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Inferring the solution space of microscope objective lenses using deep learning

Geoffroi Côté, Yueqian Zhang, Christoph Menke, Jean-François Lalonde, and Simon Thibault

Open Access Open Access

Abstract

Lens design extrapolation (LDE) is a data-driven approach to optical design that aims to generate new optical systems inspired by reference designs. Here, we build on a deep learning-enabled LDE framework with the aim of generating a significant variety of microscope objective lenses (MOLs) that are similar in structure to the reference MOLs, but with varied sequences—defined as a particular arrangement of glass elements, air gaps, and aperture stop placement. We first formulate LDE as a one-to-many problem—specifically, generating varied lenses for any set of specifications and lens sequence. Next, by quantifying the structure of a MOL from the slopes of its marginal ray, we improve the training objective to capture the structures of the reference MOLs (e.g., Double-Gauss, Lister, retrofocus, etc.). From only 34 reference MOLs, we generate designs across 7432 lens sequences and show that the inferred designs accurately capture the structural diversity and performance of the dataset. Our contribution answers two current challenges of the LDE framework: incorporating a meaningful one-to-many mapping, and successfully extrapolating to lens sequences unseen in the dataset—a problem much harder than the one of extrapolating to new specifications.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

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Supplementary Material (1)

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Supplement 1       Supplemental Document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. Upon receiving a given set of specifications and lens sequence, the model outputs $K = {8}$ different lenses that share the same sequence but differ in structure. Here, the lens sequence is composed of single lenses (SLs) and cemented doublets (CDs) separated by air gaps (–). In all layout plots, the aperture stop is shown in orange, and the scale bar indicates the size of the designs in units of effective focal length (EFL).

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Fig. 2.
Fig. 2. Subset of the 34 reference MOLs and their corresponding patent numbers (the scale is in units of EFL). Multiple lens structures can be recognized, such as Double-Gauss (a–c), Lister (d–f) and retrofocus (g–j) lenses.

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Fig. 3.
Fig. 3. Training domain used in our experiments, represented by the boxed area.

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Fig. 4.
Fig. 4. Random subset of designs inferred from a model trained without the augmented training objective ($K=1$), for 5–10 glass elements (the scale is in units of EFL). The designs do not capture the structural diversity of the dataset. Most importantly, only one design can be obtained for a given lens sequence and set of specifications.

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Fig. 5.
Fig. 5. Illustration of the process to compute the structural distance, using the 10 reference MOLs of Fig. 2. In (a), the PMR coordinates are first scaled to unit TTL (top), then passed through a derivative-of-Gaussian filter (bottom), giving us an estimation of each structure $s$. In (b), from the PMR slopes of every selected reference MOL, we compute the pairwise structural distance using Eq. (1). Designs with similar structures are generally close to one another (a–c, d–f, g–j).

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Fig. 6.
Fig. 6. The PMR slopes criterion is used to compute (a) the average structures $\overline{s_k}$ and (b) the reference structures $s^{*}_{k}$—both are required when computing the structural loss functions. In (a), the illustrated mean and standard deviation correspond to the fully trained model. The distance between branches is enforced by the structural diversity loss $L_\mathrm{SD}$, while the deviations within a branch are controlled by the structural adherence loss $L_\mathrm{SA}$. In (b), the reference structures each represent a subset of the 34 reference MOLs. The dataset representation loss $L_\mathrm{DR}$ encourages the average structures to be similar to the reference structures, hence the similarities between both graphs.

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Fig. 7.
Fig. 7. Subset of designs inferred from all branches $k$ of the trained model, using the nominal specifications (the scale is in units of EFL). For each branch, only six lens sequences are shown out of 7432 candidates. The selected lens sequences are those that minimize the OLF for the given branch and number of elements.

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Fig. 8.
Fig. 8. Distribution of the on-axis RMS spot size of all inferred designs, grouped by number of elements and output branch, for the nominal case (lower is better). For reference, the Airy disk diameter at the "d" Fraunhofer line is shown with a dashed line.

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Fig. 9.
Fig. 9. Distribution of the WD of all inferred designs, grouped by number of elements and output branch, for the nominal case. Longer WD enables more applications.

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Fig. 10.
Fig. 10. Tradeoff between on-axis RMS spot size and WD for all 8-element designs, for the nominal case. The Pareto front is populated by designs from different branches.

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Fig. 11.
Fig. 11. Uniformity of the spot size over the field across all output branches, obtained from the distribution of all inferred designs in the nominal case (lower is better). The Airy disk diameter at the "d" Fraunhofer line is shown with a dashed line.

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Equations (9)

Equations on this page are rendered with MathJax. Learn more.

s d ( s 1 , s 2 ) = 1 d 1 / 2 ‖ s 1 − s 2 ‖ 2   .
L U = exp ⁡ ( 1 b U ∑ i = 1 b U ln ⁡ O L F ( N A i , H F O V i , l i , k ( i ) ) )   .
s k ¯ = 1 b U ∑ i = 1 b U s i , k   .
L S D = 2 K 2 − K ∑ k = 1 K − 1 ∑ k ′ = k + 1 K R ( t S D − s d ( s k ¯ , s k ′ ¯ ) )   .
L S A = 1 b U K ∑ i = 1 b U ∑ k = 1 K s d ( s i , k , s k ¯ ) − min k ′ s d ( s i , k ′ , s k ′ ¯ )   .
L D R = 1 b U K ∑ i = 1 b U ∑ k = 1 K R ( s d ( s i , k , s k ∗ ) − t D R )   .
L S = 1 b S ∑ i = 1 b S 1 | l i ′ | ‖ l i ′ − l ′ j ( i ) ∗ ‖ 2 2   .
k ( j ) = a r g m i n k   s d ( s j , s k ∗ )   .
L = L U + λ S L S + λ S D L S D + λ S A L S A + λ D R L D R   .
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