Superresolution reflection microscopy via absorbance modulation: a theoretical study

Robert Kowarsch; Claudia Geisler; Alexander Egner; Christian Rembe

doi:10.1364/OE.26.005327

1. Introduction

In far-field optical nanoscopy, on- and off-switching of molecular fluorescence [1–5] overcomes Abbe’s diffraction limit [6]. Though the underlying concept of using optically driven molecular transitions is not limited to switch between fluorescent and non-fluorescent states [7,8], the hitherto realized superresolution-imaging modalities almost exclusively rely on fluorescence contrast [9]. Overcoming the diffraction limit in more general imaging contrasts (without relying on fluorescence) is usually enabled by near-field optical microscopy, which employs probes such as subwavelength apertures or sharp tips at close proximity to the sample surface in order to generate highly localized light fields or collect near-field information [10]. Since precise distance control between surface and probe is necessary, these implementations are challenging and difficult to parallelize.

Dynamic subwavelength apertures in direct contact with the sample surface have been realized via optically saturable transitions in photochromic compounds for resolution enhancement in optical data storage [11] and optical lithography [12–15]. The latter is known in literature as absorbance-modulation optical lithography (AMOL) [16]. The absorbance of the absorbance-modulation layer (AML) [15], which is placed on top of the sample surface, is modulated by light. Upon irradiation with an intensity distribution of wavelength λ₁ featuring a central region of zero intensity, the spectral properties of the AML are changed for a second wavelength λ₂ from transparent to opaque. Only in the central region, the film remains transparent, thereby generating an aperture for wavelength λ₂.The application of this concept to nanoscale imaging was introduced as absorbance-modulation imaging (AMI) [17]. So far, AMI has only been demonstrated in transmission mode [18], where imaging of structures spaced by distances as small as λ/10 has been indicated [19].

Several simulation studies and theoretical models have been published either about the photochromic process in the AML alone [20–23] or in combination with a beam-propagation model [14,24,25]. However, all published models, being angled at lithographic applications, only consider the forward photon flux and the transmitted light distribution after forward propagation through the AML.

In this paper, AMI in reflection mode is studied for the first time by employing an augmented simulation model. In our simulations, we especially consider the effect of the back-reflected photon flux on the photochromic processes in the AML and account for diffraction losses in the subwavelength aperture for the reflected measurement light. As a new microscopy technique requires simplified equations to estimate the dependence of the resolution enhancement on the relevant design parameters, and because complex simulation models are only available to a limited number of experts, we also derived a simplified analytical equation to easily calculate the expected resolution. This equation shows a clear similarity to the one known for stimulated emission depletion (STED) microscopy [26]. We further analyze figure of merits for AMI and draw conclusions on its feasibility for confocal reflection nanoscopy. This imaging modality has the potential to make superresolution imaging with visible light and far-field optics available for a wide range of samples in material science, micro- or nanoelectromechanical systems (MEMS or NEMS) or even life science, since it is independent of the fluorescence contrast and is not limited to transparent substrates.

2. Materials and methods

2.1 Confocal microscope modeling

For enhancement of the lateral resolution, a subwavelength aperture is created in the AML directly attached to the sample surface (see Fig. 1) by an axially symmetric confining spot with a central region of zero intensity. This subwavelength aperture absorbs the out-of-center regions of the diffraction-limited, incident measurement spot, thus, confining its lateral dimension below the diffraction limit. The application of an annular-shaped spot is common in STED microscopy and can be attained by a vortex phase plate [27]. The incorporation for lateral superresolution was already proposed by Hell [7] and demonstrated experimentally for absorbance modulation by Tsai [19] or Wei [13]. The optimal intensity distributions are generally dependent on the application. In lithography, for example, the employment of standing optical waves via interference provides uniaxial superresolution [12,23,24].

Fig. 1 Scheme for the generation of a dynamic subwavelength aperture in an absorbance-modulation layer by the far-field radiation of a confining beam (creating an annular spot on the AML, in blue) and a diffraction-limited measurement beam (in red). Therefore, the optically saturable absorbance at the measurement wavelength λ₂ of a unimolecular photochrome between a state A (transparent for λ₂) and state B (opaque for λ₂) is exploited. The intended radiative transition from state A to B is induced by absorption of a photon at the confining wavelength λ₁ and for the transition from state B to A at measurement wavelength λ₂. The wavelengths strongly depend on the photochemical properties of the utilized photochrome and are typically in the near-UV to visible wavelength range [12,19].

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For modeling the imaging properties of a confocal microscope, the point spread functions (PSF) of both measurement and confining beam in the focal plane on the AML are approximated with Laguerre-Gaussian polynomials. The approximation of the PSFs as Laguerre-Gaussian polynomials is well met, if the obscured power at the system aperture is neglectable, with the drawback of not fully exploiting the numerical aperture (NA) of the microscope objective [28]. Even if diffraction to an Airy-shaped PSF occurs for the measurement beam, the intensity distribution near the symmetry axis can be fit by a Gaussian beam (TEM₀₀ mode) [29] (see appendix). For the confining beam the diffraction-limited, annular-shaped PSF is approximated by a first-order Laguerre-Gaussian beam (TEM_01* mode) [30]. Near the zero-intensity minimum (at the symmetry axis) this approximation fits well, which is crucial for the estimation of the lateral resolution enhancement.

The confocal detection was modeled as the multiplication of the intensity distribution exiting the AML with the detection PSF (at the AML surface) formed by the confocal pinhole according to Wilson [31]. For our model, we chose a confocal pinhole size of 1 Airy unit ( = 1.22 λ / NA), which corresponds to the diameter of the first dark fringe of an Airy-shaped intensity distribution in the pinhole plane. However, for real superresolution capability, the lateral resolution enhancement by the subwavelength aperture dominates over the confocalization. We employed the confocalization due to its additional suppression of out-of-center signals [28] and as a benchmark for lateral resolution in conventional reflection microscopy.

2.2 Generation of subwavelength aperture

For modeling the generation of the subwavelength aperture in the AML, the interdependency between photochrome concentration and photon flux density needs to be considered. For our model a unimolecular photochromism of type p is assumed [32]. This means that a molecule undergoes a reversible transformation between two stable states A and B, which is induced photochemically (see Fig. 1). The radiative transition from state A to state B is induced by the absorption of photons at the confining wavelength λ₁, whereas the transition B→A is triggered by photons at wavelength λ₂. Since wavelength λ₂ is used for imaging purposes in this paper, it is further referred as measurement wavelength. The stable states A and B possess different absorbance spectra due to the specific transition energies, which allows to reversibly switch the AML from transparent at λ₂ (state A) to opaque at λ₂ (state B). As a parasitic effect, the absorption of photons at λ₁ in state B also induces a transition to state A and vice versa. Thus, under irradiation with both wavelengths, the photochrome in the AML undergoes a dynamic process until a photostationary state (equilibrium) is reached [32]. The following rate equation of the photochemical reaction in a bistable photochrome has been used in a simplified form in previous models of AMOL processes [12,20,22,33]. The photochrome-concentration change $d c_{A} (r, z, t) / d t$ in the transparent state A at radial and axial positions r and z and at time t is

\begin{array}{l} \frac{d}{d t} c_{A} (r, z, t) = - φ_{λ 1}^{} (r, z, t) [σ_{A \to B}^{λ 1} c_{A} (r, z, t) - σ_{B \to A}^{λ 1} c_{B} (r, z, t)] \\ + φ_{λ 2}^{} (r, z, t) [σ_{B \to A}^{λ 2} c_{B} (r, z, t) - σ_{A \to B}^{λ 2} c_{A} (r, z, t)] + k_{B} c_{B}^{} (r, z, t) \end{array}

where

c_{A} (r, z, t)

and

c_{B} (r, z, t)

are the photochrome concentrations in state A or B respectively (in mol/m³) and

ϕ_{λ 1}^{} (r, z, t)

and

φ_{λ 2}^{} (r, z, t)

are the local photon flux densities at the confining and measurement wavelength respectively (in photons/s/m²). The concentration change depends on the interaction cross section

σ_{A \to B}^{λ 1}

and

σ_{A \to B}^{λ 2}

for the transition A→B and vice versa (in m²). The thermal rate k_B (in s⁻¹) accounts for the spontaneous relaxation of state B. All photochromes within the AML with the total homogeneous concentration c_total are assumed to be incorporated into the process. Assuming a fast transition between the states, the local concentration changes follow the equation

d c_{A} / d t = - d c_{B} / d t

at each time t.

For modeling AMI in reflection, we consider the influence of the photon flux reflected at the (flat) sample surface to the photochromic process, which has never been incorporated to published AMOL or AMI models before. For simplification, we assume the light fields to be unpolarized and the photochromic molecules in the AML to be arbitrarily oriented. Thus, the electromagnetic field near the focus can be described by an axially symmetric scalar field [34] and the absorption and interaction cross sections are given by effective (orientation-averaged) cross sections (given in appendix). Thus, we introduce a local omnidirectional photon flux density $φ_{λ}^{\leftrightarrow} (r, z, t)$ at both incorporated wavelengths as a superposition of all local photon flux densities. According to Lambert-Beer’s law [35], the photon flux decays exponentially along its penetration path within the AML of total thickness D. For the generation of the absorbing aperture, it is assumed that the incoming photon flux and the reflected photon flux at both wavelengths dominate the process (oblique components are neglectable). Hence, the local superposition of the photon flux density $φ_{λ 2}^{\to} (r, z, t)$ in direction of the impinging beams and of the photon flux density $φ_{λ 2}^{\leftarrow} (r, z, t)$ reflected from the flat sample surface (at radial and axial positions r and z and at time t) is

\begin{array}{l} φ_{λ 2}^{\leftrightarrow} (r, z, t) = φ_{λ 2}^{\to} (r, z, t) + φ_{λ 2}^{\leftarrow} (r, z, t) \\ = φ_{λ 2, in}^{} (r) \exp [- \int_{0}^{z} d ζ A_{λ 2} (r, ζ, t)] . \\ + φ_{λ 2, in}^{} (r) R_{λ 2} \exp [- \int_{0}^{D} d ζ A_{λ 2} (r, ζ, t) - \int_{z}^{D} d ζ A_{λ 2} (r, ζ, t)] \end{array}

The local absorbance

A_{λ 2} (r, ζ, t)

at the measurement wavelength λ₂ is

A_{λ 2}^{} (r, ζ, t) = ε_{B}^{λ 2} c_{B} (r, ζ, t) + ε_{A}^{λ 2} c_{A} (r, ζ, t)

with the incident photon flux density

φ_{λ 2, in}^{} (r)

on the surface of the AML, the (natural logarithmic) absorption cross sections

ε_{B}^{λ 2}

and

ε_{A}^{λ 2}

, the spectral reflectivity of the flat sample surface R_λ₂, and the integration variable ζ for the axial position. Analogous equations apply to the confining wavelength λ₁.

As the photon fluxes decrease over the depth, the local photochrome concentrations in state A and B are influenced. This nonlinear interdependency of the Eqs. (1) and (2) points out the complexity of the model, which can only be evaluated numerically.

2.3 Imaging through the subwavelength aperture

Diffraction losses at the measurement wavelength occur unpreventably at the subwavelength AML aperture. Therefore, we incorporated a beam-propagation model, which propagates the incoming (axially symmetric) complex electric field E_λ_2,in, through the AML in photostationary state by applying the Huygens-Fresnel principle [35]. For the calculation, the propagation through the AML with ideal reflection at the sample surface was replaced by propagation through an AML with double thickness (first the original AML and then its mirrored version at the sample surface). It considers the modified local photostationary absorbance A_stat,_λ₂ along the path, which is thus defined for 0 ≤ z ≤ 2D. The exiting complex electric field at z = 2D, which is therefore calculated by the superposition of spherical waves originating from the AML entrance plane (z = 0), is

\begin{array}{l} E_{λ 2, out}^{} (r, ϕ, z = 2 D) = \\ \int_{0}^{\infty} r^{'} d r^{'} E_{λ 2, in}^{} (r^{'}, z = 0) [\int_{0}^{2 π} d ϕ^{'} \frac{\exp (j k | s |)}{| s |} \cos (θ) \exp (- \frac{1}{2} \int_{s}^{} d s A_{stat, λ 2})] \end{array}

with radial coordinates r and rʹ, angular coordinates ϕ and ϕʹ, wave number

k = 2 π n_{AML} / λ_{2}

with refractive index of the AML n_AML, the straight path s between

S^{'} = (r^{'}, ϕ^{'}, z = 0)

and

S = (r, ϕ, z = 2 D)

, the distance |s| between points S and Sʹ, and the angle θ between vector

S - S^{'}

and the inward normal to the AML. The obliquity factor cos(θ) is the same as in the Rayleigh-Sommerfeld diffraction formula [35].

We model the whole behavior of the AML as a sequence of aperture generation and measurement through the subwavelength aperture. Only the latter considers diffraction losses within the heterogeneously absorbing AML (in photostationary state). We therefore neglect that the diffracted photon flux feeds back to the photochromic process and, thus, influences the absorbance distribution. We estimated that this interdependency is neglectable for strong absorptions and thin AML. This approach was also taken by Foulkes [24] for modelling absorbance modulation in lithography.

At the boundary between the dielectric ambient medium with the refractive index n_amb and the AML with n_AML, reflection according to Fresnel occurs [35]. For AMOL and AMI in transmission, the influence of the reflection at the boundary is minor and can remain unconsidered. On the contrary, for AMI in reflection it is critical, since the reflected signal is superimposed to the measurement signal. Considering the significant imaginary part of the refractive index of strongly absorbing media [35], the power reflectance $R_{stat, λ 2}^{AML}$ at the measurement wavelength λ₂ in the photostationary state is

R_{stat, λ 2}^{AML} (r, z = 0) = {| \frac{n_{amb} - n_{AML} + j \frac{λ_{2}}{4 π} A_{stat, λ 2} (r, z = 0)}{n_{amb} + n_{AML} - j \frac{λ_{2}}{4 π} A_{stat, λ 2} (r, z = 0)} |}^{2}

when assuming a fully transparent ambient medium. Thus, the reflectance at this boundary increases with the absorbance A_stat,_λ₂ of the AML due to boundary conditions of the strongly absorbed electromagnetic field, especially when the penetration depth l_λ (inverse of absorbance:

l_{λ} = A_{stat, λ}^{- 1}

) is in the range of the wavelength and below. The entering power into the AML (after reflection at the boundary) is further denoted as P_1,in for the confining wavelength and P_2,in for the measurement wavelength respectively. Despite the absorbance-induced reflection, it seems desirable for us to obtain a strong absorption within the AML to diminish the influence of the sample-surface reflectance on the absorption distribution. However, for imaging the power reflectance at the AML aperture is a constant offset on the detector, which can be compensated for. Further, the application of immersion to the imaging decreases the overall reflectance by matching of the refractive indices at the boundary. For instance for a penetration depth of l_λ₂ = λ₂ / 8, the power reflectance is nearly 10% without the application of immersion (n_AML = 1.5 and n_amb = 1.0). With a perfect matching of the refractive indices (n_AML = n_amb = 1.5) a reduction to 4.3% is possible.

For the simplification of the aperture-generation simulation, it was assumed that the entering powers of both beams into the AML remain constant during the process despite of the varying absorption-induced reflectivity at the boundary. This seems adequate due to the minor influence on the absorbance distribution. However, for the beam propagation through the photostationary absorbance distribution (especially for the noise analysis), the reflectance at the boundary was considered.

2.4 Choice of simulation parameters

The back-reflected measurement light exiting the AML is composed of two contributions: light from the central zero-intensity region of the confining beam (wanted signal) and the residual (not fully suppressed) light from out-of-center regions (unwanted signal). The relative transmission of these out-of-center regions and the central region is an important parameter for the confinement of the measurement beam and the quality of the imaging. Since, ideally the central region is fully transparent, while the out-of-center regions are fully opaque for the measurement light, a transmission contrast

C T_{λ} = \frac{T_{transparent} (λ)}{T_{opaque} (λ)}

is introduced following Pariani [23]. It is the ratio of the transmission of the transparent and the opaque state of the photochromic layer. CT_λ₂ gives an estimate for the power contribution of out-of-center regions to the total power of the back-reflected measurement light. For interferometric detection the parasitic light components are especially critical. Given three-wave mixing, a contrast of CT_λ_2,target = 1000 is necessary [36] to suppress the amplitude errors in coherent detection with a confocal scanning microscope to less than 5%. For incoherent detection, an even lower suppression may be suitable. From Eqs. (2) and (3) for a complete population of state A (fully transparent) and B (fully opaque), the expected transmission contrast at the measurement wavelength can be estimated as

ln (C T_{λ 2}^{}) = 2 D (ε_{B}^{λ 2} - ε_{A}^{λ 2}) c_{total} .

Note that in contrast to Pariani's definition [23], the AML thickness is considered twice due to the reflection at the sample surface. This equation shows that a high difference in absorption cross sections between the two photochromic states at the measurement wavelength is highly beneficial.

For the simulation, the photochromic diarylethene molecule 1,2-bis(5,5′-dimethyl-2,2′-bithiophen-yl) perfluorocyclopent-1-en (also denoted as BTE) in a polymethylmethacrylate (PMMA) matrix was chosen. Andrew successfully applied this photochrome to lithography and published the photochromic parameters [12] (see appendix). The choice of the wavelength λ₁ and λ₂ strongly depends on the wavelength-dependency of these photochromic parameters in both states. BTE provides a high transmission contrast CT_λ₂ for a measurement wavelength of λ₂ = 633 nm and a high interaction cross section in state A at the confining wavelength λ₁ = 325 nm. These photochromic properties qualify BTE for superresolution in AMI. Andrew realized BTE concentrations of 2990 mol/m³ in a PMMA matrix for a AML thickness of 410 nm [12]. The depth of focus (≈ n_AML λ / NA²) for the incorporated beams has to be larger than twice the AML thickness. Thus, for AMI in reflection, a reduction of the AML thickness is necessary for a preferably high NA of the microscope objective.

A higher absorbance can be realized in respect to the state of the art by increasing either the photochrome concentration c_total or the absorption cross sections ε for the different photochromic states and wavelengths. We introduce a parameter γ by

ε_{2} c_{total, 2} = γ ε_{1} c_{total, 1}

to adjust the absorbance of the AML in our simulations. Gamma is γ = 1 for absorption cross sections

ε_{A}^{λ 1}

,

ε_{A}^{λ 2}

,

ε_{B}^{λ 1}

, and

ε_{B}^{λ 2}

of BTE and c_total = 1000 mol/m³. Thus, for a concentration of 2990 mol/m³ the γ-value is 2.99 for BTE. In Fig. 2, the transmission contrast CT_λ₂ is depicted over γ-values for different AML thicknesses D [according to Eq. (7)]. From the required transmission contrast CT_λ_2,target = 1000, the necessary γ-values can be estimated to be γ ≥ 7.3 for layer thickness of 100 nm and γ ≥ 3.7 for 200 nm.

Fig. 2 Transmission contrast CT_λ₂ at the measurement wavelength over the γ-value for different AML thicknesses. For a 100 nm thick AML, the target transmission contrast CT_λ_2,target = 1000 is achieved for γ-values of γ ≥ 7.3.

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Furthermore, for a constant suppression of unwanted light from out-of-center regions, a constant ratio between confining and measurement intensity over radial position is necessary. This requirement is met by choosing similar waist radii for the PSFs of the measurement and confining beam.

3. Results and discussion

3.1 Dynamic analysis of aperture generation

For superresolution imaging (of an extended field of view), the pixel dwell time is an important figure of merit, since it determines the image acquisition speed. Since the AML aperture is reformed at each pixel, the temporal behavior of aperture generation needs to be considered. For AMOL, Warner [20] already published advanced analytical solutions, which are not applicable since they neglect the reflection at the sample and imaging purposes. We show that the time to equilibrium can be expressed by an analytical solution of Eqs. (1) and (2) with the assumption of low absorbance over thickness and a thermally stable photochrome (k_B = 0). The photochromic process follows a limited growth similar to STED microscopy [8,26]. At the time to equilibrium

t_{stat} > 5 {[(1 + R_{λ 1}) φ_{λ 1, in}^{} (σ_{A \to B}^{λ 1} + σ_{B \to A}^{λ 1}) + (1 + R_{λ 2}) φ_{λ 2, in}^{} (σ_{B \to A}^{λ 2} + σ_{A \to B}^{λ 2})]}^{- 1}

the difference in photochrome concentration to the asymptotic concentration is less than 1% and equilibrium is assumed. However, this criterion for equilibrium is strongly dependent on the requirements for the measurement technique.

From Eq. (9) follows that the higher the photon flux densities are, the faster the photochromic process converges to equilibrium, which is fully consistent with our simulations. With the low quantum efficiency for the transition B→A in BTE, the time to equilibrium is mainly dependent on the photon flux at the confining wavelength λ₁. A time to equilibrium of t_stat > 0.1 µs can be estimated for a peak confining intensity of 0.5 mW/µm² at the maximum of the annular confining PSF (NA₁ ≈ 0.4 and 1 mW light power). However, due to the vanishing photon fluxes for increasing penetration depth into highly absorbing AML, the time increases significantly when considering aperture generation over the full depth of the AML [20]. For example, at the maximum of the confining beam, the simulated time to equilibrium increases to 0.2 µs (for γ = 3.7, D = 200 nm, and equal light power of both beams).

Shorter pixel dwell times and, thus, faster image acquisition might be feasible with sequential and pulsed irradiation at the incorporated wavelengths. Published data rates for superresolution optical discs with absorbance modulation [37] indicate an image acquisition speed for AMI in reflection comparable to STED microscopy.

3.2 Lateral resolution enhancement

Over a radial cross section, the absorption distribution provides a transparent, axially symmetric channel within the AML [38] (see Fig. 1). It confines the impinging diffraction-limited measurement beam to superresolution due to absorption in the out-of-center regions. Between the transparent central region (where photochromes in state A dominate) and the opaque aperture (where state B dominates), the diameter 2r_AML of the subwavelength AML aperture is defined at 50% transmission. For resolution enhancement, this aperture diameter has to be smaller than the diffraction-limited spot size (2r_AML < Δx). Thus, the influence of the incident measurement PSF and the confocal detection can be neglected, and the confined lateral spot size Δx_AMI is solely determined by the FWHM (full width at half maximum) of the AML transmission profile yielding

Δ x_{AMI} \approx 2 r_{AML} .

At the radial position r_AML, the photochromes in the opaque state B (in equilibrium) exist with the ratio β_HM (0 ≤ β_HM ≤1) for 50% transmission defined by

c_{B, stat} (r_{AML}) = c_{B} (r_{AML}, t \geq t_{stat}) = β_{HM} c_{total} and

c_{A,stat} (r_{AML}) = c_{A} (r_{AML}, t \geq t_{stat}) = (1 - β_{HM}) c_{total}

with the total concentration c_total and the time to equilibrium t_stat [see Eq. (9)]. Further, a homogeneous concentration over the AML thickness is assumed [

c_{A,stat} (r, z) = c_{A,stat} (r)

and

c_{B,stat} (r, z) = c_{B,stat} (r)

]. The actual value for β_HM at 50% transmission is mainly dependent on the absorption cross sections of the incorporated photochrome. The rate equation [Eq. (1)] of a thermally stable photochrome (k_B=0) in the photostationary state [

d c_{A} (t) / dt = d c_{B} (t) / dt = 0

], therefore, is

\begin{array}{l} φ_{λ 1, stat}^{\leftrightarrow} (r_{AML}) [σ_{A \to B}^{λ 1} (1 - β_{HM}) c_{total} - σ_{B \to A}^{λ 1} β_{HM} c_{total}] \\ = φ_{λ 2, stat}^{\leftrightarrow} (r_{AML}) [σ_{B \to A}^{λ 2} β_{HM} c_{total} - σ_{A \to B}^{λ 2} (1 - β_{HM}) c_{total}] \end{array}

with the omnidirectional photon fluxes

φ_{λ, stat}^{\leftrightarrow}

at both wavelengths. For simplification, it is assumed that the (parasitic) interaction cross sections

σ_{B \to A}^{λ 1}

and

σ_{A \to B}^{λ 2}

are neglectable against the (wanted) interaction cross sections

σ_{A \to B}^{λ 1}

and

σ_{B \to A}^{λ 2}

. These assumptions simplify Eq. (13) to

\frac{φ_{λ 2, s t a t}^{\leftrightarrow} (r_{A M L})}{φ_{λ 1, s t a t}^{\leftrightarrow} (r_{A M L})} = \frac{1 - β_{H M}}{β_{H M}} \frac{σ_{A \to B}^{λ 1}}{σ_{B \to A}^{λ 2}} .

Furthermore, we introduce the efficacy α to consider the actual variations of both the photon-flux ratio and the concentrations over the radial and axial position r and z. Thus, the effective ratio of the omnidirectional photon fluxes

φ_{λ, stat}^{\leftrightarrow}

is expressed by the ratio of the incident photon fluxes φ_λ_,in into the AML as

α \frac{φ_{λ 2, in}^{} (r_{AML})}{φ_{λ 1, in}^{} (r_{AML})} \approx \frac{φ_{λ 2, stat}^{\leftrightarrow} (r {}_{AML})}{φ_{λ 1, stat}^{\leftrightarrow} (r {}_{AML})} = \frac{1 - β_{H M}}{β_{H M}} \frac{σ_{A \to B}^{λ 1}}{σ_{B \to A}^{λ 2}} .

With the parabolic approximations of the incorporated PSFs near the symmetry axis (small r) [see Eqs. (23) and (25) in appendix], it follows

\frac{φ_{λ 2, in}^{} (r_{AML})}{φ_{λ 1, in}^{} (r_{AML})} \approx \frac{λ_{2}}{λ_{1}} \frac{\frac{2}{π} \frac{P_{2}}{W_{2}^{2}} [1 - 2 {(\frac{r_{AML}}{W_{2}})}^{2}]}{\frac{4}{π} \frac{P_{1}}{W_{1}^{2}} {(\frac{r_{AML}}{W_{1}})}^{2}} \approx \frac{1}{α} \frac{(1 - β_{HM})}{β_{HM}} \frac{σ_{A \to B}^{λ 1}}{σ_{B \to A}^{λ 2}}

with the waist radius W₁ and the total power P₁ of the confining PSF. W₂ and P₂ apply accordingly to the measurement PSF. Hence, with Eq. (10) the confined AMI spot size is

Δ x_{AMI} \approx 2 r_{AML}^{} = \frac{\sqrt{2} W_{_{2}}^{}}{\sqrt{1 + \frac{1}{α} \frac{(1 - β_{HM})}{β_{HM}} \frac{σ_{A \to B}^{λ 1}}{σ_{B \to A}^{λ 2}} \frac{λ_{1}}{λ_{2}} \frac{W_{2}^{4}}{W_{1}^{4}} P R}}

with the power ratio PR = P₁ / P₂ of confining to measurement power, which is the decisive figure to influence the resolution enhancement [16,20,22,24,33].

The lateral resolution factor κ is defined as the confined spot size Δx_AMI compared to the diffraction-limited spot size Δx at FWHM as

κ = \frac{Δ x_{AMI}}{Δ x} .

Note that a better lateral resolution corresponds to a lower κ-value, e.g. κ = 0.5 means that the resolution increases twofold. For normalization, the diffraction-limited spot size is assumed as $Δ x \approx \sqrt{2} W_{2}$ . Thus, in analogy to the resolution enhancement in STED microscopy [26], where Hell introduced a saturation intensity as a characteristic system constant, the lateral resolution factor κ can be expressed with Eq. (17) as

κ = \frac{Δ x_{AMI}}{Δ x} \approx \frac{1}{\sqrt{1 + \frac{P R}{P R_{sat}}}}

with the characteristic saturation power ratio

P R_{sat} = Γ \frac{σ_{B \to A}^{λ 2}}{σ_{A \to B}^{λ 1}} \frac{λ_{2}}{λ_{1}} {(\frac{W_{1}}{W_{2}})}^{4}

with

Γ = α \cdot β_{HM} / (1 - β_{HM})

. This equation shows that the resolution enhancement can be influenced by the ratio of interaction cross sections at the wavelengths λ₁ and λ₂ and, therefore, by the wavelength-dependent properties of the photochromic molecule. The resolution further depends on the ratio of the beam waist radii. We fitted Eqs. (19) and (20) to our simulation results with BTE and obtained Γ ≈ 1.2 for thin AML (D ≤ 200 nm) for a theoretic saturation power ratio of

P R_{sat} = Γ \cdot 4.6 \times 10^{- 3}

. Variations of additional parameters (AML thickness and γ-value) [see Figs. 3(a) and 3(b)] have only moderate influence on the Γ-value (Γ is in the range of 1.2 to 2.0). Thus, the influence of these parameters can be neglected in a first design step. Due to the simplifications, the introduced estimation is even applicable for predesign of AMI in transmission and even AMOL with a diffraction-limited or Gaussian measurement spot and an annular-shaped confinement.

Fig. 3 Simulated lateral resolution factor κ as a function of the power ratio PR of the incorporated beams for BTE (NA₂ = 0.6, W₁ = W₂, and confocal detection). The resolution enhancement is depicted for different AML thicknesses D (at constant γ = 2.99) (a) and varying γ-values (at constant AML thickness of D = 100 nm) (b). The (dashed) line depicts the derived estimation (Eqs. (19) and (20) with $P R_{sat} = Γ \cdot 4.6 \times 10^{- 3}$ and Γ ≈ 1.2).

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Figures 3(a) and 3(b) show that the general behavior of the simulated resolution enhancement follows the theoretic curve (dashed line). For low PR-values (PR < 10⁻²), deviations from theory appear for AML thicknesses which are comparatively large with respect to the focal depth of the beams [Fig. 3(a)]. When the power ratio PR increases, the AML aperture gets more confined and the resolution enhancement improves. However, even when the absorbance saturates in out-of-center regions, the incident radiation is not fully suppressed due to the finite absorbance and thickness of the AML. This residual radiation causes a background to the confined spot. Additionally, light is diffracted out of the subwavelength aperture into out-of-center regions. For high PR-values (dependent on the AML thickness and γ-value), these background contributions dominate the transmitted spot profile and a minimum for the resolution factor κ emerges [see Figs. 3(a) and 3(b)], which corresponds to the maximum superresolution capability. The superresolution capability can be maximized by the choice of a thick AML at state-of-the-art γ-values [see Fig. 3(a)] or an increase of the γ-value for thin AML [Fig. 3(b)].

The expected power ratios necessary for lateral resolution improvement are significant lower than those typically published for AMOL with BTE [23,33]. For lithography, the visible wavelength is utilized as confining wavelength to enhance the lateral resolution of the UV intensity pattern for exposure of the resist. We reverse the function of the wavelengths for diarylethenes, which has already been proposed for superresolution discs [37] and for AMOL [14,23].

3.3 Signal-to-background and noise analysis

An important criterion for image quality is the suppression of out-of-center signal. The back-reflected power from the sample, which is transmitted through the subwavelength AML aperture (wanted signal), therefore, has to exceed the residual, non-suppressed light from out-of-center regions (background) combined with noise. We defined a signal-to-background ratio as

S B R = \frac{P_{det, 1 / e²}}{P_{det} - P_{det, 1 / e²}}

with the confocally detected power $P_{det, 1 / e²}$ (at the measurement wavelength) enclosed within the 1/e² diameter of the confined spot and the confocally detected total power P_det. We estimate that a SBR-value larger than 2 is necessary to enable superresolution microscopy. The simulated dependency between the SBR-value and the resolution enhancement is shown in Figs. 4(a) and 4(b) (for homogeneous ideal reflectivity of the sample). Due to the necessary absorption and inevitable diffraction, less power is detected at increased resolution enhancement (smaller κ). Concurrently, residual background light therefore increases. As high absorption of residual light improves the SBR, an intermediate maximum SBR-value may arise. Figure 4(a) shows that for γ = 2.99 an AML thickness below 300 nm is not suitable to deliver a resolution factor better than 1/5 at acceptable SBR-values (larger than 2). However, with high γ ≥ 7.3, for an AML thickness of 100 nm a sufficient SBR is achieved at high resolution (κ = 1/5). This behavior is consistent with the introduced transmission contrast (Fig. 2). Therefore, the proposed Eq. (7) enables the choice of suitable photochromes for AMI prior to extensive simulation efforts.

Fig. 4 Simulated signal to background ratio SBR (a,b) and AML transmittance T_AML (c,d) over the resolution factor κ for BTE (NA₂ = 0.6, W₁ = W₂, and confocal detection). The SBR-value and the AML transmittance are shown for different AML thicknesses D (at constant γ = 2.99) (a,c) and varying γ-values (at constant AML thickness of D = 100 nm) (b,d). The theoretical transmittance in (c,d) is estimated by a simple circular-aperture model (dashed reference curve) with a diameter corresponding to the resolution factor κ.

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Beside the required suppression of the background, the (confocally detected) total power P_det has to exceed noise level. The simulated AML transmittance (T_AML = P_det / P_2,in) at the measurement wavelength λ₂ is shown in Figs. 4(c) and 4(d) (assuming ideal reflection at sample surface). The necessary absorption (for high SBR) causes a drop of the AML transmittance for increasing resolution enhancement (smaller κ). This is also shown with a simple model of a circular aperture with a diameter corresponding to the confined spot size [dashed reference in Figs. 4(c) and 4(d)]. For thick AML (at γ = 2.99), divergence and diffraction effects produce high losses, therefore, thin AML are desirable [Fig. 4(c)]

The noise floor on the (non-saturated) detector is formed by shot-noise due to the absorbance-induced reflection at the AML surface [see Eq. (5)]. As an example, for a lateral resolution factor κ = 0.1, AML thickness of 100 nm, and a γ-value of 7.3, the power reflectance from the AML surface even with immersion is 11%. However, the resulting signal-to-noise ratio between detected light power and light shot-noise power remains almost 85 dB (at 1 mW measurement power, detector quantum efficiency of 70% at 633 nm and exposure time of 1 ms). Thus, despite the high losses due to absorption and diffraction, superresolution imaging is feasible.

The results in this paper indicate that an AML thickness of 100 nm in combination with a high γ-value above 6 should perform well in experiments. Our findings demonstrate that power ratios PR below 1 are sufficient, which is advantageous to limit the energy influx to the AML. A signal-to-background ratio SBR larger than 3 and signal-to-noise ratio of better than 80 dB enable superresolution reflection microscopy with a resolution enhancement capability of up to 5 compared to the diffraction limit.

4. Conclusion

We composed a model for absorbance-modulation imaging in reflection for confocal scanning microscopy, which contains the reflection on the sample. This model includes the photochromic process, diffraction at the subwavelength aperture, reflection at the boundary and confocal microscope characteristics. We derived simple equations, which estimate the necessary saturation power ratio of the incorporated beams to achieve superresolution and the time to equilibrium for the photochromic process. These estimations clearly draw the analogy of AMI to STED microscopy. Further, an equation for the transmission contrast supports the selection of a suitable photochrome.

We conclude from the numerical simulations with our model that thin absorbance-modulation layers of 100 nm with high photochrome concentrations and high absorption cross sections (γ ≥ 6) allow to limit the effect of diffraction losses and to improve lateral resolution of more than a factor of five compared to diffraction limit with sufficient background suppression. Despite the high losses, the signal remains well above shot-noise limit even for strong absorbance-induced reflections at the surface of the absorbance-modulation layer. We show that our simple estimations agree well with the numerical simulation results and are helpful for the predesign of AMI microscopes.

Thus, AMI microscopy in reflection can be the key technology to bring unprecedented superresolution to optical analysis of technical surfaces, e.g. in material science, micro- and nanoelectromechanical systems (MEMS and NEMS) and life science.

Appendix

A. Spot profiles for aperture generation

The point spread functions (PSF) or spot profiles of the employed beams in the focal plane (of the microscope objective) are approximated with Laguerre-Gaussian polynomials. Especially if the exit-pupil diameter of the microscope objective is not wholly illuminated, a Gaussian intensity distribution is preserved in the focal plane [28].

In the article, we use the following equations for modeling the PSFs as photon flux densities (in photons/s/m²) with the photon energy E = h c / λ (with the Planck constant h, velocity of light c and wavelength λ). In Table 1 the employed equations for the photon flux densities for both axially symmetric PSF with their parabolic approximation near the symmetry axis (at radial position r = 0) are shown.

Table 1. Equations and approximations for the photon flux densities of the PSFs.

View Table | View all tables in this article

Here, r is the radial position, P₁ and P₂ are the enclosed power of each beam and W₁ and W₂ are the waist radii. For an Airy-shaped intensity distribution, an approximation can be found for a fitted Gaussian intensity distribution (area-preserving) with the waist diameter [29]

2 W_{2}^{} = \frac{2 \sqrt{2}}{π} \frac{λ_{2}}{N A_{2}} \approx 0.9 \frac{λ_{2}}{N A_{2}}

with the numerical aperture NA₂ at the wavelength λ₂.

B. Photochromic parameters

For the simulation, we used the photochromic parameters of diarylethene 1,2-bis(5,5′-dimethyl-2,2′-bithiophen-yl) perfluorocyclopent-1-en (referred as BTE) published by Andrew [12] (listed in Tables 2 and 3). Since Andrew measured these photochromic parameters with unpolarized radiation (Xenon arc lamp) in a solution with hexanes, we assumed them to be orientation-averaged cross sections. Note that the natural logarithmic (Napierian) parameters are given in the following tables. The chosen wavelengths are λ₁ = 325 nm and λ₂ = 633 nm.

Table 2. Photochromic parameters for BTE.

View Table | View all tables in this article

Table 3. Photochromic parameters for BTE (continued) .

View Table | View all tables in this article

The interaction cross sections can be calculated by multiplication of the specific quantum efficiency with the cross section of the initial state, which are enlisted in Table 3.

Andrew concludes [12] that due to the necessary photon energy for cyclization in BTE the transition A→B does not occur with irradiation at 633 nm ( $σ_{A \to B}^{λ 2} = 0$ ).

Further, we investigated the influence of the thermal rate of BTE with published values [12] in our simulation model and found a neglectable effect on our results due to the high rates from the intense photon fluxes.

Acknowledgment

We thank Andreas Schmidt, Wolfgang Maus-Friedrichs, Jörg Adams, Sebastian Dahle, and Christian Otto of Clausthal University of Technology for the constructive discussions about photochromism and thin film technology.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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PSF of confining beam (TEM01*)		Parabolic approximations (near the symmetry axis)
$φ_{λ 1, in}^{} (r) = \frac{λ_{1}}{h c} \frac{4}{π} \frac{P_{1}}{W_{1}^{2}} {(\frac{r}{W_{1}})}^{2} \exp [- 2 {(\frac{r}{W_{1}})}^{2}]$	(22)	$φ_{λ 1, in}^{} (r) \approx \frac{λ_{1}}{h c} \frac{4}{π} \frac{P_{1}}{W_{1}^{2}} {(\frac{r}{W_{1}})}^{2}$	(23)
PSF of measurement beam (TEM00)		Parabolic approximations (near the symmetry axis)
$φ_{λ 2, in}^{} (r) = \frac{λ_{2}}{h c} \frac{2}{π} \frac{P_{2}}{W_{2}^{2}} \exp [- 2 {(\frac{r}{W_{2}})}^{2}]$	(24)	$φ_{λ 2, in}^{} (r) \approx \frac{λ_{2}}{h c} \frac{2}{π} \frac{P_{2}}{W_{2}^{2}} [1 - 2 {(\frac{r}{W_{2}})}^{2}]$	(25)

Photochromic parameters of BTE	Value
Absorption cross section $ε_{A}^{λ 1}$ of state A for λ₁	$1.19 \times 10^{- 20} m^{2}$
Absorption cross section $ε_{B}^{λ 1}$ of state B for λ₁	$4.02 \times 10^{- 21} m^{2}$
Absorption cross section $ε_{A}^{λ 2}$ of state A for λ₂	$6.04 \times 10^{- 23} m^{2}$
Absorption cross section $ε_{B}^{λ 2}$ of state B for λ₂	$7.66 \times 10^{- 21} m^{2}$
Photoreaction quantum efficiency $Q E_{A \to B}^{}$	$0.24$
Photoreaction quantum efficiency $Q E_{B \to A}^{}$	$0.00088$

Photochromic parameters of BTE	Value
Interaction cross section $σ_{A \to B}^{λ 1}$	$2.86 \times 10^{- 21} m^{2}$
Parasitic interaction cross section $σ_{B \to A}^{λ 1}$	$3.54 \times 10^{- 24} m^{2}$
Interaction cross section $σ_{B \to A}^{λ 2}$	$6.74 \times 10^{- 24} m^{2}$
Parasitic interaction cross section $σ_{A \to B}^{λ 2}$	$0 m^{2}$
Thermal rate for spontaneous relaxation $k_{B}^{}$	$3 \times 10^{- 6} s^{- 1}$

PSF of confining beam (TEM01*)		Parabolic approximations (near the symmetry axis)
$φ_{λ 1, in}^{} (r) = \frac{λ_{1}}{h c} \frac{4}{π} \frac{P_{1}}{W_{1}^{2}} {(\frac{r}{W_{1}})}^{2} \exp [- 2 {(\frac{r}{W_{1}})}^{2}]$	(22)	$φ_{λ 1, in}^{} (r) \approx \frac{λ_{1}}{h c} \frac{4}{π} \frac{P_{1}}{W_{1}^{2}} {(\frac{r}{W_{1}})}^{2}$	(23)
PSF of measurement beam (TEM00)		Parabolic approximations (near the symmetry axis)
$φ_{λ 2, in}^{} (r) = \frac{λ_{2}}{h c} \frac{2}{π} \frac{P_{2}}{W_{2}^{2}} \exp [- 2 {(\frac{r}{W_{2}})}^{2}]$	(24)	$φ_{λ 2, in}^{} (r) \approx \frac{λ_{2}}{h c} \frac{2}{π} \frac{P_{2}}{W_{2}^{2}} [1 - 2 {(\frac{r}{W_{2}})}^{2}]$	(25)

Photochromic parameters of BTE	Value
Absorption cross section $ε_{A}^{λ 1}$ of state A for λ₁	$1.19 \times 10^{- 20} m^{2}$
Absorption cross section $ε_{B}^{λ 1}$ of state B for λ₁	$4.02 \times 10^{- 21} m^{2}$
Absorption cross section $ε_{A}^{λ 2}$ of state A for λ₂	$6.04 \times 10^{- 23} m^{2}$
Absorption cross section $ε_{B}^{λ 2}$ of state B for λ₂	$7.66 \times 10^{- 21} m^{2}$
Photoreaction quantum efficiency $Q E_{A \to B}^{}$	$0.24$
Photoreaction quantum efficiency $Q E_{B \to A}^{}$	$0.00088$

Superresolution reflection microscopy via absorbance modulation: a theoretical study

Abstract

1. Introduction

2. Materials and methods

2.1 Confocal microscope modeling

2.2 Generation of subwavelength aperture

2.3 Imaging through the subwavelength aperture

2.4 Choice of simulation parameters

3. Results and discussion

3.1 Dynamic analysis of aperture generation

3.2 Lateral resolution enhancement

3.3 Signal-to-background and noise analysis

4. Conclusion

Appendix

A. Spot profiles for aperture generation

B. Photochromic parameters

Acknowledgment

Disclosures

References and links

Cited By

Figures (4)

Tables (3)

Equations (22)

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