3.1.

Solution Generation

The initial step in the Monte Carlo search process is to generate first-order solutions. First, random values are chosen for the group focal lengths, TTL, and back focal length (BFL). The sign of power for the group focal lengths is also chosen at random, except in the case of type II configurations where the compensator focal lengths are constrained by the anamorphic ratio. The magnitude of these randomly selected values is based on boundary conditions provided for allowable group EFL, TTL, and BFL values.

VFO solutions that continue in the Monte Carlo search are those that meet three criteria: (1) they produce a real image, (2) there are no internal images, and (3) there are no zoom group crashes. These requirements are to ensure a properly oriented and accessible image plane and realizable zoom motions. Given the randomly selected group EFLs, TTL, and BFL, a solution’s group zoom motions can be calculated using paraxial imaging equations. This is done differently depending on whether a type I or type II configuration is being examined, but both types rely on a four-group zoom motion being applied independently in the $x-z$ and $y-z$ planes. These two four-group zooms are connected by having system focal lengths related by the anamorphic ratio and having the same stationary front and rear groups.

For type I configurations, a standard four-group zoom layout (see Fig. 5) is applied separately in the $x-z$ and $y-z$ planes. The infinite conjugate four-group zoom motion can be derived by repurposing a finite conjugate two-group zoom¹⁷ as the two internal moving groups over an internal distance $L$. The full four-group zoom is then obtained by adding the stationary front and rear groups to the two-group finite zoom to achieve infinite conjugate imaging, as done by Yee et al.¹⁹ Given a system EFL zoom range, TTL, BFL, and group focal lengths $f_1$, $f_2$, $f_3$, and $f_4$, the zoom motions $t_1$, $t_2$, and $t_3$ of a four-group zoom system are calculated as

Fig. 5

Standard four-group zoom layout to be applied to type I anamorphic zoom configurations. To obtain a type I configuration, two four-group zooms are obtained in the $x-z$ and $y-z$ planes with different system focal lengths related by the anamorphic ratio and with the same stationary front and rear groups. Real internal images are shown for first-order illustration but are not permitted in the type I configuration.

Focusing on Eq. (7), a valid zoom solution is only possible if the radical is real valued, meaning $L^2\ge 4c$ necessarily for VFO solutions. Also in Eq. (7), the plus-or-minus term means there are potentially two solutions for each configuration for both the positive and negative roots, although it is rarely the case that both solutions have valid zoom motions.

For a type I anamorphic zoom, a first-order design is created using two separate four-group solutions obtained using Eqs. (2)–(8). One four-group system uses the randomly selected values for group EFLs $f_1$, $f_{2,x}$, $f_{3,x}$, and $f_4$ while the system in the orthogonal plane of symmetry uses $f_1$, $f_{2,y}$, $f_{3,y}$, and $f_4$. These two four-group systems are related in several ways. First, the EFL zoom range of these systems is related by the anamorphic ratio, Eq. (1). Second, both four-group systems apply the same stationary front and rear group focal lengths and positions to model the rotationally symmetric focus and relay groups in a type I configuration. Third, applying the same BFL in both planes of symmetry results in a stigmatic image on-axis, as discussed in Sec. 2. With these considerations in mind, these two four-group designs can be joined in the orthogonal planes of symmetry to form a type I first-order solution. The zoom group motions $t_{1,x}$, $t_{1,y}$, $t_{2,x}$, $t_{2,y}$, $t_{3,x}$, and $t_{3,y}$ can now be analyzed to see if there are any group crashes. Examples of a type I zoom motion for both VFO and crashing solutions can be seen in Fig. 6.

Fig. 6

Example (a) valid and (b) crashing zoom motions of randomly generated type I first-order solutions in the PNPP-XYYX space. Group position in $z$ (horizontal axis) is shown as a function of system EFL through zoom in $x$ and $y$ (vertical axes) with an anamorphic ratio of 2. The position $z=0\mathrm{mm}$ corresponds to the image plane. Stationary groups are shown in blue and moving groups in green. Rotationally symmetric groups are shown with solid lines while anamorphic groups in $X$ are dotted lines and anamorphic groups in $Y$ are dashed. Zoom crashes are circled in red.

The type II anamorphic zoom configuration relies on a modified four-group zoom (see Fig. 7) applied separately in the $x-z$ and $y-z$ planes. The modifications of the four-group zoom are to meet the requirements on the shared spherical variator discussed in Sec. 2, namely, the marginal ray bundle is collimated into the relay group and the ratio of the compensators in $x$ and $y$ must equal the anamorphic ratio. Given a system EFL zoom range, TTL, BFL, and group focal lengths $f_1$, $f_2$, $f_3$, and $f_4$, the zoom motions $t_1$, $t_2$, and $t_3$ for this modified four-group system can similarly be obtained from paraxial imaging equations:

Fig. 7

Modified four-group zoom layout to be applied to type II anamorphic zoom configurations. To achieve a type II configuration, two modified four-group zooms are obtained in the $x-z$ and $y-z$ planes with different system focal lengths related by the anamorphic ratio and with the same stationary front and rear groups. Real internal images are shown for first-order illustration but are not permitted in the type II configuration.

Unlike with the standard four-group zoom for type I configurations, there is only one possible zoom motion solution for the modified four-group zoom. There is also no constraint due to the realness of a radical.

For a type II anamorphic zoom, a first-order design is created using two separate modified four-group solutions obtained using Eqs. (9)–(13). One modified four-group system uses the randomly selected values for group EFLs $f_1$, $f_2$, $f_{3,x}$, and $f_4$ while the system in the orthogonal plane of symmetry uses $f_1$, $f_2$, $f_{3,y}$, and $f_4$. As for the type I configuration, the system EFL zoom range of these two four-group systems are related by the anamorphic ratio. Again, both four-group systems apply the same stationary front and rear group focal lengths and positions, including BFL to ensure stigmatic imaging on-axis. Finally, these two modified four-group designs can be joined in orthogonal planes of symmetry to form a type II first-order solution. The zoom group motions $t_1$, $t_{2,x}$, $t_{2,y}$, $t_{3,x}$, and $t_{3,y}$ can now be analyzed to see if there are any group crashes. A different aspect of type II configurations is that the compensators in $x$ and $y$ have the same zoom motion with a constant offset equal to their difference in group focal length. This is a consequence of the added constraints on applying a shared variator. As a result, in assembly the two groups can conveniently be mounted together with the same zoom motion mechanics. As will be seen, however, these added constraints for using a shared variator significantly reduce the design space. Examples of a type II zoom motion for both VFO and crashing solutions can be seen in Fig. 8.

Fig. 8

Example (a) valid and (b) crashing zoom motions of randomly generated type II first-order solutions in the NPNP-XY space. Group position in $z$ (horizontal axis) is shown as a function of system EFL through zoom in $x$ and $y$ (vertical axes) with an anamorphic ratio of 2. The position $z=0\mathrm{mm}$ corresponds to the image plane. Stationary groups are shown in blue and moving groups in green. Rotationally symmetric groups are shown with solid lines while anamorphic groups in $X$ are dotted lines and anamorphic groups in $Y$ are dashed. Zoom crash is circled in red.

3.2.

Ray Tracing

After obtaining VFO solutions defined by their group EFLs, TTL, BFL, and zoom motions, the next step in the Monte Carlo search process is to identify successfully RT solutions. A central assumption of the Monte Carlo process is that, for each solution space, the number of found VFO-RT solutions correlates directly with the “size” of the solution space. The size of a solution space is important because it allows for greater flexibility in optimization when attempting to obtain a thick lens, color corrected design. As will be seen, although important, the size of a solution space is not the only factor in determining the best first-order starting point. For example, although a solution space may be very large, this does not necessarily mean that it has better imaging performance than a smaller one.

Ray tracing is performed using the optical design software CODE V^® via an automated process where a VFO zoom solution is modeled using thin lenses. For consistency across designs, each group in the model is composed of three thin lenses in contact with each thin lens contributing one-third of the total group power. By splitting group power, multiple thin lenses in a group improve aberration correction and offer additional variables in optimization. Three thin lenses per group were found to be adequate for analyzing and filtering starting points, although this should be adjusted as necessary for designing beyond first-order, as in Sec. 4.4. In addition, the model is evaluated with real glasses, Schott N-BK7 for positive elements, and Schott N-SF4 for negative elements, although the first-order system is evaluated monochromatically.

For all zoom positions, with the system operating at the design aperture and field-of-view, rays are traced across the entire pupil and field while checking for ray trace failures. Ray trace failures occur for predominantly two reasons. First, severe aberrations may result in rays failing to intersect a surface. These highly aberrated rays may also result in reflection or total internal reflection at a surface. Second, severe pupil aberrations may prevent a ray from successfully being traced between the centers of the pupils and the aperture stop. Pupil aberrations can be mitigated by changing the stop position, but this is not possible under the current requirement of having the stop positioned after all moving groups. If ray trace failure occurs, design optimization and evaluation cannot take place without adjustment, making these VFO solutions very unlikely to be useful starting points. As a result, VFO solutions that fail to ray trace are discarded. On the other hand, VFO solutions that successfully ray trace for all zoom positions, pupil coordinates, and fields-of-view are saved and classified as RT. These VFO-RT solutions continue on to the final step in the search process.

3.3.

Optimization and Evaluation

The identified VFO-RT solutions now reach the third and final step in the Monte Carlo process: optimization and evaluation. Each thin lens design is optimized to minimize geometrical spot size across the field-of-view and across five zoom positions. The only optimization variables are thin lens bending²⁶ and image refocusing while the group EFLs, TTL, BFL, and zoom motions are all held constant to remain consistent with the original VFO-RT solution. These limited variables were found to possess adequate degrees of freedom to analyze and filter solutions. When attempting a final, thick lens design (see Sec. 4.4), additional variables can be added for further correction such as the use of different glasses, doublets, or aspheric surfaces.

After optimization, each design is evaluated for geometrical spot size across all fields-of-view and zoom positions, third-order aberration content, and element clear aperture dimensions. These performance metrics provide ample information about both the imaging performance and packaging size of a first-order design and, combined with first-order values such as average group EFL, TTL, and BFL, offer a thorough picture of the VFO-RT solution. By evaluating a large population of solutions, the set of performance metrics can be applied, by extension, to a solution space as a whole.

4.1.

System Specifications

The outlined Monte Carlo search process was applied to all possible solution spaces for both type I and type II anamorphic zoom configurations. As discussed by Dodoc,^15,18 some solution spaces are mathematically incapable of meeting the requirements for solution generation laid out in Sec. 3.1. For example, a solution type with only negatively powered groups is not capable of forming a real image, one of the requirements of a VFO solution. Nevertheless, all solution spaces were considered in the search to demonstrate this fact with the only consequence being lower solution yield per Monte Carlo trial.

The system specifications for all designs in the search can be seen in Table 2. The system specifications are based on those of standard anamorphic cinema zoom lenses currently available on the market. For example, an anamorphic ratio of 2 is common in the cinema industry and produces desirable elliptical bokeh and differential depth-of-field.^11,12

Table 2

System specifications used for the Monte Carlo search. The zoom ratio is the ratio of the longest zoom focal length to the shortest. The projected image aspect ratio of 2.39 is the cinematic anamorphic widescreen format.

Furthermore, the Monte Carlo search boundary values are listed in Table 3. These chosen boundary values are based on several factors. First, the TTL and BFL packaging boundaries were based on the established system specifications. Some allowance was issued for the packaging boundaries due to the later transition from thin to thick lenses. For the group EFL boundary, a large range is required to find diverse solution spaces. Ideally, group EFL values will be large in magnitude to reduce thin lens aberrations, as defined by the G-sums.²⁶ Similarly, the minimum group focal length boundary is set to exclude highly aberrated, short focal length groups that are unlikely to lead to RT solutions. An additional consideration is how large of a range should be used for each boundary. Larger boundary ranges entail a more extensive design space will be explored; however, if a range is too large, many Monte Carlo trials will be spent looking at unpromising first-order designs.

Table 3

Monte Carlo search boundaries for randomly generated values. The group EFL boundary is unsigned since the power sign for each group EFL is dependent on a trial’s randomly assigned solution space.

4.2.

Type I Configuration

The type I anamorphic zoom configuration was explored with the Monte Carlo search process using 1 billion trials to examine the 384 possible solution spaces. The total computation time was $\sim 41\mathrm{h}$ using a Dell XPS Desktop (8-core i7-9700 @ 3.0 GHz, 32 GB RAM). The results of the search can be seen in Table 4. It was found that 98.2% of RT solutions had the same design form in the $x-z$ and $y-z$ planes of symmetry. This is considering that prior to ray tracing, the number of VFO solutions was approximately even with 56.6% having the same $x$ and $y$ solution types. For example, a PNNPPN-XYXY solution has the same design form, PNPN, in both $x$ and $y$ since the variators and compensators have the same sign power in the orthogonal planes of symmetry.

Table 4

Monte Carlo search results of the type I configuration based on whether the obtained solutions have the same design form in x and y. For this Monte Carlo search, 109 trials were performed.

Due to the highly skewed findings in favor of solutions with the same design form in the orthogonal planes of symmetry, a revised Monte Carlo search was performed using 1 billion trials and a narrowed scope to exclusively examine solution spaces with the same design form in $x$ and $y$. Based on the overarching assumption that the number of VFO-RT solutions correlates with the size and overall potential of a solution space, the few outlier cases of VFO-RT solutions that had different design forms in $x$ and $y$ were neglected. The reduction in scope from six independent group power signs to four means there is a $4\times$ reduction in the number of solution spaces, going from 64 to 16 power combinations. The number of cylinder orientation orderings remains unchanged at six, making a total of 96 solutions spaces possible for the type I configuration with the same design form in $x$ and $y$. In addition to increasing solution quality, this narrowing of scope also increased the rate of found RT solutions by $4\times$.

Considering only solution spaces with the same design form in $x$ and $y$, the revised Monte Carlo search for type I configurations found 41,898 VFO solutions, and of those, 1942 were found to ray trace successfully. This means, of the 1 billion total trials, only 0.002% resulted in RT solutions, demonstrating just how demanding the design space is for combined anamorphic zooms. For comparison, using a similar method for a standard rotationally symmetric four-group zoom, Bruggeman found²⁰ $\sim 0.75\%$ of Monte Carlo trials yielded a RT solution.

The distribution of solution spaces for the type I search presents many relevant findings. The progression of the search process can be seen in Fig. 9, and the breakdown of the quantitative results by solution type can be seen in Fig. 10. Interestingly, there was very little correlation between the power combinations that produced the most VFO solutions and the combinations that produced the most RT solutions. The top three VFO solution spaces are PNPN, NNPN, and NNPP while the top three RT spaces are NPPP, PNPN, and PNPP. Moreover, originally thought to be a weak variable, the order of the cylindrical groups in $x$ and $y$ has a significant impact on what solutions are found. For a certain power ordering, VFO and RT solutions are highly dependent on the cylinder ordering, as opposed to being equally likely for all cylinder orderings. For example, the most common RT solution space NPPP only has RT solutions with cylinder orderings YYXX and YXYX. These findings indicate that the validity of a first-order starting point is equally dependent on the cylinder ordering for a given power ordering as it is on the power ordering itself.

Fig. 9

Monte Carlo search process for the type I anamorphic zoom configuration. Only solutions having the same sign power in $x$ and $y$ were considered for this revised search. VFO solutions are shown in blue, and RT solutions are shown in green. The widths of bands for invalid solutions are not to scale. The quantitative results for found solution types can be found in Fig. 10.

Fig. 10

Monte Carlo search results for the type I anamorphic zoom configuration. Only solutions having the same sign power in $x$ and $y$ were considered for this revised search. VFO solutions (a) and RT solutions (b) are presented. Solution types are classified by power order (horizontal axis) and orientation ordering (hatch).

The obtained VFO-RT solutions were optimized as described in Sec. 3.3 and evaluated for a variety of image performance and packaging metrics. The geometrical spot size was calculated and averaged across all field-of-view and zoom positions. The distribution of average spot size by solution space can be seen in Fig. 11. The solution space with the smallest median spot size was NNPP while the space with the smallest absolute spot size was PNPP. The packaging quantities for the designs, namely TTL, BFL, and element clear aperture, were also evaluated on a solution space basis, as shown in Fig. 12. All RT solution spaces had relatively similar TTL and average element clear aperture distributions. On the other hand, BFL distributions varied significantly between spaces.

Fig. 11

Average spot size distribution for RT solution spaces of type I configuration. The spot size is averaged over all field-of-view and zoom positions. White points indicate the median while black boxes represents the 25th to 75th percentile range. Solution spaces are sorted by median average spot size.

Fig. 12

Packaging distributions for RT solution spaces of type I configuration. Included in packaging is TTL, BFL, and average element clear aperture. TTL is measured from front surface vertex to image. White points indicate the median while black boxes represents the 25th to 75th percentile range. Solution spaces are sorted by median values.

Another interesting result of the Monte Carlo search is how VFO and RT solutions were distributed relative to the search boundary conditions for group EFL, TTL, and BFL (see Fig. 13). A histogram of group EFL presented a peak for VFO solutions that shifted to longer focal lengths for RT solutions, as you would expect with longer group focal lengths introducing less aberration. For both VFO and RT solutions, there was a trend of more solutions for longer systems, although this trend was less pronounced for RT solutions. Lastly, VFO solutions presented a trend of more solutions for shorter BFL values while RT solutions had the inverse trend of more solutions for longer BFLs.

Fig. 13

Distribution of VFO and RT type I solutions according to the search boundary values (see Table 3). The Monte Carlo boundary values for group EFL, TTL, and BFL are shown in gray. The peak of group EFL values shifts larger for RT solutions. For BFL, the trend switches from favoring short values for VFO solutions to long values for RT solutions.

In summary, the type I configuration offers far more promising results when looking at cases where the design form is the same in both the $x-z$ and $y-z$ planes. Of this set, there were six promising RT solution spaces. Although all six are worth evaluating for a final design, the most promising solution spaces based on size (number of found solutions), imaging performance, and packaging size are PNPP, NPPP, and NNPP. For each of these solution spaces, there are specific cylinder orientation orderings that are required in order to be successful.

4.3.

Type II Configuration

The type II anamorphic zoom configuration offers a much more confined design space than the type I configuration. As discussed in Sec. 3.1, due to the shared spherical variator, there are several additional constraints on the group EFLs and zoom motions of the type II configuration. A consequence of these additional constraints is that there are also $24\times$ fewer solution spaces possible for type II configurations than type I (see Table 1).

The Monte Carlo search was performed for the type II configuration using 100 million trials. From the search, 64,060 valid-first order (VFO) solutions were found; however, solutions in only two spaces were found, 42,537 for NPNP and 21,523 for PNPP. Unfortunately, of these VFO solutions, zero designs ray-traced successfully (see Fig. 14).

Fig. 14

Monte Carlo search process for the type II anamorphic zoom configuration. Band widths for invalid solutions are not to scale.

The limited number of VFO solution spaces can be attributed to two factors. First, the constraints due to the shared spherical variator limit the number of valid solution spaces. As noted by Dodoc,¹⁵ of the 16 possible power and cylinder orientation combinations, there are only five solutions spaces that can produce valid zoom motions while satisfying the mathematical framework outlined in Eqs. (9)–(13) for the type II configuration. Second, only two of the five VFO solution spaces were found in the Monte Carlo process due to the search boundary values (Table 3). For the three valid but not found solution spaces, NNPP designs required BFL $\gtrsim 300\mathrm{mm}$, NPPP designs required BFL $\gtrsim 200\mathrm{mm}$ and $|f_1|\lesssim 20\mathrm{mm}$, and PPNP designs required BFL $\lesssim 20\mathrm{mm}$. All of these requirements are beyond the search boundary values as well as the system specifications.

Of the two identified VFO solution spaces, NPNP and PNPP, neither had a single solution that successfully ray traced across all zooms, pupil coordinates, and fields-of-view. This inability to successfully ray trace for the given system specifications is due to the reasons laid out in Sec. 3.2. Finding a VFO solution that satisfies both the field and pupil conjugates with minimal aberration is a challenging problem and, due to the extra constraints on the type II configuration, was not successful for the given system specifications.

Without any RT solution spaces, the type II configuration is not a viable option to consider for a first-order starting point given the system specifications. The type I solution spaces identified in Sec. 4.2 will be used exclusively in obtaining a thick lens, color-corrected final design.

4.4.

Final Design

A final anamorphic design was obtained using a type I configuration in the PNPP-XYYX solution space. Although there are several promising type I solution spaces, the chosen thin lens starting point was a PNPP-XYYX design due to the space’s large number of VFO-RT solutions and its good imaging performance (see Figs. 10 and 11). An example PNPP-XYYX first-order design and zoom motion can be seen in Fig. 6(a).

A final design was realized by thickening the thin lenses and performing color correction with different glass types. Starting with three thin lenses per group, elements were removed, split, or compounded into doublets, as necessary. The final design employs 15 elements including three doublets, and each moving group consists of two singlets (see Fig. 15). No aspheric surfaces were used due to their undesirable effect on the appearance of bokeh.²⁷ The design satisfies the system specifications in Table 2, including a TTL of 350 mm and a BFL of 35 mm. The final design was optimized to meet a set of MTF resolution specifications as well. The quality of this design is in large part due to the first-order Monte Carlo search for classifying different solution spaces. This design is one of several obtained and is the first thick lens combined anamorphic zoom presented since the introduction of the first-order configuration by Dodoc.¹⁵

Fig. 15

Final thick lens, polychromatic design using a type I anamorphic zoom configuration in the PNPP-XYYX solution space. The design is shown in the $x-z$ and $y-z$ planes at the short, middle, and longest EFL zoom positions. The design satisfies the system specifications in Table 2 including a TTL of 350 mm and a BFL of 35 mm.