Approaching direct optimization of
`as-built lens performance
`
`McGuire, James, Kuper, Thomas
`
`James P. McGuire Jr., Thomas G. Kuper, "Approaching direct optimization of
`as-built lens performance," Proc. SPIE 8487, Novel Optical Systems Design
`and Optimization XV, 84870D (19 October 2012); doi: 10.1117/12.930568
`Event: SPIE Optical Engineering + Applications, 2012, San Diego, California,
`United States
`
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`PROCEEDINGS OF SPIE
`
`SPIEDigitalLibrary.org/conference-proceedings-of-spie
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
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`Novel Optical Systems Design and Optimization XV, edited by G. Groot Gregory, Arthur J. Davis,
`Proc. of SPIE Vol. 8487, 84870D · © 2012 SPIE CCC code: 0277-786X/12/$18 · doi: 10.1117/12.930568
`
`Proc. of SPIE Vol. 8487 84870D-1
`
`
`
`Approaching direct optimization of as-built lens performance
`
`James P. McGuire, Jr. and Thomas G. Kuper
`
`Synopsys Inc., 3280 E. Foothill Blvd., Pasadena, CA, USA 91017
`
`ABSTRACT
`
`We describe a method approaching direct optimization of the rms wavefront error of a lens including tolerances. By
`including the effect of tolerances in the error function, the designer can choose to improve the as-built performance with
`a fixed set of tolerances and/or reduce the cost of production lenses with looser tolerances. The method relies on the
`speed of differential tolerance analysis and has recently become practical due to the combination of continuing increases
`in computer hardware speed and multiple core processing We illustrate the method’s use on a Cooke triplet, a double
`Gauss, and two plastic mobile phone camera lenses.
`
`Keywords: Optimization, tolerance analysis, objectives
`
`
`1. INTRODUCTION
`
`Typically, a lens designer will generate a number of potential optical design forms (possibly using global search methods
`such as those introduced in the late 80’s and the early 90’s1-11) and then select the “best” lens form based on the error
`function value and whether the lens looks to be easily manufactured. The designer will then proceed to assign tolerances
`for manufacturing. This process relies on the skill of the designer to achieve the best results.
`
`To assist the designer in desensitizing a lens to manufacturing errors, a number of additions to the error function have
`been proposed. Tiziani and Gray proposed desensitizing the system to axial coma by including a differential variance in
`the wavefront produced by an angular tilt or decenter of the surface 12-13. This method has been implemented in a
`commercial lens design package for many years14. Several authors have discussed cost-effective manufacturing in terms
`of a local optimum15-17. More recently, Jeffs has described a desensitization method based on reducing the angles of
`incidence at the lens surfaces and minimizing the optical powers of the individual optical elements18. Jeff’s method is
`very fast and we have found that a particularly useful variant is to minimize the difference between the angle of the
`incidence and the sine of the angle of incidence. Aberrations arise from real elements not being paraxial (angles not
`being the same as the sines of the angle) and this metric is often better at capturing the aberration contribution. All of
`these methods can be effective tools, but they do not correlate directly to the as-built wavefront error.
`
`Catalan used an analytic, not numeric, approach19. He derived the sensitivity of a Ritchey Chretien telescope to errors in
`the tilt, decenter, and despace of the secondary mirror based on the construction parameters and then analytically
`optimized the design. This technique provides great insight, but also requires a lengthy derivation for every new system
`that needs to be desensitized.
`
`More recently, a few suggestions have been made to desensitize the model by creating a zoom model with various values
`of tolerances. Fuse used two configurations for every tolerance: one for a perturbation in the positive direction, and one
`for a perturbation in the negative direction20. Fuse’s method requires a very large number of configurations (upwards of
`6 per surface). Rogers used a smaller number of zoom positions and randomly perturbed the parameters for each of these
`surfaces21. The number of configurations is independent of the number of surfaces and tolerances. Each zoom position
`is essentially a Monte Carlo realization of an as-built system. With a sufficient number of Monte Carlo positions, the
`optimizer can work to find tolerance insensitive forms. The disadvantage of the methods of Fuse and Rogers is slower
`optimization due to the increase in the number of configurations and often to the increase in the number of fields. A
`
`
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`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 2
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`Proc. of SPIE Vol. 8487 84870D-2
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`
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`rotationally symmetric system analyzed with three fields must be analyzed over 5 y-fields, if only tolerances that affected
`y-fields are used and must be analyzed over a grid of 5x5 fields, if tolerances that affected both x- and y-fields are used.
`
`In an earlier method, the author used global optimization to generate many local minimum and then sorted based on their
`predicted as-built performance using CODE V’s fast differential tolerance analysis (the TOR option) 22. This technique
`allows the designer to sort through hundreds of local minima that can be easily generated in a modern lens design
`program based on the as-built rms wavefront or MTF performance. However, it does not allow one to directly optimize
`on the as-built performance.
`
`In 2010, two authors independently proposed computing the changes in the wavefront for selected tolerances types using
`Rimmer’s differential ray trace techniques. Bates used the Tziani and Gray method to include contributions for off-axis
`aberrations due to surface displacements by using selected rays (5 rays for the mobile phone camera that he optimized)23.
`The minimal ray set to compute only one type of manufacturing error made the evaluation very fast, but the
`implementation works for only one type of manufacturing error and the user must carefully select the right ray set.
`Yabe’s technique was more general, adding the square root of the increase of the variance of the wavefront aberration for
`both decenter and curvature errors24.
`
`This paper describes the direct inclusion of the differential tolerance analysis into the error function and thus the direct
`optimization of the as-built wavefront (mean + 2sigma) for all the tolerances types supported by CODE V (decenter,
`curvature, index, thickness, aspheric errors, wedge, tilt, etc.) including the effects of compensators in the alignment
`procedure. Section 2 briefly outlines the inclusion of the differential tolerance sensitivities into the error function.
`Section 3 discusses the optimization of the as-built performance of example photographic objectives: a simple Cooke
`triplet, a double Gauss lens, and two mobile phone camera lenses.
`
`
`2. DIFFERENTIAL TOLERANCE ANALYSIS
`
`Damped Least Square (DLS) optimizers typically used in lens design software work best when the error function
`comprises many contributions that are affected approximately linearly by the variables. For example, in the case of spot
`size error functions, the transverse errors for every ray are entered into the error function, not the single number
`describing the RMS of the errors (which carries less information than the set of individual ray errors). Thus, we will
`build an error function that is composed of the contributions from each individual tolerance on the wavefront error
`evaluated using differential tolerance analysis to minimize the computational burden.
`
`The differential tolerance analysis is based on real ray tracing and predicts the effect of the various tolerances on RMS
`wavefront error. It is based on a wavefront differential ray trace12,25 that provides the derivative of the OPD with respect
`to the tolerances. This provides an extremely efficient way to calculate the changes in RMS wavefront error26-27 used in
`the statistical calculations28.
`
`In differential tolerance analysis, grids of rays are traced through the lens as needed. For each ray, the wave aberration
`derivatives are calculated for each perturbation and appropriately summed. The final result is a set of coefficients
`defining a function that describes the expansion of the variance of the wavefront with respect to each of the parameters
`of interest (perturbations).
`
`
`
`.
`
`
`
`(1)
`
`Δvariance(cid:3404)(cid:3533) (cid:3435)A(cid:2919)T(cid:2919)(cid:2870)(cid:3397)B(cid:2919)T(cid:2919)(cid:3397)∑
`(cid:3439)
`C(cid:2919)(cid:2920)T(cid:2919)T(cid:2920)
`(cid:3015)(cid:3036)(cid:2880)(cid:2869)
`N(cid:3037)(cid:2880)(cid:2869)
`where A(cid:2919) ,B(cid:2919) and C(cid:2919)(cid:2920) are expansion coefficients, and T(cid:2919), and T(cid:2920) are the tolerance values, and N is the number of tolerances.
`C(cid:2919)(cid:2920) is a strictly upper triangular matrix. This is a simple second order Taylor series expansion. CODE V provides the A(cid:2919) ,
`B(cid:2919) , and C(cid:2919)(cid:2920) coefficients through the AS_BUILT_ABC macro function (the A(cid:2919) and B(cid:2919) coefficients are also listed in the
`
`
`
`TOR option output).
`
`
`
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`IPR2020-00896
`Exhibit 2006
`Page 3
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`Proc. of SPIE Vol. 8487 84870D-3
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`
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`Once the differential expansion of the wavefront variance is computed, we need a way to include this sensitivity
`information in the error function. As a starting point, consider the square of the wavefront squared plus the mean plus 2σ
`value of the wavefront variance as an as-built error function
`
`
`
`
`(cid:1827)(cid:1828)(cid:3404)(cid:4666)(cid:1849)(cid:2868)(cid:2870)(cid:3397)μ(cid:3024)(cid:3397)2(cid:2026)(cid:3024)(cid:4667)(cid:2870)
`where W(cid:2868) is the nominal wavefront error, μ(cid:3024) is the mean of the change in the wavefront variance and (cid:2026)(cid:3024) is the standard
`
`(2)
`
`
`
`deviation of the change in the wavefront variance. If the system probability distribution is Gaussian (often a good
`approximation due to the central limit theorem), 97.7% of the as-built lenses will have an RMS value that is better than
`the mean plus 2σ.
`
`Equation (2) can be evaluated using the expression derived by Koch for the mean and standard deviation for
`symmetrical, zero mean probability densities
`
`
`
`
`
`
`
`
`
`(3)
`
`(4)
`
`
`
`
`
`
`
`
`
`
`
`
`μ(cid:3024)(cid:3404)(cid:3533)A(cid:2919) (cid:2026)(cid:2919)(cid:2870)
`(cid:3015)
`(cid:3036)(cid:2880)(cid:2869)
`(cid:3015)
`(cid:4679)
`(cid:2026)(cid:3024)(cid:2870)(cid:3404)(cid:3534)(cid:4678)(cid:1827)(cid:3036)(cid:2870) (cid:4666)(cid:2029)(cid:2872)(cid:2919)(cid:3398)(cid:2026)(cid:2919)(cid:2872)(cid:4667)(cid:3397)(cid:1828)(cid:3036)(cid:2870)(cid:2026)(cid:2919)(cid:2870)(cid:3397)(cid:2026)(cid:2919)(cid:2870)(cid:3533) (cid:1829)(cid:3036)(cid:3037)(cid:2870) (cid:2026)(cid:2920)(cid:2870)
`N(cid:3037)(cid:2880)(cid:2869)
`(cid:3036)(cid:2880)(cid:2869)
`where (cid:2029)(cid:2872)(cid:2919) is the fourth moment of the tolerance probability distribution and (cid:2026)(cid:2919)(cid:2870) is the variance of the tolerance probability
`distribution28 ( A(cid:2919) , B(cid:2919) , and C(cid:2919)(cid:2920) are the differential expansion coefficients in (1)). The resulting expression is not easily
`(cid:1845)(cid:1827)(cid:1828)(cid:3404)(cid:3533)Aberration(cid:2919)(cid:2870)
`(cid:3015)
`(cid:3036)(cid:2880)(cid:2869)
`Aberration(cid:2919)(cid:2870)(cid:3404)(cid:1849)(cid:2868)(cid:2872)(cid:3397)4(cid:2026)(cid:3024)(cid:1849)(cid:2868)(cid:2870)
`(cid:3397)2(cid:1849)(cid:2868)(cid:2870)A(cid:2919)(cid:2026)(cid:2919)(cid:2870)(cid:3397)4(cid:2026)(cid:3024)A(cid:2919)(cid:2026)(cid:2919)(cid:2870)(cid:3397)4A(cid:2919)(cid:2870)(cid:4666)(cid:2029)(cid:2872)(cid:2919)(cid:3398)(cid:2026)(cid:2919)(cid:2872)(cid:4667)
`N
`(cid:3397) 4B(cid:2919)(cid:2870)(cid:2026)(cid:2919)(cid:2870)(cid:3397)4(cid:2026)(cid:2919)(cid:2870)(cid:3533)C(cid:2919)(cid:2920)(cid:2870)(cid:2026)(cid:2920)(cid:2870)
`(cid:3397)A(cid:2919) (cid:2026)(cid:2919)(cid:2870)(cid:3533)A(cid:2920) (cid:2026)(cid:2920)(cid:2870)
`(cid:3015)
`(cid:3015)
`(cid:3037)(cid:2880)(cid:2869)
`(cid:3037)(cid:2880)(cid:2869)
`wavefront, W(cid:2868), and the standard deviation of the wavefront, (cid:2026)(cid:3024) (not to be confused with standard deviations of the
`tolerance probability distributions (cid:2026)(cid:2919)(cid:2870) and (cid:2026)(cid:2920)(cid:2870)). The number of tolerances and the nominal wavefront error are constants
`and do not depend on the values of the tolerances. The standard deviation (cid:2026)(cid:3024) depends on the individual tolerances, but
`the error function (5) allows the straightforward incorporation of the A(cid:2919) , B(cid:2919) , and C(cid:2919)(cid:2920) expansion coefficients for each
`
`incorporated into a DLS optimizer, because the resulting expression is not a simple sum of squares of aberrations.
`Therefore, we built a Simplified As-Built (SAB) error function composed of a sum of components that depend on only
`one tolerance
`
`
`
`
`
`
`where the square of the aberration for each tolerance is
`
`
`
`
`
`
`
`
`
`(5)
`
`(6)
`
`
`
`
`
`
`Note that there are three terms without tolerance subscripts in the above equation: the number of terms, N, the nominal
`
`we choose to simplify the calculation of the merit function by computing (4) and taking the square root. The choice of
`
`tolerance with the minimal loss of information. The SAB error function is easily added to the existing error functions
`and has proven to be successful in desensitizing a wide range of systems.
`
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
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`Proc. of SPIE Vol. 8487 84870D-4
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`
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`the standard transverse aberration or wavefront error functions. Because SAB is based on a second order Taylor series
`expansion and has a few simplifications, SAB was not used as a standalone merit function. Best results were most
`frequently obtained when the relative weighting of the SAB and standard error functions were the same order of
`magnitude, although the optimum weights varied for each lens (as one might expect). While the SAB error function can
`
`matrix size in the DLS optimization does increase the computational burden. The designer can improve the speed by
`using only the most sensitive tolerances from the lens. Because SAB is simplified, and not an exact description of the
`wavefront, it can “push” the optimizer in a useful direction. SAB has often been found to work better as a complement to
`a standard error function, rather than as a standalone error function.
`
`
`The designs described in the remainder of this paper utilize the (cid:1845)(cid:1827)(cid:1828) error function described above, in conjunction with
`reuse the rays traced for the standard error function, computation of the A(cid:2919) , B(cid:2919) , and C(cid:2919)(cid:2920) coefficients and the increased
`It is important to note that the sensitivity coefficients A(cid:2919) , B(cid:2919) , and C(cid:2919) (cid:2920)include the effects of any tolerance compensators in
`
`the system. That is, SAB estimates the as-built performance under the condition that the specified compensators are
`used. The compensators could be as simple as setting focus or could include multiple layers of compensation, for
`instance decentering a lens to correct axial coma, and then readjusting focus. The implementation of this method in
`CODE V allows the user to enter different compensators for different sets of tolerances (through labeling tolerances and
`compensators). This may be used, for example, on a relay system, consisting of two sub-assemblies, each of which is
`built and assembled separately. (Importantly, handling the tolerances in this way prevents the two sub-assemblies from
`“cross-correcting” each other’s aberrations.) It can also be used to model transverse compensators for coma and air
`space adjustments for spherical aberration, in for example, a microscope.
`
`
`3. EXAMPLES
`
`In general there are two types of design problems: design-to-performance and design-to-cost. In design to performance,
`we want to find the global minimum for manufacturing cost to meet a particular set of performance requirements. This
`involves designing the most manufacturable lens for the requirements and then choosing the most appropriate set of
`tolerances and compensators to minimize manufacturing cost of that lens. In design-to-cost, we want to find the highest
`performing lens for a given set of manufacturing parameters. In this section we will describe three simple design-to-cost
`examples: a Cooke triplet, a double Gauss lens, and a mobile phone camera lens.
`
`3.1 Cooke triplet
`
`The Cooke triplet is a photographic lens designed and patented in 1893 by Dennis Taylor, who was chief engineer of
`T. Cooke & Sons of York, England29. It consists of two positive singlet elements and one negative singlet element. The
`negative flint element is located in the middle of the positive crown elements, thus maintaining a large amount of
`symmetry. It has enough effective degrees of freedom (6 radii, 2 air spaces, 3 indices) to affect all the primary
`aberrations (longitudinal color, lateral color, field curvature, astigmatism, coma, and spherical aberration). At the time,
`the Cooke triplet was a major advancement in lens design. It was superseded by later designs in high-end cameras, but is
`still widely used in inexpensive cameras and other applications.
`
`As a first example of the use of this technique, we start with the Cooke1 lens design shipped with CODE V. Table 1 lists
`the specifications. Figure 1 shows the design locally optimized with a 100 mm focal length constraint and a) transverse
`error function, b) wavefront error function, c) SAB and transverse error function, and d) SAB and wavefront error
`function. For the SAB optimization, we used the “commercial quality” tolerances in Table 2 and only a focus
`compensator. All tolerances in the SAB optimization are assumed to have uniform probability distributions. The
`optimization with transverse aberration error functions resulted in larger, weaker lenses with larger air spaces between
`elements. (There was no length constraint.) The glasses for the first two elements changed from moderate index crowns
`and flints (n ≈ 1.63) to high index crows and flints (n ≈ 1.73). The SAB error function led to more “relaxed” designs.
`Figure 2 shows the nominal wavefront aberrations for the four cases. All designs have large residual chromatic
`aberrations, which the SAB error function cannot fix. (There are not enough degrees of freedom with “normal” glasses to
`correct secondary color and the variations of aberrations with wavelength.) Both SAB designs have larger residual
`
`
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`Proc. of SPIE Vol. 8487 84870D-5
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`
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`aberrations, but the residual aberrations are lower order. Lower order aberrations are often less sensitive to alignment; it
`is often a sign that there is less “balancing” between surfaces and elements. The field weights were unity in all examples
`to provide the simplest comparison of the performance without and with SAB in the error function (in a real design the
`weights would likely be iterated to give the desired field performance). Comparison of the as-built performance in
`Figure 3 shows the addition of the SAB improved the as-built performance of the worst field by more than 20%. In the
`figure, it is clear that the two versions that included SAB in the merit function (Cases (c) and (d)) are superior, as-built,
`to the two versions that were optimized without SAB (Cases (a) and (b)). Comparing Cases (c) and (d), Case (d) has the
`lowest as-built RMS wavefront at the worst field. Case (c) has lower as-built RMS wavefront values on axis and at the
`edge of the field, but is slightly worse than Case (c) at the intermediate field. All three fields are better, as-built, for
`Cases (c) and (d) than for Cases (a) and (b). The balance between the as-built performances at each field can be adjusted
`through the field weights (they were identically weighted in this example) and/or the weight between SAB and the
`standard error functions.
`
`Parameter
`Wavelengths (nm)
`Focal length (mm)
`F-number
`Full field (deg, diagonal)
`Vignetting
`
`Table 1. Cooke triplet example specifications
`Value
`656, 588, 486 with 1:2:1 weights
`100
`4.5
`40
`None
`
`cooke1_ta.len
`
`14.71 MM
`
`Scale:
`
`1.70
`
`ORA 06-Jun-12
`
`
`(a) (b)
`
`cooke1_wfr.len
`
`14.71 MM
`
`Scale:
`
`1.70
`
`ORA 06-Jun-12
`
`
`
`14.71 MM
`
`14.71 MM
`
`
`
`
`
`cooke1_ta_sab.len
`
`Scale:
`
`1.70
`
`ORA 06-Jun-12
`
`cooke1_wfr_sab.len
`
`
`(c) (d)
`Figure 1 Cooke triplet locally optimized with the transverse error function (a), wavefront error function (b), SAB and
`transverse (c), and SAB and wavefront (d).
`
`Scale:
`
`1.70
`
`ORA 07-Jun-12
`
`
`
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`IPR2020-00896
`Exhibit 2006
`Page 6
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`Proc. of SPIE Vol. 8487 84870D-6
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`
`
`
`TANGENTIAL
` 4.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 20.00 )O
`
`SAGITTAL
` 4.0
`
`TANGENTIAL
` 4.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 20.00 )O
`
`SAGITTAL
` 4.0
`
`0.69 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`cooke1_ta.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`06-Jun-12
`
` 656.3000 NM
` 546.1000 NM
` 486.1000 NM
`
`0.69 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`cooke1_wfr.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`06-Jun-12
`
`
`(a) (b)
`
`TANGENTIAL
` 4.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 20.00 )O
`
`SAGITTAL
` 4.0
`
`TANGENTIAL
` 4.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 20.00 )O
`
`SAGITTAL
` 4.0
`
`0.69 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`cooke1_ta_sab.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`06-Jun-12
`
` 656.3000 NM
` 546.1000 NM
` 486.1000 NM
`
`0.69 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
` 4.0
`
`-4.0
`
`cooke1_wfr_sab.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`07-Jun-12
`
`
`(c) (d)
`
` 656.3000 NM
` 546.1000 NM
` 486.1000 NM
`
`
`
` 656.3000 NM
` 546.1000 NM
` 486.1000 NM
`
`
`
`Figure 2. Wavefront aberrations for a Cooke triplet locally optimized with the transverse error function (a), wavefront error
`function (b), SAB and transverse (c), and SAB and wavefront (d).
`
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 7
`
`

`

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`
`Proc. of SPIE Vol. 8487 84870D-7
`
`
`
`
`Table 2. Tolerances assuumed for the Coooke triplet exammple
`
`Valuee
`
`
`
`
`Tolerannce
`
`
`Test plate radii (on curvedd surfaces only)
`
`
`Power erroor relative to testt plate
`
`Irregularity
`
`Thickness errors
`
`Spacing errrors
`
`Element wwedge
`
`0.2%
`
`5 fringees
`
`2 fringees
`
`150 umm
`75 um
`at
`total indicated
`50 um
`run out (TIR)
`
`
`ace f clear aperturee of rear surfa
`edge o
`relative
`to front surface
`
`50 um TTIR
`
`100 umm
`0.001
`0.8%
`
`
`Element tiilt
`
`Element ddecentration
`
`Index erroor (nd)
`
`Abbe nummber
`
`
`
`
`
`Figure 3. EExpected nominnal and as-built pperformance of aa Cooke triplet ddesigned using foour different merrit
`
`
`
`
`
`
`
`
`
`
`
`
`functionss
`
`
`
`
`
`
`
`
`
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 8
`
`

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`
`Proc. of SPIE Vol. 8487 84870D-8
`
`
`
`3.2 Double Gauss
`
`The double Gauss is a significant step up in complexity from a Cooke triplet. It, and its variants, have dominated the
`class of photographic lenses for many years.
`
`The original two element Gauss was a telescope objective lens consisting of closely spaced positive and negative
`menisci, invented in 1817 by Carl Friedrich Gauss. Alvan Clark further refined the design in 1888 by taking two of these
`lenses and placing them back to back making a "double Gauss" with unimpressive photographic results. Current double
`Gauss lenses can be traced back to an 1895 improved design, when Paul Rudolph corrected for chromatic aberration by
`thickening the interior negative menisci and converting them to cemented doublets of two elements with equal refraction
`but differing dispersion in the Zeiss Planar of 1896. This design was the original six element symmetric f/4.5 Double
`Gauss lens. Horace Lee added a slight asymmetry to the Planar in 1920, and created the Taylor, Taylor & Hobson Series
`0 f/2 lens. This slight asymmetry is the basis of the modern double Gauss. It has 8 curvatures and 6 indices, which
`provides a very diverse design space, with literally thousands of adaptations of this form depending on the aperture and
`field.
`
`We started with US Patent 2,532,751 and the specifications listed in Table 3. Figure 4 shows the design locally
`optimized with a 100 mm focal length constraint and a) transverse error function, b) wavefront error function, c) SAB
`and transverse error function, and d) SAB and wavefront error function. The flat surfaces were not allowed to vary. Just
`as in the previous example, we used the tolerances in Table 2 and only a focus compensator. All fields were weighted
`identically. Both SAB optimized lenses are larger than the conventionally optimized lenses (there were no length
`constraints). The refractive index of the glasses changed less than about 0.03 (some increased and some decreased).
`Figure 5 shows the wavefront aberrations of the four locally optimized lenses. We see that the wavefronts of both the
`lenses optimized with transverse aberrations (with and without SAB) are more similar to each other than the two lenses
`optimized with wavefront aberrations. The wavefront optimized lenses have less coma at the edge of the field and
`slightly less spherochromatism on-axis. Figure 6 summarizes the as-built performance of the four lenses. In this case,
`the SAB plus wavefront optimization yielded the best result. The on-axis field degraded by 10% from the best case
`without SAB (transverse ray aberration optimized lens), but the 0.7 and full fields improved by 13% and 18%,
`respectively. The variation in as-built MTF across the field could easily be changed by adjusting the field weights. This
`simple example shows the SAB error function component provides direct control over the as-built wavefront errors. For
`a practical application, the designer would have to take into account other factors such as size, cost, weight, etc.
`
`Table 3. Double Gauss example specifications
`Value
`656, 588, 486 with 1:2:1 weights
`100
`2
`28
`None
`
`Parameter
`Wavelengths (nm)
`Focal length (mm)
`F-number
`Full field (deg, diagonal)
`Vignetting
`
`
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 9
`
`

`

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`
`Proc. of SPIE Vol. 8487 84870D-9
`
`dbgauss_ta.len
`
`Scale:
`
`1.40
`
`ORA 12-Aug-12
`
`dbgauss_wfr.len
`
`17.86 MM
`
`17.86 MM
`
`Scale:
`
`1.40
`
`ORA 26-Jul-12
`
`(a)
`
`(b)
`
`dbgauss_ta_sab.len
`
`Scale:
`
`1.40
`
`ORA 12-Aug-12
`
`dbgauss_wfr_sab.len
`
`17.86 MM
`
`17.86 MM
`
`Scale:
`
`1.40
`
`ORA 12-Aug-12
`
`
`
`
`
`(d)
`(c)
`Figure 4. Double Gauss lens locally optimized with the transverse error function (a), wavefront error function (b), SAB and
`transverse (c), and SAB and wavefront (d).
`
`
`
`
`
`
`
`
`
`
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 10
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`Proc. of SPIE Vol. 8487 84870D-10
`
`TANGENTIAL
` 5.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`SAGITTAL
` 5.0
`
`TANGENTIAL
` 5.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`SAGITTAL
` 5.0
`
`0.71 RELATIVE
`FIELD HEIGHT
`( 10.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`
`
`0.71 RELATIVE
`FIELD HEIGHT
`( 10.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`dbgauss_ta.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`12-Aug-12
`
`
`
` 656.3000 NM
` 587.6000 NM
` 486.1000 NM
`
`
`
`dbgauss_wfr.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`12-Aug-12
`
` 656.3000 NM
` 587.6000 NM
` 486.1000 NM
`
`
`
`(a)
`
`(b)
`
`TANGENTIAL
` 5.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`SAGITTAL
` 5.0
`
`TANGENTIAL
` 5.0
`
`1.00 RELATIVE
`FIELD HEIGHT
`( 14.00 )O
`
`SAGITTAL
` 5.0
`
`
`
`
`
`
`
`
`
`0.71 RELATIVE
`FIELD HEIGHT
`( 10.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`dbgauss_ta_sab.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`12-Aug-12
`
`
`
`0.71 RELATIVE
`FIELD HEIGHT
`( 10.00 )O
`
`0.00 RELATIVE
`FIELD HEIGHT
`( 0.000 )O
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
` 5.0
`
`-5.0
`
`dbgauss_wfr_sab.len
`
`OPTICAL PATH DIFFERENCE (WAVES)
`ORA
`12-Aug-12
`
` 656.3000 NM
` 587.6000 NM
` 486.1000 NM
`
` 656.3000 NM
` 587.6000 NM
` 486.1000 NM
`
`
`
`
`(d)
`(c)
`Figure 5. Wavefront aberrations for a double Gauss lens locally optimized with the transverse error function (a), wavefront
`error function (b), SAB and transverse (c), and SAB and wavefront (d).
`
`APPLE V. COREPHOTONICS
`IPR2020-00896
`Exhibit 2006
`Page 11
`
`

`

`Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 11 Mar 2021
`Terms of Use: https://www.spiedigitallibrary.org/terms-of-use
`
`Proc. of SPIE Vol. 8487 84870D-11
`
`
`
`
`Figure 66. Expected nomminal and as-buillt performance oof a double Gauss lens designed uusing four differrent merit functi
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`ons
`
`
`
`
`3.3 Mobile pphone camera
`
`
`
`
`
`
`
`Mobile phonne cameras lensses are a commmercially interesting subject bbecause annual
`production voolumes exceed
`1 billion
`
`
`
`
`
`sold units. FFrom an opticaal design persppective, they arre interesting bbecause of the
`
`
`exceedingly hhigh number off degrees
`
`
`of freedom; eeach surface iss an asphere w
`
`
`
`
`
`ith the numberr of orders onlyy limited by thhe designer’s reestraint. Theyy are also
`
`
`
`
`especially intteresting subjeects for this stuudy, as it is nottoriously easy
`
`
`
`to design lensses with very hhigh nominal MMTF that
`
`degrade quitee rapidly with
`
`
`
`tolerances. Leenses are reputtedly routinelyy produced in
`
`
`quantities of aa million a moonth with
`
`
`them scrappedd due to poor performance.
`over 20% of
`
`
`The complexxity of mobile
`
`
`
`
`phone cameraa lenses has steeadily increaseed. Initial lensses were only
`singlets. Four
`element
`
`
`
`
`
`
`
`
`designs oftenn looked like aa triplet, with aa field elementt to correct thee residual distoortion errors annd match the cchief ray
`
`angles of thee optics to the
`
`
`
`
`
`
`sensor (CMOSS sensors havee microlenses tthat are