Multi-Mode Gap Lens

• Multi-Mode Gap Lens is an engineered gap region repurposed as an active lens to control multiple optical or electron modes.
• It enables tunable focusing and reduced imaging aberrations across various implementations including momentum microscopy, integrated photonics, and plasmonics.
• Its cross-domain designs leverage electrostatic tuning, transformation optics, and nonlinear effects to optimize mode selection and performance.

Searching arXiv for papers directly relevant to “multi-mode gap lens” and closely related usages. “Multi-Mode Gap Lens” is not a single standardized term in the arXiv literature. Instead, it denotes a family of concepts in which a gap region is promoted from a passive spacing or transition zone into an active lensing element that can operate across multiple modes, multiple operating states, or multiple transport regimes. In the most explicit usage, the term refers to a tunable electrostatic front lens for momentum microscopy and XPEEM in which one or more annular electrodes convert the sample–extractor gap into a lens with several operating modes (Tkach et al., 2024). Closely related but distinct usages appear in multimode integrated photonics, where a square Maxwell’s fisheye lens functions as a compact multimode waveguide crossing medium (Badri et al., 2019); in nonlinear waveguide lasers, where a pump-induced soft aperture provides a distributed Kerr-lens-like mechanism across several transverse modes (Demesh et al., 2022); and in plasmonics, where a metal–dielectric–metal gap-plasmon lens performs passive focusing and routing within a planar waveguide (Dennis et al., 2015). By contrast, “multi-gap” superconductivity and “multi-gap” monocentric lens assemblies use superficially similar wording but refer to different physical problems rather than to a multi-mode gap lens in the optical or electron-optical sense (Katsumi et al., 2024, Kala et al., 18 Aug 2025).

1. Terminological scope and conceptual core

The most precise common denominator is the treatment of a gap as an optical or electron-optical degree of freedom rather than as an unavoidable separation. In the photoelectron-microscopy literature, the gap between sample and extractor becomes an additional converging electrostatic lens whose behavior is changed by electrode voltages, yielding extractor mode, gap-lens mode, zero-field mode, and repeller mode (Tkach et al., 2024). In integrated photonics, the central crossing region is replaced by a square Maxwell’s fisheye lens designed to reroute several orthogonal guided modes with minimal mode leakage, inter-mode crosstalk, and reflection (Badri et al., 2019). In a multimode Cr:ZnS waveguide laser, the pump-induced Gaussian gain landscape acts as a soft aperture and therefore as a distributed lens/loss filter that can reshape modal competition and potentially support distributed Kerr-lens mode-locking (Demesh et al., 2022).

This suggests an Editor’s term, “active-gap lensing,” for the shared design principle: the gap region itself is engineered to perform focusing, filtering, or modal selection. That shorthand is interpretive rather than conventional. The literature does not present a single unified theory linking all of these devices, but it repeatedly assigns functional significance to regions that in conventional designs are merely transitional.

A common misconception is to treat every occurrence of “gap lens” or “multi-gap” as equivalent. The monocentric long-wave infrared system is a gap-optimized multi-element lens assembly in which the air gaps are design parameters for transmission and thermal robustness, not a modal lensing mechanism in the waveguide or cathode-lens sense (Kala et al., 18 Aug 2025). Similarly, MgB$_2$ is a multi-gap superconductor, where “gap” refers to superconducting energy gaps and collective excitations rather than to a spatial lens element (Katsumi et al., 2024).

2. Electrostatic multi-mode front lens in momentum microscopy and XPEEM

The clearest explicit realization of a multi-mode gap lens is the front-lens configuration for momentum microscopes and photoemission electron microscopes described in “Multi-Mode Lens for Momentum Microscopy and XPEEM: Theory” (Tkach et al., 2024). The design inserts one or two concentric annular electrodes into the sample–extractor gap, transforming that region into a tunable electrostatic lens. The physical motivation is to mitigate the complications associated with conventional cathode-lens extractor fields of about $3$–$8$ kV/mm, especially field emission or flashovers at sharp sample features and space-charge effects caused by slow background electrons.

The paper distinguishes four operating modes. In extractor mode, all front electrodes are held at high positive potential relative to the sample, producing the conventional strongly accelerating field. For the case shown with $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$ kV and a 6 mm gap, the field is $F \approx 3.3\ \text{kV/mm}$ (Tkach et al., 2024). In gap-lens mode, lowering the ring potentials relative to the extractor produces an additional converging lens in the gap; the example $U_{\mathrm{Ex}} = 12\ \text{kV}$, $U_{R1}=U_{R2}=0$ reduces the field at the sample to about $F \approx 1\ \text{kV/mm}$. In zero-field mode, sufficiently negative ring voltages tune the field at the sample to $F=0$, as in the example $U_{\mathrm{Ex}} = 8\ \text{kV}$, $3$0. In repeller mode, the field becomes retarding; the example $3$1, $3$2 yields $3$3 and a saddle point $3$4, so electrons with energies below about $3$5 eV are repelled (Tkach et al., 2024).

The significance of the design is not only operational flexibility but also its effect on imaging aberrations. In momentum microscopy, the added converging action near the sample substantially flattens the $3$6-image curvature in the backfocal plane. At $3$7 eV, for example, the paper reports that with $3$8 kV the field at the sample drops from $3$9 to $8$0 kV/mm and the full $8$1–$8$2 angular range becomes usable with only minor degradation near the rim (Tkach et al., 2024). At $8$3 eV with $8$4 kV, the field at the sample is about $8$5 kV/mm and the full $8$6 field is imaged well; at $8$7 keV with $8$8 kV and $8$9 kV, the sample field can be as low as $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$0 V/mm and the full $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$1 momentum field is imaged with high quality. The same concept also supports large-field-of-view XPEEM, including a usable FoV larger than $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$2 mm at $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$3 eV in gap-lens mode and repeller-mode simulations indicating about $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$4 nm resolution under favorable conditions at $U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$5 eV (Tkach et al., 2024).

In this context, “multi-mode” refers literally to multiple front-lens operating states accessible without changing the overall microscope geometry. The system is switched between extractor, gap-lens, zero-field, and repeller modes simply by changing electrode voltages (Tkach et al., 2024).

3. Multimode integrated-photonic lens crossing

In silicon photonics, a related but non-identical concept appears in “Multimode waveguide crossing based on square Maxwell’s fisheye lens” (Badri et al., 2019). Here the central challenge is multimode routing in mode-division multiplexing, where several orthogonal spatial modes act as independent data streams but waveguide crossings cause mode leakage, inter-mode crosstalk, and reflection. The proposed solution is a square Maxwell’s fisheye lens used as a multimode waveguide crossing medium on a silicon-on-insulator platform.

The design originates in quasi-conformal transformation optics. In the virtual domain, the circular Maxwell’s fisheye lens has refractive-index profile

$U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$6

with virtual-domain constitutive parameters

$U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$7

To avoid the impractically high center index that would result from direct circular implementation on SOI, the circular lens is mapped into a square lens in the physical domain, using orthogonal grids found by solving the Laplace equation with Dirichlet-Neumann boundary conditions. The transformed material tensors follow the compact notation written in the paper as

$U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$8

The square lens is then truncated so that its side boundaries more closely match the waveguide core index, reducing interface reflection (Badri et al., 2019).

Two realization routes are reported. The first uses a graded photonic crystal, where the continuous refractive-index profile is approximated by subwavelength cells containing silicon rods in air or SiO$U_{\mathrm{Ex}} = U_{R1} = U_{R2} = 20$9, with effective permittivity

$F \approx 3.3\ \text{kV/mm}$0

The second uses varying thickness of a silicon slab waveguide so that the effective TE-mode index follows the square MFE profile; the authors describe this as closer to the ideal continuous GRIN lens because the thickness changes smoothly rather than in discrete cells (Badri et al., 2019).

The crossing is a $F \approx 3.3\ \text{kV/mm}$1 device with waveguides of width $F \approx 3.3\ \text{kV/mm}$2 and thickness $F \approx 3.3\ \text{kV/mm}$3 nm, chosen to support at least three modes. The square lens footprint is $F \approx 3.3\ \text{kV/mm}$4, and the truncated square lens side length is $F \approx 3.3\ \text{kV/mm}$5, reducing the footprint by $F \approx 3.3\ \text{kV/mm}$6 compared with the circular MFE lens (Badri et al., 2019). The design is evaluated for TE$F \approx 3.3\ \text{kV/mm}$7, TE$F \approx 3.3\ \text{kV/mm}$8, and TE$F \approx 3.3\ \text{kV/mm}$9, and for the 3D thickness-varying Si slab implementation it reports average insertion loss of $U_{\mathrm{Ex}} = 12\ \text{kV}$0, $U_{\mathrm{Ex}} = 12\ \text{kV}$1, and $U_{\mathrm{Ex}} = 12\ \text{kV}$2 dB; crosstalk of less than $U_{\mathrm{Ex}} = 12\ \text{kV}$3, $U_{\mathrm{Ex}} = 12\ \text{kV}$4, and $U_{\mathrm{Ex}} = 12\ \text{kV}$5 dB; and maximum return loss of $U_{\mathrm{Ex}} = 12\ \text{kV}$6, $U_{\mathrm{Ex}} = 12\ \text{kV}$7, and $U_{\mathrm{Ex}} = 12\ \text{kV}$8 dB, respectively (Badri et al., 2019). The bandwidth extends from $U_{\mathrm{Ex}} = 12\ \text{kV}$9 to $U_{R1}=U_{R2}=0$0 nm, an ultrawide bandwidth of $U_{R1}=U_{R2}=0$1 nm covering the O, E, S, C, L, and U bands, and the fidelity factors for low-distortion pulse transmission are $U_{R1}=U_{R2}=0$2, $U_{R1}=U_{R2}=0$3, and $U_{R1}=U_{R2}=0$4 for TE$U_{R1}=U_{R2}=0$5, TE$U_{R1}=U_{R2}=0$6, and TE$U_{R1}=U_{R2}=0$7 (Badri et al., 2019).

This is not termed a “gap lens” in the paper, but it belongs to the broader family of multimode lensing structures in which a compact central region images and reroutes multiple modes. A plausible implication is that it represents the integrated-photonic analogue of gap-based multifunctional lensing: a small central interaction region is engineered to preserve modal integrity at a crossing.

4. Distributed Kerr-lens and soft-aperture interpretation in multimode waveguide lasers

A third usage appears in the Cr:ZnS waveguide-laser literature, where the phrase “Multi-Mode Gap Lens” is best understood as a pump-induced, spatially graded nonlinear focusing element acting across several transverse modes in a multimode waveguide (Demesh et al., 2022). The work does not directly demonstrate stable Kerr-lens mode-locking. Instead, it reports experimental signs and simulations that motivate a distributed Kerr-lens picture.

The experimental platform is a Cr:ZnS depressed-cladding channel waveguide fabricated by ultrafast laser inscription in a 7 mm single crystal, with core diameter about $U_{R1}=U_{R2}=0$8, pumped by a $U_{R1}=U_{R2}=0$9 nm Er-fiber laser focused to a spot of about $F \approx 1\ \text{kV/mm}$0 inside the waveguide (Demesh et al., 2022). With a $F \approx 1\ \text{kV/mm}$1 output coupler, the device produced $F \approx 1\ \text{kV/mm}$2 mW output power, $F \approx 1\ \text{kV/mm}$3 slope efficiency, and emission around $F \approx 1\ \text{kV/mm}$4–$F \approx 1\ \text{kV/mm}$5 nm. With $F \approx 1\ \text{kV/mm}$6 OC transmission, the device was damaged at only about $F \approx 1\ \text{kV/mm}$7 mW output, corresponding to about $F \approx 1\ \text{kV/mm}$8 W intracavity power, which the authors argue was likely associated with transient switching between CW and mode-locked regimes, that is, Q-switching or Q-switched-mode-locking-like bursts (Demesh et al., 2022).

The central mechanism is the combination of multimode propagation and large Kerr nonlinearity, with $F \approx 1\ \text{kV/mm}$9. The optical field can occupy multiple transverse modes, interact through nonlinear refraction and self-phase modulation, redistribute power among modes, and self-focus toward the lowest-order mode (Demesh et al., 2022). This modal condensation is assisted by a pump-shaped gain profile rather than by a hard physical aperture. The gain is written as

$F=0$0

and the soft-aperture increment is given as

$F=0$1

The authors describe this pump-induced spatial gain profile as a distributed dissipative trap (Demesh et al., 2022).

The spatial dynamics are modeled with a driven nonlinear Helmholtz-type equation and, for low numerical aperture, a Lugiato-Lefever-like equation,

$F=0$2

capturing confinement, gain/loss, and Kerr nonlinearity (Demesh et al., 2022). Simulations indicate that when pump width $F=0$3 is close to the waveguide trapping width $F=0$4, the beam profile becomes cleanest and higher-order modes are minimized. When $F=0$5 or $F=0$6, the system shows Q-switching behavior and stronger excitation of higher-order modes (Demesh et al., 2022).

The connection to a multi-mode gap lens is conceptual rather than terminological. The waveguide “aperture” is the pump-induced gain profile rather than a material iris, and Kerr self-focusing changes the spatial mode content while the soft aperture converts that change into differential gain/loss. This suggests a distributed lensing mechanism that acts on multiple interacting transverse modes and may favor pulsed operation. The paper further discusses spatiotemporal regimes including a chirped dissipative soliton in normal GDD with estimated intracavity DS energy of about $F=0$7, and a shorter nearly chirp-free soliton in anomalous GDD with energy around $F=0$8 nJ (Demesh et al., 2022).

5. Gap-plasmon lensing in metal–dielectric–metal waveguides

A planar plasmonic realization is reported in “Diffraction limited focusing and routing of gap plasmons by a metal-dielectric-metal lens” (Dennis et al., 2015). The device is a planar Au/SiO$F=0$9/Au waveguide on glass in which the optical mode is confined in a nanoscale dielectric gap between two gold layers. It contains an in-coupler grating, MDM waveguide sections, a plano-convex lens region made by patterning and undercutting SiO$U_{\mathrm{Ex}} = 8\ \text{kV}$0, and out-coupler slits in the bottom Au layer (Dennis et al., 2015).

The operating principle is passive gap-plasmon focusing and routing in two dimensions. A gap plasmon launched from the grating propagates through the waveguide, passes the lens, and becomes a surface plasmon polariton in the out-coupler region, where slit sampling allows microscope imaging of the focus (Dennis et al., 2015). The lens is therefore a waveguide lens rather than a free-space lens: it changes the local effective index of the propagating gap plasmon so that the phase front bends and focuses.

The reported structure uses a 220 nm gap thickness; lens radii $U_{\mathrm{Ex}} = 8\ \text{kV}$1 and $U_{\mathrm{Ex}} = 8\ \text{kV}$2 for the overall plano-convex lens; an effective SiO$U_{\mathrm{Ex}} = 8\ \text{kV}$3 lens portion after undercut with $U_{\mathrm{Ex}} = 8\ \text{kV}$4; an undercut of about $U_{\mathrm{Ex}} = 8\ \text{kV}$5; and out-coupler slit spacing of $U_{\mathrm{Ex}} = 8\ \text{kV}$6 (Dennis et al., 2015). For the SiO$U_{\mathrm{Ex}} = 8\ \text{kV}$7 gap lens at $U_{\mathrm{Ex}} = 8\ \text{kV}$8 nm, with $U_{\mathrm{Ex}} = 8\ \text{kV}$9 for SiO$3$00 and $3$01 for Au, the effective index is $3$02; for the air-gap meniscus lens region, $3$03 (Dennis et al., 2015). The thin-lens estimate gives $3$04, the weakly negative meniscus contribution gives $3$05, and the combined focal length is $3$06 (Dennis et al., 2015).

Optical characterization reports an effective numerical aperture of $3$07, a calculated diffraction-limited beam diameter $3$08, and a measured focal spot with $3$09. After correcting for slit-related broadening of about $3$10, the plasmonic focal spot is estimated as $3$11, corresponding to a $3$12 beam diameter of about $3$13 (Dennis et al., 2015). The measured SPP propagation length is $3$14, substantially shorter than the ideal flat-interface estimate of about $3$15, which the authors attribute to surface roughness (Dennis et al., 2015).

Although the device is centered on a single guided gap-plasmon mode rather than on multimode routing, it is relevant because it demonstrates that lensing can be integrated directly into a guided gap region. The routing functionality achieved by under-filling different parts of the lens aperture anticipates later designs in which modal selectivity and lensing are co-designed.

6. Boundaries of the term: adjacent but non-equivalent “gap” usages

Two papers in the supplied corpus are best treated as boundary cases. The first concerns MgB$3$16, a canonical two-gap $3$17-wave superconductor studied with terahertz two-dimensional coherent spectroscopy (Katsumi et al., 2024). There, the central topic is the amplitude mode in a multi-gap superconductor. The measured nonlinear response includes a peak near $3$18 at low temperature, first- and third-harmonic signals under narrow-band THz driving at $3$19, and a damping parameter $3$20, much larger than the $3$21 value previously found in NbN (Katsumi et al., 2024). None of these uses of “gap” refer to a spatial lens or a modal gap-lensing mechanism. The paper is relevant mainly because it illustrates how “multi-gap” can be mistaken for “multi-mode gap lens” despite describing a different class of collective excitation.

The second boundary case is the monocentric long-wave infrared lens assembly with optimized air gaps between a ball lens and two hemispherical shell lenses (Kala et al., 18 Aug 2025). The architecture consists of Ge shell lenses L1 and L3 and an IG6 central ball lens L2, designed for $3$22–$3$23 operation with baseline $3$24-transmitting ARC on all surfaces (Kala et al., 18 Aug 2025). The study reports that at $3$25 wavelength transmittance peaks at about $3$26 when the air gap is $3$27, while the $3$28–$3$29 average transmittance reaches about $3$30 near a $3$31 gap; it also tracks thermal gap evolution from an initial $3$32 gap to about $3$33 at $3$34C and about $3$35 at $3$36C (Kala et al., 18 Aug 2025). This is a gap-optimized multi-element optical design rather than a multi-mode lens in the waveguide or electron-optical sense.

These distinctions matter because the phrase “multi-mode gap lens” can otherwise collapse unrelated notions of mode, gap, and lens into a single category. In the stricter technical usage, the term is most defensible when the gap region itself participates directly in mode control, field shaping, or tunable focusing.

7. Comparative significance and research outlook

Across these literatures, several recurring design themes emerge. First, the gap is treated as a controllable functional element. In momentum microscopy, annular electrodes redistribute refractive power within the sample–extractor region, enabling lower sample fields, zero-field operation, retarding fields, and flatter $3$37-image curvature over energies from a few eV to $3$38 keV (Tkach et al., 2024). In silicon photonics, a compact square Maxwell’s fisheye crossing medium uses transformation optics to preserve multimode imaging behavior within a $3$39 footprint and over $3$40–$3$41 nm (Badri et al., 2019). In Cr:ZnS waveguide lasers, the pump-induced soft aperture acts as a distributed nonlinear selector that links Kerr self-focusing to multimode spatial dynamics and possible spatiotemporal dissipative-soliton formation (Demesh et al., 2022). In plasmonics, effective-index engineering within a metal–dielectric–metal gap achieves diffraction-limited planar focusing and passive routing (Dennis et al., 2015).

Second, multimodality takes different forms in different subfields. In the cathode-lens case it means multiple operating modes of a single front-lens geometry. In silicon photonics it means simultaneous support for multiple guided transverse-electric modes. In nonlinear laser dynamics it means interaction among multiple transverse cavity modes under a distributed soft aperture. A plausible implication is that the phrase is most useful when qualified by domain: multi-mode gap lens for momentum microscopy, multimode gap-lensing crossing, or distributed multimode Kerr-gap lensing.

Third, the main technical advantages are domain-specific but structurally analogous: suppression of unwanted coupling or background, improvement of focusing or imaging over larger phase-space regions, and compact integration. In photoelectron microscopy this appears as reduced field emission, space-charge suppression, and larger usable momentum fields of view (Tkach et al., 2024). In integrated photonics it appears as low insertion loss, low crosstalk, low reflection, compact footprint, and scalability to more modes by increasing lens size and waveguide width (Badri et al., 2019). In waveguide lasers it appears as pump-geometry-dependent mode cleaning with the possibility of distributed Kerr-lens mode-locking (Demesh et al., 2022).

Taken together, the literature supports treating “Multi-Mode Gap Lens” not as a singular established device class but as a cross-domain motif: a deliberately engineered gap region that performs lensing or mode selection across more than one operative state or more than one supported mode.