## Abstract

A geometric optics model for evaluating end separation and transverse offset losses at a graded-index (GRIN) rod lens to lens coupler is extended to the case of angular tilt misalignment loss. Two different fiber–lens combinations are examined, and the model is verified by comparison to experimental measurements, which show close agreement to the theoretical predictions. In addition, invariance of the angular tilt loss to differing optical launch and receive conditions is shown, and a possible explanation for this phenomenon is provided. Finally, the theoretical model is used to predict fiber–lens combinations which minimize possible angular tilt losses.

© 1986 Optical Society of America

Full Article |

PDF Article
### Equations (14)

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

(1)
$$p\phantom{\rule{0.1em}{0ex}}(r)=p\phantom{\rule{0.1em}{0ex}}(0)\phantom{\rule{0.1em}{0ex}}[1-{(r\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}a)}^{\alpha}]\phantom{\rule{0.1em}{0ex}},$$
(2)
$$P\phantom{\rule{0.1em}{0ex}}(\theta )=P\phantom{\rule{0.1em}{0ex}}(0)exp\phantom{\rule{0em}{0ex}}[\phantom{\rule{0em}{0ex}}-\phantom{\rule{0em}{0ex}}2\phantom{\rule{0.1em}{0ex}}{(\theta \phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}{\theta}_{m})}^{2}]\phantom{\rule{0.2em}{0ex}},$$
(3)
$$\text{N}.\phantom{\rule{0.1em}{0ex}}\text{A}.\phantom{\rule{0.1em}{0ex}}(r)=n\phantom{\rule{0.1em}{0ex}}(r)\phantom{\rule{0.2em}{0ex}}sin\phantom{\rule{0em}{0ex}}{\theta}_{m}={n}_{1}\sqrt{2\Delta}\phantom{\rule{0em}{0ex}}{[1-{(r\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}a)}^{\alpha}]}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2},$$
(4)
$$\left(\begin{array}{c}{r}_{2}\\ {a}_{2}\end{array}\right)=\left[\begin{array}{ll}cos\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill & {{n}_{a}}^{-1}{A}^{-1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}\phantom{\rule{0.2em}{0ex}}sin\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill \\ -\phantom{\rule{0em}{0ex}}{n}_{a}\phantom{\rule{0.1em}{0ex}}{A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}\phantom{\rule{0.2em}{0ex}}sin\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill & \phantom{\rule{0em}{0ex}}cos\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill \end{array}\right]\phantom{\rule{0.2em}{0ex}}\left(\begin{array}{c}{r}_{1}\\ {a}_{1}\end{array}\right)\phantom{\rule{0.2em}{0ex}},$$
(5)
$${P}_{E}\phantom{\rule{0.1em}{0ex}}\mathit{\text{\alpha}}\sum _{\beta \phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}0}^{2\pi}\phantom{\rule{0.2em}{0ex}}\sum _{r\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.1em}{0ex}}-\phantom{\rule{0.1em}{0ex}}a}^{a}\phantom{\rule{0.3em}{0ex}}\sum _{\theta \phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.1em}{0ex}}-\phantom{\rule{0.1em}{0ex}}{\theta}_{m}}^{{\theta}_{m}}P\phantom{\rule{0.1em}{0ex}}(\theta +\frac{\delta \theta}{2})\phantom{\rule{0.1em}{0ex}}p\phantom{\rule{0.1em}{0ex}}(r+\frac{\delta r}{2})\phantom{\rule{0.1em}{0ex}}r\mathit{\text{\delta}}\theta \delta \mathit{\text{r}}\delta \beta \phantom{\rule{0.2em}{0ex}}.$$
(6)
$$\left(\begin{array}{c}{r}_{4}\\ {a}_{4}\end{array}\right)=\left[\begin{array}{ll}cos\phantom{\rule{0em}{0ex}}({A}^{1/2}L)\hfill & {{n}_{a}}^{-1}{A}^{-1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}sin\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill \\ -\phantom{\rule{0em}{0ex}}{n}_{a}\phantom{\rule{0.1em}{0ex}}{A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}sin\phantom{\rule{0em}{0ex}}({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill & \phantom{\rule{0em}{0ex}}cos({A}^{1\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}2}L)\hfill \end{array}\right]\phantom{\rule{0.2em}{0ex}}\left(\begin{array}{c}{r}_{3}\\ {a}_{3}\end{array}\right)\phantom{\rule{0.2em}{0ex}},$$
(7)
$${r}_{3}\simeq {r}_{2}\phantom{\rule{0.2em}{0ex}}.$$
(8)
$${a}_{3}={a}_{2}cos\phantom{\rule{0em}{0ex}}\beta \phantom{\rule{0.2em}{0ex}}.$$
(9)
$$\begin{array}{ll}{a}_{3}\hfill & \phantom{\rule{0em}{0ex}}={a}_{2}cos\phantom{\rule{0em}{0ex}}\beta \phantom{\rule{0.1em}{0ex}}\pm \phantom{\rule{0.1em}{0ex}}\tau \phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}{a}_{2}\phantom{\rule{0.1em}{0ex}}\u2a7e0,\hfill \\ {a}_{3}\hfill & \phantom{\rule{0em}{0ex}}={a}_{2}cos\phantom{\rule{0em}{0ex}}\beta \phantom{\rule{0.1em}{0ex}}\mp \phantom{\rule{0.1em}{0ex}}\tau \phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}{a}_{2}<0,\hfill \end{array}$$
(10)
$$|\phantom{\rule{0.1em}{0ex}}{r}_{3}\phantom{\rule{0.1em}{0ex}}|\phantom{\rule{0.1em}{0ex}}>R,$$
(11)
$$|\phantom{\rule{0.1em}{0ex}}{r}_{4}\phantom{\rule{0.1em}{0ex}}|\phantom{\rule{0.2em}{0ex}}>a,$$
(12)
$$|\phantom{\rule{0.1em}{0ex}}{a}_{4}\phantom{\rule{0.1em}{0ex}}|\phantom{\rule{0.2em}{0ex}}>\text{N}.\phantom{\rule{0.1em}{0ex}}\text{A}\phantom{\rule{0.1em}{0ex}}.\phantom{\rule{0.1em}{0ex}}({r}_{4})\phantom{\rule{0.2em}{0ex}},$$
(13)
$${P}_{L}\phantom{\rule{0.2em}{0ex}}\alpha \phantom{\rule{0.2em}{0ex}}\sum _{\beta}\phantom{\rule{0.2em}{0ex}}\sum _{r}\phantom{\rule{0.2em}{0ex}}\sum _{\theta}\phantom{\rule{0.2em}{0ex}}P\phantom{\rule{0.1em}{0ex}}(\theta +\frac{\delta \theta}{2})\phantom{\rule{0.1em}{0ex}}p\phantom{\rule{0.1em}{0ex}}(r+\frac{\delta r}{2})\phantom{\rule{0.1em}{0ex}}r\mathit{\text{\delta}}\theta \delta \mathit{\text{r}}\delta \beta ,$$
(14)
$$\text{loss}=\phantom{\rule{0em}{0ex}}-\phantom{\rule{0em}{0ex}}10log\phantom{\rule{0em}{0ex}}[1-({P}_{L}\phantom{\rule{0em}{0ex}}/\phantom{\rule{0em}{0ex}}{P}_{E})\phantom{\rule{0.1em}{0ex}}]\phantom{\rule{0.2em}{0ex}}\text{dB}\phantom{\rule{0.1em}{0ex}}.$$