## Abstract

We report on the wavelength-multiplexing diffractive phase element (WMDPE) capable of generating independent spot patterns for different wavelengths. The iterative method proposed by Bengtsson [Appl. Opt. **37**, 1998] for designing a kinoform that produces different patterns for two wavelengths is extended to the WMDPE for multiple wavelengths (more than two wavelengths). Effectiveness of the design algorithm is verified by design and computer simulations on the WMDPE’s for four and nine wavelengths. The WMDPE for three wavelengths (441.6, 543.5, and 633 nm) is designed with five phase levels and is fabricated by electron-beam lithography. We observed that the individual spot patterns are reconstructed for the design wavelengths correctly. Performance of the WMDPE is evaluated by computer simulations on the uniformity error, the light efficiency, and the contrast. On the basis of the results, the characteristics of the WMDPE’s are discussed in terms of various conditions of fabrication and usage.

© 2001 Optical Society of America

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### Equations (24)

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(1)
$${U}_{\mathit{lm}}={U}_{\mathit{ol}}exp(j\mathrm{\Delta}{\phi}_{l}){A}_{\mathit{lm}}exp(j{\phi}_{\mathit{lm}}),$$
(2)
$${A}_{\mathit{lm}}exp(j{\phi}_{\mathit{lm}})=\frac{1}{\mathrm{\lambda}L}exp\left\{\frac{\mathit{jk}}{2L}[({u}_{m}-{x}_{l}{)}^{2}+({v}_{m}-{y}_{l}{)}^{2}]\right\}.$$
(3)
$$\mathrm{\Delta}{\phi}_{l}^{(Y)}=\frac{{\mathrm{\lambda}}_{A}}{{\mathrm{\lambda}}_{Y}}\mathrm{\Delta}{\phi}_{l}^{(A)}.$$
(4)
$$\hspace{1em}\hspace{1em}{U}_{m}\equiv {A}_{m}exp(j{\phi}_{m})$$
(5)
$$=\sum _{l}{U}_{\mathit{lm}}=\sum _{l}{A}_{\mathit{ol}}{A}_{\mathit{lm}}exp(j{\phi}_{\mathit{lm}})exp\{j[{\phi}_{\mathit{ol}}+\mathrm{\Delta}{\phi}_{l}^{(A)}]\}.$$
(6)
$$\mathrm{\Delta}{U}_{m}\cong {A}_{\mathit{lm}}cos[{\varphi}_{\mathit{lm}}-\delta {\phi}_{l}^{(A)}]-{A}_{\mathit{lm}}cos{\varphi}_{\mathit{lm}},$$
(7)
$${\varphi}_{\mathit{lm}}={\phi}_{m}-[{\phi}_{\mathit{lm}}+{\phi}_{\mathit{ol}}+\mathrm{\Delta}{\phi}_{l}^{(A)}].$$
(8)
$$f[\delta {\phi}_{l}^{(A)}]=\sum _{X}{W}_{X}{S}_{X}cos\left[\frac{{\mathrm{\lambda}}_{A}}{{\mathrm{\lambda}}_{X}}\delta {\phi}_{l}^{(A)}-{\alpha}_{l}^{(X)}\right]+\mathrm{const}.,$$
(9)
$${S}_{X}=\mathrm{sgn}({S}_{1X})({S}_{1X}^{2}+{S}_{2X}^{2}{)}^{1/2},$$
(10)
$${S}_{1X}=\sum _{m\in {M}_{X}}{w}_{m}{A}_{\mathit{lm}}cos{\varphi}_{\mathit{lm}},$$
(11)
$${S}_{2X}=\sum _{m\in {M}_{X}}{w}_{m}{A}_{\mathit{lm}}sin{\varphi}_{\mathit{lm}},$$
(12)
$${\alpha}_{l}^{(X)}=\mathrm{arctan}\left(\frac{{S}_{2X}}{{S}_{1X}}\right),$$
(13)
$${w}_{m}={w}_{m}^{\mathrm{old}}{\left(\frac{{I}_{X}^{\mathrm{ave}}}{{I}_{m}}\right)}^{p},$$
(15)
$${W}_{Y}={W}_{Y}^{\mathrm{old}}{\left(\frac{{I}_{A}^{\mathrm{ave}}}{{\beta}_{Y}{I}_{Y}^{\mathrm{ave}}}\right)}^{q},$$
(16)
$${\beta}_{X}=({\mathrm{\lambda}}_{X}/{\mathrm{\lambda}}_{A}{)}^{2}.$$
(17)
$$\mathrm{\Delta}{\phi}_{l}^{(A)}=\forall \gamma \in \mathrm{\Gamma},$$
(18)
$$(\mathrm{\Gamma}=\{\gamma |0\u2a7d\gamma \u2a7d(\mathrm{maximum}\hspace{0.5em}\mathrm{phase}\hspace{0.5em}\mathrm{modulation})\}),$$
(21)
$$\mathrm{Unif}.\hspace{0.5em}\mathrm{Err}.=\frac{{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{max}}({I}_{m})-{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{min}}({I}_{m})}{{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{max}}({I}_{m})+{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{min}}({I}_{m})},$$
(22)
$$\mathrm{Light}.\hspace{0.5em}\mathrm{Eff}.=\frac{(\mathrm{summation}\hspace{0.5em}\mathrm{of}\hspace{0.5em}\mathrm{power}\hspace{0.5em}\mathrm{within}\hspace{0.5em}\mathrm{all}\hspace{0.5em}\mathrm{spots}\hspace{0.5em}\mathrm{for}\hspace{0.5em}\mathrm{one}\hspace{0.5em}\mathrm{wavelength})}{(\mathrm{total}\hspace{0.5em}\mathrm{power}\hspace{0.5em}\mathrm{of}\hspace{0.5em}\mathrm{illuminating}\hspace{0.5em}\mathrm{light}\hspace{0.5em}\mathrm{for}\hspace{0.5em}\mathrm{one}\hspace{0.5em}\mathrm{wavelength})}.$$
(23)
$$\mathrm{Contrast}=\frac{{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{min}}({I}_{m})-{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{max}}({I}^{\mathrm{ghost}})}{{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{min}}({I}_{m})+{\displaystyle \underset{{\mathrm{\lambda}}_{X}}{max}}({I}^{\mathrm{ghost}})},$$
(24)
$$\frac{1}{{\mathrm{\lambda}}_{A}}-\frac{1}{{\mathrm{\lambda}}_{B}}=\frac{1}{{\mathrm{\lambda}}_{B}}-\frac{1}{{\mathrm{\lambda}}_{C}}=\cdots ,$$