^{1,a)}and Nir Davidson

^{2}

### Abstract

In this article the design, optimization and characterization of diffractive optical elements formed on a curved surface are reviewed. For such curved diffractive optical elements not only the phase function, but also the surface shape are free parameters that can be used for optimization, yielding much better performances than both flat diffractive optical elements and reflective/refractive optical elements when operating with quasimonochromatic light. We present a new analytic design approach for the surface shape that ensures uniform collimation of a light source with any angular distribution. We demonstrate the usefulness of this design also for ideal (brightness conserving) collimation and concentration of diffuse light, aberration-free imaging, and optical Fourier transform. We present experimental results that confirm our theoretical analysis.

The authors would like to thank Ami Ishaaya for his helpful comments.

I. INTRODUCTION

II. BASIC CONCEPTS AND LIMITATIONS FOR DIFFUSE LIGHT CONCENTRATION

III. UNIFORM COLLIMATION AND IDEAL COLLIMATION AND CONCENTRATION IN ONE DIMENSION WITH CDOEs

IV. IDEAL CONCENTRATION IN 1D WITH CDOEs AT FINITE DISTANCE

V. CDOEs SATISFYING THE ABBE CONDITION

A. The Abbe condition

B. Aplanatic CDOEs for concentration and collimation of diffuse light

C. Aplanatic CDOEs for imaging

D. Aplanatic CDOEs for Fourier transform

VI. EXTENSION TO 2D CONCENTRATION/COLLIMATION WITH 3D CDOE SHAPES

VII. CONCLUSION

### Key Topics

- Collimation
- 22.0
- Optical aberrations
- 20.0
- Collimators
- 17.0
- Nonlinear acoustics
- 15.0
- Mirrors
- 14.0

## Figures

(a) Recording and (b) readout geometry of a holographic CDOE of shape , working as a collimator. is the angle between the direction of a reconstructing ray and the axis. The corresponding coordinates of the CDOE surface are and .

(a) Recording and (b) readout geometry of a holographic CDOE of shape , working as a collimator. is the angle between the direction of a reconstructing ray and the axis. The corresponding coordinates of the CDOE surface are and .

Schematic diagram of an elongated Lambertian source whose angular intensity distribution is proportional to , the projected diameter of the cross section of the source as seen from direction .

Schematic diagram of an elongated Lambertian source whose angular intensity distribution is proportional to , the projected diameter of the cross section of the source as seen from direction .

Sections of two collimator shapes for an isotropic source (illustrated by a uniform angular density of emerging rays) with the same paraxial focal length: (a) a CDOE described by Eq. (11) for which collimated rays are equally spaced, indicating uniform intensity of the collimated beam; (b) a PM for which the spacing between collimated rays increases for larger , indicating a reduction in intensity.

Sections of two collimator shapes for an isotropic source (illustrated by a uniform angular density of emerging rays) with the same paraxial focal length: (a) a CDOE described by Eq. (11) for which collimated rays are equally spaced, indicating uniform intensity of the collimated beam; (b) a PM for which the spacing between collimated rays increases for larger , indicating a reduction in intensity.

Calculated intensity distributions for the collimated beam produced by a uniform collimator CDOE (solid line) and a PM (dotted curve), showing a uniform collimated intensity in the case of the CDOE and a nonunifom intensity in the case of the PM (as shown schematically in the ray diagrams in Fig. 3).

Calculated intensity distributions for the collimated beam produced by a uniform collimator CDOE (solid line) and a PM (dotted curve), showing a uniform collimated intensity in the case of the CDOE and a nonunifom intensity in the case of the PM (as shown schematically in the ray diagrams in Fig. 3).

Diffuse light concentration onto a small cylindrical target by: (a) a uniform collimator CDOE and (b) a PM. For the PM the reflected ray pencils have the same angular spread as the incoming ray pencils, resulting in poor concentration for large . For the CDOE, for the diffracted ray pencils are narrower than the incoming ray pencils , resulting in ideal concentration for all .

Diffuse light concentration onto a small cylindrical target by: (a) a uniform collimator CDOE and (b) a PM. For the PM the reflected ray pencils have the same angular spread as the incoming ray pencils, resulting in poor concentration for large . For the CDOE, for the diffracted ray pencils are narrower than the incoming ray pencils , resulting in ideal concentration for all .

Calculated normalized concentration ratio onto a small cylindrical target as a function of for the uniform collimator CDOE (solid line) and the PM (dotted curve) with incoming half-diffusive angle , indicating that the CDOE reaches the thermodynamic limit of concentration for , whereas the optimum performance of the PM is worse than the thermodynamic limit.

Calculated normalized concentration ratio onto a small cylindrical target as a function of for the uniform collimator CDOE (solid line) and the PM (dotted curve) with incoming half-diffusive angle , indicating that the CDOE reaches the thermodynamic limit of concentration for , whereas the optimum performance of the PM is worse than the thermodynamic limit.

Setup for measuring the concentrating performance of the uniform collimator CDOE. The CDOE is illuminated with a monochromatic plane wave inclined at variable angle with respect to the axis. A cylindrical black target is placed at the focal point of the CDOE and the unblocked light intensity is measured by integrating the unblocked light intensity that hits a white diffuse screen placed behind the target and imaged with the help of a beam splitter (BS) onto a calibrated CCD camera.

Setup for measuring the concentrating performance of the uniform collimator CDOE. The CDOE is illuminated with a monochromatic plane wave inclined at variable angle with respect to the axis. A cylindrical black target is placed at the focal point of the CDOE and the unblocked light intensity is measured by integrating the unblocked light intensity that hits a white diffuse screen placed behind the target and imaged with the help of a beam splitter (BS) onto a calibrated CCD camera.

Experimental data for characterizing the performance of the uniform collimator CDOE as a diffuse light concentrator onto a small cylindrical target, obtained using the setup of Fig. 7. The measured light intensity not blocked by the cylindrical target as a function of the illumination angle (diamonds) shows good agreement with the theoretical calculations for the CDOE (solid curve). Also included is the theoretical graph for the PM (broken curve), showing poor performance of the PM as a diffuse light concentrator. Note that any measured intensity below the ideal cutoff angle of 4.3° results from nonideal concentration.

Experimental data for characterizing the performance of the uniform collimator CDOE as a diffuse light concentrator onto a small cylindrical target, obtained using the setup of Fig. 7. The measured light intensity not blocked by the cylindrical target as a function of the illumination angle (diamonds) shows good agreement with the theoretical calculations for the CDOE (solid curve). Also included is the theoretical graph for the PM (broken curve), showing poor performance of the PM as a diffuse light concentrator. Note that any measured intensity below the ideal cutoff angle of 4.3° results from nonideal concentration.

The effect of noninfinitesimal diffusive angles on the concentration efficiency of the uniform collimator CDOE, obtained from numerical calculations for two cases: for (corresponding to the optimal geometry for the PM) and . The CDOE has nearly ideal concentration even for large diffusivities.

The effect of noninfinitesimal diffusive angles on the concentration efficiency of the uniform collimator CDOE, obtained from numerical calculations for two cases: for (corresponding to the optimal geometry for the PM) and . The CDOE has nearly ideal concentration even for large diffusivities.

Uniform collimator CDOE profiles for small Lambertian sources of different shapes located at , . The CDOE profiles correspond to the following source shapes: one sided (vertical) flat source (solid curve), double sided (horizontal), flat source (dashed curve), and isotropic source (dotted curve). All CDOEs have the same paraxial focal length and extend to (except for the one sided flat source which is limited to ).

Uniform collimator CDOE profiles for small Lambertian sources of different shapes located at , . The CDOE profiles correspond to the following source shapes: one sided (vertical) flat source (solid curve), double sided (horizontal), flat source (dashed curve), and isotropic source (dotted curve). All CDOEs have the same paraxial focal length and extend to (except for the one sided flat source which is limited to ).

Design geometry of a CDOE for concentration at finite distance working in: (a) reflection and (b) transmission. Source and target are at a distance from each other, is the angle between a central ray emitted by and the line connecting and , is the angle between the central ray intercepted by and the line connecting and , and is the distance between and the given point on the CDOE.

Design geometry of a CDOE for concentration at finite distance working in: (a) reflection and (b) transmission. Source and target are at a distance from each other, is the angle between a central ray emitted by and the line connecting and , is the angle between the central ray intercepted by and the line connecting and , and is the distance between and the given point on the CDOE.

CDOE profiles for a cylindrical source and cylindrical targets of different sizes , , , and . The corresponding magnifications are , 2, 3, and , respectively. The CDOE shape corresponding to is the uniform collimator/concentrator shown in Fig. 3(a).

CDOE profiles for a cylindrical source and cylindrical targets of different sizes , , , and . The corresponding magnifications are , 2, 3, and , respectively. The CDOE shape corresponding to is the uniform collimator/concentrator shown in Fig. 3(a).

Calculated normalized spot sizes as a function of for a CDOE (dotted line) and for reflective elliptical cavities having different eccentricities e (solid curves), indicating that the CDOE yields a uniform intensity distribution on the target, whereas the concentration profiles of the elliptical cavities are highly nonuniform.

Calculated normalized spot sizes as a function of for a CDOE (dotted line) and for reflective elliptical cavities having different eccentricities e (solid curves), indicating that the CDOE yields a uniform intensity distribution on the target, whereas the concentration profiles of the elliptical cavities are highly nonuniform.

Double lamp pumping configurations using: (a) two reflective CDOEs and (b) two reflective elliptical cavities. In this case the CDOEs have cylindrical shapes. The CDOEs yield smaller geometrical loss and a more uniform pump density profile than the elliptical cavities.

Double lamp pumping configurations using: (a) two reflective CDOEs and (b) two reflective elliptical cavities. In this case the CDOEs have cylindrical shapes. The CDOEs yield smaller geometrical loss and a more uniform pump density profile than the elliptical cavities.

Phase-space spot diagrams obtained from numerical ray tracing at the cylindral target of: (a) the double CDOE cavity of Fig. 14(a), and (b) the double elliptical mirror cavity of Fig. 14(b). Rays coming from the left side source and the right side source are represented by filled circles and open circles, respectively. The double CDOE device has a much more uniform phase space than the double elliptical mirror, and achieves uniform concentration at the thermodynamic limit.

Phase-space spot diagrams obtained from numerical ray tracing at the cylindral target of: (a) the double CDOE cavity of Fig. 14(a), and (b) the double elliptical mirror cavity of Fig. 14(b). Rays coming from the left side source and the right side source are represented by filled circles and open circles, respectively. The double CDOE device has a much more uniform phase space than the double elliptical mirror, and achieves uniform concentration at the thermodynamic limit.

Reflective CDOE concentrator shape designed using Eqs. (13) and (14) for a one sided flat Lambertian source and a cylindrical Lambertian target, having practical significance, e.g., for the side pumping of cylindrical solid-state laser rods with laser diode arrays that have a flat radiating surface.

Reflective CDOE concentrator shape designed using Eqs. (13) and (14) for a one sided flat Lambertian source and a cylindrical Lambertian target, having practical significance, e.g., for the side pumping of cylindrical solid-state laser rods with laser diode arrays that have a flat radiating surface.

(a) Transmission CDOE concentrator shape satisfying the Abbe condition with magnification , and (b) phase-space diagrams of the light distributions at the input (source) and output (target), where is the location of a particular ray on the source/target, and is its inclination angle of the ray from the optical axis. Although the target size is times as large as the source size, the sine of the angular range of rays hitting the target is times as large as the sine of the angular range of rays emitted by the source, hence phase-space area is conserved.

(a) Transmission CDOE concentrator shape satisfying the Abbe condition with magnification , and (b) phase-space diagrams of the light distributions at the input (source) and output (target), where is the location of a particular ray on the source/target, and is its inclination angle of the ray from the optical axis. Although the target size is times as large as the source size, the sine of the angular range of rays hitting the target is times as large as the sine of the angular range of rays emitted by the source, hence phase-space area is conserved.

Experimental concentrated spots on a flat target obtained from: (a) a cylindrical aplanatic CDOE and (b) a flat DOE. For both cases the DOEs have an aperture of 6 cm, and a focal length of 3.5 cm. A diffuse beam with half-divergence angle is used for the illumination.

Experimental concentrated spots on a flat target obtained from: (a) a cylindrical aplanatic CDOE and (b) a flat DOE. For both cases the DOEs have an aperture of 6 cm, and a focal length of 3.5 cm. A diffuse beam with half-divergence angle is used for the illumination.

(a) Measured horizontal intensity cross sections of the concentrated spots of Fig. 18 for the cylindrical aplanatic CDOE (dotted line) and the flat DOE (solid line), showing that the CDOE yielded a narrower spot with sharper edges than the flat DOE, and (b) the corresponding calculated intensity cross sections obtained from numerical ray tracing.

(a) Measured horizontal intensity cross sections of the concentrated spots of Fig. 18 for the cylindrical aplanatic CDOE (dotted line) and the flat DOE (solid line), showing that the CDOE yielded a narrower spot with sharper edges than the flat DOE, and (b) the corresponding calculated intensity cross sections obtained from numerical ray tracing.

Calculated normalized concentration ratio for the cylindrical aplanatic CDOE (solid line), the flat DOE (dotted line), and the PM (dashed line) as a function of NA. Also shown are the experimental normalized concentration ratio values for the CDOE and the flat DOE , with error bars estimated from the data. The aplanatic CDOE reaches the thermodynamic limit at high NA, whereas the maximum achievable concentration ratio for the PM and the flat DOE is 50% (at ) and 38% (at ) of the thermodynamic limit, respectively.

Calculated normalized concentration ratio for the cylindrical aplanatic CDOE (solid line), the flat DOE (dotted line), and the PM (dashed line) as a function of NA. Also shown are the experimental normalized concentration ratio values for the CDOE and the flat DOE , with error bars estimated from the data. The aplanatic CDOE reaches the thermodynamic limit at high NA, whereas the maximum achievable concentration ratio for the PM and the flat DOE is 50% (at ) and 38% (at ) of the thermodynamic limit, respectively.

Measured horizontal spot sizes (defined as the diameter of the region containing 90% of the total energy) of a cylindrical aplanatic CDOE (diamonds) and a flat DOE (squares), as a function of the off-axis angle of the illuminating plane wave. As seen, the flat DOE has large aberrations at large off-axis angles, whereas the CDOE has an essentially fixed spot size. For both cases the DOEs have an aperture of 6 cm, and a focal length of 3.5 cm.

Measured horizontal spot sizes (defined as the diameter of the region containing 90% of the total energy) of a cylindrical aplanatic CDOE (diamonds) and a flat DOE (squares), as a function of the off-axis angle of the illuminating plane wave. As seen, the flat DOE has large aberrations at large off-axis angles, whereas the CDOE has an essentially fixed spot size. For both cases the DOEs have an aperture of 6 cm, and a focal length of 3.5 cm.

The ratio between the combined aberrations and (defined as the rms deviations of the output ray positions from the desired focal point), for the flat DOE and the aplanatic CDOE, respectively, as a function of NA, for off-axis illuminating angles of (solid curve) and (dotted curve), obtained from ray tracing simulations. As seen, the CDOE performs especially well compared with the flat DOE for high NAs.

The ratio between the combined aberrations and (defined as the rms deviations of the output ray positions from the desired focal point), for the flat DOE and the aplanatic CDOE, respectively, as a function of NA, for off-axis illuminating angles of (solid curve) and (dotted curve), obtained from ray tracing simulations. As seen, the CDOE performs especially well compared with the flat DOE for high NAs.

(a) a CDOE having a “2D shape” (a shape with zero curvature), achieving concentration/collimation in 1D along the axis, and (b) a CDOE having a cylindrically symmetric “3D shape” (a shape with nonzero curvature), achieving 2D concentration/collimation along and . For the 1D concentration/collimation case of (a) the target/source is elongated in the direction and both the CDOE and the target/source are symmetrical with respect to the plane and the plane, and for the 2D concentration/collimation case of (b) both the CDOE and the target/source are symmetrical about the axis.

(a) a CDOE having a “2D shape” (a shape with zero curvature), achieving concentration/collimation in 1D along the axis, and (b) a CDOE having a cylindrically symmetric “3D shape” (a shape with nonzero curvature), achieving 2D concentration/collimation along and . For the 1D concentration/collimation case of (a) the target/source is elongated in the direction and both the CDOE and the target/source are symmetrical with respect to the plane and the plane, and for the 2D concentration/collimation case of (b) both the CDOE and the target/source are symmetrical about the axis.

3D collimator CDOE shape designed from Eq. (18) with . The rays are equally spaced in the transverse plane, indicating uniform intensity of the collimated plane wave.

3D collimator CDOE shape designed from Eq. (18) with . The rays are equally spaced in the transverse plane, indicating uniform intensity of the collimated plane wave.

The effect of skew rays (rays that are not in the same plane with the symmetry axis). The diffuse ray pencils hitting the CDOE in the skew direction (dotted lines) are unaffected by the LG and are directed toward the target as identical diffuse ray pencils, similarly to simple reflections. This is in contrast with diffuse ray pencils that are in the same plane with the symmetry axis (dashed lines). Such ray pencils undergo diffraction on the local grating as they are deflected toward the target, hence their diffusive angle will change according to the diffraction relation, as described in Secs. III and IV.

The effect of skew rays (rays that are not in the same plane with the symmetry axis). The diffuse ray pencils hitting the CDOE in the skew direction (dotted lines) are unaffected by the LG and are directed toward the target as identical diffuse ray pencils, similarly to simple reflections. This is in contrast with diffuse ray pencils that are in the same plane with the symmetry axis (dashed lines). Such ray pencils undergo diffraction on the local grating as they are deflected toward the target, hence their diffusive angle will change according to the diffraction relation, as described in Secs. III and IV.

Calculated normalized concentration ratio for the spherical aplanatic CDOE (solid line), the “3D” flat DOE (dotted line), and the 3D PM (dashed line) as a function of NA (see Fig. 20 for the corresponding graphs of the 2D concentrator shapes). The aplanatic CDOE still reaches the thermodynamic limit at high NA, whereas the maximum achievable concentration ratio for the PM and the flat DOE is 25% (at ) and 15% (at ) of the thermodynamic limit, respectively.

Calculated normalized concentration ratio for the spherical aplanatic CDOE (solid line), the “3D” flat DOE (dotted line), and the 3D PM (dashed line) as a function of NA (see Fig. 20 for the corresponding graphs of the 2D concentrator shapes). The aplanatic CDOE still reaches the thermodynamic limit at high NA, whereas the maximum achievable concentration ratio for the PM and the flat DOE is 25% (at ) and 15% (at ) of the thermodynamic limit, respectively.

Uniform collimator CDOE working as a concentrator. The target is placed at the origin. is the (full) diffusive angle of the illuminating diffuse beam, is the (full) diffusive angle of the diffracted beam directed toward the target located at a distance from the given CDOE point, and the angle between the tangent to the CDOE and the axis.

Uniform collimator CDOE working as a concentrator. The target is placed at the origin. is the (full) diffusive angle of the illuminating diffuse beam, is the (full) diffusive angle of the diffracted beam directed toward the target located at a distance from the given CDOE point, and the angle between the tangent to the CDOE and the axis.

CDOE working as a concentrator for finite distances. The source and the target are at a distance from each other. is the (full) diffusive angle of the beam emitted by the source, is the (full) diffusive angle of the diffracted beam directed toward the target, and the angle between the tangent to the CDOE and the axis. and are the distances of the target and the source from the given CDOE point, respectively.

CDOE working as a concentrator for finite distances. The source and the target are at a distance from each other. is the (full) diffusive angle of the beam emitted by the source, is the (full) diffusive angle of the diffracted beam directed toward the target, and the angle between the tangent to the CDOE and the axis. and are the distances of the target and the source from the given CDOE point, respectively.

Circle-shaped cross section of a spherical aplanatic CDOE working as a concentrator for finite distance. Both the source and the target are flat. Diffraction occurs in the plane and the skew direction is along the axis. Since the spot sizes are identical in the diffractive and in the skew directions, the performance of such a spherical aplanatic CDOE is not affected by skew rays.

Circle-shaped cross section of a spherical aplanatic CDOE working as a concentrator for finite distance. Both the source and the target are flat. Diffraction occurs in the plane and the skew direction is along the axis. Since the spot sizes are identical in the diffractive and in the skew directions, the performance of such a spherical aplanatic CDOE is not affected by skew rays.

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