## Abstract

Design considerations are presented for pairs of spherically aberrating elements that produce diffractionless beams of high efficiency and nearly constant size and intensity over specified ranges of axial position. An approximate design, assuming an aberration quadratic in radial ray positions, is followed by a final lens (mirror) specification that is verified through meridional ray tracing. This then provides an accurate determination of beam characteristics. Examples are presented that represent a variety of applications. As a design aid, a simple prescription is given for generating families of substantially different Bessel-like beams from any given pair of elements under small changes in element separation. Pattern sizes and ranges are compared with those of Gaussian beams, shadow lengths are examined for Bessel-type beams, and beam efficiencies are fully analyzed.

© 1994 Optical Society of America

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### Equations (54)

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(1)
$$S=\mathrm{\pi}/(k\hspace{0.17em}\text{sin}\hspace{0.17em}\mathrm{\beta}).$$
(2)
$$F={F}_{c}(1-B{\mathrm{\rho}}^{2}),$$
(3)
$${\mathrm{\rho}}_{2}={\mathrm{\rho}}_{1}({F}_{1}-Q)/{F}_{1},$$
(4)
$$q=({r}_{2}+{r}_{1})/({r}_{2}-{r}_{1}),$$
(5)
$$q\cong 0.7\pm 0.9{(B{{F}_{c}}^{2}-1.0)}^{1/2},$$
(6)
$$B{{F}_{c}}^{2}\cong 1.0+1.2{(q-0.7)}^{2}.$$
(7)
$$\mathrm{\eta}=-\frac{({\mathrm{\rho}}^{2}/f)(\text{d}f/\text{d}\mathrm{\rho})}{[1-(\mathrm{\rho}/f)(\text{d}f/\text{d}\mathrm{\rho})]},$$
(8)
$$\mathrm{\xi}=\frac{f}{[1-(\mathrm{\rho}/f)(\text{d}f/\text{d}\mathrm{\rho})]}.$$
(9)
$${\mathrm{\xi}}_{1}+{\mathrm{\xi}}_{2}=Q.$$
(10)
$${\mathrm{\eta}}_{2}/{\mathrm{\eta}}_{1}={\mathrm{\xi}}_{2}/(Q-{\mathrm{\xi}}_{1}).$$
(11)
$$\mathrm{\eta}\cong 2B{\mathrm{\rho}}^{3}/(1+B{\mathrm{\rho}}^{2}),$$
(12)
$$\mathrm{\xi}\cong {F}_{c}{(1-B{\mathrm{\rho}}^{2})}^{2}/(1+B{\mathrm{\rho}}^{2}).$$
(13)
$$\mathrm{\xi}\cong {F}_{c}{(1-B{\mathrm{\rho}}^{2})}^{3}.$$
(14)
$$S\cong -\frac{\mathrm{\pi}{F}_{1}}{k{\mathrm{\rho}}_{1}}\frac{{F}_{2}}{{x}_{2}},$$
(15)
$${x}_{2}=Q-{F}_{1}-{F}_{2},$$
(16)
$$Z={{F}_{2}}^{2}/{x}_{2}+{F}_{2},$$
(17)
$$Q={F}_{1}\left(1-\frac{\mathrm{\pi}}{k{\mathrm{\rho}}_{1}}\frac{Z}{S}\right),$$
(18)
$${\mathrm{\rho}}_{2}=(\mathrm{\pi}/k)(Z/S).$$
(19)
$${F}_{2c}={[({{\mathrm{\rho}}_{22}}^{2}\hspace{0.17em}\sqrt[3]{{\mathrm{\xi}}_{21}}-{{\mathrm{\rho}}_{21}}^{2}\hspace{0.17em}\sqrt[3]{{\mathrm{\xi}}_{22}})/({{\mathrm{\rho}}_{22}}^{2}-{{\mathrm{\rho}}_{21}}^{2})]}^{3},$$
(20)
$${B}_{2}=(\sqrt[3]{{\mathrm{\xi}}_{21}}-\sqrt[3]{{\mathrm{\xi}}_{22}})/({{\mathrm{\rho}}_{22}}^{2}\hspace{0.17em}\sqrt[3]{{\mathrm{\xi}}_{21}}-{{\mathrm{\rho}}_{21}}^{2}\hspace{0.17em}\sqrt[3]{{\mathrm{\xi}}_{22}}).$$
(21)
$${B}_{2}/{B}_{1}\cong {(-{F}_{1c}/{F}_{2c})}^{p}$$
(22)
$${{F}_{1c}}^{\prime}/{F}_{1c}={{B}_{1}}^{\prime}/{B}_{1}={Z}_{i}/({Z}_{i}+{F}_{1c}),$$
(23)
$${{f}_{1c}}^{\prime}/{f}_{1c}={({{q}_{1}}^{\prime}/{q}_{1})}^{2/3}={Z}_{i}/({Z}_{i}+{f}_{1c}).$$
(24)
$$\begin{array}{l}(-2.50,2.50,0.30,0.34)0.93(2.944,0.52)\\ (-2.36,-2.835)1.18(2.918,-0.720)\\ (1.330,0.20,0.636,0.42)1.18(1.754,0.41,-10.776,0.78).\end{array}$$
(25)
$$\begin{array}{l}(30.00,0.00122,3.50,4.10)25.75(-5.253,0.927)\\ (30.334,0.496)26.60(-4.584,-5.45)\\ (20.966,0.935,-62.233,4.70)26.60(-0.735,0.20,-1.065,0.42).\end{array}$$
(26)
$$\begin{array}{l}(-9.20,0.2222,0.50,0.60)28.87(37.792,0.002197)\\ (-9.02,-3.29)28.95(37.799,-0.6425)\\ (4.073,0.20,2.174,0.80)28.95(23.796,0.568,-109.326,3.60).\end{array}$$
(27)
$$\begin{array}{l}(-10.00,0.0288,0.60,0.80)200(210,2.835\times {10}^{-6})\\ (-10.00,-0.50)190(\text{mirror}\text{'}\text{s}\hspace{0.17em}\text{radius}\hspace{0.17em}\text{of}\hspace{0.17em}\text{curvature}=400)\\ (-20.68,0.20,6.893,1.10)190(\text{radius}\hspace{0.17em}\text{of}\hspace{0.17em}\text{curvature}=400,{\mathrm{\rho}}_{m}=22.6).\end{array}$$
(28)
$$S=\frac{\mathrm{\pi}Z}{k{\mathrm{\rho}}_{2}}.$$
(29)
$${W}_{B}=\frac{1.75Z}{k{\mathrm{\rho}}_{2}}.$$
(30)
$$w={w}_{0}{[1+{(Z/{Z}_{R})}^{2}]}^{1/2}\cong \frac{2Z}{k{w}_{0}},$$
(31)
$${W}_{G}={w}_{0}=\frac{2Z}{kw}.$$
(32)
$${Z}_{B}=\frac{kN{S}^{2}}{\mathrm{\pi}},$$
(33)
$${Z}_{B}\cong \frac{\mathrm{\pi}N{Z}^{2}}{k{{\mathrm{\rho}}_{2}}^{2}}.$$
(34)
$${Z}_{R}=\frac{k{{w}_{0}}^{2}}{2}\cong \frac{2{Z}^{2}}{k{w}^{2}}.$$
(35)
$$L\cong \frac{R}{(\mathrm{\beta}+\mathrm{\lambda}/4R)}.$$
(36)
$$S\cong \frac{\mathrm{\lambda}}{2}{\left(\frac{{B}_{1}{{F}_{1c}}^{3}}{Q-{F}_{1c}-{F}_{2c}}\right)}^{1/2}\frac{{F}_{2c}}{(Q-{F}_{1c}-{F}_{2c})}{\left(\frac{{\mathrm{\zeta}}^{3}}{\mathrm{\zeta}-1}\right)}^{1/2},$$
(37)
$$\mathrm{\zeta}=\frac{z(Q-{F}_{1c}-{F}_{2c})}{{{F}_{2c}}^{2}}.$$
(38)
$$(z+\mathrm{\delta}z)=\frac{{f}_{2}{({\mathrm{\rho}}_{2})}^{2}}{{x}_{2}+\mathrm{\delta}Q}.$$
(39)
$$(z+\mathrm{\delta}z)\cong z{\left[1+\frac{z\mathrm{\delta}Q}{{f}_{2}{({\mathrm{\rho}}_{2})}^{2}}\right]}^{-1}.$$
(40)
$$(S+\mathrm{\delta}S)\cong S{\left[1+\frac{z\mathrm{\delta}Q}{{f}_{2}{({\mathrm{\rho}}_{2})}^{2}}\right]}^{-1}.$$
(41)
$${I}_{1}({\mathrm{\rho}}_{1})=2P\frac{\text{exp}[-2{({\mathrm{\rho}}_{1}/{w}_{1})}^{2}]}{\mathrm{\pi}{{w}_{1}}^{2}},$$
(42)
$${P}_{\text{ring}}=\frac{4{\mathrm{\pi}}^{2}Z{I}_{2}({\mathrm{\rho}}_{2})}{k}\left(\frac{\text{d}{\mathrm{\rho}}_{2}}{\text{d}{r}_{2}}\right).$$
(43)
$$\frac{\text{d}{\mathrm{\rho}}_{2}}{\text{d}{r}_{2}}=\frac{Z}{{\mathrm{\rho}}_{2}}\frac{\text{d}{\mathrm{\rho}}_{2}}{\text{d}Z}.$$
(44)
$$\frac{Z}{{\mathrm{\rho}}_{2}}\frac{\text{d}{\mathrm{\rho}}_{2}}{\text{d}Z}=S\frac{\text{d}}{\text{d}Z}\left(\frac{Z}{S}\right)=1-\frac{Z}{S}\frac{\text{d}S}{\text{d}Z},$$
(45)
$${P}_{\text{ring}}=\frac{4{\mathrm{\pi}}^{2}Z{I}_{2}({\mathrm{\rho}}_{2})}{k}\left(1-\frac{Z}{S}\frac{\text{d}S}{\text{d}Z}\right).$$
(46)
$${I}_{2}({\mathrm{\rho}}_{2})\cong 2P\frac{\text{exp}[-2{({\mathrm{\rho}}_{2}/{w}_{2})}^{2}]}{\mathrm{\pi}{{w}_{2}}^{2}},$$
(47)
$${w}_{2}\cong \left|\frac{{f}_{2c}}{{f}_{1c}}\right|{w}_{1}.$$
(48)
$$\mathrm{\varepsilon}=\frac{8\mathrm{\pi}Z}{k{{w}_{2}}^{2}}\text{exp}[-2{({\mathrm{\rho}}_{2}/{w}_{2})}^{2}]\left(1-\frac{Z}{S}\frac{\text{d}S}{\text{d}Z}\right).$$
(49)
$$\mathrm{\varepsilon}=\frac{8\mathrm{\pi}Z}{k{{w}_{2}}^{2}}\text{exp}[-2{(\mathrm{\pi}Z/kS{w}_{2})}^{2}]\left(1-\frac{Z}{S}\frac{\text{d}S}{\text{d}Z}\right).$$
(50)
$$\mathrm{\varepsilon}(Z;{Z}_{w})=\frac{8k{{S}_{w}}^{2}}{\mathrm{\pi}{{Z}_{w}}^{2}}\text{exp}[-2{(Z{S}_{w}/S{Z}_{w})}^{2}]Z\left(1-\frac{Z}{S}\frac{\text{d}S}{\text{d}Z}\right).$$
(51)
$$\mathrm{\varepsilon}\cong \frac{8k{{S}_{w}}^{2}}{\mathrm{\pi}{Z}_{w}}\left(\frac{Z}{{Z}_{w}}\right)\text{exp}[-2{(Z/{Z}_{w})}^{2}].$$
(52)
$$\frac{Z}{S}\frac{\text{d}S}{\text{d}Z}=\frac{\mathrm{\zeta}-{\scriptstyle {}^{3}/_{2}}}{\mathrm{\zeta}-1},$$
(53)
$$\mathrm{\varepsilon}(Z;{Z}_{w})\cong \left(\frac{4k{{S}_{w}}^{2}}{\mathrm{\pi}{Z}_{w}}\right)\frac{Z}{{Z}_{w}}\text{exp}[-2{(Z{S}_{w}/S{Z}_{w})}^{2}]\frac{1}{(\mathrm{\zeta}-1)}.$$
(54)
$$\mathrm{\varepsilon}{(Z)}_{\text{max}}=\frac{2k{S}^{2}}{\mathrm{\pi}eZ(\mathrm{\zeta}-1)}.$$