Gaussian curvature and stigmatic imaging relations for the design of an unobscured reflective relay

E. M. Schiesser; Seung-Whan Bahk; Jake Bromage; Jannick P. Rolland

doi:10.1364/OL.43.004855

Unobscured mirror systems require breaking symmetry. Most optical designs employ an offset in aperture, a bias in the field-of-view, or both, to an otherwise rotationally symmetric design to avoid obscuration [1]. The parent optical system of these designs is composed of the parent optical surfaces oriented along a common optical axis. In optical design softwares, the parent optical surfaces are specified using their axial parameters (e.g., vertex radius of curvature [RoC], conic constant). This is a convenient way to specify the design due to the rotational symmetry of the parent system, but the unobscured subset of the parent design does not share rotational symmetry about the effective apertures of each optical surface. Similarly, the distances between the effective apertures of unobscured designs are different from the distances between the parent surfaces’ vertices, especially for a system of confocal conics. Packaging constraints are specified relative to the used apertures, not the parent apertures. In contrast to the co-axial rotationally symmetric designs in this Letter, other recent design techniques for freeform telescopes employ a system of non-coaxial off-axis confocal conics combined with XY polynomials to correct field aberrations, breaking the axial symmetry of the design [2]. Here we parameterize the properties of the coaxial parent surfaces in terms of the off-axis parameters to enable the rapid iteration of the design according to changing packaging requirements and specifications.

An unobscured, achromatic image relay (AIR) is required by the multi-terawatt optical parametric amplifier line (MTW-OPAL), a research laser system under construction at the Laboratory for Laser Energetics (LLE) [3]. The beam has a broadband of 200 nm, which drives the need for reflective optics, since chromatic effects degrade the focused laser intensity. In addition, long working distances on the input and output are required to accommodate beam steering mirrors, filter optics, and a beam dump. To satisfy these imaging requirements, a four-mirror system consisting of two back-to-back telescopes is necessary. Due to the extremely small field-of-view requirements for this relay (0.14 deg), each telescope can use a Cassegrain configuration, which consists of a positive parabolic reflector and a negative hyperbolic reflector [4,5]. A Cassegrain is a system of coaxial confocal conics, which is a system of Cartesian reflectors whose stigmatic foci overlap.

To avoid obscuration, an offset in the aperture is used. For a system of confocal conics, any ray passing through the stigmatic foci can be considered to be a paraxial ray. Therefore, paraxial first-order imaging equations can be used to compute the imaging properties along that ray. Consequently, we can choose a ray that passes through the center of the offset aperture, and determine the exact distances between the centers of the used apertures throughout the system. These distances are the “off-axis parameters,” which we must specify for layout purposes of the MTW-OPAL relay.

To compute the axial parameters, we determine the relationship between the specified off-axis parameters and the axial parameters (Fig. 1). Next, we determine the relationships that will allow us to use paraxial equations (e.g., the paraxial matrix method [6]) with the required off-axis distances. These relationships allow us to translate the off-axis parameters to the parents’ axial parameters. Specifically, we take the off-axis distances, plug them into the normal paraxial equations to determine the (off-axis) focal lengths and, finally, translate those (off-axis) focal lengths to the axial focal lengths and distances. Thus, the off-axis distances and off-axis “effective” radii determine the axial distances and axial radii required of the parent system given some offset in aperture. Then we apply these relationships to the design of an off-axis reflective image relay.

Fig. 1. Layout of a back-to-back Cassegrain relay with an off-axis chief ray traced through the system. The intersection points of the ray with each $n$ th mirror are labeled $P_{n}$ . The off-axis distances are labeled $d_{n}$ . The vertices are labeled $V_{n}$ , and the stigmatic imaging points of each mirror are labeled $S_{n}$ . The stigmatic imaging points overlap for each successive mirror to stigmatically relay between a collimated object space to a collimated image space.

Download Full Size | PDF

Coddington’s equations describe the conjugate distances for the tangential and sagittal rays at a given oblique angle of incidence [7,8]. They are given as

\frac{n^{'}}{s^{'}} - \frac{n}{s} = \frac{n^{'} \cos I^{'} - n \cos I}{R_{s}},

\frac{n^{'} \cos^{2} I^{'}}{t^{'}} - \frac{n \cos^{2} I}{t} = \frac{n^{'} \cos I^{'} - n \cos I}{R_{t}},

where

s

and

t

, respectively, are the sagittal and tangential object distances;

s^{'}

and

t^{'}

, respectively, are the sagittal and tangential image distances;

n

and

n^{'}

, respectively, are the indices in the object and image spaces;

I

and

I^{'}

, respectively, are the angle of incidence and refraction; and

R_{s}

and

R_{t}

, respectively, are the sagittal and tangential RoCs at the point of intersection of the chief ray and the surface.

In the case of a mirror, we make the substitutions $n^{'} = - 1, n = 1, I^{'} = - I$ , and $s = t = l$ . In the case of stigmatic imaging, $s^{'} = t^{'} = ℓ^{'}$ , where we define $ℓ, ℓ^{'}$ as the object and image conjugate distances (see Fig. 2). Equations (1) and (2) then simplify as [9]

\frac{1}{ℓ^{'}} + \frac{1}{ℓ} = \frac{2 \cos I}{R_{s}},

\frac{1}{ℓ^{'}} + \frac{1}{ℓ} = \frac{2}{R_{t} \cos I} .

With these simplifications, the relationship between

R_{s}

and

R_{t}

for a given angle of incidence

I

is given as [9]

\frac{R_{s}}{R_{t}} = \cos^{2} I .

This is the relationship between the sagittal and tangential RoCs for astigmatism-free imaging for a given angle of incidence. Cartesian reflectors are surfaces of revolution, which means that these RoCs are the principal RoCs because the plane of reflection lies on a meridional plane [10].

Fig. 2. Geometry of a hyperboloid used as a Cartesian reflector. The incoming ray (solid red) is directed toward the point $F$ , the front focal point of the hyperboloid. The point $A$ is the point of intersection of the ray and the surface. The dashed blue line is the surface normal at $A$ . $C$ is the geometric center of the hyperboloid. $R_{S}$ is the local sagittal RoC.

Download Full Size | PDF

Equations (3) and (4) resemble the familiar Gaussian imaging equation for a mirror. Since $ℓ$ and $ℓ^{'}$ are known from the first-order properties, we can consider an effective RoC, $R_{eff}$ , for the stigmatic off-axis conjugates given as

\frac{1}{ℓ^{'}} + \frac{1}{ℓ} = \frac{2}{R_{eff}} = \frac{2 \cos I}{R_{s}} = \frac{2}{R_{t} \cos I} .

From Eq. (6), we can write the effective RoC for a given angle of incidence of the chief ray to the Cartesian surface as

R_{eff} = \frac{R_{s}}{\cos I} = R_{t} \cos I .

This effective RoC is the RoC to be used in the first-order imaging equations.

The Gaussian curvature is the product of the principal curvatures, which can be found from $κ_{1} = 1 / R_{t}$ and $κ_{2} = 1 / R_{s}$ . Applying the relationship in Eq. (5), the Gaussian curvature for a Cartesian reflector is given as

K_{G} = κ_{1} κ_{2} = \frac{1}{R_{s} R_{t}} = \frac{\cos^{2} (I)}{R_{s}^{2}} = \frac{1}{R_{t}^{2} \cos^{2} (I)} .

Comparing Eqs. (7) and (8), we see that the effective RoCs are related to the Gaussian curvature by

R_{eff} = \pm \frac{1}{\sqrt{K_{G}}} .

Note that, except for planar surfaces,

K_{G} > 0

for Cartesian reflectors. With this relation, the simple first-order imaging equations can be used to compute the imaging properties along any stigmatic ray in a stigmatic optical system made of Cartesian reflectors.

The required relay for MTW-OPAL can be achieved with two Cassegrain telescopes placed back-to-back. Figure 1 shows that this configuration in an example layout with a chief ray traced through the system. The distances along the axis of symmetry (between the vertices $V_{1}, V_{2}$ , etc.) are used to specify the system in the usual paraxial imaging equations (and in ray-tracing software). However, we now have a framework to use those same equations with the off-axis distances we wish to specify in our design.

For a function $z = F (x, y)$ , the Gaussian curvature is given as

K_{G} = \frac{F_{x x} \cdot F_{y y} - F_{x y}^{2}}{{(1 + F_{x}^{2} + F_{y}^{2})}^{2}},

where

F_{i} \equiv \partial F / \partial i

and

F_{i j} \equiv \partial^{2} F / (\partial i \partial j)

[11]. The sag of a conic surface of revolution is given by

z (r) = a [\sqrt{1 + \frac{r^{2}}{a R_{v}}} - 1],

where

a

is the distance from the geometric center to the surface vertex (see Fig. 2),

R_{v}

is the vertex RoC,

z

is the axial coordinate or sag, and

r

is the radial (cylindrical) coordinate. This is not the typical sag equation for conics in optics, but the geometric parameter

a

lends itself better to computation. The conic constant is related to

a

by

k = - (R_{v} / a + 1)

.

Using Eq. (10) with Eq. (11), the Gaussian curvature $K_{G}$ of a conic surface as a function of cylindrical coordinate $r$ relates to the vertex RoC $R_{v}$ and center-to-vertex distance $a$ as

K_{G} = {[\frac{a R_{v}}{r^{2} R_{v} + a (r^{2} + R_{v}^{2})}]}^{2} .

Equation (12), combined with Eq. (9), gives the relationship between the off-axis “effective curvature” we require and the associated axial curvature for a conic defined by

a

.

In this Letter, we are concerned with $K_{G}$ for a hyperboloid, but Eq. (12) can be generalized to any conic of revolution. For an ellipsoid, $a$ is the semi-major axis measured from the geometric center to the vertex. For a paraboloid, $a \to \infty$ , so the limit of Eqs. (12) and (11) must be taken. Note that for an ellipsoid, $a$ and $R_{v}$ are always opposite in sign, and for a hyperboloid they are the same sign.

For a hyperboloid, $a$ can be calculated from the known quantities $ℓ$ and $ℓ^{'}$ , which are the off-axis conjugate distances, as shown in Fig. 2. In fact, that is the definition of a hyperboloid: it is the set of points such that $| | ℓ^{'} | - | ℓ | | = 2 | a |$ .

Solving Eq. (12) and substituting $K_{G}$ according to Eq. (9) for the effective RoC $R_{eff}$ , the vertex RoC for a conic with an effective RoC $R_{eff}$ at radial coordinate $r$ is given as

R_{v} (r; a, R_{eff}) = \frac{a R_{eff} - r^{2} + \sqrt{{(a R_{eff} - r^{2})}^{2} - 4 a^{2} r^{2}}}{2 a} .

By expressing

a

and

R_{eff}

in terms of the angle of incidence

I

, the angle

η

, and

r

, Eq. (13) reduces to a more compact expression for

R_{v}

:

R_{v} = r \frac{\cos I}{\sin η} .

Referring to Fig. 2,

R_{s}

can be determined by extending the local surface normal to the axis of symmetry of the surface of rotation. The center of curvature of the sphere tangent to the surface at

A

in the sagittal direction is located at the intersection of the surface normal and the axis of symmetry. We can then compute the RoC of this sphere to obtain

R_{s}

as

R_{s} = \frac{r}{\sin (η)} = \frac{r}{\sin (\frac{α - β}{2})} .

With this result, Eq. (14) can be further simplified to

R_{v} = r \frac{\cos I}{\sin η} = R_{s} \cos I = R_{eff} \cos^{2} I .

Note that this result is independent of any parameter regarding the specific shape of the conic surface and, therefore, applies to any Cartesian reflector.

The relationship between the Gaussian curvature of a Cartesian surface and the stigmatic imaging condition, along with first-order parameterization, allows us to directly design the relay mirror system for the MTW-OPAL upgrade. The layout constraints inside the grating compressor chamber (GCC) in which this relay resides are typically in flux and require constant shifting of the optical layout to conform to the new requirements. In addition, MTW-OPAL is a research laser system designed to support engineering technologies applicable to the larger beam size of the OMEGA Laser System at LLE; therefore, requirements would need to be scaled for use in the OMEGA system.

The relay is required to take collimated light from the N5 plane to the G4 plane with $2 \times$ magnification (see Fig. 3) while imaging the N5 plane onto the G4 plane. As a result, we have two different imaging conditions: the “far-field” (FF) collimation and the “near-field” (NF) pupil imaging. The pupil planes of the FF become the image conjugates of the NF, and the image conjugates of the FF become the pupil planes of the NF. The terminology of this simultaneous NF and FF requirement can be found in [12].

Fig. 3. GCC layout showing locations of the N5 crystal plane and G4 grating plane. The available space for the final design of the AIR is shown. All optics between N5 and the AIR are flat mirrors. All optics between the AIR and G4 are flat optics with regard to imaging.

Download Full Size | PDF

To parameterize the first-order constraints of the NF and FF imaging conditions simultaneously, we set up a system of equations using a $2 \times 2$ rotationally symmetric matrix following the first-order optical matrix method and then impose the required imaging conditions on the system matrix [6]. The matrices span from the N5 plane to the G4 plane and contain nine variables: the five off-axis distances $d_{0}, d_{1}, d_{2}, d_{3}, d_{4}$ and the off-axis effective focal lengths (EFLs) of the mirrors $f_{1}, f_{2}, f_{3}, f_{4}$ . The five distances are specified to conform to the spatial constraints inside the GCC (see Fig. 3). The EFL of the first mirror $f_{1}$ is chosen to maintain a certain angle of incidence given the aperture offset. The other three focal lengths are determined by the NF imaging, FF imaging, and magnification conditions. The EFLs determine the effective RoC $R_{eff}$ for each mirror for use in Eq. (16), which determines the vertex radius $R_{v}$ for each mirror.

The shape parameter $a$ is calculated from $| | ℓ^{'} | - | ℓ | | = 2 | a |$ for the hyperboloids. Therefore, shape parameter $a$ is not a degree of freedom, as it is imposed by the combined distances $ℓ$ and $ℓ^{'}$ . With $a$ in hand, we use Eq. (13) to obtain $R_{v}$ . For the paraboloids we use the limit of Eq. (13) as $a \to \infty$ . Once the vertex radii $R_{v}$ and conic shape parameters are determined for all mirrors, the geometry is entirely specified due to the imposed stigmatic imaging condition. The locations of the stigmatic foci relative to the vertices can be calculated from $R_{v}$ and $a$ ; then the axial distances between the vertices can be calculated from the foci locations.

A suite of programs was written in MATLAB to quickly iterate the design using the above calculations. Two examples with different layout requirements (one long and one short) are shown in Figs. 4 and 5. The off-axis requirements and solved axial parameters for both designs are given in Data File 1 and Data File 2, respectively. The FF entrance pupil diameter is 45 mm, and the FF full field-of-view is 0.14 deg.

Fig. 4. Layout of the “long” relay, with paraboloids in green and hyperboloids in red. See Data File 1 for off-axis requirements and Data File 2 for the computed axial values.

Download Full Size | PDF

Fig. 5. Layout of the “short” relay, with paraboloids in green and hyperboloids in red. See Data File 1 for off-axis requirements and Data File 2 for the computed axial values.

Download Full Size | PDF

Both example relays have Gaussian image planes at the required NF and FF locations simultaneously through the imposed constraints on the first-order equations. Stigmatic imaging for the FF is ensured through the use of the conic surfaces with coincident stigmatic foci. The distances along the central chief ray match the specified distances in Data File 1, while avoiding obscuration using an offset aperture. The RMS wavefront performance of these examples was computed in CODE V. The RMS WFE is less than 0.000068 waves and 0.000041 waves for the short and long relays, respectively, at a wavelength of 910 nm.

We have shown the connection between the Gaussian curvature and stigmatic imaging. This connection was revealed using Coddington’s equations and the definition of the Gaussian curvature. This relationship was then exploited to directly design an unobscured image relay for a high-power laser using Cartesian reflectors. To directly express the axial parameters of the relay in terms of the off-axis constraints, we used the relationship between the Gaussian curvature and the effective RoC of the off-axis imaging system. We included an example of a long and a short relay with different distance constraints to illustrate the flexibility of the algorithm.

Funding

National Nuclear Security Administration (NNSA) (DE-NA0001944); University of Rochester; New York State Energy Research and Development Authority.

Acknowledgment

The authors thank Jonathan Papa and Aaron Bauer for technical discussions, and to the Laboratory for Laser Energetics for support through the Horton Fellowship. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The support of the DOE does not constitute an endorsement by DOE of the views expressed in this Letter.

REFERENCES

1. J. M. Rodgers, Proc. SPIE 4832, 33 (2002). [CrossRef]

2. D. Reshidko and J. Sasian, Opt. Eng. 57, 1 (2018). [CrossRef]

3. S.-W. Bahk, J. Bromage, and J. D. Zuegel, Opt. Lett. 39, 1081 (2014). [CrossRef]

4. Mssr. de Berce, Supplement du Journal des Sçavans (1672–1674), pp. 70–71.

5. A. Baranne and F. Launay, J. Opt. 28, 158 (1997). [CrossRef]

6. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

7. R. Kingslake, Opt. Photon. News 5(8), 20 (1994). [CrossRef]

8. H. Coddington, A Treatise on the Reflexion and Refraction of Light; Being Part I. of a System of Optics (Cambridge University, 1829).

9. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), p. 191.

10. S. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Springer, 2015).

11. A. Pressley, Elementary Differential Geometry (Springer, 2001).

12. D. Y. Wang, D. M. Aikens, and R. E. English, Opt. Eng. 39, 1788 (2000). [CrossRef]

Name	Description
Data File 1	Required off-axis parameters for the long and short image relay. Used to compute the axial parameters given in Data File 2.
Data File 2	Computed axial parameters corresponding to the required off-axis parameters in Data File 1. These parameters describe the designs in Figs. 4 and 5.

Gaussian curvature and stigmatic imaging relations for the design of an unobscured reflective relay

Abstract

Funding

Acknowledgment

REFERENCES

Supplementary Material (2)

Cited By

Figures (5)

Equations (16)

Optics Letters