## Abstract

This study explores how interference manipulation breaks through the diffraction limit and induces super-resolution nano-optical hot spots through the nonlinear Fabry–Perot cavity structure. The theoretical analytical model is established, and the numerical simulation results show that when the thickness of the nonlinear thin film inside the nonlinear Fabry–Perot cavity structure is adjusted to centain value, the constructive interference effect can be formed in the central point of the spot, which causes the nanoscale optical hot spot in the central region to be produced. The simulation results also tell us that the hot spot size is sensitive to nonlinear thin film thickness, and the accuracy is required to be up to nanometer or even subnanometer scale, which is very large challenging for thin film deposition technique, however, slightly changing the incident laser power can compensate for drawbacks of low thickness accuracy of nonlinear thin films. Taking As_{2}S_{3} as the nonlinear thin film, the central hot spot with a size of 40nm is obtained at suitable nonlinear thin film thickness and incident laser power. The central hot spot size is only about $\lambda /16$, which is very useful in super-high density optical recording, nanolithography, and high-resolving optical surface imaging.

© 2014 Optical Society of America

## 1. Introduction

Overcoming the Abbe diffraction limit and obtaining a super-resolution spot have been hot topics because of the important demands in the fields of nanolithography and ultrahigh density data storage [1–3], nanoscale-resolved optical imaging and detection [4], and enhanced cell transfection [5]. Numerous methods and techniques have been proposed to overcome Abbe limit, including near-field probe [6], plasmonic nanolens [7, 8], superlens [9, 10], nanoantenna [11], hyperlens [12], solid immersion lens [13], phase filter [14–16], metamaterial lens [17], microsphere based microscopic-lens [18], and fluorescence labeling [19] etc.

Obtaining super-resoluiton spot through optical nonlinear effect is a good alternative method, which includes two aspects, one is nonlinear absoption induced super-resolution, and the other is nonlinear refraction induced super-resolution. For nonlinear absorption, the nano-optical data storage was demonstrated experimentally [20, 21], and nanofabrication and nanolithography can be also found in lots of work [22–24]. For nonlinear refraction, the self-focusing induced super-resolution effects were observed experimentally [25–28], where the nonlinear refraction samples were thick films and the sample thickness exceeded light wavelength. However, one interesting question is that if the nonlinear refraction sample thickness is reduced to below about light wavelength, or even to nanoscale, can the super-resolution spot be obtained? It is well known that for sample thickness below about light wavelength, the self-focusing can be ignored, and the internal multi-interference effect is dominant. If the nonlinear sample is sandwiched by two thin films (where one is deposied on the top of the nonlinear sample, and the other is deposited on the bottom of nonlinear sample), the sandwiched sample structures are typical nonlinear Fabry–Perot cavity structures [29–31]. Compared with the linear Fabry–Perot cavity, the nonlinear Fabry–Perot cavity structures can be designed to be super-resolution optical devices. Kreuzer et al observed the dynamic exiting spot pattern from the nonlinear Fabry-Perot resonators, compared with the incident spot patterns the exiting spot patterns are obviously reduced when the incident laser beam induces a suitable refractive index profile [32]. The Fabry–Perot cavity is a typical interference manipulation device, and Kreuzer’s experiments indicated that the nonlinear Fabry–Perot cavity structures could be used as super-resolution optical devices by the interference manipulation. That is, the interference manipulation can break through the optical diffraction limit and obtain super-resolution spot. Recently, Mosk and colleagues also obtained experimentally a single sharp optical focus by constructive interference [33]. Sentenac et al. exploited the constructive interference effect in simulating the sub-diffraction spot by a grating substrate [34]. Based on the thin film optical theory, one of authors also reported internal multi-interference reshaping induced super-resolution effect with a nonlinear thin film [35]. In this work, a detailed theoretical model and simulation results are presented to analyze the the super-resolution spot formation through nonlinear Fabry–Perot cavity structures by considering the interference manipulation effect. This work is useful for nanolithography, ultrahigh density data storage, and nanoscale-resolved optical imaging.

## 2. Theoretical analytical model of interference manipulation

The designed nonlinear Fabry–Perot cavity structure is shown in Fig. 1, where region 2 is filled with a nonlinear thin film material, which is sandwiched between dielectric mirrors situated in region 1 and region 3. Figure 1 shows the schematic of the light beam traveling through the nonlinear Fabry–Perot cavity, where the self-action and diffraction effects of the light beam are negligible because the thickness of nonlinear Fabry–Perot cavity structure is only less than $1\mu m$ . The nonlinear Fabry–Perot cavity is made up of three regions. Region 1 and region 3 are filled with dielectric materials and form two dielectric mirrors. The nonlinear thin film is situated in region 2, sandwiched between region 1 and region 3. The interface between region 1 and region 2 is F_{12}, and the interface between region 2 and region 3 is F_{23}.

Assuming that a light beam with an electric field intensity of ${A}_{1}$ is normally incident in the nonlinear Fabry–Perot cavity structure from region 1, ${A}_{1}$ can be represented as

where ${w}_{0}$ is the spot radius and${E}_{0}$is the electric field intensity at the center of the spot $r=0$. Accordingly, the light intensity isThe amplitude transmission and reflection at interface F_{12} are assumed to be ${\tau}_{12}$ and ${\rho}_{12}$, respectively. At interface F_{23}, they are assumed to be ${\tau}_{23}$ and ${\rho}_{23}$, respectively [36].

The reflectance $R$ and transmittance $T$ can be simply written as [36]

_{12}and travels to interface F

_{23}, and the light field becomes ${A}_{2}^{(1)}$, which is calculated aswhere $k=2\pi n/\lambda $ denotes the propagation constant, ${e}^{[ikL-(\alpha L/2)]}$ the variation of the phase and intensity of the light field traveling from F

_{12}and F

_{23}. Notably, for the exponent ${e}^{[ikL-(\alpha L/2)]}$ in formula (6), the mean-field approximation is made. In other words, $n$ and $\alpha $ are assumed to be constant along the thin film thickness position $z$. If such is not the case, the exponent $[(i{\scriptscriptstyle \frac{2\pi}{\lambda}}n-{\scriptscriptstyle \frac{1}{2}}\alpha )L]$ in formula (6) should be replaced by ${\int}_{0}^{L}[i{\scriptscriptstyle \frac{\pi}{\lambda}}n(z)-{\scriptscriptstyle \frac{1}{2}}\alpha (z)]}dz$ [37].

${A}_{2}^{(1)}$ is partially reflected in interface F_{12} after traveling a distance of $L$, and the light field becomes ${A}_{2}^{\text{'}(1)}$,

_{23}after traveling a distance of $L$, and the light field becomes ${A}_{2}^{(2)}$

_{12}after traveling a distance of $L$, and the light field becomes ${A}_{2}^{\text{'}(2)}$,

Similar to ${A}_{2}^{(2)}$, the ${3}^{th}$ forward light wave ${A}_{2}^{(3)}$ should be written as

The ${i}^{th}$ forward light wave ${A}_{2}^{(i)}$ should be written as

After coming and going _{$m$} times ($m$ is an integer), the ${m}^{th}$ forward light wave ${A}_{2}^{(m)}$ should be written as

After the light beam is reflected m times between interface F_{12} and interface F_{23}, a stable effective field is obtained. The effective field consists of a forward light wave and a backward light wave. The forward light wave is marked as ${A}_{2}$ and the backward light wave is marked as ${A}_{2}^{\text{'}}$. Figure 2 presents the schematic, where ${A}_{3}$ and ${A}_{1}^{\text{'}}$ are the exiting and reflected light fields from the nonlinear Fabry–Perot cavity structure, respectively. In this work, we only consider the exiting light field ${A}_{3}$.

The forward light field ${A}_{2}$ is a superposition that can be calculated as

Formula (16) describes the forward light field of the nonlinear Fabry–Perot cavity structure. The intensity of the forward light wave ${A}_{2}$ is

As shown in Fig. 2, the exiting light field from the nonlinear Fabry–Perot cavity structure is ${A}_{3}$, which can be calculated as ${A}_{3}=\tau {A}_{2}$. Accordingly, the intensity can be calculated as

In formula (19), *n* and *a* are induced by the internal cavity field inside region 2, called ${A}_{cavity}$. ${A}_{cavity}$ consists of two parts, namely, the forward light wave ${A}_{2}$ and the backward light wave ${A}_{2}^{\text{'}}$, such that ${A}_{cavity}={A}_{2}+{A}_{2}^{\text{'}}$. For the sake of simplicity, we ignore the standing wave within the cavity structure. Accordingly, the light intensity ${I}_{cavity}$ in region 2 is

According to formula (17), ${I}_{2}^{\text{'}}$ can be calculated as

Thus,

If $n$ and $\alpha $ remain unchanged along the film thickness direction $z$ and are only functions of the radial coordinate $r$, which is reasonable because the self-action and diffraction effects are ignored, then $n$ and $\alpha $ are expressed as $n(r)$ and $\alpha (r)$, respectively. The internal cavity-field-induced refractive index and absorption coefficient are

From formula (19), we can obtain the intensity distribution of the exiting spot from the nonlinear Fabry–Perot cavity structure. The thickness of the dielectric mirror is assumed to be less than 100nm. The light intensity distribution in region 2 can be directly mapped onto the exiting surface of the nonlinear Fabry–Perot cavity structure because the thickness of the dielectric mirror is in the optical near-field range. Based on formula (2), formula (19) can be rewritten as

## 3. Nanoscale optical hot spot formation by constructive interference manipulation

In the theoretical model, both the nonlinear absorption and nonlinear refraction are considered. However, for thin film materials with both nonlinear absorption and nonlinear refraction, generally speaking, the super-resolution effect caused by nonlinear (saturation) absorption is stronger than that caused by nonlinear refraction because nonlinear (saturation) absorption easily generates an optical pinhole channel [38]. In addition, the strong linear absorption or multi-photon absorption causes the interference effect to weaken inside the nonlinear Fabry–Perot cavity structure, which results in some difficulties in generating the nanoscale optical spot with constructive interference manipulation. Thus, the selection of nonlienar thin films is very important, and some requirements need to be met. The first is that the linear and nonlinear absorption coefficients should be very small and ignorable, and the second is that nonlinear refractive coefficient ought to be large, which can result in constructive interference with a low laser power of milliwatt magnitude. In real applications, the low laser power can avoid strong thermal expansion and sample damage, is also enough for direct laser writing nanolithography and high-density optical recording. Therefore, in the following, we consider a special case of nonlinear thin film without absorption, that is, $\alpha =0$, to analyze the interference manipulation effect inside the nonlinear Fabry–Perot cavity structure.

Arsenic trisulfide (As_{2}S_{3}) thin film is a feasible material due to very large third order nonlinear susceptibility when the laser wavelength falls in the Urbach tail region. In Urbach tail region, the linear absorption coefficient decreases exponentially with the energy difference between the band gap energy ${E}_{g}$ and the illumination photon energy $\text{h}\nu $. The linear absorption of As_{2}S_{3} thin film in Urbach tail region is very small (1~10/cm) and easily saturated by low power illumination [39]. Thus, based on Ref [27], we take the As_{2}S_{3} thin film in region 2 as an example because of the large positive nonlinear refraction coefficient of $\gamma =8.65\times {10}^{-10}{m}^{2}/W$ and negligible linear and nonlinear absorption coefficient ${\alpha}_{0}\approx \beta \approx 0$ at a wavelength of $\lambda =633nm$. ${n}_{0}\approx 2.5$ is chosen because of the large linear refraction for chalcogenide glass materials. The incident laser power is fixed at *P =* 0.2*mW*, and ${w}_{0}=300nm$. The incident light intensity can be obtained from ${I}_{0}=2P/(\pi {w}_{0}^{2})$.

Let us first analyze the effective stable cavity field intensity and the refractive index distribution of the nonlinear thin film. Based on formula (25), to obtain the nanoscale spot, ${I}_{3}(r)$ should be maximum at $r=0$, which requires $4\pi nL/\lambda =2q\pi $ with $q=0,1,2\cdot \cdot \cdot $. After the incident laser power is fixed, we can change the nonlinear thin film thickness to obtain the strongest interference field at $r=0$. Thus

The internal cavity field intensity is roughly estimated by assuming L = 100nm, as shown in Fig. 3(a). We find that ${I}_{cavity}=8.55\times {10}^{8}W/{m}^{2}$ at $r=0$. The internal cavity field-induced refractive index is plotted in Fig. 3(b), where $n\approx 3.24$ at $r=0$.

According to formula (22), ${I}_{cavity}$ maintains minimal fluctuation at $4\pi nL/\lambda =2q\pi $. Thus, the *n* value at $r=0$ also maintains little fluctuation with the nonlinear thin film thickness. To obtain the interference enhancement at $r=0$, we substitute $n\approx 3.24$ into formula (26) and optimize the calculation to obtain a series of *L* values as follows: L = 286, 386, 484, 586, 687nm, $\cdot \cdot \cdot $ .

Let us analyze the size and intensity distribution of exiting spot from nonlinear Fabry-Perot cavity structure for different nonlinear thin film thicknesses. Figure 4 shows normalized two-dimensional spot intensity distributions.

Figure 4(a) is the incident spot itself, which has a typical Gaussian profile. Figure 4(b) is the exiting spot at L = 286nm. The spot is obviously smaller than the incident spot. When the thickness is increased to L = 386nm, the spot becomes smaller than the incident spot. At L = 484nm, a central spot with a size of about 100nm occurs, as shown in Fig. 4(d). Let us further increase the nonlinear thin film thickness to L = 586nm, a central hot spot of about 70nm appears in the central region (also see Fig. 4(e)). A very small central hot spot with a size of 40nm occurs in the central region when L = 687nm (as shown in Fig. 4(f)), and the central hot spot size is only about $\lambda /16$, which is very useful in nanolithography and high-resolving light imaging etc. To further analyze the spot characteristics, Fig. 5 presents the normalized three-dimensional spot intensity distributions.

Figure 5(a) is the incident spot. Figure 5(b) is the exiting spot at L = 286nm, which shows that the spot becomes sharp when compared with Fig. 5(a). Figure 5(c) is the exiting spot at L = 386nm, the spot becomes shape, and a central hot spot occurs at the position with normalized intensity of about 2/3 Fig. 5(d) shows that at L = 484nm, the central hot spot becomes sharper, the size of central hot spot is also smaller than that in Fig. 5(c). Figure 5(e) shows that, at L = 586nm, the central hot spot size markedly decreases compared with that in Fig. 5(d). The central hot spot size is further reduced at L = 687nm, as shown in Fig. 5(f), in which two inflection points occur at the exiting spot intensity distribution. One is at about 1/2 of the normalized intensity, and the other is at about 2/3 of the normalized intensity. At about 2/3 of the normalized intensity, a very sharp central hot spot occurs, which resembles the tip of an optical probe. The central hot spot size is far smaller than the incident spot itself.

Figure 6(a) shows the cross-section profile of the normalized spot intensity along the radial direction for different nonlinear thin film thickness. The increase of nonlinear thin film thickness can sharpen the exiting spot, and the central optical hot spot occurs at $L\ge 386nm$. The central hot spot size is reduced to about 40nm at L = 687nm, which is a nanoscale optical hot spot applicable in nano-optical data storage, high-resolving imaging, and direct laser writing lithography.

In spot simulation, one can find that the central hot spot size is very sensitive to nonlinear thin film thickness. In order to understand the sensitivity of central hot spot size to nonlinear thin film thickness, we take the L = 687nm as an example, and analyze the dependence of central hot spot size on nonlinear thin film thickness at L = 687~690nm.

Figure 6(b) presents the calculated results, one can see that the central hot spot size linearly increases with nonlinear thin film thickness for L = 687~690nm, which indicates that the hot spot size is sensitive to nonlinear thin film thickness, and the accuracy is required to be up to nanometer or even subnanometer scale, which is very large challenging for thin film deposition method and process. Fortunately, some advanced thin film deposition technqiue can prepare the thin film materials with nanoscale (or even subnanometer scale) accurancy. For example, the atomic layer deposition technqiue, also called as atomic layer epitaxy, is a good method to produce high-quality large-area thin film materials with perfect structure and process controllability, and the accurancy can reach up to single atom layer, accordingly [40].

The advanced thin film deposition technqiue, such as atomic layer epitaxy, is indeed good method for obtaining high accuracy nonlinear thin films. Actually, there is also an alternative method for realizing nanoscale central optical hot spot, that is, slightly changing the incident laser power can obtain the same performance as the high accuracy manipulation of nonlinear thin film thickness due to super-resolution spot is also sensitive to the incident laser intensity. Here we take the Fig. 4(f) with L = 687nm and P = 0.2mW as an example. The central hot spot with a size of about 40nm can be obtained for L = 687nm and P = 0.2mW, however, the central hot spot immediately becomes large and the spot top becomes flat when the nonlinear thin film thickness becomes L = 690nm, as is seen in the two-dimensional intensity distribution of Fig. 7(a). Comapred with Fig. 4(f) one can find that the central hot spot obviously becomes large although the thickness difference is only 3nm. The cross-section profile curve is shown with red curve in Fig. 7(b), and the central hot spot becomes about 80nm, accordingly. In order to obtain the same performance as Fig. 4(f), one can slightly tune the incident laser power from P = 0.2mW to P = 0.196mW. Figure 7(c) gives the two-dimensional intensity distribution accordingly. Compared with Fig. 7(a) one can see that the central hot spot obviously becomes small by tuning incident laser power of P = 0.196mW, and the cross-section profile curve is shown with blue curve in Fig. 7(b), and the central hot spot becomes about 40nm, accordingly. Figure 7(b) also gives the intensity profile of Fig. 4(f) with dark cyan curve (with L = 687nm and P = 0.2mW), where the dark cyan curve is almost coincident with the bule curve (with L = 690nm and P = 0.196mW). Therefore, Fig. 7 indicates that the slightly changing the incident laser power is also an alternative method for compensating for drawback of low thickness accuracy of nonlinear thin films.

The nanoscale optical hot spot can be reversibly generated because the nonlinear characteristics of thin film in region 2 is reversible. For practical applications, such as nanolithography and optical surface imaging, generation and movement of the nanoscale central hot spot are possible by scanning the incident light beam through a galvanometer mirror or by moving the nonlinear Fabry–Perot cavity structure. Similar to the nonlinear thin film super-resolution applications [3], the resist thin films or the samples to be imaged are directly deposited on the surface of the nonlinear Fabry–Perot cavity structure. The nano-central hot spot is directly coupled into the resists or samples to be imaged because the thickness of the dielectric mirror is less than 100nm, which is in the near-field range.

However, the central hot spot size cannot decrease infinitely with increasing L. That is, the q value of formula (26) cannot increase infinitely. Let us further increase the q value such that L = 781nm at P = 0.2mW. Figure 8 illustrates the optimized exiting spot characteristics. Figure 8(a) shows the radial distribution of the normalized spot intensity. The central hot spot becomes coarse and the size becomes large compared with the case at L = 687nm and P = 0.2mW. Figure 8(b) shows the normalized two-dimensional intensity image. Compared with Fig. 4(f) where L = 687nm and P = 0.2mW, the spot morphology is no longer a single central hot spot but is an annular spot, and the annular central spot size is about 80nm. Such size is obviously larger than the central hot spot at L = 687nm. Figure 8 indicates, to obtain a nanoscale central hot spot by interference manipulation, an optimum nonlinear thin film thickness for the nonlinear Fabry–Perot cavity structure must occur. In our example, the optimum thickness is L = 687nm for P = 0.2mW and the nano-optical hot spot size is 40nm.

Here one needs to notice that, based on the theoretical model, different laser power can induce different refractive index profile, thus the super-resolution performance is also different for different laser power irradiation. That is, the super-resolution effect can be regulated by changing the laser power.

It needs to be pointed out that in the theoretical model some reasonable approximations are used, such as mean-filed approximation or ignoring the standing wave effect. To verify the simulation results, the FDTD analysis will be investigated in our next work, and the test and application schematic design are presented in section 4. In addition, Ref [32]. also gave some calculation and experimental results for reducing the spot through the nonlinear Fabry-Perot resonators.

## 4. Test and application schematic designs of nonlinear Fabry-Perot cavity structures

According to the simulation analysis in section 3, the super-resolution spot can be obtained through the constructive manipulation in the nonlinear Fabry-Perot cavity structures when the As_{2}S_{3} thin film is chosen as the nonlinear thin film. Figure 9 presents a test schematic design for the super-resolution spot. The nonlinear thin film is sandwiched in between the dielectric mirrors, where the thickness of dielectric mirrors is less than 100nm. A collimated laser beam passes through the neutral density optical filter and is focused by a converging lens. The laser power can be fine regulated by the neutral density optical filter. The focused spot is normally incident into the nonlinear Fabry-Perot cavity structure and induces the formation of super-resolution spot. The super-resolution spot only exists at the surface of nonlinear Fabry-Perot cavity structure, and is difficult to propgagte in the air. Therefore, in order to test the super-resoluiton spot, a near-field spot scanning method needs to be designed. The super-resolution spot can be scanned and tested by near-field spot scanning head, where the distance regulation between the fiber tip of scanning head and the surface of the sample is controlled by utilizing a tuning fork, and the regulated distance should be maintained at ~50nm, which is a typical near-field range.

The application schematics can be roughly designed as follows. The resists and optical recording materials with obvious threshold effect (such as structural change and molten ablation etc) are directly deposited onto the nonlinear Fabry-Perot cavity structure, as shown in Fig. 10(a). The thickness of dielectric mirrors in the nonliear Fabry-Perot cavity structure is less than 100nm. A collimated laser beam is focused and normally incident onto the nonlienar Fabry-Perot cavity structure, and the super-resolution spot can be generated and directly coupled into the resists or optical recording media in the near-field range, where the near-field distance can be regulated by the thickness of dielectric mirror. The laser power can be fine adjusted by the neutral density optical filter. When the spot intensity of 70% central maximum induces the threshold effect of resists or recording media the nanoscale lithography or optical recording can take place, as shown in Fig. 10(b).

Here it should be explained that in the example the photo-induced or thermally-induced expansion effect is very weak due to the low laser power of milliwatt magnitude, which is enough for laser direct writing lithography, optical recording and imaging. If there is thermal expansion, which induces the thickness change of nonlinear thin films, one can regulate the laser power a little by the neutral density optical filter to eliminate the influence on super-resolution resulting from expansion.

It should be noted that when the resists or reocrding media are deposited onto the nonlinear Fabry-Perot cavity structures the interface loss may occur and have a little influence on the super-resolution spot performance. The influence can be reduced or eliminated by fine adjusting the neutral density optical filter to changing the laser power a little.

## 5. Conclusion

In summary, one can manipulate the nonlinear thin film thickness of the nonlinear FabryPerot cavity structure to make the position at $r=0$ become constructive interference. The nanoscale optical hot spot can be produced in the central region. The hot spot size is sensitive to nonlinear thin film thickness, and the accuracy is required to be up to nanometer, or even subnanometer scale, which is very large challenging for thin film deposition technique, however, slightly changing the incident laser power can compensate for drawbacks of low thickness accuracy of nonlinear thin films. Taking As_{2}S_{3} as the nonlinear thin film, the central hot spot with a size of 40nm is obtained at suitable nonlinear thin film thickness and incident laser power. The central hot spot size is only about $\lambda /16$, which is very useful in super-high density optical recording, nanolithography and high-resolving optical surface imaging.

## Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 51172253 and 61137002), the Instrument Developing Project of the Chinese Academy of Sciences (Grant No. YZ201140), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 11JC1412700 and 11JC1413300).

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