Phase singularities occur when the amplitude of light reflected by an object is zero and its phase becomes undefined1,2. Singular optics, the field of optics concerned with the exploitation of phase singularities3, has primarily studied optical vortices in electromagnetic field profiles in real space4. Phase singularities can also be observed in photonic dispersion plots, where a response function (such as phase change on reflection) is plotted in parameter space (e.g. as a function of energy and incident angle5 or geometric parameter6). These phase singularities, which lead to sharp phase jumps in reflection spectra7, could find application in sensing8,9,10,11, potentially enabling a sensitivity three orders of magnitude greater than commercial amplitude-based sensing technology12,13, and in flat optics, where it is otherwise difficult to induce strong optical phase variations in materials with deeply subwavelength thicknesses4,14. Phase singularities have been observed using the Brewster angle15, surface plasmon resonances7,8,9,13, plasmonic lattices5,12,16, transition metal dichalcogenides14, optical Tamm states10 and Fabry-Pérot microcavities6,11. The realisation of phase singularities requires careful design of complicated structures using lithography5,6,12, self-assembly16 or multilayers10,11,13. In these structures, phase singularities can only be observed under a very specific set of conditions, such as one particular incident angle of light.

In this article, we show that phase singularities can be created using strong light-matter coupling. An ensemble of molecular resonators and a confined electromagnetic field can become coupled if the matter and light modes can be excited under the same conditions17,18. When the coupling strength exceeds losses to the environment the light and matter modes enter the strong coupling regime, forming hybrid states known as polaritons19. Strong coupling could find use in applications such as lasing20 or the modification of chemical processes21; to the best of our knowledge, the generation of phase singularities using strong coupling has not previously been reported. While most strong coupling experiments rely on external structures (such as planar microcavities17,22 or plasmonic nanostructures19) to generate confined electromagnetic fields, our results build on recent work which shows that such structures are not always needed23,24,25,26. We use such a cavity-free design here to observe phase singularities associated with each newly created polariton state. We use a photochemical reaction to control the number of coupled molecules, and hence light-matter coupling strength, by simple irradiation of light22. This allows one to tune into and detune from the phase singularities associated with each polariton branch, modifying the phase sensitivity of the films. Our results demonstrate what we believe to be the first application of strong light-matter coupling to the creation of phase singularities in dispersion plots in a remarkably simple structure.


Sample design and characterisation

We studied silicon substrates coated with a thin film of organic molecules (see diagram in Fig. 1a and film fabrication details in Methods). We used the photochromic molecule spiropyran (SPI), which is transparent to visible light and can be converted to merocyanine (MC) by exposure to ultraviolet light. MC has a strong absorption peak at energy E = EMC = 2.22 eV (see Supplementary Information for SPI/MC molecular structures and MC transmission spectrum) and can be reconverted back into SPI after exposure to visible radiation27. Reversible SPI/MC photoisomerisation has been exploited for all-optical control of strong coupling in Fabry-Pérot microcavities22 and lasing in plasmonic lattices28. The permittivites of SPI and MC (determined using ellipsometry—see Methods) are plotted in Fig. 1c.

Fig. 1: Cavity-free strong light-matter coupling.
figure 1

a Sample design: SPI/MC film on silicon substrate. b Ellipsometer schematic. c Complex permittivities ϵ of SPI and MC, as derived from ellipsometry for a film of thickness 109 nm. d, e Dispersion plots constructed using the ellipsometric parameter Ψ for d SPI and e MC films over a range of thicknesses at fixed angle θ = 65°. The dashed lines in d, e indicate the positions of the uncoupled TE (black) and TM (white) leaky modes. The green dashed line in e at E = 2.22 eV indicates the position of the MC molecular resonance. The solid lines show the predicted positions of polariton branches using the 2N coupled oscillator model. The coupled TE polariton branches (black) were fit with a coupling strength g = 225 meV. The TM2 and TM3 polariton branches (white) were fit with g = 185 meV and g = 200 meV, respectively. We did not perform a coupled oscillator fit for the TM1 mode since it does not show any clear anticrossing.

We characterised our samples using spectroscopic ellipsometry (schematic in Fig. 1b; see also Methods). Ellipsometry measures the ratio of the p- and s-polarised reflection coefficients, ρ, expressed as Ψ and Δ29:

$$\rho =\frac{{r}_{p}}{{r}_{s}}=\tan ({{\Psi }}){e}^{i{{\Delta }}}.$$

rp and rs are respectively the p- and s-polarised Fresnel amplitude reflection coefficients. The amplitude of ρ (\(\tan ({{\Psi }})\)) is the ratio of the moduli of rp and rs; the phase of ρ (Δ) gives the difference between the phase shifts experienced by p- and s-polarised light on reflection. Our ellipsometer’s xenon light source emits low-intensity ultraviolet radiation, allowing us to monitor the conversion of SPI to MC.

The Ψ spectra for SPI films with a range of thicknesses L (84 nm < L < 680 nm) measured at an incident angle θ = 65° are plotted in Fig. 1d to create a dispersion plot. The large impedance mismatches at the Si/SPI and SPI/air interfaces give rise to a series of transverse electric (TE) and transverse magnetic (TM) leaky modes (sometimes called quasi-normal modes30). For TE modes, rs < rp, giving maxima in Ψ; the positions of uncoupled TE leaky modes are indicated in Fig. 1d, e by dashed black lines. For TM modes, rp < rs, giving minima in Ψ; the positions of uncoupled TM leaky modes are indicated in Fig. 1d, e by dashed white lines. In Fig. 1e we plot Ψ spectra for the films in Fig. 1d after photoconversion to MC. Leaky modes can strongly couple to MC’s molecular resonance. There is clear anticrossing of the TE leaky modes around E = EMC26, a signature of strong coupling between the TE leaky modes and MC resonance19. TM modes are generally weaker than TE modes since rp is much lower than rs at the SPI/MC-Si interface31 at θ = 65°. A coupled oscillator fit (Methods) suggests that the TM2 and TM3 modes have coupling strengths of gTM2 = 185  meV and gTM3 = 200 meV, meaning that they fulfil the strong coupling resolution criterion19:

$${g}_{{{{{{{\rm{TM2,3}}}}}}}} \; > \; \frac{1}{4}\big({\gamma }_{{{{{{{\rm{MC}}}}}}}}+{\gamma }_{{{{{{{\rm{TM2,3}}}}}}}}\big),$$

where γMC ≈ 360 meV is the MC resonance linewidth and γTM2 ≈ 350 meV and γTM3 ≈ 240 meV are the TM2,3 leaky mode linewidths, respectively. The TM1 mode (for which γTM1 ≈ 430 meV) does not fulfil this criterion and there is no clear anticrossing of the TM1 mode in Fig. 1e.

Observation of phase singularities

In Fig. 2 we show how the ellipsometric parameters Ψ, Δ and ρ change as a thin SPI film is converted to MC. For this thickness of film (L = 407 nm) and incident angle of light (θ = 65°) the energy of the TM2 leaky mode matches that of the MC resonance. Fig. 2a shows how Ψ changes with increasing MC concentration. After 400 s of UV exposure two distinct modes appear; as the MC concentration increases, the mode splitting increases. This behaviour is consistent with the transition from the weak to strong coupling regime32. The minimum value of Ψ drops dramatically in the strong coupling regime, with one point on each branch going below 0. 1° (comparable with experimental error). In Fig. 2b we plot Δ, which shows the same phase behaviour between the two polariton branches that we have previously interpreted as a signature of strong coupling32. Shortly after entering the strong coupling regime, we observe two phase singularities: the first, at t = 470 s, on the lower polariton branch; the second, at t = 510 s, on the upper polariton branch. They correspond to positions on the upper and lower polariton branches in Fig. 2a at which Ψ < 0. 1°. When a loop is traced around a phase singularity in two-dimensional parameter space a phase of 2πC is accumulated, where C is the topological charge associated with each phase singularity. The phase singularities in Fig. 2b have topological charges of C = + 1 and C = −1, indicating that the total topological charge of the system is preserved6,14.

Fig. 2: Creation of phase singularities in an SPI/MC thin film.
figure 2

The ellipsometric parameters (a) Ψ, (b) Δ and (c) ρ plotted for a thin SPI film (thickness L = 407 nm, measured at an incident angle θ = 65°) while exposed to UV radiation. As UV exposure time increases, SPI undergoes photoisomerisation to MC and the system transitions from the weak coupling regime to the strong coupling regime.

Figure 2c shows ρ at four times: t = 10 s, when the film is almost entirely composed of SPI molecules; t = 220 s, when some SPI has undergone conversion to MC, but the MC density in the film remains too low for strong coupling; t = 510 s, coincident with the upper polariton phase singularity, when the MC density in the film is high enough to place the system within the strong coupling regime; and t = 1100 s, the final measurement when most SPI has been converted into MC. The evolution of ρ as the system enters the strong coupling regime resembles what we have observed in other strongly coupled systems32: the initial TM2 leaky mode is represented by an arc; in the weak coupling regime this arc is perturbed by the presence of MC; in the strong coupling regime a new loop is observed. As the MC concentration increases, the loop both increases in size and touches the point ρ = 0 for t = 510 s (green loop) and t = 470 s (not plotted for clarity). Provided the strong coupling loop passes over ρ = 0 as the system evolves, we are guaranteed to observe two phase singularities.

In Fig. 3 we plot Δ for SPI/MC films with thicknesses 84 nm < L < 680 nm. The equivalent Ψ data is plotted in Supplementary Fig. S2. Each film was exposed to UV radiation under the same conditions (with θ = 65°). We plot the phase response after (a) t = 10 s, (c) t = 330 s, (e) t = 440 s and (g) t = 1, 100 s of UV irradiation. The dashed black lines indicate the positions of the uncoupled TM modes in the SPI films. These results are well reproduced in Fresnel calculations (Fig. 3b, d, f, h) by varying the Lorentz oscillator strength of the MC resonance. Initially, in Fig. 3a, b, we observe no phase singularities associated with any TM leaky modes. We first observe the splitting of the TM3 leaky mode (the best-confined of the plotted TM modes) in Fig. 3c, d, followed by splitting of the TM2 mode in Fig. 3e, f. The topological charges of the phase singularities associated with each anticrossing are C = + 1 and C = −1. Therefore, as in Fig. 2b, the topological charge is preserved as each pair of phase singularities is created. We did not observe clear anticrossing of the TM1 mode in Fig. 1e; likewise, in Fig. 3g, h we do not unambiguously observe the creation of phase singularities associated with the TM1 mode. Phase singularities can also be created by coupling to TE modes under slightly different conditions; see Supplementary Section S3.

Fig. 3: Creation of phase singularities for a range of film thicknesses.
figure 3

a, c, e, g Experimental and b, d, f, h calculated (Fresnel approach) dispersion plots constructed using the ellipsometric parameter Δ for SPI/MC films over a range of thicknesses at fixed angle θ = 65°. The MC concentrations were varied in experimental plots by UV exposure time (a t = 10 s, c t = 330 s, e t = 440 s, g t = 1100 s) and in calculated plots by varying the Lorentz oscillator strength of MC (b fMC = 0 eV2, d f = 0.25 eV2, f f = 0.36 eV2 and h f = 0.56 eV2). The positions of the uncoupled TM modes are indicated by the black dashed lines in a, c, e, g.


Exciton-polariton condensates arising from strong light-matter coupling have been used to explore a wide range of phenomena33, including the manipulation of real space phase singularities34,35,36,37. These phase singularities (and their associated optical vortices) are typically observed in electromagnetic field profiles and located at particular spatial coordinates4. The phase singularities in Fig. 3 do not occur in physical space but in a response function in parameter space: they can be observed in dispersion plots but not in real-space plots. They are perhaps most similar to the work of Berkhout and Koenderink6, where phase singularities in plasmon antenna array etalons were observed by plotting reflectivity and phase spectra over a range of etalon spacings. It is also possible to identify phase singularities by plotting reflectivity and phase spectra for a range of incident angles5,11,14. We have shown that it is possible to manipulate phase singularities in photonic dispersion plots in the time domain over the course of seconds, similar to previous observations of phase singularities in the spatiotemporal domain over the course of femtoseconds38,39,40,41. The qualitative similarities between our results and those in the spatiotemporal domain highlight the universal nature of phase singularities. Note that we have only studied the forward photochemical reaction (SPI to MC), but since this process can be reversed under exposure to high-intensity visible light22,28, our system in principle allows one to both create and annihilate phase singularities in dispersion plots.

In conclusion, we have shown that molecules in a thin film can undergo strong coupling to leaky modes in the same thin film of which they are a part, and that entering the strong coupling regime creates pairs of phase singularities. The phase sensitivity of a thin film of a particular thickness can be varied by modifying the incident angle of light or the concentration of molecules in the film. The use of photochromic molecules allowed us to demonstrate all-optical control of phase singularities. Our results highlight what we believe to be both a new application of strong coupling and a new, extraordinarily simple platform for singular optics.


SPI/MC film fabrication

Polymethyl methacrylate (PMMA, molar weight 450,000) was used as a host matrix for SPI (Sigma-Aldrich; see Supplementary Section 1 for SPI and MC molecular structures and MC transmission spectrum). PMMA was dissolved in toluene. SPI was then dissolved in the PMMA-toluene solution with a weight ratio of 3:2 SPI to PMMA. SPI/PMMA films were deposited on a silicon wafer by spin-coating three layers each with a spin speed of 2000 rpm. This produced film thicknesses in the range 84–681 nm.

Optical constants of SPI and MC

Optical constants of SPI and MC were determined using CompleteEASE® 6.51 (J.A. Woollam Co., Inc.). The permittivity of SPI (ϵSPI) was modelled as a Cauchy dielectric:

$${\epsilon }_{{{{{{{\rm{SPI}}}}}}}}={\left(A+B{\omega }^{2}+C{\omega }^{4}\right)}^{2}$$

where A = 1.584, B = −4.171 × 10−34 rad−2 s2 and C = 2.105 × 10−64 rad−4 s4. The permittivity of MC (ϵMC) was modelled with a Lorentz oscillator (ϵL) and a pole in the ultraviolet (ϵUV):

$${\epsilon }_{{{{{{{\rm{MC}}}}}}}}={\epsilon }_{\infty }+{\epsilon }_{{{{{{{\rm{UV}}}}}}}}+{\epsilon }_{L}$$
$$={\epsilon }_{\infty }+\frac{{A}_{{{{{{{\rm{UV}}}}}}}}}{{E}_{{{{{{{\rm{UV}}}}}}}}^{2}-{(\hslash \omega )}^{2}}+\frac{f}{{E}_{L}^{2}-{(\hslash \omega )}^{2}-i\hslash \omega {\gamma }_{L}},$$

where ϵ = 0.839, AUV = 102.133 eV2, EUV = 7.970 eV, f = 0.7226 eV2, EL = 2.215 eV, γL = 0.3587 eV.


Spectroscopic ellipsometry was performed with a J. A. Woollam Co. M-2000XI which measures Ψ and Δ in the wavelength range 210–1690 nm, with a wavelength step of 1.5 nm for 210–1000 nm and 3.5 nm for 1000–1690 nm. Ψ is the ratio of the field reflection coefficients for p- and s-polarised light (the moduli of the complex Fresnel reflection coefficients for p- and s-polarised, rp and rs respectively); Δ is the phase difference between these coefficients such that \({r}_{p}/{r}_{s}=\tan ({{\Psi }}){e}^{i{{\Delta }}}\). Ellipsometry measures the ratio of two quantities, making it a very low-noise, sensitive technique. This makes it possible to confidently identify points at which ρ = 0 where we would expect to see phase singularities.

Coupled oscillator fit

The absence of mid-polariton bands in Fig. 1e suggests that the strong coupling we observe can be best modelled by a 2N coupled oscillator model42, meaning the coupling between the MC resonance and each43 leaky mode can be described by the following equation:

$$\left(\begin{array}{cc}{E}_{{{{{{{\rm{MC}}}}}}}}&{g}_{j}\\ {g}_{j}&{E}_{j}\end{array}\right)\left(\begin{array}{c}{a}_{{{{{{{\rm{(L,U)}}}}}}}j}\\ {b}_{{{{{{{\rm{(L,U)}}}}}}}j}\end{array}\right)={E}_{{{{{{{\rm{(L,U)}}}}}}}j}\left(\begin{array}{c}{a}_{{{{{{{\rm{(L,U)}}}}}}}j}\\ {b}_{{{{{{{\rm{(L,U)}}}}}}}j}\end{array}\right).$$

EMC is the MC resonance energy, gj is the coupling strength between MC and the jth leaky mode, Ej is the energy of the jth leaky mode, E(L,U)j are the lower and upper polariton branch energies associated with the jth leaky mode and a(L,U)j2 and b(L,U)j2 are the Hopfield coefficients describing the mixing of the MC resonance with the jth leaky mode.

Reporting Summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.