion coefficient of AuSn solder bump used in corresponding flip-chip packaging system. The hybrid lens was designed via CODE-V^{TM} professional software. The results show that the lens has good optical performance for the optical interconnection use.

©2002 Optical Society of America

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### Equations (18)

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(1)
$${x}_{f,r}=\alpha -\frac{1}{n-{n}_{i}}\left(\frac{\mathit{dn}}{\mathit{dT}}-n\frac{{\mathit{dn}}_{0}}{\mathit{dT}}\right)$$
(2)
$${x}_{f,d}=2\alpha +\frac{1}{{n}_{i}}\frac{{\mathit{dn}}_{i}}{\mathit{dT}}$$
(3)
$$f=\frac{{\mathit{nr}}_{1}{r}_{2}}{\left(n-1\right)\left[n\left({r}_{2}-{r}_{1}\right)+\left(n-1\right)d\right]}$$
(4)
$$f=\frac{{r}_{1}}{n-1}$$
(5)
$$\frac{\mathit{df}}{\mathit{dT}}=\frac{1}{n-1}\frac{\mathit{dr}}{\mathit{dT}}-\frac{r}{{\left(n-1\right)}^{2}}\frac{\mathit{dn}}{\mathit{dT}}$$
(6)
$$\frac{\mathit{df}}{\mathit{dT}}=\frac{r}{n-1}\alpha -\frac{r}{{\left(n-1\right)}^{2}}\frac{\mathit{dn}}{\mathit{dT}}$$
(7)
$${x}_{f,r}=\frac{1}{f}\frac{\mathit{df}}{\mathit{dT}}=\alpha -\frac{1}{n-1}\frac{\mathit{dn}}{\mathit{dT}}$$
(8)
$${\left({f}_{d}+m\frac{{\lambda}_{0}}{n}\right)}^{2}={r}_{m}^{2}+{f}_{d}^{2}$$
(9)
$${f}_{d}=\frac{{\mathit{nr}}_{m}^{2}}{2m{\lambda}_{0}},\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}m=\mathrm{1,2,3},\dots $$
(10)
$${r}_{m}\left(T\right)={r}_{m}\left(1+\alpha \Delta T\right)$$
(11)
$$n\left(T\right)=n+\frac{\mathit{dn}}{\mathit{dT}}\Delta T$$
(12)
$${f}_{d}\left(T\right)={f}_{d}\left[1+2\alpha \Delta T+{\alpha}^{2}{\left(\Delta T\right)}^{2}+\frac{1}{n}\frac{\mathit{dn}}{\mathit{dT}}\Delta \mathit{T+}2\frac{1}{n}\frac{\mathit{dn}}{\mathit{dT}}\alpha {\left(\Delta T\right)}^{2}+\frac{1}{n}\frac{\mathit{dn}}{\mathit{dT}}{\alpha}^{2}{\left(\Delta T\right)}^{3}\right]$$
(13)
$${x}_{f,d}=\frac{1}{{f}_{d}}\frac{{\mathit{df}}_{\mathit{d}}}{\mathit{dT}}=2\alpha +\frac{1}{n}\frac{\mathit{dn}}{\mathit{dT}}$$
(14)
$$n=3.075\left(1+2.7\times {10}^{-5}T\right)$$
(15)
$$\frac{{x}_{f}}{f}=\frac{{x}_{f,r}}{{f}_{r}}+\frac{{x}_{f,d}}{{f}_{d}}$$
(16)
$${\mathit{\Delta f}}_{\mathrm{Total}}={\mathit{\Delta f}}_{\mathrm{Lens}}+{\mathit{\Delta f}}_{\mathrm{Bump}}$$
(17)
$$z=\frac{{\mathit{cr}}^{2}}{1+\sqrt{1-\left(1+k\right){c}^{2}{r}^{2}}}+{\mathit{Ar}}^{4}{\mathit{Br}}^{6}+{\mathit{Cr}}^{8}+\dots +{\mathit{Jr}}^{20}$$
(18)
$$\Phi \left(r\right)={C}_{2}{r}^{2}+{C}_{4}{r}^{4}+{C}_{6}{r}^{6}+\dots +{C}_{20}{r}^{20}$$