## Abstract

The structure of the inhibition patterns is important to the stimulated emission depletion (STED) microscopy. Usually, Laguerre-Gaussian (LG) beam and the central zero-intensity patterns created by inserting phase masks in Gaussian beams are used as the erase beam in STED microscopy. Aberration is generated when focusing beams through an interface between the media of the mismatched refractive indices. By use of the vectorial integral, the effects of such aberration on the shape of depletion patterns and the size of fluorescence emission spot in the STED microscopy are studied. Results are presented as a comparison between the aberration-free case and the aberrated cases.

© 2009 Optical Society of America

## 1. Introduction

A spatial intensity distribution with a center region of zero intensity surrounded by regions of intense illumination plays an important role in optics. Because of its special shape, it has been recently used in laser trapping [1–2], up-conversion fluorescence depletion microscopy [3–4] and in the concept of reversible saturable optically linear fluorescence transitions (RESOLFT) microscopy [5–8]. The idea of stimulated emission depletion (STED) microscopy is one of the most prominent examples of RESOLFT. STED microscopy has been demonstrated to increase the resolution to a few tens of nanometers [9–11]. In STED microscopy, the sample is illuminated with a pump beam and almost simultaneously, such zero-intensity patterns serve as the STED beam suppressing the fluorescence process at the periphery around the center through stimulated emission [7]. The rate of fluorescence depletion depends exponentially on the intensity of the erase beam [12–13] and the effect of the STED concept is mainly dominated by the quality of the depletion patterns. Ultimately, a number of depletion patterns have been proposed by the use of the phase masks [14–15] and also a systematic survey of the optimal design of phase plates to create inhibition modes has been made [16]. Laguerre-Gaussian helical (LG) beam has a helical phase structure at the center with the intensity distribution possessing annular intensity rings around a dark core and becomes a promising alternative as an erase beam [17].

In many microscopy applications, stratified media are often used as specimens and of particular interest is the diffraction distribution when focusing beams through an interface between materials of mismatched refractive indices. The general focusing problems of electromagnetic waves has been the area of investigation for a long time. The exact solution of electromagnetic fields focused into a homogeneous medium has been previously investigated by Wolf and Richards [18–19]. Based on their research, Török *et al*. gave a detailed analysis for the case of focusing through a plane interface [20] and a stratified medium [21]. Subsequently, in the paper of [22], different diffraction theories that describe the effect of interface on focusing distribution were compared and studied in depth. And Helseth has derived the solutions of focusing problems of beams of arbitrary polarization in homogeneous medium and through the interface [23–24]. When focusing beams through an interface, the aberration is introduced due to the mismatched refractive indices [20] and the presence of the aberration will lead to the structural modification of the focused spot [25–26].

In the view of the size and the shape of the diffraction pattern playing an important role in RESOLFT microscopy, we investigate the effects of aberration induced by the presence of the interface on the inhibition patterns by using the vectorial integral. The STED generation volume of using the de-excitation pattern created by a phase plate of central *π* -phase retardation aberrated by a glass to water interface was previously mentioned in the paper of [17]. However, no detailed studies seem to have been made on the influence of this aberration on the depletion patterns and the resolution of STED microscopy. The aim of the present study is to investigate the effects of the aberration introduced by focusing beams through a plane interface between media of mismatched refractive indices on the structure of the inhibition modes. In this paper, as the common depletion patterns, *x* -polarized and left circularly polarized Laguerre-Gaussian beams, the central zero-intensity patterns generated by inserting a central [27] and a semi-circular *λ*/2 phase masks [14] in Gaussian beams are mainly studied. When these modes are applied in STED microscopy, the resolution dependence on such aberrations is also examined. Our results for the presence of aberration have been compared with those of an ideal system.

## 2. Theory

Although diffraction problem of focusing through an interface between the media of mismatched indices is a simplified and fundamental model of the complex problems of focusing into stratified medium, research on this problem of the reflection and refraction of light at an interface can help offer our general understanding to the effects of aberration introduced by the refractive index mismatch. In our calculations, an optical system which consists of two media separated by a plane interface perpendicular to the optical axis is considered, as is shown in Fig. 1. The origin of the coordinate system is positioned at the Gaussian focus in the absence of the aberration. *z* = -*d* denotes the position of the interface in units of the wavelength. The absolute value of *d* is the displacement of the interface relative to the unaberrated Gaussian focus and is often defined as the focusing depth [17, 22]. *n _{j}* is the refractive index in the first (

*j*= 1) and the second (

*j*= 2) medium.

#### 2.1 Calculations of the electric field of depletion patterns

As an effective candidate of depletion patterns in STED microscopy, theoretical study of focused LG beam has been made by many papers [28–30]. An initial field of a LG beam can be represented by the function of the conic angle *θ* and the polar coordinate *ϕ* as follows [17, 25]

where *A*
_{0} is the amplitude, *L ^{m}_{l}*(

*x*) is the Laguerre polynomial, here

*l*is the radial mode number and

*m*is called the topological charge with m = 0 denoting a Gaussian beam.

*γ*=

*a*

_{0}/

*ω*

_{0}is the truncation parameter with

*a*

_{0}as the aperture radius and

*ω*

_{0}denoting the beam size at waist.

*α*represents the maximal semi-aperture angle. Because the case of

*l*= 0 exhibits one intensity minimum, we choose the parameter

*l*= 0 in the calculations and the equation (1) can be written as

where ${A}_{1}\left(\theta \right)\text{}={A}_{0}{\left(\sqrt{2}\gamma \frac{\mathrm{sin}\theta}{\mathrm{sin}\alpha}\right)}^{\mid m\mid}\mathrm{exp}\left({-\gamma}^{2}\frac{{\mathrm{sin}}^{2}\theta}{{\mathrm{sin}}^{2}\alpha}\right)$denotes the amplitude distribution of the focusing angel *θ*.

When focusing such beams through an interface between two different media with a high numerical aperture lens, the diffracted field of the point *p* near focus is given as [19–20, 22–23]

Here *f* is the focal length of the lens, *λ*
_{0} and *k*
_{0} are the wavelength and the wave number in vacuo, respectively. *A*
_{2}(*θ*
_{1}, *ϕ*) is the apodization factor and equals to $\sqrt{\mathrm{cos}{\theta}_{1}}$ when the optical system obeys the sine condition. *P*(*θ*
_{1},*ϕ*) denotes the polarization distribution and is given as [23]

in which the coefficients *a* and *b* are the strengths of the incident *x* and *y* polarized components, respectively. *t _{p}* and

*t*are the Fresnel coefficients.

_{s}(*r _{p}*,

*θ*,

_{p}*ϕ*) are the spherical polar coordinates of the point

_{p}*p*in the focal region.

*ψ*denotes the aberration function introduced due to the refractive-index mismatch and its definition tells that for constant

_{d}*d*, the larger the difference of refractive indices is, the more significant the aberration becomes.

*ψ*(

_{s}*θ*

_{1},

*ϕ*) represents the phase change induced by inserting the phase plate. For the case of

*x*polarized beam (

*a*= 1,

*b*= 0), the integration of the optical field components in equation (3) can be obtained as the following expressions when

*ψ*(

_{s}*θ*

_{1},

*ϕ*) is only the function of

*θ*

_{1}:

$${E}_{y}=\frac{-i{n}_{1}f}{2{\lambda}_{0}}\left\{i2\pi {i}^{m+2}{I}_{m+2}\mathrm{exp}\left[i\left(m+2\right){\varphi}_{p}\right]-i2\pi {i}^{m-2}{I}_{m-2}\mathrm{exp}\left[i\left(m-2\right){\varphi}_{p}\right]\right\}$$

$${E}_{z}=\frac{-i{n}_{1}f}{2{\lambda}_{0}}\left\{i2\pi {i}^{m+1}{I}_{m+1}\mathrm{exp}\left[i\left(m+1\right){\varphi}_{p}\right]+2\pi {i}^{m-1}{I}_{m-1}\mathrm{exp}\left[i\left(m-1\right){\varphi}_{p}\right]\right\}$$

where *E _{x}*,

*E*,

_{y}*E*are the optical components in the

_{z}*x*,

*y*,

*z*directions. After we substitute the coordinates

*ρ*=

_{p}*r*sin

_{p}*θ*,

_{p}*z*=

_{p}*r*cos

_{p}*θ*into equation (3) and the integrals

_{p}*I*,

_{m}*I*

_{m±1}and

*I*

_{m±2}are given by

$${I}_{m\pm 1}={\int}_{0}^{\alpha}{A}_{1}\left({\theta}_{1}\right)\sqrt{\mathrm{cos}{\theta}_{1}}{t}_{p}{J}_{m\pm 1}\left({k}_{0}{n}_{1}\mathrm{sin}{\theta}_{1}{\rho}_{p}\right)\mathrm{exp}\left\{i{k}_{0}\left[{\psi}_{d}+{\psi}_{s}\left({\theta}_{1}\right)\text{}+{z}_{p}{n}_{2}\mathrm{cos}{\phantom{\rule{.2em}{0ex}}\theta}_{2}\right]\right\}\mathrm{sin}{\phantom{\rule{.2em}{0ex}}\theta}_{1}\mathrm{sin}{\phantom{\rule{.2em}{0ex}}\theta}_{2}d{\phantom{\rule{.2em}{0ex}}\theta}_{1}$$

$${I}_{m\pm 2}=\frac{1}{2}{\int}_{0}^{\alpha}{A}_{1}\left({\theta}_{1}\right)\sqrt{\mathrm{cos}{\theta}_{1}}\left({t}_{s}-{t}_{p}\mathrm{cos}\phantom{\rule{.2em}{0ex}}\mathrm{cos}{\theta}_{2}\right){J}_{m\pm 2}\left({k}_{0}{n}_{1}\mathrm{sin}{\theta}_{1}{\rho}_{p}\right)\mathrm{exp}\left\{i{k}_{0}\left[{\psi}_{d}+{\psi}_{s}\left({\theta}_{1}\right)\text{}+{z}_{p}{n}_{2}\mathrm{cos}{\phantom{\rule{.2em}{0ex}}\theta}_{2}\right]\right\}\mathrm{sin}{\theta}_{1}d{\theta}_{1}$$

where *J _{m}* denotes a Bessel function of the first kind of order

*m*. The intensity point spread function (I-PS) is therefore given as

*I*(

*r*⃗) ∝ ∣

*E*∣

_{x}^{2}+ ∣

*E*∣

_{y}^{2}+ ∣

*E*∣

_{z}^{2}.

Using the electric field distribution of the *x* - and *y* - polarized incident LG beam, we obtain the filed distribution of the circularly polarized light in the focal plane [25]. For left-handed circularly polarized LG beam, the components of the electric field at focus are expressed as

$${E}_{y}=\frac{-i{n}_{1}f}{2{\lambda}_{0}}\left\{2\pi {i}^{m}{I}_{m}\mathrm{exp}\left(im{\varphi}_{p}\right)+i4\pi {i}^{m\text{}+2}{I}_{m+2}\mathrm{exp}\left[i\left(m+2\right){\varphi}_{p}\right]\right\}$$

$${E}_{z}=\frac{i{n}_{1}f}{2{\lambda}_{0}}\left\{4\pi {i}^{m+1}{I}_{m+1}\mathrm{exp}\left[i\left(m+1\right){\varphi}_{p}\right]\right\}$$

#### 2.2 Calculations of the electric field of STED microscopy

In STED microscopy, the subdiffraction dimensions of the fluorescence spot results from nearly exponential depletion of the population of a typical fluorescent mark with the energy of the inhibition pattern. The remained fluorescence is confined to the local zero. Scanning this spot across the sample and registering the signal renders a subdiffraction image. With *h _{exc}*(

*r*⃗) and

*h*(

_{sted}*r*⃗) denoting the normalized excitation probability of the molecule to be excited by the excitation pulse and the normalized I-PSF of the STED pulse, the probability of the fluorescence emission is given by [9, 12, 31]

Here, *σ* is the fluorophore cross section of the stimulated emission and Φ_{max} denotes the maximum photon flux per area. When Φ_{max} > *I _{s}*, the saturation intensity, the fluorescence in the zero-intensity region of the depletion pattern remains unaffected, whereas that from other area is suppressed and a subdiffraction spot is obtained.

In the following calculations, the excitation beam is assumed as a Gaussian beam and the polarization of the excitation field is parallel to that of the STED field. The wavelengths *λ _{exc}* = 633

*nm*and

*λ*= 785

_{sted}*nm*are employed. For the cells are often maintained in a watery environment, focusing from an immersion lens through the materials of deferent refractive indices is of particular interest in biological application. In the practical STED imaging system, typically oil-immersion objective lens is used to focus the light through the coverglass and then into the watery medium. Under the approximation of that the refractive index of glass coverslip (

*n*= 1.54) is thought equal to that of the oil (

*n*= 1.515), we give the special attention to the case of focusing beams by an oil immersion objective lens with the numerical aperture

*NA*= 1.25 (

*α*=55.6°) for the case of an oil (

*n*

_{1}= 1.515) to water (

*n*

_{2}=1.33) interface.

## 3. Results and discussions

#### 3.1. Intensity distributions of fluorescence depletion patterns

Firstly, the influence of the interface on the structure of the fluorescence depletion patterns is studied. The complex amplitude at the focus can be obtained using Eqs. (3), (7) and (9), respectively. And all the results are normalized by the maximum intensity of the aberration-free case. A program to evaluate these equations is written in MATLAB and has been verified with the conclusions in other published paper. Results for a linearly polarized Gaussian beam with the zero-aberration agree well with the work of Chon *et al*. [32]. Our calculations on the focal shape of linearly polarized LG beam focused by a high numerical-aperture objective in free space accord well with the results in reference paper of [28]. The results of focusing problems of LG beams in the presence of the primary spherical aberration agree well with those of Singh *et al*. [25].

Figure 2 shows intensity distributions in the focal plane (*xy* plane) and through focus (*yz* plane) for an *x* -polarized LG beam with *m* = 1 for different positions of the interface: *d* = 0, 10 and 20, which is in units of the wavelength of the excitation. Two high-intensity lobes with residual intensity in the center and a focal spot elongated along the polarization direction are apparent. For *m* = 1 , the linearly polarized LG beam possesses a nonzero electronic field at the focal point due to the longitudinal component dominating the center of the point-spread function [27]. With an increase in the value of *d*, Fig. 2(c) shows that the maximum intensity of this pattern decreases. An increment of the peak-to-peak separation is also seen, which is increased from about 430*nm* with *d* =10 [Fig. 2(b)] to 481*nm* with *d* = 20 [Fig. 2(c)]. The shape of this pattern is generally remained and the calculation results in Figs. 2(a–c) show that this pattern in the focal plane is not sensitive to the position of the interface. It can be seen from Fig. 2(d–f) that the aberration induced by the refractive index mismatch caused the elongation of the focal spot along the optical axis. In the presence of aberration, the position of the peak intensity shifts from the geometrical focal point towards the position of the interface, and the shift increases with an increase in *d*.

Similar intensity plots for a left-circularly (LC) polarized LG beam with *m* = 1 are given in Fig. 3. The intensity distribution exhibits an annular intensity profile with a dark core. The shape of LC polarized LG beam remains reasonably stable and the symmetry of the dark hole around the center is maintained as the value of *d* increases. The presence of the aberration reduces the intensity at the periphery, along with an increase of the size of the dark hole [Fig. 3(b–c)] and the stretching along *z* axis [Fig. 3(d–f)]. In this plot, the diameter of dark hole increases from 506nm with d = 0 to 563*nm* with *d* = 20 . The stretching along *z* axis becomes more obvious as the distance of the interface to the origin becomes larger. As this figure shows, the effect of aberration is small for LC polarized LG beam.

The effective inhibition patterns are regularly created by inserting appropriate phase plates in the Gaussian beams [16]. Figure 4 shows, respectively, the intensity distribution for an aberration-free and for the aberrated cases of a focused Gaussian beam inserted with a central and a semi-circular *λ*/ 2 phase plate. As a result a narrow line-shaped valley oriented in the *x* axis is formed and this model is expected to predict efficient fluorescence depletion along *y*, which can be seen from Fig. 4(a). Figure 4 (b–c) show its dependence of the intensity distribution in the focal plane on the presence of the interface. When a refractive-index mismatch occurs, the induced aberration has a broadening effect on the intensity distribution and a decrease in the maximum intensity.

A central *π* phase retardation of the wavefront in a circularly polarized Gaussian beam produces an inhibition pattern featuring two maxima on the optical axis and employing this pattern is interesting for obtaining a narrower fluorescence spot in *z* direction [Fig. 4(d)]. Figure 4 (e–f) reveal a degradation of the PSF by the aberration induced by the oil-water interface. In the presence of the aberration, there is a redistribution of the intensity along with its maximum value in the (*y*, *z*) plane and the symmetric dark core along *z* axis in the pattern around the focal point for an aberration-free case vanishes. With increase of the aberration, the intensity maximum of the two separated spots becomes asymmetric along *z* axis and the distance between them increases [Fig. 4(f)], which in turn produces an asymmetric and extended fluorescence spot along the optical axis in STED microscopy. The presence of interface in the system also leads to a positional displacement of the dark core.

For STED microscopy it is essential to optimize the depletion modes both for the a sharp hole and high power at the periphery. As a contrast, the typical line profiles through central zero intensity of different patterns are depicted in Figure 5 for various interface positions: *d* = 0 , 1, 10 and 20. As this figure shows, all the dark areas are broadened as the aberration increases. The separation increment between the peak side lobes and the normalized dip depth increment (radio of the residual intensity at the focal hole to the peak intensity) result in reduction in the sharpness of the dark core. For *x* -polarized LG beam, the hole depth rises at beginning and then falls with increasing the value of *d*. For example, the residual intensity at the focal point in homogeneous media (*d* = 0) of Fig. 5(a) is about ≈ 30.6% of the maximum intensity and increases to 41.3% with *d* = 1. It then falls to ≈ 32.9% with *d* = 20. Since the central spot of this beam is not perfectly zero, it counteracts the enhancement of the resolution for the suppression of fluorescence also happens in the center. Fig. 5(b–c) shows that the zero intensity in the center of the I-PSF is maintained regardless of the aberration introduced by the presence of the interface for the LC polarized LG beam and the pattern produced by a semicircular *λ*/2 phase plate. It is noticed in Fig. 5(b–c) that, up to *d* = 10, the broadening effect is marginal for LC polarized LG beam and the pattern produced by the semi-circular *λ*/2 phase plate. The effects of the presence of interface on the inhibition pattern of center-*π* retardation phase mask are shown in Fig. 5(d). Because the aberration causes the focal shift, for better contrast, all the profiles are dealt with to keep the zero-intensity point at the origin. The profiles in Fig. 5(d) show obviously that the dark hole are degraded to be asymmetric with increasing *d* and the residual intensity elevates, which correspondingly weakens the fluorescence in the center. As an example, the local minima are 0, 1.28 and 6.71% of the maximum intensity for *d* = 0, 10 and 20, respectively. It is also noticed that for smaller *d* , it’s maximum intensity of the focus increases [Fig. 4(e)] and the distance of the brighter portions decreases compared with the case of *d* = 0 [Fig. 5(d)]. Consequently, the fluorescence spot of STED microscopy decreases than that of the aberration-free case when this pattern is applied at beginning.

#### 3.2. Intensity distributions of STED microscopy

Next, we illustrate the effects of aberration induced by the refractive-index mismatch on the fluorescence spot of STED microscopy when these inhibition patterns are applied. The calculations are operated using Eq. (10). With a typical value *π* ≈ 10^{-16}
*cm*
^{2}, in our calculations the maximum erase beam photo flux is limited to a value slightly above the saturation intensity: Φ_{max} =2×10^{16}
*photons* / *cm*
^{2}. The polarization state of the STED beam is assumed to be parallel with the pump beam. The pump beam is assumed to be a Gaussian beam (*m* = 0). The normalized total intensity distribution are shown in Fig. 6 and in Fig. 7 for *d* = 1 and *d* = 20, respectively. In comparison with the excitation spot in the first column, the focal spot in the right column is substantially reduced by employing an erase beam. When aberration occurs, the size of the fluorescence spot has been increased for *d* = 20 in comparison with the case of *d* = 1 because of the broadening effects as is mentioned above. Using the full-width-at-half-maximum (FWHM) of the fluorescence spot as a gauge, the size for the case of LC-polarized LG beam with *d* = 1 is 165nm and it slightly increases up to 190nm with *d* = 20. While for the pattern of using central *λ*/2 phase mask, it rises from 437nm to 798nm and the widening of a factor of two is almost obtained. For better comparison, the fluorescence maximum is also listed in the images.

Since the presence of the interface influences the local minimum of the focal hole, the relevant maximum fluorescence emission in STED microscopy is also studied. These findings are summarized in Fig. 8, where the maximum fluorescence emission and the FWHM of the fluorescence spot are plotted as a function of different values of the interface position. As this figure shows, the FWHM of the fluorescence spot of STED microscopy is increased for all the inhibition patterns with the increment of *d* [Fig. 8(b)]. However, up to *d* = 10 , the FWHM of the focal spot along *z* axis is smaller than the aberration free case for the light pattern using a central *λ*/2 phase mask. As a result of the increased residual intensity for this pattern, its maximum fluorescence emission is getting smaller with the increment of *d* [Fig. 8(a)]. For the LC-polarized LG beam and the light pattern generated by semi-circular *λ*/2 phase mask, the maximum fluorescence emission keeps unity (data not shown) and changes of their FWHM values are small [Fig. 8(b)]. For the linearly polarized LG beam with *m* = 1, the maximum fluorescence emission rises with an increasing in *d* (*d* ≠ 0).

## 4. Conclusion

We have presented a comparative theoretical study of the effects of an interface on the inhibition patterns focused by high numerical aperture lens and their applications in STED microscopy. With the increment of the aberration induced by the refractive-index mismatch, the black hole in the focal plane for all the patterns is broadened, which leads to the increased size of the fluorescence spot in STED microscopy. The intensity distribution is stretched in *z* direction, along with a positional shift of the peak and also a reduction in the peak intensity. The effects of the aberration are small for the LC polarized LG beam and the depletion pattern generated by the half-circular *λ*/2 phase plate, which proves that these beams are mostly preferred for the STED purpose. The depletion pattern created by using the central *λ*/2 phase mask is mostly degraded to produce a modification in the shape of the pattern with increasing the residual intensity in the center when aberration of interface exists and its application in STED microscopy is limited to some extent. Because of the non-zero intensity in the center, linearly polarized LG beam (*m* = 1) is not very suitable for the use in the STED microscopy. In addition, the influence of the aberration caused by the interface will become more severe with an increasing difference of the refractive indices of the two media.

## Acknowledgments

This work is financially supported by the National Natural Science Fund (NO. 60527004) of China.

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