## Abstract

A laser reflection-confocal focal-length measurement (LRCFM) is proposed for the high-accuracy measurement of lens focal length. LRCFM uses the peak points of confocal response curves to precisely identify the lens focus and vertex of the lens last surface. LRCFM then accurately measures the distance between the two positions to determine the lens focal length. LRCFM uses conic fitting, which significantly enhances measurement accuracy by inhibiting the influence of environmental disturbance and system noise on the measurement results. The experimental results indicate that LRCFM has a relative expanded uncertainty of less than 0.0015%. Compared with existing measurement methods, LRCFM has high accuracy and a concise structure. Thus, LRCFM is a feasible method for high-accuracy focal-length measurements.

© 2013 Optical Society of America

Full Article |

PDF Article
### Equations (21)

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

(1)
$${l}_{F}^{\prime}=2|{z}_{A}-{z}_{B}|,$$
(2)
$${I}_{A}(u)={|{\int}_{0}^{\infty}{\{{\int}_{0}^{1}{P}_{t}(\rho ){P}_{c}(\rho )\mathrm{exp}\left(\frac{iu{\rho}^{2}}{2}\right){J}_{0}({\nu}_{1}\rho )\rho \mathrm{d}\rho \}}^{2}{\nu}_{1}\mathrm{d}{\nu}_{1}|}^{2},$$
(3)
$$\{\begin{array}{c}{\nu}_{1}=\frac{\pi}{\lambda}\frac{D}{{f}_{t}^{\prime}}{r}_{1}\\ u=\frac{\pi}{2\lambda}z\frac{{D}^{2}}{{f}_{t}^{\prime 2}}\end{array}.$$
(4)
$${I}_{A}(u)={\left[\frac{\mathrm{sin}(u/2)}{u/2}\right]}^{2}.$$
(5)
$${I}_{B}(u)={\left[\frac{\mathrm{sin}(u)}{u}\right]}^{2}.$$
(6)
$${I}_{A}^{\prime}(u,{u}_{\delta M})={\left\{\frac{\mathrm{sin}(u/2-{u}_{\delta M}/4)}{u/2-{u}_{\delta M}/4}\right\}}^{2},$$
(7)
$${I}_{B}^{\prime}(u,{u}_{\delta M})={\left\{\frac{\mathrm{sin}(u-{u}_{\delta M}/4)}{u-{u}_{\delta M}/4}\right\}}^{2},$$
(8)
$${u}_{\delta M}=\frac{\pi}{2\lambda}{\delta}_{M}\frac{{D}^{2}}{{f}_{c}^{\prime 2}},$$
(9)
$$\mathrm{\Delta}{u}_{1}=\frac{{u}_{\delta M}}{2}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\mathrm{\Delta}{u}_{2}=\frac{{u}_{\delta M}}{4}.$$
(10)
$$\mathrm{\Delta}{l}_{1}=\frac{1}{2}\frac{{f}_{t}^{\prime 2}}{{f}_{c}^{\prime 2}}{\delta}_{M}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\mathrm{\Delta}{l}_{2}=\frac{1}{4}\frac{{f}_{t}^{\prime 2}}{{f}_{c}^{\prime 2}}{\delta}_{M}.$$
(11)
$${\mathrm{\Delta}}_{\text{deviation}}=2(\mathrm{\Delta}{l}_{1}-\mathrm{\Delta}{l}_{2})=\frac{1}{2}\frac{{f}_{t}^{\prime 2}}{{f}_{c}^{\prime 2}}{\delta}_{M}.$$
(12)
$${u}_{1}=\frac{{\mathrm{\Delta}}_{\text{deviation}}}{\sqrt{3}}=\frac{1}{2\sqrt{3}}\frac{{f}_{t}^{\prime 2}}{{f}_{c}^{\prime 2}}{\delta}_{M}.$$
(13)
$${\mathrm{\Delta}}_{\text{axial}}\approx {f}_{t}^{\prime}(\frac{\mathrm{cos}\text{\hspace{0.17em}}\beta}{\mathrm{cos}\text{\hspace{0.17em}}\alpha}-1),$$
(14)
$${u}_{2}=\frac{{\mathrm{\Delta}}_{\text{axial}}}{\sqrt{3}}=\frac{{f}_{t}^{\prime}}{\sqrt{3}}(\frac{\mathrm{cos}\text{\hspace{0.17em}}\beta}{\mathrm{cos}\text{\hspace{0.17em}}\alpha}-1),$$
(15)
$${I}_{A}^{\prime \prime}(u)=\left|{\int}_{0}^{\infty}\right[{\int}_{0}^{1}{U}_{LC}(\rho ){P}_{t}(\rho ){P}_{c}(\rho )\mathrm{exp}\left(\frac{iu{\rho}^{2}}{2}\right){J}_{0}({\nu}_{1}\rho )\rho \mathrm{d}\rho ]\phantom{\rule{0ex}{0ex}}{\times [{\int}_{0}^{1}{P}_{t}(\rho ){P}_{c}(\rho )\mathrm{exp}\left(\frac{iu{\rho}^{2}}{2}\right){J}_{0}({\nu}_{1}\rho )\rho \mathrm{d}\rho ]{\nu}_{1}\mathrm{d}{\nu}_{1}|}^{2},$$
(16)
$${I}_{B}^{\prime \prime}(u)=|{\int}_{0}^{\infty}[{\int}_{0}^{1}{U}_{LC}(\rho ){P}_{t}(\rho ){P}_{c}(\rho )\mathrm{exp}(iu{\rho}^{2}){J}_{0}({\nu}_{1}\rho )\rho \mathrm{d}\rho ]\times [{\int}_{0}^{1}{P}_{t}(\rho ){P}_{c}(\rho )\mathrm{exp}(iu{\rho}^{2}){J}_{0}({\nu}_{1}\rho )\rho \mathrm{d}\rho ]{\nu}_{1}\mathrm{d}{\nu}_{1}{|}^{2},$$
(17)
$${U}_{LC}(r)=\{\begin{array}{ll}a\text{\hspace{0.17em}}\mathrm{exp}(-\frac{{r}^{2}}{{124.12}^{2}}),& |r|\le 50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}\\ 0,& |r|>50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}\end{array},$$
(18)
$${U}_{LC}(\rho )=\{\begin{array}{ll}a\text{\hspace{0.17em}}\mathrm{exp}(-\frac{{\rho}^{2}}{6.16}),& |\rho |\le 1\\ 0,& |\rho |>1\end{array}.$$
(19)
$$u=\sqrt{{u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}}.$$
(20)
$$u=\sqrt{{u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}}\phantom{\rule{0ex}{0ex}}=\sqrt{{0.12}^{2}+{0.02}^{2}+{0.7}^{2}}\phantom{\rule{0ex}{0ex}}=0.71\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}.$$
(21)
$$\delta =\frac{U}{{\overline{l}}_{F}^{\prime}}\times 100\%=\frac{1.42}{147.4198\times 1000}\times 100\%\approx 0.001\%,$$