1. INTRODUCTION

In sequential optical system design, it is often useful to begin by using a paraxial version for some, or perhaps all, of the lens elements. Such a first-order layout helps define what the performance of a system looks like when aberrations are neglected for the idealized elements. However, there are instances when the properties of a paraxial lens are insufficient to mimic a real, well-corrected compound lens. The foremost limitation stems from the fact that a paraxial lens, when used outside of the paraxial regime, does not satisfy the Abbe sine condition, whereas many actual lenses do obey the sine condition, at least approximately, and are known as aplanatic lenses. They are free from, or have minimal, spherical aberration and coma (for small off-axis fields). If a lens is free from astigmatism over its field of view, while also having no spherical aberration and no coma, it is referred to as being a stigmatic lens. On a related historical note, the well-known lens designer Paul Rudolph, while working for the Carl Zeiss company in Germany, coined the term “anastigmatic” to describe a lens for which the astigmatism at one off-axis field angle is zero ( [1], p. 236). More recently, this term has come to mean a lens that is designed to have reduced astigmatism over its field of view [2]. An interesting example of an actual stigmatic lens is the Luneburg lens based on a spherical gradient of the refractive index (see e.g., [3,4], p. 712). However, for modeling an idealized perfect lens, we can ignore the physical structure of the lens and instead treat it as a black box that has properties consistent with Fermat’s principle.

When capturing images with a planar sensor, one would also like to have minimal field curvature. Distortion, on the other hand, may or may not be a problem. For perfect imaging there should be no distortion, but a perfect Fourier transform lens has intrinsic distortion when viewed from an imaging perspective [5]. So, it is beneficial to have access to a generalized perfect lens model that (a) is free from spherical aberration, coma, astigmatism, and field curvature; (b) provides a specified form for the distortion; and (c) accurately captures the constraints associated with Fermat’s principle and hence satisfies the sine condition for arbitrary numerical apertures. Additional flexibility arises if this perfect lens model can mimic a thick lens with its two principal planes separated from one another.

Commercial ray-tracing software packages provide various types of idealized lens models. For example, Zemax OpticStudio [6] includes a thin paraxial lens surface but has no version of a perfect lens satisfying the sine condition. Other software such as CODE V [7,8] and OSLO [9] offer versions of perfect lenses that are based on so-called “mock ray tracing” using an eikonal formulation that typically requires an iterative numerical solution when the lens is represented by a specific eikonal function ( [10], Ch. 31).

In this paper a relatively straightforward perfect lens ray-tracing model is presented that complies with Fermat’s principle and satisfies a generalized sine condition, but does not require eikonal function theory. Two versions are developed, one that has no distortion, characterized by an $f\tan (\theta)$ response, as needed to model perfect imaging, and a second version that has intrinsic $f\sin (\theta)$ distortion, which is representative of an ideal Fourier transform lens.

We refer to our particular formulation of this model as a “Cardinal Lens” in order to distinguish it from other embodiments of idealized lenses [11]; it has been implemented as a user-defined surface for sequential ray tracing in Zemax OpticStudio, written in C++ and compiled as a 64-bit DLL (dynamic link library) and made available in Code 1, Ref. [12]. Along with the form of the distortion, this model requires the user to specify the effective focal length and the paraxial magnification at which the lens provides optimal performance. The lens can be embedded in media with different refractive indices before and after the lens and can be used at large numerical aperture (NA), thereby allowing, for example, simulation of (a) air or immersion microscope objectives, (b) Fourier transform lenses outside of the paraxial regime, and (c) wide-angle, high-speed imaging lenses.

The remainder of the paper is structured as follows. Section 2 provides an overview of the key features associated with modeling a perfect lens. Section 3 describes the basic formulation of the Cardinal Lens model. The details associated with its implementation in OpticStudio follow in Section 4. A few examples are provided in Section 5, while additional example layouts are included in Supplement 1. Finally, Section 6 offers a brief summary and conclusion.

2. MODELING AN IDEALIZED PERFECT LENS

A. Problem Statement

Consider a ray propagating in a sequential optical system and passing through an axially symmetric perfect lens as shown in Fig. 1. The lens effective focal length is $f$, and the thickness of the lens, $t$, corresponds to the separation between the principal planes. The input space to the left of the lens is homogeneous with refractive index ${n_1}$, while the output space on the right is homogeneous with index ${n_2}$. The ray $z$ coordinates are measured locally relative to the principal planes, with the input and output spaces being referred to the first and second principal planes, respectively. A $z$ coordinate lying to the left of either principal plane is negative, and to the right it is positive. A ray segment is defined by a unit vector pointing along the direction of travel and a point in space through which the ray passes. We denote a ray vector by $\vec s = (L,M,N)$, where $(L,M,N)$ are direction cosines such that $N = (1 - {L^2} - {M^2}{)^{1/2}}$.

 

Fig. 1. Local coordinate system for ray tracing through an axially symmetric perfect lens of thickness $t$. For a given input ray, we seek to find the corresponding output ray that complies with Fermat’s principle.

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In Fig. 1 an incoming ray segment ${\vec s_1} = ({L_1},{M_1},{N_1})$ is incident on the first principal plane of the lens at point $({u_1},{v_1})$. These values are readily available from the ray-tracing software. Our task is to find the outgoing ray segment coordinates $({u_2},{v_2})$ on the second principal plane along with the direction cosines ${\vec s_2} = ({L_2},{M_2},{N_2})$ in such a way that this outgoing ray complies with Fermat’s principle, and then send these results back to the main program so the ray can continue to propagate. Of course proper analysis of a general optical system that incorporates a perfect lens requires tracing many rays, so this process is repeated for each ray launched into the system and arriving at the perfect lens.

B. Localized Imaging Perspective and the Principal Reference Spheres

While the perfect lens may be just one element in a more complicated sequential lens train, for purposes of ray tracing we can analyze it in isolation, based on a localized imaging viewpoint since determination of an outgoing ray from the lens only relies on knowledge of the incoming ray, independent of any other specific optical elements that may come before or after the lens. Therefore, let us consider the imaging properties of a perfect lens taken by itself, as doing so is key to construction of the ray-tracing model. The local object and image planes for the perfect lens need not correspond to the actual physical object or image planes for the overall system unless, of course, the system is constructed that way. Instead, the localized conjugate planes are, in general, virtual constructs used solely to determine the ray path through the lens.

Sharp imaging, also known as stigmatic imaging, implies that all rays leaving any given point on an object surface will arrive at a single corresponding point on a conjugate surface in image space. For homogeneous object and image spaces, it is well-established that a rotationally symmetric perfect lens can stigmatically image just one object surface to one image surface (neglecting certain degenerate cases, such as perfect virtual imaging of the whole object space via reflection by a plane mirror, which are of little practical value) ( [1], p. 149; [10], Ch. 22). Here we are interested in object/image surfaces that are planar. Therefore, a perfect lens can only provide stigmatic imaging at one specific transverse magnification (or paraxial magnification when distortion is present). This is distinctly different from a paraxial lens, which provides aberration-free imaging at any magnification and is therefore inconsistent with Fermat’s principle. In our perfect lens model, the optimal (paraxial) magnification is a user-specified quantity.

Because we adopt a localized imaging model, we are not concerned here with an overall system stop or the location of the pupils. Instead, while the lens can have a limiting aperture that restricts ray transmission, the principal planes of the lens are of primary interest. More specifically, it is helpful to utilize the concept of principal reference spheres as described next.

When tracing a given ray, we first find the corresponding local object and image points, determined by the locations where the ray pierces the object and image planes of the lens as illustrated in Fig. 2. The first and second principal reference spheres are centered on the object and image points, respectively, and intersect the optical axis at the first and second principal points (see, e.g., [10], p. 194; [4], p. 430; and [13], p. 15). In some limited respects the principal reference spheres are analogous to the entrance and exit pupil reference spheres commonly used in aberration theory. However, the principal reference spheres pertain solely to the lens of interest and have no connection to the system stop. In the paraxial regime, the principal spheres become conjugate planes with unity magnification, while the pupil spheres become conjugate planes with a magnification that depends on the stop location. So these two sets of reference spheres are fundamentally different entities.

 

Fig. 2. Schematic of a perfect lens showing the localized object and image points for a given ray that is being traced through the lens. The principal reference spheres are centered on these points. The principal rays, which connect the object/image points to their respective principal points, are utilized internally within the lens model.

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While the principal reference spheres are conceptually very important, for general ray-tracing implementation it is preferable to calculate the ray coordinates at the principal planes and to display the rays in a layout diagram such that they terminate at the first principal plane and resume propagation at the second principal plane (as shown in Fig. 1). If the user would like to draw the principal reference spheres for one or more localized object/image points, and optionally turn off drawing of the ray segments to the principal planes, they can do so manually as illustrated in Section 5.A, Fig. 8.

C. Optical Path Length Considerations and the Cosine Rule

For a perfect lens, the optical path length (OPL) is the same for all rays connecting any given object point to its image point. A subtler, but vitally important, property of a perfect lens derives from Hamiltonian optics, in particular from the cosine rule as first derived by Smith in 1922 [14] and subsequently discussed in other references; for example, see [10,15,16]. For the sake of completeness, we present here a short derivation of the cosine rule, by adopting a slight variation of the description provided by Walther [10], Ch. 6. With reference to Fig. 3, we begin by showing an object point ${P_1}$ that is stigmatically imaged to its conjugate point ${P_2}$. Therefore the OPL from ${P_1}$ to ${P_2}$, denoted as $[{P_1}{P_2}]$, is the same for all rays connecting ${P_1}$ and ${P_2}$. Now, consider a second object point ${Q_1}$ that resides within a small neighborhood of ${P_1}$ and is located by an infinitesimal displacement vector $d{\vec r_1}$, and assume that it too is stigmatically imaged to point ${Q_2}$ in the neighborhood of ${P_2}$ with displacement vector $d{\vec r_2}$. This assumption requires that $[{Q_1}{Q_2}]$ be the same for all rays connecting ${Q_1}$ and ${Q_2}$. An equivalent requirement is that the optical path length difference, given by $[{Q_1}{Q_2}] - [{P_1}{P_2}]$, be constant.

 

Fig. 3. Geometry for derivation of the cosine rule.

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To quantify the situation, consider a single arbitrary ray segment that begins at ${P_1}$, pointing along a unit vector ${\vec s_1} = ({L_1},{M_1},{N_1})$ and intersecting the first principal reference sphere at ${A_1}$. This ray then goes on to intersect the second principal reference sphere at ${A_2}$, finally terminating at ${P_2}$ via ${\vec s_2} = ({L_2},{M_2},{N_2})$. Its OPL is $[{P_1}{P_2}] = [{P_1}{A_1}{A_2}{P_2}]$. To find $[{Q_1}{Q_2}]$ we can apply Fermat’s principle. Instead of using an actual ray path from ${Q_1}$ to ${Q_2}$, we take a path that is closely adjacent to the ray path from ${P_1}$ to ${P_2}$ (going through ${A_1}$ and ${A_2}$), which is possible because the two object points and their respective image points are infinitesimally close to one another. Fermat’s principle of stationarity ensures that the adjacent path OPL from ${Q_1}$ to ${Q_2}$, denoted $[{Q_1}{A_1}{A_2}{Q_2}]$, is approximately equal to the actual ray OPL, which we write as $[{Q_1}{Q_2}]$, at least to first order in the small displacements. We therefore have

D. Generalized Sine Condition

The cosine rule applies to displacement vectors $d{\vec r_1}$ and $d{\vec r_2}$ residing in small three-dimensional volumes centered on ${P_1}$ and ${P_2}$, respectively. Of primary interest here is imaging from an object plane to an image plane, both planes being normal to the optical axis. In this case we restrict the displacement vectors to lie in infinitesimal two-dimensional circles on these planes.

In general, for a lens with distortion, the displacement vectors need not be parallel to one another. For example, a small displacement of an arbitrary object point in the $x$ direction may yield a shift of the image point having both $x$ and $y$ components. However, for a lens with axial symmetry, as we assume here, this coupling can be avoided by constraining the object point to lie on either the $x$ or $y$ axis. We choose the $y$ axis, so the object and image points both reside in the $yz$ plane, which we refer to as the tangential (or meridional) plane. Doing so simplifies calculation of the differential magnification components. To accommodate an arbitrary off-axis object point, the lens system (including the incoming ray) can first be rotated about the $z$ axis to bring this point into the tangential plane, then the ray of interest can be traced from the new object point location through the lens, and finally the lens coordinate system containing the outgoing ray segment can be rotated back to its original orientation. More detail about this process is discussed in Section 3.A.

Now, assuming the object point lies in the tangential $yz$ plane, if we choose one set of displacement vectors to be oriented along the $x$ direction and a second set along the $y$ direction, the cosine rule yields the following two relations, which constitute the generalized sine condition (GSC) (for related versions see [14], p. 37; [17], Eq. 11):

This formulation of the GSC makes use of the ray direction cosines, but these $(L,M)$ values for a ray segment can be found by projecting the ray segment onto the $xz$ and $yz$ planes, respectively, and then computing the sines of the angles that these two projected vectors make with respect to the $z$ axis. For this reason, Eq. (3) is considered a “sine” condition. In addition, note that Eq. (3) uses ${m_{\textit{dx}}}$ and ${m_{\textit{dy}}}$, which can be positive or negative, whereas Eq. (2) uses $|{m_d}|$. With a little effort, the interested reader can confirm the signs of ${m_{\textit{dx}}}$ and ${m_{\textit{dy}}}$ are needed when converting the cosine rule based on the angles ${\phi _1}$, ${\phi _2}$ (now measured with respect to differential displacement vectors confined to the $xy$ plane) to the sine condition based on $(L,M)$ values. Lastly, Eq. (3) is referred to as the “generalized” sine condition because traditionally the Abbe sine condition applies to on-axis points with the constants ${C_x} = {C_y} = 0$ and ${m_{\textit{dx}}} = {m_{\textit{dy}}} = {m_p}$, while the generalized version allows us to explicitly consider off-axis conjugate points, either with or without distortion. For a perfect lens, the GSC is satisfied for all object/image conjugate points throughout the local field of view of the lens, with each field point having its own set of constants ${C_x}$ and ${C_y}$.

Enforcement of the GSC at any given field point also ensures local stigmatic imaging in the near vicinity of that point. However, we do not rely on the GSC to yield stigmatic imaging; instead, as will be seen in Section 3, we implement a model that by design has no ray or wavefront aberrations, independent of the GSC, when the lens is used at its specified optimal magnification. We do, though, utilize the GSC to calculate the direction cosines of the rays leaving the lens. Doing so ensures that the angular composition of a ray bundle (e.g., an elliptic cone) coming to focus at an imaging point is properly related to the angular composition of the corresponding ray bundle leaving the object point. It is in this sense that the model complies with Fermat’s principle because the GSC derives from the cosine rule, which in turn derives from the application of Fermat’s principle.

E. Distortion Considerations

The image from a perfect lens may or may not suffer from distortion, but, regardless, it yields stigmatic imaging with a one-to-one mapping of object-plane points to image-plane points. Without distortion, perfect imaging is characterized by a constant transverse magnification that is applicable across the entire image plane. (So again there is no need to constrain the object point to the $yz$ plane via a lens system rotation as discussed in the previous section.) Therefore, when imaging at infinity, the image point location follows an ${f_1}\tan ({\theta _1})$ relationship in which ${f_1}$ is the object-space focal length and ${\theta _1}$ is the incoming field angle. However, if distortion is present, then the conventional transverse magnification, as well as the differential magnification values, vary with the object point location (i.e., they are field dependent). In this case we can specify the paraxial magnification for object points very near the optical axis, but the change in transverse magnification with field depends on what type of perfect lens the user would like to simulate. For example, a perfect Fourier transform lens imparts ${f_1}\sin ({\theta _1})$ distortion (see [5]; and [13], p. 36). Our Cardinal Lens model simulates a perfect imaging lens as well as a perfect Fourier transform lens. Additional details related to these two distortion relationships are discussed in Sections 3.D and 3.E.

F. Conjugate Plane Locations

Given the effective focal length $(f)$ of the lens and the paraxial transverse magnification $({m_p})$ at which stigmatic imaging occurs, we can easily determine the locations, ${z_1}$ and ${z_2}$, of the local object and image conjugate planes (as shown in Fig. 2). Beginning with the lens equation,

3. RAY PROPAGATION ANALYSIS: THE CARDINAL LENS MODEL

We now describe how any given ray incident on the lens is traced through the lens; the details comprise our Cardinal Lens model. It is a two-step process in which we first find the principal ray segments and then second use these principal ray segments to facilitate tracing the ray of interest via the GSC. For purposes of discussion, assume the object plane is located to the left of the lens (negative ${z_1}$) and the image plane is on the right (positive ${z_2}$) as depicted in Fig. 2. In this case the lens is positive, so both the object and image are real and the magnification is negative. In general, though, the lens can be either positive or negative, so the object and image planes can reside on either side of the lens. This means the object, as well as the image, may be either real or virtual in accordance with Eq. (6). The ray-tracing formulation developed here is applicable to any of these combinations (illustrated in Table S1 of Supplement 1).

A. Conjugate Point Locations

To determine the object point location $({x_1},{y_1})$, we simply calculate where the incoming ray being traced crosses the object plane. For the perfect imaging lens we can proceed directly to calculation of the corresponding image point coordinates as discussed below. For the perfect Fourier transform lens, as noted in Section 2.D, we need to first apply a rotation to the lens coordinate system to bring the object point onto the $y$ axis unless, of course, the object point already happens to reside in the $yz$ plane. The angle of rotation is given by $\alpha = - \text{atan2}({y_1},{x_1}) + \pi /2$, where $\text{atan2}$ is the four-quadrant inverse tangent function (yielding the azimuthal angle of the object point as measured from the ${+}x$ axis). The minus $\text{atan2}$ term takes the object point to the ${+}x$ axis, and the $\pi /2$ term transfers it to the ${+}y$ axis. By using the standard 2D rotation matrix, $R(\alpha) = [\cos \alpha , - \sin \alpha ;\sin \alpha ,\cos \alpha]$, the following vectors are transformed: $({x_1},{y_1})$, $({L_1},{M_1})$, and $({u_1},{v_1})$. For simplicity we use the same notation for the vectors after rotation, with it being implicitly understood that for the Fourier lens the ray analysis is conducted in the rotated coordinate system, and the results are subsequently transformed back into the original coordinate system via $R(- \alpha)$.

To determine the image point location, we make use of the input/output principal ray segments as shown in Fig. 4. These principal rays are not seen by the user; instead, they are only used internally within the model. The incoming principal ray is directed from the object point to the center of the first principal plane; its direction cosines are denoted ${\vec s_{1p}} = ({L_{1p}},{M_{1p}},{N_{1p}})$. From Fig. 4 we see ${M_{1p}} = \sin ({\theta _{1y}})$. For the perfect imaging lens we similarly have ${L_{1p}} = \sin ({\theta _{1x}})$, while for the perfect Fourier lens ${L_{1p}} = 0$. In either case, ${N_{1p}} = (1 - L_{1p}^2 - M_{1p}^2{)^{1/2}}$.

 

Fig. 4. Cross-sectional diagram of the tangential $yz$ plane showing the principal ray segments in object/image spaces. The object-space principal ray ${\vec s_{1p}}$ connects the object point (associated with a given input ray) to the center of the first principal plane. The image-space principal ray ${\vec s_{2p}}$ propagates from the center of the second principal plane toward a point on the back focal plane determined by the user’s choice for the form of the lens distortion. The intersection of the image-space principal ray with the image plane determines the image point location.

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Next, the form of the distortion is used to find the outgoing principal ray in image space via its intersection with the back focal plane. For the imaging lens, the back focal plane coordinates are $({\tilde x_2},{\tilde y_2})$, which then allow us to find the image point location $({x_2},{y_2})$ by extending the ray segment to the image plane. The details are presented below in Section 3.4. In this case the conjugate points are related via the paraxial magnification, and the differential magnification components are equal to ${m_p}$. However, for a Fourier lens (after rotation), the $x$ coordinates in both the back focal plane and image plane are nominally zero. However, to calculate the differential magnification components, we need to examine small $x$ and $y$ displacements of the object point (e.g., using $\delta {x_1} = \delta {y_1} = {z_1}/1000$) and find the corresponding image point perturbations ($\delta {x_2}$, $\delta {y_2}$), from which we can readily calculate the differential magnification values ${m_{\textit{dx}}} = \delta {x_2}/\delta {x_1}$ and ${m_{\textit{dy}}} = \delta {y_2}/\delta {y_1}$. So our analysis for the Fourier lens case (Section 3.E) accommodates the necessary $\delta {x_1}$ (and corresponding $\delta {x_2}$) displacements out of the $yz$ plane.

B. Managing Infinite Conjugates

At this point we digress briefly to address the issue of infinite-conjugate imaging. For small values of magnification, the input beam is close to being collimated, while for large magnification the output beam is approximately collimated. The limiting case of ${m_p} \to 0$ (with ${z_1} \to \pm \infty$) corresponds to an infinite front conjugate case, so that all incoming rays are parallel to the principal ray in object space. A similar situation applies to the infinite rear conjugate case in which ${m_p} \to \pm \infty$ (with ${z_2} \to \mp \infty$). Numerically, we assume the infinite front and rear conjugate scenarios correspond to $|{m_p}| \le 1\text{e} - 10$ and $|{m_p}| \ge 1\text{e} + 10$, respectively. In these two limiting cases, we apply a long-conjugate approximation and somewhat arbitrarily set ${z_1} = - 1\text{e}10$ (infinite front conjugate) and ${z_2} = + 1\text{e}10$ (infinite rear conjugate), both measured in lens units (typically millimeters). We then internally set the paraxial magnification so that it equals ${z_2}/{z_1}$ and, for the infinite front conjugate case, temporarily modify the incoming ray direction cosines to make them consistent with the finite long-conjugate approximation. Next, as will be discussed in Section 3.F, we apply the GSC to obtain the output ray direction cosines, from which we calculate the outgoing ray coordinates $({u_2},{v_2})$ (see Fig. 1). Because either the incoming or the outgoing ray direction cosines are only approximate, we lastly make the necessary minor adjustments to the direction cosines to ensure the beam is properly collimated in either object or image space. For the infinite front conjugate case, this simply entails restoring the original direction cosines of the incoming ray. For the infinite rear conjugate case, we set the outgoing ray direction cosines equal to those of the outgoing principal ray.

C. General Approach and Role of the Paraxial Magnification

Our general approach utilizes the form of the distortion to propagate the outgoing principal ray from the center of the second principal plane into image space. As mentioned previously, we consider two cases: (a) a perfect imaging lens without distortion, characterized by an $f\tan (\theta)$ response, and (b) a perfect Fourier transform lens with $f\sin (\theta)$ distortion. For each case we consider two separate ranges of magnification, $|{m_p}| \le 1$ and $|{m_p}| \gt 1$, which helps facilitate handling of the front and rear infinite-conjugate scenarios. After deriving equations for the outgoing principal ray direction cosines and corresponding image point coordinates in Section 3.D (imaging lens) and Section 3.E (Fourier lens), we turn our attention to tracing the ray of interest in Section 3.F. The reader who is primarily interested in using the Cardinal Lens model in OpticStudio, and is less concerned with the mathematical details, may go directly to Sections 4 and 5.

D. Perfect Imaging Lens

For a perfect imaging lens without distortion, the output principal ray leaves the center of the second principal plane and intersects the back focal plane at coordinates

1. Imaging Lens: $|\textsf{m}_{\textsf p}| \le $ 1

Note that $|{m}_{p}| \le 1$ includes the case of infinite front conjugate imaging for which ${m_p} = 0$, so in this regime we use an approach that does not rely on the object plane location. From similar triangles in the image space of Fig. 4 we see ${y_2}/{z_2} = {\tilde y_2}/({n_2}f)$, with a corresponding result for ${x_2}$ and ${\tilde x_2}$, which are then combined with Eq. (7) to determine the image point coordinates,

2. Imaging Lens: $|\textsf{m}_{\textsf p}| \gt $ 1

For $|{m_p}| \gt 1$ (which includes the case of infinite rear conjugate with ${z_2} \to \infty$) we proceed by first calculating the outgoing principal ray direction cosines. To do so we use Eq. (8) and write

It is straightforward to show ${N_{2p}} = (1 + {a^2} + {b^2}{)^{- 1/2}}$. So the object point coordinates yield values for $a$ and $b$, and hence ${N_{2p}}$, from which we determine the remaining direction cosines:

E. Perfect Fourier Transform Lens

For a perfect Fourier transform lens, in order to apply the GSC, the object and image conjugate points are assumed to nominally reside in the $yz$ plane (no $x$ components). However, the following analysis is not subject to this constraint; it includes both $x$ and $y$ coordinates and is generally applicable to arbitrary conjugate point locations. Therefore, it can be used to analyze ($\delta {x_1},\delta {x_2}$) displacements of the nominal object/image points out of the $yz$ plane as needed to calculate the differential magnification in the $x$ direction. Bearing this important detail in mind, we can proceed in a fashion similar to that for the perfect imaging lens, and begin by noting the output principal ray intersects the back focal plane at coordinates

1. Fourier Transform Lens: $|\textsf{m}_{\textsf p}| \le $ 1

As before, from Fig. 4 we have ${y_2}/{z_2} = {\tilde y_2}/({n_2}f)$, with a corresponding result for ${x_2}$ and ${\tilde x_2}$, which are combined with Eq. (14) to determine the image point coordinates as follows:

2. Fourier Transform Lens: $|\textsf{m}_{\textsf p}| \gt $ 1

For $|{m_p}| \gt 1$ (which includes the case of infinite rear conjugate with ${z_2} \to \infty$) we proceed to calculate the outgoing principal ray direction cosines by combining Eq. (15) with ${\tilde y_1}/({n_1}f) = {y_1}/{z_1}$ (along with a corresponding result for ${x_1}$ and ${\tilde x_1}$), as seen from similar triangles in object space, to yield

F. Output Ray Direction Cosines from the Generalized Sine Condition

Having found the image point coordinates through which the output ray being traced must pass, we can turn our attention to finding the direction cosines $({L_2},{M_2},{N_2})$ of this output ray segment. This is done by application of the GSC derived in Section 2.D, so the result is consistent with Fermat’s principle. Once we have the output direction cosines, we can determine the intersection point $({u_2},{v_2})$ of the outgoing ray with the second principal plane of the lens, which completes our task.

 

Fig. 5. Cross-sectional diagram showing the input and output ray segments, along with the principal ray segments. The output ray segment being traced is found from application of the generalized sine condition.

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Fig. 6. Example of the Cardinal Lens user-defined DLL surface (with its three input parameters) in the OpticStudio Lens Data Editor. The thickness of the Cardinal Lens surface corresponds to the spacing between the principal planes.

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Consider first the perfect imaging lens case. With reference to Fig. 5, we restrict attention to object points lying in the ${z_1}$ plane and image points in the ${z_2}$ plane. We begin with the known principal ray segment direction cosines, $({L_{1p}},{M_{1p}})$ and $({L_{2p}},{M_{2p}})$, for the specific set of conjugate points associated with the ray that is being traced. These principal ray direction cosines are found as outlined in the previous section. Next we apply the generalized sine condition to these principal ray segments and compute values for ${C_x}$ and ${C_y}$:

For the case of a perfect Fourier transform lens, rotation of the lens system to place the conjugate points in the $yz$ plane means ${x_1} = {x_2} = 0$, and the principal ray segments have ${L_{1p}} = {L_{2p}} = 0$, which in turn yields ${C_x} = 0$. Additionally, the ${m_d}$ components are found using the approach described in Section 3.A. Otherwise, Eqs. (21) and (22) remain applicable in the rotated coordinate system, and the resulting vectors $({L_2},{M_2})$ and $({u_2},{v_2})$ can be subsequently transformed back into the original lens orientation via the inverse rotation.

G. Optical Path Length Modeling

Calculating the proper optical path length (OPL) of the ray as it transits the lens (i.e., the OPL assigned to the ray in between the principal planes) requires a few steps. First, the ray OPL is internally computed as the sum of the OPL from the local object point to the first principal plane, plus the OPL from the second principal plane to the local image point—call this sum ${\text{OPL}_1}$. Second, this process is repeated using the principal ray for the same local object/image points, yielding ${\text{OPL}_0}$. Last, the final OPL value assigned to the ray being traced equals ${\text{OPL}_0} - {\text{OPL}_1}$. Doing so forces the optical path difference (OPD) between the ray being traced and the principal ray to equal zero. This allows analysis features in OpticStudio that depend on the OPD to function properly (e.g., wavefront map, point spread function, etc.). When a Cardinal Lens is used in isolation at the specified paraxial magnification, no aberrations are introduced, so the OPD is zero and the lens response is diffraction-limited. A similar situation occurs when modeling a more complicated optical train that includes glass elements, but the local object/image planes for the Cardinal Lens happen to correspond to intermediate conjugate planes in the optical system (either real or virtual). In this case the ray aberrations arriving at a Cardinal Lens are simply relayed through the Cardinal Lens with the appropriate magnification—no additional aberrations are generated. However, if a Cardinal Lens is used in a way that deviates from its optimal magnification, then aberrations associated with the lens itself arise; an example is provided in Section 5.D.

4. IMPLEMENTATION IN ZEMAX OPTICSTUDIO

The Cardinal Lens ray-tracing procedure has been implemented in Zemax OpticStudio v23.2 by using a custom user-defined surface. It is written in C++ and compiled as a 64-bit DLL file. The user must supply a value for the lens thickness (zero for a thin lens), along with values for three parameters: (1) the effective focal length, (2) the paraxial magnification for which the lens is aberration-free, and (3) a binary switch value (0/1) to indicate whether the lens is distortion-free with an $f\tan (\theta)$ response, or it operates as a Fourier transform lens with intrinsic $f\sin (\theta)$ distortion. An example of these parameters entered in the Lens Data Editor is shown in Fig. 6.

A. Cardinal Lens Model: Key Features

Here we summarize the key features of the Cardinal Lens DLL implementation:

B. Cardinal Lens Model: User Guidelines

In order to use the Cardinal Lens DLL in Zemax OpticStudio, there are a few important guidelines that the user should follow:

 

Fig. 7. Ray-trace layouts for imaging at infinity with an immersion objective, showing a comparison between (a) a paraxial thin lens with ${\text{NA} = {0.80}}$ and (b) a thin Cardinal Lens with $\text{NA} = {1.0}$. Both lenses have an $\text{EFL} = {5}\;\text{mm}$ and an $\text{EPD} = {10}\;\text{mm}$, but yield different numerical apertures. Field angles of 0° and 20° are shown.

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5. EXAMPLES

In this section, we provide a few examples that illustrate the use of the Cardinal Lens in Zemax OpticStudio via our user-defined surface, Cardinal_Lens.dll. Both the DLL and the example models are provided in Code 1, Ref. [12].

A. Infinite-Conjugate High-NA Immersion Objective

First, consider the case of imaging at infinity (${m_p} = 0$) with a high-NA immersion lens. In Fig. 7 we compare a paraxial lens to a thin Cardinal Lens, both having an effective focal length (EFL) of 5 mm and no aberrations. The object space is air $({n_1} = 1.0)$, and the image space is water $({n_2} = 1.3)$. The field angles are 0° and 20°. The aperture is set to an entrance pupil diameter (EPD) of 10 mm. For the Cardinal Lens, the paraxial magnification parameter is zero as appropriate for this infinite front conjugate case. Even though both lenses have the same EPD and EFL, the focused beam NAs are quite different. For the paraxial lens the $\text{NA} = 0.80$, while for the Cardinal Lens it is 1.0. As a result, diffraction-related quantities such as the Airy disk diameter and the modulation transfer function cutoff frequency are also different for the two models. Clearly, if one were trying to model a well-corrected microscope objective, the Cardinal Lens version would provide a more realistic option. In OpticStudio, the “Image Space NA” is taken to be a paraxial quantity; the actual NA based on a real marginal ray should be calculated as one over two times the “Working F/#.” We note for the Cardinal Lens with an object at infinity, the relation $\text{EPD} = {2\cdot \text{NA}\cdot\text{EFL}}$ is generally applicable.

 

Fig. 8. Modified version of the Cardinal Lens shown in Fig. 7(b) to make it a thick lens. Also, for each of the two incoming field angles (0° and 10°), the corresponding output principal reference spheres have been added to the model. The drawing options available in OpticStudio have been used to suppress rays in between the first principal plane and the second principal reference spheres.

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It is also interesting to compare the value of the “Offense against the Sine Condition” (OSC) for the two lenses of Fig. 7. For an actual lens, the OSC is a dimensionless quantity equal to the relative sagittal coma [18]. If spherical aberration is negligible, and the OSC assumes a very small value, then linear coma is also negligible for small off-axis fields. This is typically the case for well-corrected objectives. In our example here, both lenses yield aberration-free focusing by design, so elimination of linear coma does not come into play, but nonetheless the OSC value provides a measure of how closely a lens follows the sine condition. To compare the paraxial and Cardinal lenses, we use a simplified version of the $\text{OSC} = ({u_2}/\sin {U_2}) - 1$, where ${u_2}$ is the paraxial marginal ray slope and ${U_2}$ is the real marginal ray angle, both taken in image space. The paraxial lens has $\text{OSC} = {0.261}$, while the Cardinal Lens has $\text{OSC} = {0}$. In passing, we should mention that OpticStudio provides a merit function operand (OSCD) that calculates a more general version of the OSC for imaging an object at infinity (based on [18], Eq. 9.41).

If desired, the user can simulate a thick version of the Cardinal Lens. For example, Fig. 8 shows a 5 mm gap in between the principal planes, obtained by simply setting the thickness of the Cardinal Lens surface to 5 mm. In addition, the second principal reference spheres can be manually added to the model and displayed in the layout as shown in Fig. 8. In this case we use two configurations, one for each of the two fields, and display both configurations simultaneously. For a given field the corresponding principal reference sphere radius and tilt angle, which correspond to the output principal ray segment length and angle, can be found via the merit function and then applied to a corresponding dummy spherical surface via the multi-configuration editor. The ray drawing options can be used to suppress the rays in between the input principal plane and the output principal reference spheres.

B. Microscope Objective with Tube Lens

When combining a Cardinal objective lens with a glass tube lens, both the ray and wave aberrations associated with the tube lens are apparent. Figure 9 shows the layout for an immersion objective with an effective focal length of 5 mm and an object-space index of 1.5, operating at $\text{NA} = {1.3}$, followed by a simple doublet tube lens having a focal length of 200 mm. This creates a simple microscope system having a magnification of ${-}{40}$. In this case the system aperture stop is placed on the first principal plane of the Cardinal objective lens. The figure shows rays from three field points (0, 0.25, and 0.50 mm). The principal planes of the objective are separated by 10 mm. The objective lens has its paraxial magnification parameter set to ${-}{1.0\text{e}10}$, which the DLL interprets as infinite rear conjugate. The ray aberrations across the visible spectrum for the 0.5-mm field point, arising solely from the tube lens, are clearly observed in the spot diagram of Fig. 10(a). The OPD fans of Fig. 10(b) show the associated wavefront aberrations.

 

Fig. 9. Simple ${40} \times$ microscope system based on a 10-mm-thick Cardinal Lens objective ($\text{EFL} = {5}\;\text{mm}$, $\text{NA} = {1.3}$) and a doublet tube lens ($\text{EFL} = {200}\;\text{mm}$). Field points with $y$ coordinates of 0, 0.25, and 0.50 mm are shown.

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Fig. 10. Aberrations for the 0.5-mm off-axis field point of the microscope system shown in Fig. 9, which are solely attributable to the tube lens. (a) Spot diagram showing transverse ray aberrations, and (b) OPD curves (plotted on a scale from ${-}{5}$ to ${+}{5}$ waves) representing the wavefront aberrations.

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C. Fourier Transform Lens: One- and Two-Element Versions

A Cardinal Lens can function as a perfect Fourier transform (FT) lens. This is done in OpticStudio by setting the $f\sin (\theta)$ distortion switch in the lens data editor (Parameter 3 for the lens surface) to a value of 1, which ensures the incoming angular distribution of rays is mapped to the proper spatial distribution of ray intercept points across the back focal plane. In a coherent optical system, the back focal plane serves as the Fourier plane. The focused spots in the Fourier plane have $(x,y)$ coordinates that are directly proportional to the $(L,M)$ direction cosines of the collimated beams, or angular spectrum components, leaving the front focal plane, with the constant of proportionality being the front focal distance ([13], Ch. 2). Of course an optical FT is a diffraction phenomenon and should therefore be simulated using wave optics (i.e., physical optics propagation, or POP, in OpticStudio). However, from a ray-tracing perspective, an ideal FT lens should provide the proper angle-to-spatial mapping as well as the correct focusing NA, consistent with the generalized sine condition, which is what we demonstrate here.

Figure 11(a) shows a Cardinal Lens configured as a thin Fourier transform (FT) lens in air with f/1.0 and $\text{EFL} = {10}\;\text{mm}$. The aperture stop is located in the front focal plane, so this is an image-space telecentric configuration. The plot of Fig. 11(b) illustrates the desired distortion.

 

Fig. 11. Example of a thin Cardinal Lens used in Fourier transform mode (in air with f/1.0 and $\text{EFL} = {10}\;\text{mm}$). (a) Ray-tracing layout for field angles of 0° and 20° and (b) corresponding distortion plot showing the $f\sin (\theta)$ response.

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Fig. 12. Example of a compound FT lens using two Cardinal lenses, each having a thickness of 2 mm, with the lenses separated by 2 mm. The optical path as constructed in the OpticStudio lens data editor is shown with the Cardinal Lens surfaces highlighted.

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A somewhat more complicated version of a compound FT lens in air using two separate Cardinal lenses is shown in Fig. 12. In this case each lens has an $\text{EFL} = {50}\;\text{mm}$, is 2 mm thick, and the gap between them is 2 mm. The EFL of this lens assembly is 25.51 mm. The first lens operates in infinite-front-conjugate mode with a paraxial magnification of zero, and its distortion switch is set to 1, corresponding to an $f\sin (\theta)$ mapping as required for the FT operation, where $f$ is the front focal length of this first lens (note: in air this equals the EFL of the lens). The second lens operates in finite-conjugate perfect-imaging mode and applies no additional distortion. The appropriate magnification and back focal distance for the second lens are found by setting those values as variables and optimizing for minimum focused spot size. The distortion associated with the first Fourier lens is scaled by the magnification of the second imaging lens such that the compound lens provides the correct FT angle-to-spatial mapping based on the front focal length of the compound lens assembly.

 

Fig. 13. Ray-trace layouts for finite-conjugate imaging with a Cardinal Lens in air ($\text{EFL} = {10}\;\text{mm}$, ${m_p} = - 2.0$, ${\text{NA}_1} = 0.80$, ${\text{NA}_2} = 0.40$, $t = 5\;\text{mm} $). Two fields are shown: (a) on-axis imaging and (b) off-axis imaging with ${y_1} = 2\;\text{mm} $ and ${y_2} = - 4\;\text{mm} $. Because both the image space and object space have the same refractive index, the upper and lower marginal rays each cross the two principal reference spheres at the same distance from the optical axis as shown by the red dashed lines.

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Fig. 14. Aberrations arise when using the Cardinal Lens away from its optimal magnification value. Shown are results for the lens of Fig. 13 when ${m_p} = - 2.01$. (a) On-axis spot diagram, (b) on-axis wavefront map, (c) spot diagram for the 2-mm off-axis field point, and (d) wavefront map for the 2-mm off-axis field point. The wavelength is 0.50 µm.

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We conclude this section by mentioning that if two Cardinal FT lenses are used for coherent imaging in a 4f configuration, then an object in the form of a grating can be properly imaged ( [13], p. 21), whereas using two paraxial lenses leads to an image having the wrong period; an example of this is shown in Supplement 1.

D. Finite-Conjugate Imaging

In this section we illustrate a Cardinal Lens used for finite-conjugate imaging, first at its optimal aberration-free magnification; then we next show how aberrations arise when the magnification deviates from the optimal value. Figure 13 shows the optimal case with a 5-mm-thick lens used by itself, and the layout magnification is the same as that of the user-specified value (${m_p} = - 2.0$), so no aberrations are present. The diffraction-limited Airy disk radius is 0.76 µm. The ratio of the object-space to image-space numerical apertures equals the paraxial magnification, or ${\text{NA}_1}/{\text{NA}_2} = {m_p}$. Also, because both the object and image spaces have the same refractive index, the upper and lower marginal rays each individually cross the two principal reference spheres at the same radial distance from the optical axis ( [13], p. 15). If the object/image-space refractive indices are different, then this simple geometrical result is no longer exact, but instead only holds approximately.

 

Fig. 15. Biconvex lens OPD and the change in OPD when using a relay lens. (a) Biconvex lens with 1:1 imaging and (b) its OPD fan. (c) Relay imaging with a Cardinal Lens and (d) resulting OPD fan. (e) Relay imaging with a paraxial lens and (f) corresponding OPD fan.

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If the lens of Fig. 13 is used at a magnification other than the designated optimum value, significant aberrations can arise. For example, if the paraxial magnification specification of the lens is changed to ${m_p} = - 2.01$, but we retain the same object and image plane locations, the resulting spot diagrams and wavefront maps are shown in Fig. 14. In this case, for both fields, the RMS spot radius is ${\sim}10\;{\unicode{x00B5}\text{m}}$ and the RMS wavefront error is ${\sim}1$ wave (with tilt removed from the off-axis field), so the lens is now operating far outside of the diffraction limit. From Eq. (6) we have ${z_1} = - 14.975\;\text{mm}$ and ${z_2} = 30.10\;\text{mm}$, while the layout object and image plane locations remain at ${-}{15.0}\;\text{mm}$ and 30.0 mm, respectively, in accord with a layout magnification of ${-}{2.00}$.

E. Ray and Wavefront Aberrations for the Cardinal Lens

As a final example, it is instructive to consider how both transverse ray aberrations and wavefront aberrations propagate through the Cardinal Lens and to assess their self-consistency with one another, which serves as a check as to whether or not the lens complies with Fermat’s principle. Figure 15(a) shows a biconvex lens operating with an object-space $\text{NA} = {0.23}$ and a magnification of ${-}{1}$. The first lens surface is selected as the stop surface, and the aperture size is set via the specified NA value. The marginal ray suffers from significant aberration as seen in the associated OPD fan of Fig. 15(b) with a value of approximately 100 waves. Now consider, as shown in Fig. 15(c), a Cardinal Lens used to relay this image with a magnification of ${-}{1}$. Similarly, Fig. 15(e) shows a paraxial lens used for the relay. For both cases the transverse ray aberrations are identical, but the two corresponding OPD fans [Figs. 15(d) and 15(f)] are quite different. The Cardinal Lens version has a maximum OPD of 500 waves, while the paraxial lens imparts only 50 waves. In OpticStudio, various modes exist for computing the OPD through a paraxial lens; here we use $\text{Mode} = {1}$, which provides the highest degree of accuracy.

The OPD is measured with respect to a reference sphere that is centered on a point where the chief ray intersects the image plane. In OpticStudio this reference sphere surface is constructed so it contains the point where the chief ray crosses the exit pupil plane. We expect there to be a change in the OPD when adding a relay lens because the location of the exit pupil changes relative to the new image plane; hence the radius of the reference sphere changes. However, for both the Cardinal and paraxial relays the exit pupil location is the same [specifically, ${-}{6.6}\;\text{mm}$ from the final image plane in Figs. 15(c) and 15(e)]. So which OPD curve is correct? To answer this question, we can employ the following well-established relations ( [1], p. 206) linking the transverse ray aberrations to the OPD, which is denoted by the function $\Phi ({x_p},{y_p})$:

To proceed, we use a custom macro in OpticStudio to trace a tangential fan of rays spanning the entrance pupil and record the following data for each of the two relay lens options: ${y_p}$, $\Delta y$, $\Phi ({y_p})$, and ${R^\prime}({y_p})$. Using Eq. (23) we calculate $\Delta y$ from the derivative of $\Phi ({y_p})$ and compare this to the result found directly from ray tracing as shown in Fig. 16. The ray fan computed using the Cardinal Lens relay OPD agrees precisely with the ray fan from direct ray tracing, while the paraxial lens relay OPD leads to ray fan errors at the edges of the pupil (for $|{P_y}| \gt 0.9$). Therefore, the OPD in the Cardinal Lens case is correct as it is fully consistent with the transverse ray aberration, which is another indicator that the Cardinal Lens complies with Fermat’s principle.

 

Fig. 16. Comparison of ray fans based on direct ray tracing as well as computed from the OPD fans for the Cardinal Lens and paraxial lens relay cases [per Eq. (23)].

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F. Summary of Additional Examples in the Supplemental Document

Other examples involving the Cardinal Lens are provided in Supplement 1, including (a) virtual imaging with a negative Cardinal Lens, (b) its use in mirror space, (c) placing the system stop at or behind the Cardinal Lens output principal plane in which case ray aiming is required, (d) quantitative evaluation of spherical aberration when the lens is not used at the prescribed magnification, including comparison to analytical results as reported by Walther ( [10], Sec. 31.2), (e) comparisons of the Cardinal and paraxial lenses via 4f coherent imaging of a grating, and (f) a zoom lens example that includes a discussion of how to convert a paraxial lens to a Cardinal Lens. We also include a table summarizing various configurations in which the Cardinal Lens can be used, along with the associated paraxial magnification values.

6. CONCLUSION

This paper presents a ray-tracing model for a perfect thick lens, referred to as the Cardinal Lens, that satisfies the generalized sine condition and is therefore consistent with Fermat’s principle, but its formulation does not require the use of an eikonal function to represent the lens. When simulating a well-corrected lens, this model provides a more accurate alternative compared to use of a simple paraxial thin lens. Two versions of the model are described, one being a distortion-free $f\tan (\theta)$ lens for imaging applications, and the other having $f\sin (\theta)$ distortion as appropriate for a Fourier transform lens. In both cases, the lens is free of aberration only when used at its prescribed (paraxial) magnification. A version of the Cardinal Lens suitable for use in Zemax OpticStudio has been coded as a user-defined surface DLL, and a variety of ray-tracing examples using the DLL have been provided (see Code 1, Ref. [12]). It is hoped that this model will be of some use to the lens design community as a relatively straightforward means to mimic well-corrected compound lenses for which detailed prescription data are not available.