Ray-based optical design tool for freeform optics: coma full-field display

Aaron Bauer; Jannick P. Rolland; Kevin P. Thompson

doi:10.1364/OE.24.000459

1. Introduction

To meet the increasingly demanding performance specifications and package requirements of the next generation of optical systems, optical designers must delve into their toolboxes to find increasingly complex solutions, of which freeform optical surfaces are an important subset. The fabrication of freeform surfaces [1], defined here as surfaces that are rotationally non-symmetric, has evolved to a level of optical quality where fabrication errors no longer prohibit their use in optical imaging systems [2, 3]. As a result, there has been increased interest in the field of freeform optics, including the development of design principles and system evaluation techniques.

The aberration theory for freeform surfaces, specifically surfaces described by Zernike polynomials, was developed as an extension to nodal aberration theory (NAT) [4] by Fuerschbach et al. [5] and serves as the framework for developing freeform design principles [6]. In these and other works, the aberration fields of an optical system are quantified and visualized using full-field displays (FFDs), which are instrumental to the design process [7]. A FFD displays a system metric (e.g. MTF, spot size, distortion) evaluated over a dense grid of points that samples the entire 2-D field-of-view (FOV). Specifically, aberration FFDs display the magnitude and orientation of a specific aberration(s) across the full FOV, allowing the optical designer to determine both the dominant aberration and its field dependence, thereby offering insight into how to proceed with the design. This is especially important for systems with freeform surfaces whose aberrations fields are often multinodal.

Astigmatism and coma (specifically, aperture-squared and aperture-cubed in wavefront, respectively) are the most significant contributors to the aberrations of unobscured freeform system starting designs due to their non-symmetric geometries, and controlling the nodal distribution of these two aberrations is often a top priority. As such, having the ability to isolate, visualize, and quantify these two aberrations within a FFD greatly enhances the designer’s ability to manage them. Currently, the most common technique to generate aberration FFDs is to fit Zernike polynomials to the aberrated wavefront at a pupil plane for each field point. This method generates a FFD for each aberration type, combining the higher and lower order wavefront aberrations, as detailed in [8]. Another option that is available in CODE V^® is the astigmatism FFD, which provides a FFD that plots the magnitude and orientation of the line images that would be produced at the focal plane solely from the astigmatism of the system. This generalized Coddington ray-based calculation [9, 10]offers a distinct advantage in robustness over the Zernike FFDs. Unobscured freeform starting designs are often highly aberrated in their infancy as a result of their non-symmetric geometries. The large aberrations in this stage of the design can produce errors or even failures in the wavefront fitting process used to generate the Zernike FFDs. Ray-based methods, however, avoid these errors, as they only trace rays near the optical axis, where the aberration is low, thus enabling FFDs during these crucial steps. The limitation of the Zernike FFD is illustrated in Fig. 1 for an example freeform starting design. The differential ray-based method, however, currently only exists for astigmatism.

Fig. 1 (left) An unobscured freeform starting design. This early stage system is highly aberrated as a result of the many surface tilts. (middle) The Zernike coma FFD in commercial software fails to plot due to a wavefront fitting error. (right). The ray-based method in this paper successfully plots a FFD for the same system, illustrating the value of the algorithm.

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This research builds and expands upon preliminary work briefly presented in [11] by providing motivation for the work, introducing the comatic aberrations, providing an in-depth discussion of the developed algorithm with rigorous mathematical details, describing the plotting method, and validating the results for a variety of optical systems. The ray-based algorithm is developed and shown in-detail for isolating the aperture-cubed (in wavefront) comatic aberrations – field-linear coma, field-cubed coma, and field-cubed elliptical coma (referred to as just “elliptical coma” or as “trefoil” in fabrication and testing [12]) – for any system within optical design software by tracing only a few rays. Using this method, a ray-based FFD is generated that displays the magnitude, shape, and orientation of the blur spot at the image plane due to only these comatic aberrations. A significant result is the ability to visualize comatic aberrations for highly aberrated unobscured freeform starting designs. Also, examples of the ray-based coma FFD for freeform systems are shown and validated.

2. Method

In this paper, the focus will be on the comatic aberrations that have a cubic aperture dependence in wavefront. Field-linear coma is often the most dominant of the three types and the most important for which to correct. At the image plane, this aberration creates circular blurs whose sizes vary quadratically with aperture and linearly with FOV. Similarly, field-cubed coma creates circular blurs at the image plane whose sizes also vary quadratically with aperture, but cubically with FOV. The last of the three, elliptical coma, is the result of the obliquity of the off-axis field angles at the pupil plane. It creates elliptical blurs at the image plane and when combined with circular coma, deforms the coma circles to ellipses, shown here in Fig. 2. Further details on the geometric nature of coma can be found in [13]. The blurs at the image plane are the result of transverse ray aberrations, which will be exploited to mathematically isolate these three aberrations and thereby generate the ray-based coma FFD using the procedure outlined in the following section.

Fig. 2 (left) The arrows illustrate the effect that elliptical coma has on circular coma in a rotationally symmetric optical system. (middle) A full circular coma blur is shown sampled over the pupil (shown in black) together with the corresponding blur after the addition of elliptical coma (shown in blue). (right) The final blur is the combination of circular and elliptical coma shown alone. (Image adapted from [14])

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2.1 Transverse ray aberrations for an optical system with freeform surfaces

For an imaging system, the transverse ray aberrations are defined as deviations between the real rays and their corresponding paraxial rays measured at the image plane. For a freeform imaging system, the total transverse ray aberration can be written in vector notation as

\begin{array}{l} n^{'} u^{'} \vec{ε} = 2 {[]}_{220 m} \vec{ρ} + {\vec{[]}}_{222}^{2} {\vec{ρ}}^{*} + 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot \vec{ρ}} \vec{ρ} \\ + (\vec{ρ} \cdot \vec{ρ}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}^{2}^{*} + O ({\vec{ρ}}^{3 +}), \end{array}

where n' is the refractive index in image space, u' is the paraxial marginal ray angle in image space,

\vec{ε}

is the total transverse ray aberration vector, and the * denotes a conjugate vector as defined in NAT [4]. The bracket quantities (

{[]}_{220 m}

,

{\vec{[]}}_{222}^{2}

,

{\vec{[]}}_{131}

,

{\vec{[]}}_{331 m}

, and

{\vec{[]}}_{333}^{3}

) are shorthand notations for field-dependent aberration quantities defined in NAT, and are explicitly defined in [14] and included for reference here in Appendix A. Equation (1) is written in increasing orders of

\vec{ρ}

, the normalized radial pupil coordinate vector, and assumes the system is devoid of any defocus and/or tilt aberrations.

The aberrations that have either linear or quadratic aperture dependence (in transverse aberration space) are written out explicitly in Eq. (1) – the transverse ray aberrations for an optical system are proportional to the gradient in $\vec{ρ}$ of the wavefront). The first two terms in Eq. (1) relate to the field curvature and astigmatism of the system (linear aperture dependence), while the remaining terms relate to coma (quadratic aperture dependence). The higher-order aberration terms are grouped together in the final term of Eq. (1). It is the goal of this work to isolate and solve for the comatic vectors, ${\vec{[]}}_{131}, {\vec{[]}}_{331 m}$ , and ${\vec{[]}}_{333}$ , which are the basis of the ray-based coma FFD. Towards this goal,, a key observation is made; for small values of $\vec{ρ}$ , the higher-order terms become negligible. However, because the astigmatism terms, ${[]}_{220 m}$ and ${\vec{[]}}_{222}$ , are linear in aperture, they must be isolated and solved for as a precursor to determining the coma terms.

2.2 Solving for the astigmatism terms

A transverse ray aberration measurement allows the quantification of the total amount of ray deviation from a perfect image point, but the individual contributions from each aberration type (astigmatism, coma, spherical aberration, etc.) are not accessible. However, the linear aperture dependence of the transverse aberration astigmatic terms can be exploited. A near-axis ray-trace can be performed where rays with a pupil magnitude near zero are traced through the system, making the coma and higher-order terms in Eq. (1) become negligible. This near-axis raytrace isolates the linearly dependent aperture terms, leaving (after substituting $\vec{ϵ} = n^{'} u^{'} \vec{ε}$ )

{\vec{ϵ}}_{a} = 2 {[]}_{220 m} {\vec{ρ}}_{a}^{} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{a}^{*},

where

{\vec{ϵ}}_{a}

is the transverse ray aberration due to only astigmatic aberrations, hence the subscript “a”. A unique equation is generated by tracing a ray, specified by a value of

{\vec{ρ}}_{a}

, through the optical system within ray-tracing software, yielding

{\vec{ϵ}}_{a}

. The freedom exists to choose any value for

{\vec{ρ}}_{a}

, provided that

| {\vec{ρ}}_{a} |

is near zero to satisfy the near-axis ray-trace condition. After tracing two separate rays,

{\vec{ρ}}_{a 1}

and

{\vec{ρ}}_{a 2}

, a system of two equations is generated, yielding

{\vec{ϵ}}_{a 1} = 2 {[]}_{220 m} {\vec{ρ}}_{a 1} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{a 1}^{*},

{\vec{ϵ}}_{a 2} = 2 {[]}_{220 m} {\vec{ρ}}_{a 2} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{a 2}^{*},

This system of equations can now be solved for the astigmatism terms,

{[]}_{220 m}

and

{\vec{[]}}_{222}

. More details regarding solving this system of equations can be found in Appendix B. These quantities are used in the next step when following a similar procedure to solve for the coma terms.

2.3 Solving for the coma terms

To solve for the coma terms, which are quadratic in aperture (in transverse aberration space), a slightly modified version of the near-axis ray-trace used to solve for the astigmatism terms in Section 2.2 is employed. In this step, another near-axis ray-trace is performed, but the pupil magnitude of the chosen rays must be such that the higher-order terms remain negligible, while maintaining the comatic contribution to the overall transverse ray aberration of the system. The transverse ray aberration equation can now be written as

\begin{array}{l} {\vec{ϵ}}_{a c} = 2 {[]}_{220 m} {\vec{ρ}}_{c} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{c}^{*} + 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c}} {\vec{ρ}}_{c} \\ + ({\vec{ρ}}_{c} \cdot {\vec{ρ}}_{c}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c} {^{2}}^{*}, \end{array}

where

{\vec{ϵ}}_{a c}

is the transverse ray error due to astigmatism and coma, and

{\vec{ρ}}_{c}

is the normalized pupil vector for the coma calculation, denoted by the subscript “c”.

Subtracting the known astigmatism contribution from ${\vec{ϵ}}_{a c}$ leaves only the coma terms as unknown quantities in Eq. (5). The transverse ray error due only to comatic aberrations, ${\vec{ϵ}}_{c}$ , can then be written as

{\vec{ϵ}}_{c} = 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c}} {\vec{ρ}}_{c} + ({\vec{ρ}}_{c} \cdot {\vec{ρ}}_{c}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c} {^{2}}^{*},

where,

{\vec{ϵ}}_{c} = {\vec{ϵ}}_{a c} - 2 {[]}_{220 m} {\vec{ρ}}_{c} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{c}^{*} .

Given that

{\vec{[]}}_{131}

and

{\vec{[]}}_{331 m}

always show up in tandem as a sum in Eq. (6), the sum is treated as a single quantity, leaving two unknowns,

({\vec{[]}}_{131} + {\vec{[]}}_{331 m})

and

{\vec{[]}}_{333}

. Again, two rays,

{\vec{ρ}}_{c 1}

and

{\vec{ρ}}_{c 2}

, are required to be traced to determine the two unknowns, leaving

{\vec{ϵ}}_{c 1} = 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c 1}} {\vec{ρ}}_{c 1} + ({\vec{ρ}}_{c 1} \cdot {\vec{ρ}}_{c 1}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c 1} {^{2}}^{*},

{\vec{ϵ}}_{c 2} = 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c 2}} {\vec{ρ}}_{c 2} + ({\vec{ρ}}_{c 2} \cdot {\vec{ρ}}_{c 2}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c 2} {^{2}}^{*} .

A clever method in which to solve this vector-based system of equations is to choose the rays,

{\vec{ρ}}_{c 1}

and

{\vec{ρ}}_{c 2}

, such that the complexity of the problem is greatly reduced, namely

{\vec{ρ}}_{c 1} = 0 \hat{x} + ρ \hat{y},

{\vec{ρ}}_{c 2} = ρ \hat{x} + 0 \hat{y} .

The system can now be solved more readily, yielding values for

({\vec{[]}}_{131} + {\vec{[]}}_{331 m})

and

{\vec{[]}}_{333}

. More details regarding solving this system of equations are provided in Appendix C. Using these two quantities, the comatic aberration contribution at the image plane for each point in the FOV can be calculated.

2.4 Assumptions

In Section 2.2, by starting with the full transverse ray aberration equation (Eq. (1)) for a freeform optical system and performing a near-axis ray-trace, the aperture linearity of the astigmatism terms was leveraged. By choosing rays, ${\vec{ρ}}_{a 1}$ and ${\vec{ρ}}_{a 2}$ , such that their magnitudes are sufficiently low, the quadratic and higher order aperture dependent terms become negligible. In practice, choosing $| {\vec{ρ}}_{a 1} | = | {\vec{ρ}}_{a 2} | = 0.001$ , satisfies this requirement, making the astigmatic contribution to the transverse ray aberrations 1000x larger than the next largest contributor. However, the choice of ${\vec{ρ}}_{c 1}$ and ${\vec{ρ}}_{c 2}$ is not as clear.

As stated earlier, it is necessary to choose ${\vec{ρ}}_{c}$ such that $| {\vec{ρ}}_{c} |$ > $| {\vec{ρ}}_{a} |$ and also to maintain the condition that the higher-order aperture terms are negligible. In practice, one could, for example, choose $| {\vec{ρ}}_{c 1} | = | {\vec{ρ}}_{c 2} | = 0.01$ , thus making the quadratic terms a significant contributor to the transverse ray aberrations, however, at this value, the next largest contributor (the cubic terms) is not insignificant. For this reason, the near-axis ray trace does not completely isolate the quadratic terms for optical systems with large aperture-cubed aberrations. To compensate for this fact, a third pair of rays, ${\vec{ρ}}_{c 3}$ and ${\vec{ρ}}_{c 4}$ , with pupil magnitude $| {\vec{ρ}}_{c 3, 4} | = 2 | {\vec{ρ}}_{c 1, 2} |$ is traced through the system. If the aperture-squared aberrations of interest are sufficiently isolated, the resulting transverse ray aberration, ${\vec{ϵ}}_{c 3, 4}$ , would increase by a factor of 4. Any ray error in excess of the factor of 4 is therefore not due to comatic aberrations and is subsequently subtracted from the overall calculation. This subsequent ray trace helps further isolate the quadratic terms.

3. Generating the ray-based coma FFD

The result of Section 2 was to solve for the coma terms, $({\vec{[]}}_{131} + {\vec{[]}}_{331 m})$ and ${\vec{[]}}_{333}$ . To fully visualize this result within a FFD for a freeform system, this process is repeated for an $(m \times n)$ grid of points that fully samples the FOV, thus generating an $(m \times n)$ matrix of coma vectors. This information is then plotted by determining the magnitude, shape, and orientation of the comatic blur at the image plane for each sampled field point for the optical system operating at its full pupil specification ( $| \vec{ρ} | = 1$ ).

3.1 Determination of the magnitude and orientation of the FFD symbols

The symbols plotted in the ray-based coma FFD are shaped to mimic the blur spot that would be seen at the image plane due to the comatic aberrations. The horizontal and vertical axes of a typical FFD are X-FOV and Y-FOV, respectively, and the corresponding symbol for a given point in the FOV, $(H_{x}, H_{y})$ , is plotted. A typical circular coma symbol consists of an “ice-cream cone” type shape formed by cascading circles, where the tip of the cone is the location of the chief ray ( $| \vec{ρ} | = 0$ ) and the topmost circle is formed by $| \vec{ρ} | = 1$ rays. The inclusion of elliptical coma into the FFD squeezes the circular symbols into ellipses, as was shown in Fig. 2. To plot each $(m \times n)$ symbol, one needs the location of the tip of the “cone”, the magnitude of the largest ellipse (major and minor axes lengths), of which a circle is a special case, the location of the center of that ellipse, and the orientation of the ellipse axes.

The location of tip of the “cone” is found by noting that for $\vec{ρ} = 0$ there is no aberration contribution, so, by default, the tip of each cone is located directly at $(H_{x}, H_{y})$ . To find the magnitude of the largest ellipse, Eq. (6) is considered again. However, now that the values of $({\vec{[]}}_{131} + {\vec{[]}}_{331 m})$ and ${\vec{[]}}_{333}$ are known, the equation is instead used to find the transverse ray aberrations at the full pupil ( $| \vec{ρ} | = 1$ ). Multiple rays with $| \vec{ρ} | = 1$ are then traced through the system to determine the major and minor axis lengths of the largest ellipse. The center of this ellipse is found by leveraging the fact that even though elliptical coma squeezes the circles into ellipses, it does not change the location of the center of each individual circle/ellipse. It is known that the diameter of the largest circle for circular coma is equal in magnitude to the distance from the tip of the cone to the center of the largest circle [15]. By considering only circular coma, the diameter of the largest circle can be found in the same manner as was described above for finding the ellipse axes.

The last quantity of interest for plotting the symbols is the orientation of each elliptical symbol. This has been previously worked out in [14]. The angle of each ellipse, $θ_{e l l}$ , is determined by the orientations of the circular coma term, $θ_{31}$ , and the elliptical coma term, $θ_{33}$ , and is shown in [14] to be

θ_{e l l} = \frac{1}{2} (3 θ_{33} - θ_{31}) .

One last feature of each plotted symbol is a pair of lines that connects the tip of the cone to the ellipse. Because only the largest ellipse will be drawn in the plotted symbol, the lines are added to represent the full extent of the blur created by the comatic aberrations for all rays that fill the pupil. These lines are tangent to the ellipse on either side. The location, magnitude, shape, and orientation of the coma symbol are now all fully prescribed and the symbol can now be plotted for each

(m \times n)

field point. To aid those interested in replicating this work for their own use, the full process is shown in the flow chart in Fig. 3.

Fig. 3 The coma FFD process is shown, from using the algorithm to plotting the symbols, in the above flow chart.

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3.2 Examples of the ray-based coma FFD

The algorithm and FFD result will now be demonstrated for several well-behaved optical systems. As a first example, the ray-based coma FFD will be demonstrated for use with a nominally rotationally symmetric Cooke triplet with a misaligned element operating at f/4 with a 20° full FOV. The system layout and associated ray-based coma FFD are shown in Fig. 4. The dominant aberration for the system is the misalignment-induced circular coma caused by a shift in the coma node, as seen in the resulting FFD. The plot axes are in units of normalized FOV.

Fig. 4 (top) A nominally rotationally symmetric f/4 Cooke Triplet with a 20° full FOV. The middle element is decentered in both X and Y by 5 microns. (bottom, left) The effect of the decentering can be seen as the node is moved off-axis. (bottom, right) The Zernike coma FFD shows good correlation between the two types of FFDs. The 180° symbol rotation is due to the transfer from the pupil plane (Zernike FFD) to the image plane (ray-based coma FFD).

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To demonstrate that a ray-based coma FFD accurately portrays the true field behavior of the comatic aberrations, it can be compared to various Zernike FFDs generated in commercial ray-tracing software (an 11x11 grid of points was used to sample the FOV for the examples shown here, and is recommended as the minimum density of points). It is noted that though this research was motivated by the need for a more robust analysis technique for freeform starting designs, Zernike FFDs work sufficiently well for well-behaved optical systems and, as such, are suitable to use for validation purposes. However, because the Zernike FFDs contain all orders of aberration, the ray-based FFDs aberration data is not physically equivalent. For this reason, only qualitative comparisons are done, which are, however, sufficient for verifying the relative magnitudes, orientations, and the location of the nodes, which are key features for FFDs. For systems in which the field-linear and field-cubed coma is dominant (such as this Cooke triplet), the Zernike coma (Z7/8) FFD is a suitable FFD to which the ray-based coma FFD can be fairly compared. A qualitative comparison of the field behavior between the two FFDs shows a good correlation for this well-behaved system, validating the results of the ray-based coma FFD for field-linear and field-cubed coma.

The next example demonstrates that the ray-based coma FFD can be used with an unobscured freeform system, shown in Fig. 5. It is an eyepiece-type design consisting of two freeform reflectors described by Zernike polynomials, which are tilted to make the system unobscured [6]. The system operates at f/4 with a 12 mm eyebox and a 25° full diagonal FOV. The resulting ray-based coma FFD in Fig. 5 shows that the residual comatic aberration is dominated by the elliptical coma term, especially at the edge of the FOV.

Fig. 5 (left) A fully unobscured freeform viewfinder with an external pupil. This design covers a 25° full diagonal FOV with a 12 mm eyebox. (right) The resulting coma FFD shows the impact of the elliptical coma and the non-symmetric nature of the optical system.

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The elliptical coma contribution for the ray-based coma FFD will now be validated for this system. Through 5th-order, the Zernike trefoil (Z10/11) FFD only shows a contribution from elliptical coma, so it serves as a good comparison FFD for the elliptical coma portion of the ray-based coma FFD. For this comparison, only the elliptical coma term will be plotted within the ray-based FFD, with the symbols plotted temporarily as lines to offer a direct comparison to the Zernike trefoil FFD from commercial software, shown in Fig. 6. There is a good qualitative correlation of the field behavior of elliptical coma between the two plots, thus validating the elliptical coma results.

Fig. 6 (left) Ray-based coma FFD showing only the elliptical coma contribution plotted as lines for a direct comparison with the (right) Zernike trefoil FFD, which shows only elliptical coma through 5th-order.

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The final example is another freeform reflective design [16]. In this f/1.9, 8°x6° full FOV system, the elliptical coma blur is much greater than the circular coma blur. So much greater, in fact, that the blur due to the elliptical coma completely overtakes the circular coma, no longer resulting in a cone shaped spot, but rather a single elliptical blur. The system layout and resulting ray-based coma FFD are shown in Fig. 7.

Fig. 7 (left) A fully unobscured freeform telescope from [16]. This f/1.9 system with an 8°x6° full FOV has a larger elliptical coma contribution than it does circular coma. (right) The resulting coma FFD for this system. The elliptical blur completely overtakes the classic “coma” shape.

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The Zernike FFDs, by definition, contain aberrations of all orders in aperture, whereas the ray-based coma FFD was developed for aberrations with a cubic aperture dependence in wavefront. The system in Fig. 7 has a large entrance aperture (30 mm) and is, therefore, subject to significant contributions from these higher-order aberrations, specifically those with a strong aperture dependence. To compare between the ray-based coma FFD and the Zernike FFDs for this system, the aberrations that are higher-order in aperture need to be mitigated so they do not contribute to the Zernike FFDs. This suppression is achieved in this example by generating the Zernike FFDs at a reduced aperture diameter (10 mm), which reduces the magnitude of the unwanted aberrations in the Zernike FFDs. For validation, the circular coma (field-linear and field-cubed) and the elliptical coma contributions of the ray-based coma FFD are split into two separate plots to be compared individually to the appropriate Zernike FFDs. The circular coma contribution of the ray-based coma FFD is compared to the Zernike coma (Z7/8) FFD, shown in Fig. 8 and exhibits a good qualitative correlation. Note that the symbol orientation flip is due to the transfer from the exit pupil, where the Zernike FFDs are computed, to the image plane, where the ray-based FFD is computed. Similarly, the elliptical coma contribution of the ray-based coma FFD (plotted as line symbols) is compared to the Zernike trefoil (Z10/11) FFD, shown in Fig. 9, which also shows a good qualitative match. Based on the validations performed for the three systems, it is concluded that the ray-based coma FFD developed here represents the aperture-cubed comatic aberrations for a freeform optical system.

Fig. 8 (left) Ray-based coma FFD showing only the circular coma contribution. (right) Zernike coma (Z7/8) FFD. At the full aperture, the Zernike coma FFD has a significant contribution from aberrations that are higher-order in aperture, which were made negligible by generating this Zernike FFD at a reduced aperture, allowing a comparison to the ray-based coma FFD.

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Fig. 9 (left) Ray-based coma FFD showing only the elliptical coma contribution plotted as lines for a direct comparison with (right) the Zernike trefoil (Z10/11) FFD. At the full aperture, the Zernike trefoil FFD has a significant contribution from aberrations that are higher-order in aperture, which were made negligible by generating this Zernike FFD at a reduced aperture, allowing a comparison to the ray-based coma FFD.

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4. Conclusion

For an optical designer using freeform surfaces, aberration FFDs are invaluable as a tool to move from a rotationally symmetric design form to a fully unobscured non-symmetric geometry. Especially when a freeform design is in its infancy, it is important to have a robust FFD algorithm such as the Coddington ray-based Astigmatism FFD in CODE V and, now, the ray-based Coma FFD described herein. For the first time, circular coma in tandem with elliptical coma can be isolated and visualized using a ray-based method enabling the use of coma FFDs for freeform starting designs, which can be a limitation of the wavefront based Zernike FFDs. As freeform optical designs become more prevalent, the need for tools such as these grows, representing a push towards making non-symmetric optical design more accessible for future designers.

Appendix A

Listed in Eq. (1) are the transverse aberration terms for a freeform optical system, with no symmetry assumptions. For each aberration term, there is an associated [ ]_klm quantity, which is a shorthand method of writing a more complex quantity containing the associated W_klm term (the wavefront aberration term as defined by H. H. Hopkins [17]), the FOV ( $\vec{H}$ ), and offset vectors ( ${\vec{σ}}_{j}$ ’s), as described in NAT [4, 18]. For reference, the [ ]_klm terms shown in Eq. (1) are included below:

{[]}_{220 m} \equiv W_{220 m} (\vec{H} \cdot \vec{H}) - 2 (\vec{H} \cdot {\vec{A}}_{220 m}) + {\vec{B}}_{220 m}

{\vec{[]}}_{222}^{2} \equiv W_{222} {\vec{H}}^{2} - 2 \vec{H} {\vec{A}}_{222} + {\vec{B}}_{222}^{2},

{\vec{[]}}_{131} \equiv W_{131} \vec{H} - {\vec{A}}_{131},

\begin{array}{l} {\vec{[]}}_{331 m} \equiv W_{331 m} (\vec{H} \cdot \vec{H}) \vec{H} - 2 (\vec{H} \cdot {\vec{A}}_{331 m}) \vec{H} + 2 B_{331 m} \vec{H} \\ - (\vec{H} \cdot \vec{H}) {\vec{A}}_{331 m} + {\vec{B}}_{331 m}^{2} {\vec{H}}^{*} - {\vec{C}}_{331 m} \end{array}

{\vec{[]}}_{333}^{3} \equiv W_{333} {\vec{H}}^{3} - 3 {\vec{H}}^{2} {\vec{A}}_{333} + 3 \vec{H} {\vec{B}}_{333}^{2} - {\vec{C}}_{333}^{3},

{\vec{A}}_{k l m} \equiv \sum_{j} W_{k l m_{j}} {\vec{σ}}_{j},

B_{k l m} \equiv \sum_{j} W_{k l m_{j}} ({\vec{σ}}_{j} \cdot {\vec{σ}}_{j}),

{\vec{B}}_{k l m}^{2} \equiv \sum_{j} W_{k l m_{j}} {\vec{σ}}_{j}^{2},

{\vec{C}}_{k l m} \equiv \sum_{j} W_{k l m}_{_{j}} ({\vec{σ}}_{j} \cdot {\vec{σ}}_{j}) {\vec{σ}}_{j},

{\vec{C}}_{k l m}^{3} \equiv \sum_{j} W_{k l m_{j}} {\vec{σ}}_{j}^{3} .

Appendix B

In Section 2.2, the linear aperture astigmatism terms (in transverse aberration space) were isolated by utilizing a near-axis ray-trace. Two unique rays, ${\vec{ρ}}_{a 1}$ and ${\vec{ρ}}_{a 2}$ , were traced in this fashion, generating a two-variable linear system in Eq. (3) and Eq. (4), which are repeated here,

{\vec{ϵ}}_{a 1} = 2 {[]}_{220 m} {\vec{ρ}}_{a 1} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{a 1}^{*},

{\vec{ϵ}}_{a 2} = 2 {[]}_{220 m} {\vec{ρ}}_{a 2} + {\vec{[]}}_{222}^{2} {\vec{ρ}}_{a 2}^{*} .

The desired astigmatism information is found within the quantities

{[]}_{220 m}

and

{\vec{[]}}_{222}

, so this system of equations must be solved.

The first step to solving this system of equations is to eliminate the conjugate vector, ${\vec{ρ}}^{*}$ , from Eq. (23) and Eq. (24) by using the property, $\vec{ρ} {\vec{ρ}}^{*} = {\vec{ρ}}^{*} \vec{ρ} = {| ρ |}^{2}$ , yielding

{\vec{ϵ}}_{a 1} {\vec{ρ}}_{a 1} = 2 {[]}_{220 m} {\vec{ρ}}_{a 1}^{2} + {\vec{[]}}_{222}^{2} {| {\vec{ρ}}_{a 1} |}^{2},

{\vec{ϵ}}_{a 2} {\vec{ρ}}_{a 2} = 2 {[]}_{220 m} {\vec{ρ}}_{a 2}^{2} + {\vec{[]}}_{222}^{2} {| {\vec{ρ}}_{a 2} |}^{2} .

To simplify this system further, the two rays,

{\vec{ρ}}_{a 1}

and

{\vec{ρ}}_{a 2}

, are cleverly chosen such that

| {\vec{ρ}}_{a 1} | = | {\vec{ρ}}_{a 2} | = | {\vec{ρ}}_{a} |

. By subtracting Eq. (25) from Eq. (26),

{\vec{[]}}_{222}^{2}

is eliminated and

{[]}_{220 m}

remains, giving

{\vec{ϵ}}_{a 2} {\vec{ρ}}_{a 2} - {\vec{ϵ}}_{a 1} {\vec{ρ}}_{a 1} = 2 {[]}_{220 m} ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2}) .

Equation (27) is then dotted on both sides by

({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2})

to give

({\vec{ϵ}}_{a 2} {\vec{ρ}}_{a 2} - {\vec{ϵ}}_{a 1} {\vec{ρ}}_{a 1}) \cdot ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2}) = 2 {[]}_{220 m} ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2}) \cdot ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2}),

which can then be solved by a scalar divide to yield

{[]}_{220 m} = \frac{({\vec{ϵ}}_{a 2} {\vec{ρ}}_{a 2} - {\vec{ϵ}}_{a 1} {\vec{ρ}}_{a 1}) \cdot ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2})}{2 ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2}) \cdot ({\vec{ρ}}_{a 2}^{2} - {\vec{ρ}}_{a 1}^{2})} .

With

{[]}_{220 m}

now known, it can then be substituted back into Eq. (25) which is then solved for

{[]}_{222}^{2}

, giving

{[]}_{222}^{2} = \frac{{\vec{ϵ}}_{a 1} {\vec{ρ}}_{a 1} - 2 {[]}_{220 m} {\vec{ρ}}_{a 1}^{2}}{{| {\vec{ρ}}_{a 1} |}^{2}} .

Appendix C

In Section 2.3, a similar method to the one in Section 2.2 was employed to isolate the quadratic (in transverse aberration space) coma terms in Eq. (1). A system of two equations and two unknowns was generated (Eq. (8) and Eq. (9)) and it will be shown here how to solve that system of equations for the two unknown coma terms. For reference, Eq. (8) and Eq. (9) are repeated here,

{\vec{ϵ}}_{c 1} = 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c 1}} {\vec{ρ}}_{c 1} + ({\vec{ρ}}_{c 1} \cdot {\vec{ρ}}_{c 1}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c 1} {^{2}}^{*},

{\vec{ϵ}}_{c 2} = 2 {({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) \cdot {\vec{ρ}}_{c 2}} {\vec{ρ}}_{c 2} + ({\vec{ρ}}_{c 2} \cdot {\vec{ρ}}_{c 2}) ({\vec{[]}}_{131} + {\vec{[]}}_{331 m}) + \frac{3}{4} {\vec{[]}}_{333}^{3} {\vec{ρ}}_{c 2} {^{2}}^{*} .

There are three unknown variables,

{\vec{[]}}_{131}

,

{\vec{[]}}_{331 m}

, and

{\vec{[]}}_{333}^{3}

. However, because

{\vec{[]}}_{131}

and

{\vec{[]}}_{331 m}

always appear together as a sum, they are treated as a single unknown. To simplify the notation as the system of equations is being solved, the following substitutions are made,

\vec{α} \equiv {\vec{[]}}_{131} + {\vec{[]}}_{331 m},

\vec{β} \equiv {\vec{[]}}_{333}^{3} .

Writing out Eq. (31) and Eq. (32) with these two substitutions gives

{\vec{ϵ}}_{c 1} = 2 (\vec{α} \cdot {\vec{ρ}}_{c 1}) {\vec{ρ}}_{c 1} + ({\vec{ρ}}_{c 1} \cdot {\vec{ρ}}_{c 1}) \vec{α} + \frac{3}{4} \vec{β} {\vec{ρ}}_{c 1} {^{2}}^{*},

{\vec{ϵ}}_{c 2} = 2 (\vec{α} \cdot {\vec{ρ}}_{c 2}) {\vec{ρ}}_{c 2} + ({\vec{ρ}}_{c 2} \cdot {\vec{ρ}}_{c 2}) \vec{α} + \frac{3}{4} \vec{β} {\vec{ρ}}_{c 2} {^{2}}^{*} .

As alluded to in Section 2.3, by cleverly choosing the rays that are traced through the system, the complexity of solving the system of equations is significantly decreased, namely

{\vec{ρ}}_{c 1} = 0 \hat{x} + ρ_{c} \hat{y},

{\vec{ρ}}_{c 2} = ρ_{c} \hat{x} + 0 \hat{y} .

To facilitate the process and simplify the math, Eq. (35) and Eq. (36) are written out in component form, so the benefits from the ray choices of Eq. (37) and Eq. (38) can be more readily seen,

\begin{array}{l} {\vec{ϵ}}_{c 1} = 2 (α_{x} ρ_{c 1 x} + α_{y} ρ_{c 1 y}) (ρ_{c 1 x} \hat{x} + ρ_{c 1 y} \hat{y}) + {| {\vec{ρ}}_{c 1} |}^{2} \vec{α} \\ + \frac{3}{4} {[β_{x} (ρ_{c 1 y}^{2} - ρ_{c 1 x}^{2}) - 2 β_{y} ρ_{c 1 x} ρ_{c 1 y}] \hat{x} + [β_{y} (ρ_{c 1 y}^{2} - ρ_{c 1 x}^{2}) + 2 β_{x} ρ_{c 1 x} ρ_{c 1 y}] \hat{y}} \end{array}

\begin{array}{l} {\vec{ϵ}}_{c 2} = 2 (α_{x} ρ_{c 2 x} + α_{y} ρ_{c 2 y}) (ρ_{c 2 x} \hat{x} + ρ_{c 2 y} \hat{y}) + {| {\vec{ρ}}_{c 2} |}^{2} \vec{α} \\ + \frac{3}{4} {[β_{x} (ρ_{c 2 y}^{2} - ρ_{c 2 x}^{2}) - 2 β_{y} ρ_{c 2 x} ρ_{c 2 y}] \hat{x} + [β_{y} (ρ_{c 2 y}^{2} - ρ_{c 2 x}^{2}) + 2 β_{x} ρ_{c 2 x} ρ_{c 2 y}] \hat{y}} \end{array}

By substituting Eq. (37) and Eq. (38) into Eq. (39) and Eq. (40) respectively, many terms vanish, leaving

{\vec{ϵ}}_{c 1} = 2 α_{y} ρ_{c}^{2} \hat{y} + ρ_{c}^{2} \vec{α} + \frac{3}{4} ρ_{c}^{2} (β_{x} \hat{x} + β_{y} \hat{y}),

{\vec{ϵ}}_{c 2} = 2 α_{x} ρ_{c}^{2} \hat{x} + ρ_{c}^{2} \vec{α} - \frac{3}{4} ρ_{c}^{2} (β_{x} \hat{x} + β_{y} \hat{y}) .

Equation (41) and Eq. (42) are then added together to give

{\vec{ϵ}}_{c 1} + {\vec{ϵ}}_{c 2} = 4 ρ_{c}^{2} \vec{α} .

\vec{β}

is now eliminated from the equation, allowing

\vec{α}

to be solved for, giving

\vec{α} = \frac{{\vec{ϵ}}_{c 1} + {\vec{ϵ}}_{c 2}}{4 ρ_{c}^{2}} .

Finally, now that

\vec{α}

is known, Eq. (41) can be solved for

\vec{β}

, yielding

\vec{β} = \frac{4}{3} (\frac{{\vec{ϵ}}_{c 1}}{ρ_{c}^{2}} - \vec{α} - 2 α_{y} \hat{y}),

thus, fully specifying the quadratic coma of the optical system.

Acknowledgments

This work was supported by the National Science Foundation (EECS-1002179) and has synergized with the Center for Freeform Optics (CeFO) (IIP-1338877 and IIP-1338898). We thank Synopsys Inc. for the student license of CODE V.

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Ray-based optical design tool for freeform optics: coma full-field display

Abstract

1. Introduction

2. Method

2.1 Transverse ray aberrations for an optical system with freeform surfaces

2.2 Solving for the astigmatism terms

2.3 Solving for the coma terms

2.4 Assumptions

3. Generating the ray-based coma FFD

3.1 Determination of the magnitude and orientation of the FFD symbols

3.2 Examples of the ray-based coma FFD

4. Conclusion

Appendix A

Appendix B

Appendix C

Acknowledgments

References and links

Cited By

Figures (9)

Equations (45)

Optics Express