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Comparative analysis of imaging configurations and objectives for Fourier microscopy

Jonathan A. Kurvits, Mingming Jiang, Rashid Zia

arXiv:1507.04037v2physics.optics

TL;DR

Quantitative Fourier microscopy requires understanding aberrations in microscopes designed for real-space imaging, yet the effects of objective and setup choices are unclear. The paper models commercial objectives and Fourier-imaging configurations, finding a preferred objective class and a beneficial Bertrand-lens placement. It also identifies alignment and tolerancing costs from added relay elements.

  • Problem

    Fourier microscopy seeks quantitative angular-spectrum measurements, but varied objective classes and BFP configurations leave unclear which corrections matter and how Fourier optics add aberrations.

    Source: §1 · pp. 1–2

  • Method

    The authors model Nikon, Olympus, and Zeiss microscope objectives and complete systems using patent-derived parameters, checked by disassembly where possible, then analyze Fourier-imaging aberrations and configurations.

    Source: §1 · p. 2

  • Results

    The analysis identifies a plan-corrected apochromat with high NA and low magnification as the best objective choice, and finds that placing the Bertrand lens between objective and tube lens can improve imaging.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Relay configurations offer flexibility but their additional optical elements significantly increase alignment difficulty and tolerancing errors.

    Source: §2 · p. 3

    The analysis assumes a standard visible-wavelength achromatic doublet as the Bertrand lens, and relay designs add alignment and tolerancing difficulty.

    Source: §2 · pp. 2–3

Abstract

from arXiv · show

Fourier microscopy is becoming an increasingly important tool for the analysis of optical nanostructures and quantum emitters. However, achieving quantitative Fourier space measurements requires a thorough understanding of the impact of aberrations introduced by optical microscopes, which have been optimized for conventional real-space imaging. Here, we present a detailed framework for analyzing the performance of microscope objectives for several common Fourier imaging configurations. To this end, we model objectives from Nikon, Olympus, and Zeiss using parameters that were inferred from patent literature and confirmed, where possible, by physical disassembly. We then examine the aberrations most relevant to Fourier microscopy, including the alignment tolerances of apodization factors for different objective classes, the effect of magnification on the modulation transfer function, and vignetting-induced reductions of the effective numerical aperture for wide-field measurements. Based on this analysis, we identify an optimal objective class and imaging configuration for Fourier microscopy. In addition, as a resource for future studies, the Zemax files for the objectives and setups used in this analysis have been made publicly available.

1. INTRODUCTION

Fourier microscopy is used to measure the angular spectrum of optical nanostructures and quantum emitters, but studies employ varied objectives and BFP-imaging configurations. The paper evaluates which objective corrections and imaging choices matter for quantitative Fourier imaging.

1. INTRODUCTION

  • Quantitative Fourier microscopy seeks angular-spectrum measurements across studies of quantum emitters, optical nanostructures, and their interactions.Applications include characterizing emitter orientation, antenna radiation patterns, and quantum-transition multipolar origins.Source: §1 · p. 1
  • Studies use objectives spanning dry 0.8 NA, 1.49 NA TIRF, and 1.65 NA high-index designs, alongside several BFP-imaging arrangements.Arrangements place a Bertrand lens before or after the image plane, or use relay optics to reimage and magnify the BFP.Source: §1 · p. 1
  • It remains unclear which objective corrections matter most for quantitative Fourier imaging and how much additional aberration Fourier-imaging optics introduce.Real-space objectives and configurations were optimized for different applications, and Fourier-space alignment can make aberrations appear as fine features.Source: §1 · pp. 1–2
  • The paper models complete microscope systems to assess objective suitability and Fourier-imaging configurations, identifying a plan-corrected apochromat with high NA and low magnification as ideal.It also finds that placing the Bertrand lens in the infinity space between objective and tube lens can improve imaging.Source: §1 · p. 2

2. INTRODUCTION TO FOURIER MICROSCOPY

In an infinity-corrected microscope, the objective and tube lens form the real-space image, while a Bertrand lens images the objective’s BFP. Four common configurations trade BFP magnification and real-space switching flexibility against space constraints and alignment difficulty.

2. INTRODUCTION TO FOURIER MICROSCOPY

  • An infinity-corrected objective and tube lens form the object image, while a Bertrand lens reimages the objective’s BFP to produce Fourier imaging.For this paper, any lens used to perform a Fourier transform is called a Bertrand lens; the assumed lens is a visible-wavelength achromatic doublet.Source: §2 · p. 2
  • A Bertrand lens before the image plane allows switching between Fourier and real-space imaging, but its limited space typically restricts focal length to ≤50 mm and BFP magnification.Placing it after the image plane permits greater BFP magnification but requires additional optics for real-space imaging.Source: §2 · p. 2
  • Relay configurations move the image plane farther from the microscope exit and provide greater flexibility, with lenses replaceable or removable for real-space imaging.Figure 1 depicts the single-lens and relay arrangements and traces rays emitted at ±30° to different detector locations.Source: §2 · p. 2, Figure 1 · p. 2
  • The extra optical elements in relay designs significantly increase alignment difficulty and tolerancing errors.The analysis weighs these trade-offs while considering another approach.Source: §2 · p. 3

3. OPTICAL MODELING OF COMMERCIAL MICROSCOPES

The authors reconstruct commercial microscope components from patent specifications, using physical disassembly where possible to check basic designs. They describe the limits of those specifications and report manufacturer-specific findings for glass and tube lenses.

3. OPTICAL MODELING OF COMMERCIAL MICROSCOPES

  • Patent specifications provide surface curvature, thickness, refractive index, and Abbe number for modeling optical aberrations, but omit coatings needed to model total system throughput.The authors use patents near objective release dates and disassembly where possible; commercial designs may differ from patent details and change over time.Source: §3 · p. 3

A. Glass Determination

  • Glass substitutions matched patent refractive indices and Abbe numbers to catalog glasses where possible, improving modeled chromatic aberrations across the visible spectrum.The authors checked likely manufacturers: Schott for nearly all Zeiss surfaces, Ohara for nearly all Olympus surfaces, and Schott plus Japanese suppliers for Nikon.Source: A. Glass Determination · p. 3

B. Tube Lens

  • Disassembly broadly supported the tube-lens designs, but the Olympus lens had a different element count and the Nikon lens differed in component thicknesses by about 1 mm.The Zeiss tube lens matched the single-element patent design; the authors considered the basic designs reliable given their recurrence across patents.Source: B. Tube Lens · p. 3

C. Objectives

The authors selected patented Nikon, Olympus, and Zeiss objectives to span numerical apertures and aberration corrections, while acknowledging that patent models may not exactly match commercial products. They validated and refined models for aberration and vignetting analyses.

C. Objectives

  • The selected objectives span manufacturers, numerical apertures, and aberration corrections, though patent-derived designs cannot be confirmed as exact commercial matches.The authors argue that qualitative comparisons can still help identify an optimal objective class for Fourier imaging.Source: C. Objectives · p. 3
  • Patent-based Zemax models were checked by reproducing patent d-line distortion and spherical-aberration figures before subsequent aberration calculations.Glass substitution was included to verify the models and build a broader-use objective library, although later calculations were performed at the d-line.Source: C. Objectives · pp. 3–4
  • Because most patents omit component semi-diameters, the authors allowed them to vary until an on-axis source emitting at the specified NA was fully collected.Comparisons with patents that specified semi-diameters supported the procedure; Fig. 2 illustrates a modeled Nikon 1.4 NA, 60x objective.Source: C. Objectives · p. 4, Figure 2 · p. 4

4. COMPARATIVE ANALYSIS OF MICROSCOPE OBJECTIVES

This section compares objective aberration corrections, magnification, numerical aperture, and collection choices for Fourier imaging. Plan-Apo and high-index Apo objectives show lower alignment-tolerance errors than TIRF and 1.25 NA water-immersion objectives.

A. Objective Class and Alignment Sensitivity

  • Plan-Apo and Apo objectives stay near ideal apodization over ±1 cm of Bertrand-lens displacement, unlike TIRF and high-NA water-immersion objectives.Source: Figure 3 · p. 4

4. COMPARATIVE ANALYSIS OF MICROSCOPE OBJECTIVES

  • The comparison uses apodization as its objective-quality metric and examines Bertrand-lens defocus because it couples with distortion and is difficult to correct.Apodization describes the increase in projected ray density with field angle; the section also compares nominal magnification, numerical aperture, and collection choices.Source: §4 · p. 4, A. Objective Class and Alignment… · p. 5

A. Objective Class and Alignment Sensitivity

  • Plan-Apo and high-index Apo objectives show lower alignment-tolerance errors than TIRF and 1.25 NA water-immersion objectives across the examined configurations.The objective-class performance trends also hold for the AIP and preferred configurations, although configuration affects the magnitude of tolerancing errors.Source: A. Objective Class and Alignment… · p. 5

B. Magnification and Fourier Space Resolution

This section asks how objective magnification affects Fourier-image size and resolution. Lower nominal magnification produces a larger back-focal-plane image while preserving comparable image quality for the compared objectives.

B. Magnification and Fourier Space Resolution

C. Vignetting Reductions of Effective NA

Vignetting is negligible for a small central confocal collection area but can reduce effective NA and distort Fourier images as the collection field expands. Wide-field collection therefore risks quantitatively modifying the measured radiation pattern.

C. Vignetting Reductions of Effective NA

  • At large angles, vignetting decreases observed intensity during full-field Fourier imaging and can change fitted radiation patterns.The effect makes collection area, often linked to excitation spot size, important to consider.Source: C. Vignetting Reductions of Effective NA · p. 6
  • For a 1 µm central confocal region, vignetting is negligible and all objectives transmit essentially 100% of rays up to their full NA.Vignetting effects remain minor within a 10 µm central collection area.Source: C. Vignetting Reductions of Effective NA · p. 6
  • Beyond 100 µm fields or 50 µm from center, vignetting reduces effective NA and can distort the Fourier image.Although wide-field excitation can improve signal-to-noise, the resulting quantitative modification should be avoided in almost all microscope objectives.Source: C. Vignetting Reductions of Effective NA · p. 6

5. COMPARATIVE ANALYSIS OF IMAGING CONFIGURATIONS

Among the four configurations, apodization factors are nearly identical, while relay configurations show greater field curvature. A lens placed between the objective and tube lens offers a way to combine configuration flexibility with reduced tolerancing errors and easier alignment.

5. COMPARATIVE ANALYSIS OF IMAGING CONFIGURATIONS

  • A plan-corrected apochromatic objective with low magnification and high NA is the preferred choice for Fourier-space imaging.For the Nikon 1.4 NA, 60x Plan-Apo comparison, all four configurations have nearly identical apodization, while M4F and AC show about 1 mm field curvature versus about 0.1 mm for AIP and BIP.Source: §5 · p. 6
  • BIP restricts added optics and Bertrand-lens focal length, AIP prevents real-space imaging at the same detector plane, and M4F/AC trade flexibility for more curvature and complexity.Additional optics can include polarizers, beamsplitters, and filters; relay configurations also bring associated tolerancing errors.Source: §5 · p. 7
  • Placing a Bertrand lens between the objective and tube lens combines one-lens simplicity with relay-like BFP magnification flexibility, without curvature or tolerancing complications.The lens directly images the objective’s BFP, forming an infinite conjugate pair with the tube lens.Source: §5 · p. 7
  • With the Bertrand lens before the tube lens, lens-position deviations cause almost pure defocus, so sharp Fourier-image features indicate good focus and proper alignment.The configuration also significantly reduces apodization tolerancing errors compared with the common configurations.Source: §5 · p. 7, Figure 7 · p. 7

6. CONCLUSION

The analysis favors a high-NA, low-magnification Plan-Apo objective and small on-axis collection, while a simple modification between the objective and tube lens can improve performance and alignment.

6. CONCLUSION

  • A high-NA Plan-Apo objective with low magnification is the recommended choice, while large apodization deviations can severely aberrate Fourier images, including with TIRF objectives.The recommendation follows modeling based on patent information for a range of microscope objectives.Source: §6 · p. 7
  • Full-field collection can improve signal-to-noise but reduces high-angle throughput through vignetting, so the conclusion recommends small-area on-axis collection..Source: §6 · p. 7
  • Placing a lens between the objective and tube lens can significantly improve imaging performance and simplify alignment on a standard commercial microscope.The paper also points to tabulated surface data and Zemax models in Ref. as resources for future work.Source: §6 · p. 7

A. TUBE LENS SURFACE SPECIFICATIONS

The appendix lists inferred tube-lens surface specifications for Nikon, Olympus, and Zeiss, based on patent literature, with clear apertures set using measurements from actual tube lenses.

A. TUBE LENS SURFACE SPECIFICATIONS

B. OBJECTIVE SURFACE SPECIFICATIONS

The section lists microscope-objective surface specifications and notes how material data are represented when exact glass matches are unavailable. The tables cover patented objectives from Nikon, Olympus, and Zeiss.

B. OBJECTIVE SURFACE SPECIFICATIONS

  • When an exact glass match was unavailable, or for immersion oil, the refractive index and Abbe number are given at the d-line.The listed specifications use the notation described in Appendix A.Source: B. OBJECTIVE SURFACE SPECIFICATIONS · p. 8
  • The cited captions identify five Nikon-patent objectives: 60x and 100x at 1.4NA from [50], 60x at 1.25NA from [54], and 60x and 100x at 1.49NA from [42].Source: Table 5 · p. 8, Table 6 · p. 8, Table 7 · p. 8
  • The remaining tables list Olympus 60×, 1.4 NA and 100×, 1.65 NA objectives, and a Zeiss 100×, 1.45 NA objective.The models are attributed to patents [56], [58], and [66], respectively.Source: Table 10 · p. 9, Table 11 · p. 9, Table 12 · p. 10
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