## Abstract

The analysis of a slitless volume holographic spectrometer is presented in detail. The spectrometer is based on a spherical beam volume hologram followed by a Fourier-transforming lens and a CCD. It is shown that the spectrometer is not sensitive to the incident angle of the input beam for the practical range of applications. A holographic spectrometer based on the conventional implementation is also analyzed, and the results are used to compare the performance of the proposed method with the conventional one. The experimental results are consistent with the theoretical study. It is also shown that the slitless volume holographic spectrometer lumps three elements (the entrance slit, the collimator, and the diffractive element) of the conventional spectrometer into one spherical beam volume hologram. Based on the unique features of the slitless volume holographic spectrometer, we believe it is a good candidate for portable spectroscopy for environmental and biological applications.

© 2006 Optical Society of America

Full Article |

PDF Article
### Equations (16)

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

(1)
$${E}_{i}\left({k}_{ix}\prime ,{k}_{iy}\prime ,{k}_{iz}\prime \right)={A}_{i}\text{\hspace{0.17em}}\mathrm{exp}\left[j\left({k}_{ix}\prime x+{k}_{iy}\prime y+{k}_{iz}\prime z\right)+{\phi}_{i}\right].$$
(2)
$${E}_{id}\left(x,y,z\right)=\frac{\mathrm{exp}\left[j\left({k}_{rx}+{k}_{ix}\prime \right)x\right]\mathrm{exp}\left(j{k}_{iy}\prime y\right)}{4{\pi}^{2}}\text{\xd7}{\displaystyle \int \int {\tilde{E}}_{id}\left({k}_{x},{k}_{y},z\right)\mathrm{exp}\left[-j\left({k}_{x}x+{k}_{y}y\right)\right]\mathrm{d}{k}_{x}\mathrm{d}{k}_{y}},$$
(3)
$${\tilde{E}}_{id}\left({k}_{x},{k}_{y},z\right)=\frac{j2{\pi}^{2}\mathrm{\Delta \u03f5}{{k}^{\prime}}^{2}L{A}_{i}\mathrm{exp}\left(j{\phi}_{i}\right)}{\u03f5{k}_{idz}\prime}\text{\hspace{0.17em}}\mathrm{exp}\left(j{k}_{idz}\prime z\right)\times \text{sinc}\left[\frac{L}{\text{2}\pi}\text{\hspace{0.17em}}\left({K}_{gz}+{k}_{iz}\prime -{k}_{idz}\prime \right)\right].$$
(4)
$${K}_{gz}={k}_{rz}-{\left({k}^{2}-{k}_{x}^{\text{\hspace{0.17em} \hspace{0.17em}}2}-{k}_{y}^{\text{\hspace{0.17em} \hspace{0.17em}}2}\right)}^{1/2},$$
(5)
$${k}_{idz}\prime ={\left[{{k}^{\prime}}^{2}-{\left({K}_{gx}+{k}_{ix}\prime \right)}^{2}-{\left({K}_{gy}+{k}_{iy}\prime \right)}^{2}\right]}^{1/2},$$
(6)
$${E}_{io}\left(u,v,z=2f\right)=\frac{{A}_{i}}{j{\lambda}^{\prime}f}\text{\hspace{0.17em}}F{\left\{{E}_{id}\left(x,y,L/2\right)\right\}|}_{{f}_{x}=u/{(\lambda}^{\prime}f)\text{\hspace{0.17em} and \hspace{0.17em}}{f}_{y}=v/{(\lambda}^{\prime}f)},$$
(7)
$$\tilde{P}\left(2\pi {f}_{x},2\pi {f}_{y},z\right)=F\left\{p\left(x,y,z\right)\right\}={\displaystyle \int \int p\left(x,y,z\right)\text{\xd7 \hspace{0.17em}}\mathrm{exp}\left[-j2\pi \left({f}_{x}x+{f}_{y}y\right)\right]}\mathrm{d}x\mathrm{d}y.$$
(8)
$$\begin{array}{c}{E}_{io}\left(u,v,2f\right)=\frac{{A}_{i}}{j{\lambda}^{\prime}f}\text{\hspace{0.17em}}F{\left\{\mathrm{exp}\left[j\left({k}_{rx}+{k}_{ix}\prime \right)x\right]\mathrm{exp}\left(j{k}_{iy}\prime y\right){F}^{-1}{\left\{{\tilde{E}}_{di}\left({k}_{x},{k}_{y},2f\right)\right\}}_{\begin{array}{l}x\to -x\\ y\to -y\end{array}}\right\}}_{{f}_{x}=u/({\lambda}^{\prime}f\text{\hspace{0.17em}})\text{and \hspace{0.17em}}{f}_{y}=v/{(\lambda}^{\prime}f)}\text{\hspace{0.17em} \hspace{0.17em}}\end{array}\begin{array}{c}=\text{\hspace{0.17em}}\frac{{A}_{i}}{j{\lambda}^{\prime}f}\text{\hspace{0.17em}}{\tilde{E}}_{id}{\left[-\left(2\pi {f}_{x}-{k}_{rx}-{k}_{ix}\prime \right),-\left(2\pi {f}_{y}-{k}_{iy}\prime \right),2f\right]|}_{{f}_{x}=u/{(\lambda}^{\prime}f)\text{\hspace{0.17em}and \hspace{0.17em}}{f}_{y}=v/{(\lambda}^{\prime}f)}\text{\hspace{0.17em} \hspace{0.17em}}\end{array}\begin{array}{c}=\text{\hspace{0.17em}}\frac{{A}_{i}}{j{\lambda}^{\prime}f}\text{\hspace{0.17em}}{\tilde{E}}_{id}\left(-{k}^{\prime}u/f+{k}_{rx}+{k}_{ix}\prime ,-{k}^{\prime}v/f+{k}_{iy}\prime ,2f\right).\end{array}$$
(9)
$$H\left(u,v,z=2f,{\lambda}^{\prime}\right)={E}_{io}\left(u,v,2f\right)/\left[{A}_{i}\text{\hspace{0.17em}}\mathrm{exp}\left(j{\phi}_{i}\right)\right]=\frac{j2{\pi}^{2}\mathrm{\Delta \u03f5}{{k}^{\prime}}^{2}L}{{\u03f5}_{0}{[{{k}^{\prime}}^{2}-{\left({k}^{\prime}u/f\right)}^{2}-{\left({k}^{\prime}v/f\right)}^{2}]}^{1/2}}\times \text{\hspace{0.17em}}\mathrm{exp}\left\{j2f{\left[{{k}^{\prime}}^{2}-{\left({k}^{\prime}u/f\right)}^{2}-{\left({k}^{\prime}v/f\right)}^{2}\right]}^{1/2}\right\}\text{sinc}\mathbf{(}L/2\pi \left\{{k}_{rz}-{\left[{k}^{2}-{\left({k}^{\prime}u/f-{k}_{rx}-{k}_{ix}\prime \right)}^{2}-{\left({k}^{\prime}v/f-{k}_{iy}\prime \right)}^{2}\right]}^{1/2}+{k}_{iz}\prime -{\left[{{k}^{\prime}}^{2}-{\left({k}^{\prime}u/f\right)}^{2}-{\left({k}^{\prime}v/f\right)}^{2}\right]}^{1/2}\right\}\mathbf{)}\mathrm{.}$$
(10)
$${I}_{o}\left(u,v,2f\right)={\displaystyle \int {A}_{i}^{\text{\hspace{0.17em} \hspace{0.17em}}2}\left({k}_{ix}\prime ,{k}_{iy}\prime \right)|H{\left(u,v,z=2f,{\lambda}^{\prime}\right)|}^{2}\mathrm{d}{k}_{x}\prime \mathrm{d}{k}_{y}\prime}$$
(11)
$$={\displaystyle \int {\left|{E}_{io}\left(u,v,2f\right)\right|}^{2}\mathrm{d}{k}_{x}\prime \mathrm{d}{k}_{y}\prime},$$
(12)
$$h\left({x}_{i},{y}_{i};\text{\hspace{0.17em}}{x}_{o},{y}_{o};\text{\hspace{0.17em}}{\lambda}^{\prime}\right)=C\text{\hspace{0.17em} sinc}\left(\frac{{L}_{1}}{2\pi}\text{\hspace{0.17em}}{k}_{1}\right)\times \hspace{0.17em}\text{sinc}\left(\frac{{L}_{2}}{2\pi}\text{\hspace{0.17em}}{k}_{2}\right)\text{sinc}\left(\frac{{L}_{3}}{2\pi}\text{\hspace{0.17em}}{k}_{3}\right),$$
(13)
$${k}_{1}=\frac{2\pi}{{\lambda}^{\prime}}\text{\hspace{0.17em}}\left[\left(\frac{-{x}_{i}}{f}+1\right)\mathrm{cos}\left(\alpha \right)-\left(\frac{{x}_{o}}{f}-1\right)\mathrm{sin}\left(\alpha \right)\right]+{K}_{g},$$
(14)
$${k}_{2}=\frac{2\pi}{{\lambda}^{\prime}}\text{\hspace{0.17em}}\left(\frac{{y}_{i}}{f}+\frac{{y}_{o}}{f}\right),$$
(15)
$${k}_{3}=n\text{\hspace{0.17em}}\frac{2\pi}{{\lambda}^{\prime}}\text{\hspace{0.17em}}\left[-\left(\frac{-{x}_{i}}{f}+1\right)\mathrm{sin}\left(\alpha \right)-\left(\frac{{x}_{o}}{f}-1\right)\mathrm{cos}\left(\alpha \right)\right],$$
(16)
$$\text{rect}\left(u\right)=\{\begin{array}{l}1\text{\hspace{1em}}\left|u\right|<1/2\\ 0\text{\hspace{1em} otherwise}\end{array}.$$