1. Introduction

Two-photon polymerization (2PP) is a well established technology for the realization of arbitrary three-dimensional microstructures with feature sizes smaller than the diffraction limit [1–6]. Ultrashort laser pulses are focused into a polymer, initiating a locally confined polymerization reaction in the focal region of the focusing optics, where the intensities are sufficiently high for two-photon absorption. Tight spatial focusing of the pulses is needed, to realize structures with features smaller than the diffraction limit. Therefore, oil immersion microscope objectives with high numerical apertures up to NA = 1.45 are commonly used for focusing the ultrashort laser pulses into the polymer. A typical setup consists of the objective, an immersion fluid and a coverslip with the polymer located on the backside.

However, these oil immersion microscope objectives suffer two main drawbacks when applied for 2PP. Firstly, they only produce a diffraction-limited spot, if applied under their specific design conditions (coverslip thickness, immersion fluid). Owing to the refractive-index-mismatch between coverslip and polymer strong spherical aberration are introduced [7,8]. The amount of the spherical aberration depends mainly on the NA of the focusing lens, the focusing depth, and the difference in the refractive indices between coverslip and polymer. Secondly, microscope objectives consist of complex lens systems that are causing a high internal group velocity dispersion (GVD) leading to a temporal broadening of the focused laser pulses. Thus, an external compression of the dispersion is needed in order to maintain a short pulse duration.

To overcome these drawbacks, we designed and manufactured a high numerical aperture hybrid optics specifically for 2PP. The design comprises a diffractive optical element (DOE) for the correction of chromatic and spherical aberrations introduced by refractive lenses. The optics is corrected for refractive-index-mismatch-induced aberrations over a large working distance range of 577 μm and causes low internal dispersion.

2. Design and simulations

Refractive-diffractive hybrid optics represent an excellent focusing concept for ultrashort laser pulses. The achromatization of a refractive lens with a DOE leads to a well corrected optics with low internal dispersion, thus to reduced temporal broadening of the focused laser pulses. Fuchs et al [9] demonstrated a hybrid optics for the focusing of 40 fs laser pulses with a NA = 0.45 into fused silica for the inscription of waveguides.

This concept of correction of a single lens with a DOE is no longer applicable for focusing laser pulses with NA >1 into a polymer located on a substrate. The strong spherical aberration that emerge at the interfaces between immersion fluid, substrate, and polymer cannot be corrected by the diffractive power of the DOE over the complete wavelength spectrum of the pulses. The hybrid optics would suffer from wavelength dependent spherical aberration (spherochromatism). Therefore, the refractive part of the intended focusing optics should exhibit minor aberrations for monochromatic light, so that the DOE is only used for correction of chromatic aberration.

To further increase the NA of a well-corrected lens or objective without introducing additional aberration a solid immersion lens can be used [10–12]. Originally developed for increasing the resolution in optical microscopy, a medium with a high refractive index is brought between objective and sample. There are two possible configurations for a SIL. In the case of a hemispherical SIL, a hemisphere made of glass with refractive index n is placed with its center of curvature in the focus of the objective. The rays enter perpendicular to the surface normal, no refraction occurs and the NA of the objective is increased by a factor n, due to the reduced wavelength inside the media. The second, more interesting configuration is the aplanatic solid immersion lens (ASIL). The ASIL consists of a hyper-hemisphere with radius R and refractive index n. The geometrical aberrations introduced by the ASIL are equal to zero, for a center thickness [13]

In our optical design, we apply an ASIL to increase the NA of an aspheric lens, which is well corrected for spherical aberration. Subsequently we use a DOE for correction of chromatic and the small remaining spherical aberration. Figure 1 shows the layout of the optical design. The hybrid optics is designed for focusing ultrashort pulses with a central wavelength of 515 nm and a spectral width of 3 nm (FWHM) into the commercially available polymer ORMOCORE^® (Micro Resist Technology GmbH, Germany).

 

Fig. 1 Layout of the specially designed hybrid optics consisting of an aspheric lens, a diffractive optical element (DOE) and a half ball lens working as aplanatic solid immersion lens (ASIL).

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The asphere used is a precision aspheric lens from Edmund Optics (NT49-097) with NA = 0.6, a diameter of 15 mm, a clear aperture of 13.5 mm, and a center thickness of 7 mm. The distance between asphere and ASIL is 1.074 mm. In order to enable different focusing depths without changing the optical function of the optics, the ASIL has to be integrated into the experimental setup, consisting of immersion fluid, substrate and polymer. This was done according to Fig. 1. The radius of curvature of the used half ball lens (HBL) is R = 2 mm and the center thickness ct = 2.5 mm. Since the ASIL no longer consists of one material with homogeneous refractive index, spherical aberration occur at the interfaces. However, they can be minimized by choosing a glass for the HBL with a refractive index adjusted to that of the polymer. In our case the HBL is made of N-SK11 with n_515nm = 1.568, almost exactly coinciding with n_515nm = 1.569 of the polymer. For the substrate, we use standard microscope coverslips (thickness 170 μm, n_515nm = 1.528). With the focus located on the interface between coverslip and polymer, the thickness of the immersion fluid has to be 577 μm according to Eq. (1), so that no additional aberrations are introduced. The thickness of the immersion fluid is simultaneously the working distance of the hybrid optics.

The achromatization of the optics for the complete spectrum of the laser pulse and the correction of the small remaining spherical aberration caused by the refractive-index-mismatch between HBL, polymer and coverslip is achieved by the DOE. The phase function was calculated for the first diffraction order using the commercial optical design software Zemax [15]. The phase added to the wavefront is described by the following polynomial expansion:

Table 1. Calculated Coefficients A_i of the Phase function of the DOE According to Eq. (2).

View Table

The necessity for correction of the chromatic aberration becomes obvious, when looking at the wavefront aberrations of the optics shown in Fig. 2. Without the DOE the optics exhibits aberrations below 0.5 λ for the central wavelength of 515 nm. These are mainly spherical aberration caused by the refractive-index-mismatch between coverslip and the remaining parts of the ASIL. The wavefront errors for 510 nm and 520 nm are significant larger with up to 3 λ. When the DOE is applied for correction, the aberrations can be corrected down to 0.05 λ for the complete spectrum of the laser pulses.

 

Fig. 2 Wavefront error of the focusing optics without a) and with the DOE b). The wavefront error is reduced below 0.05 λ for the complete spectrum of the laser pulses by the DOE.

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Finally, we use the polymer without the photoinitiator as immersion fluid. Without the photoinitiator no light induced polymerization can occur and the polymer is transparent for the laser pulses. Applying this kind of homogeneous immersion, no writing depth dependent aberrations are introduced and the focal intensity distribution is maintained for different focusing depths. When the optics, and consequently the focus, is moved along the optical axis deeper into the polymer, the gained distance inside the polymer is perfectly compensated by the decreasing thickness of the immersion fluid. Since immersion fluid and the polymer now have exactly the same refractive index, the optical path remains constant for shifting the focus along the optical axis and consequently no writing depth dependent aberrations are introduced. The correction of aberrations with the DOE in combination with the homogeneous immersion allows for diffraction-limited focusing of the laser pulses into polymer over the complete working distance range.

To illustrate the improvements for focusing into the polymer in different z-positions, we calculated the point spread function (PSF) for our hybrid optics and an oil immersion microscope objective for illumination with linearly polarized, monochromatic light at λ = 515 nm. The PSF describes the intensity distribution in the focal region of a microscope objective illuminated with a collimated beam. We used the software program PSF Lab [16] for calculating the PSFs of both objectives. This program applies vector diffraction theory to calculate the PSF of microscope objectives, thereby correctly reproducing the optical setup including refractive index and thickness of immersion fluid, coverslip, and sample. The analytic equations applied for calculation of the PSF are given in appendix A and are described in more detail in Ref. [8].

For the calculation of the PSFs, we assumed a microscope objective with NA = 1.4, corrected for a coverslip thickness of 170 μm (n_515nm = 1.528), with use of a standard immersion oil (n_515nm = 1.52). The hybrid optics was modeled with NA = 1.33, the same thickness and refractive index of the coverslip, but with the polymer as immersion liquid (n_515nm = 1.569). Figure 3 shows cross-sections of the resulting PSFs in the focal plane for different z-positions of the focus. The intensities are individually normalized for each objective to the peak intensity for the focus position located on the substrate surface (z = 0 μm). The peak intensity of the microscope objective decreases with increasing z-positions of the focus inside the polymer, caused by the refractive-index-mismatch-induced spherical aberration. At z = 170 μm, a typical working distance of immersion oil microscope objectives, the peak intensity has dropped to 14 % compared to the aberration-free case at z = 0 μm. Owing to the correction for the spherical aberration of the hybrid optics and in combination with the homogeneous immersion, the PSF of the hybrid optics remains constant for all z-positions of the focus over the complete working distance range.

 

Fig. 3 Cross-sections of the calculated PSFs in the focal plane of hybrid optics (NA = 1.33) and microscope objective (NA = 1.40) for different z-positions of the focus inside the polymer. The intensities are individually normalized for each objective to the peak intensity on the substrate surface (z = 0 μm). The peak intensity of the microscope objective decreases with increasing z-positions of the focus, due to the refractive-index-mismatch-induced aberrations. In contrast, the PSF of the hybrid optics remains constant for all z-positions over the complete working distance range.

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A further advantage of our hybrid optics is the low internal dispersion. Due to the wavelength dependence of the group velocity of light in a dispersive medium, the different spectral components of the pulse travel with different group velocities, leading to a temporal broadening of the pulse as it propagates. The temporal broadening due to GVD, assuming a Gaussian temporal pulse shape, is given by [17]:

 

Fig. 4 Temporal broadening of laser pulses caused by the GDD of the focusing optics in dependence of the initial pulse duration. The values of the introduced GDD for hybrid optics and microscope objective are 842 fs² and 2000 fs², respectively.

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3. Experimental characterization

To demonstrate the improvements for three-dimensional microstructuring, we compare our optics with an oil immersion microscope objective (Plan-Apochromat 100x, NA = 1.4, Zeiss MicroImaging GmbH, Germany). The microscope objective is corrected for a coverslip thickness of 170 μm and usage of the immersion oil Immersol^® (n_e = 1.518, Zeiss MicroImaging GmbH, Germany). The entrance pupil diameter of the microscope objective is 4.1 mm. For our experiments, we used a frequency-doubled Yb:YAG oscillator (t-Pulse 500, Amplitude Systemes, France) at 515 nm, with a fundamental laser wavelength of 1030 nm, a pulse duration of 400 fs, and a repetition rate of 10 MHz. The laser power was adjusted with a polarization beam splitter in combination with a half-wave plate. An acousto-optic modulator (AOM) was applied as shutter. The laser beam was expanded to a diameter of d = 15 mm with a telescope, to illuminate the entire entrance pupil diameter d = 12.5 mm of the hybrid optics. The average laser powers were measured with a photodiode in front of the objectives. The coverslip with the polymer located on its backside was mounted on a three-axis positioning system (ALS 130–150, Aerotech GmbH, Germany), in order to enable three-dimensional structuring of the polymer. We used the polymer ORMOCORE^®with 3 wt. % of the photoinitiator IRGACURE^® 369 (Ciba AG, Switzerland).

At first, we investigated structuring of the polymer directly on the substrate surface, where microscope objective and hybrid optics as well are designed for diffraction-limited focusing. We applied point-by-point illumination with exposure times from 100 to 2 ms for different average laser powers and determined the diameter of the cured volume pixel (voxel) from SEM pictures. The measured voxel diameters are shown in Fig. 5. The smallest achievable voxel diameters are slightly above 200 nm for both objectives. The differences in the applied average powers between Fig. 5(a) and 5(b) are due to the different entrance pupil diameters of microscope objective and hybrid optics. The larger entrance pupil diameter of the hybrid optics also leads to the different slopes of the graphs in Fig. 5(b) compared to Fig. 5(a).

 

Fig. 5 Diameter of voxels on the substrate surface written with microscope objective (a) and hybrid optics (b). In both cases the smallest feature sizes are close to 200 nm.

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In order to compare the two optics for volume structuring of the polymer, we fabricated 30 μm high walls onto the substrate. These walls acted as suspension for lines, structured in different writing depths with constant process parameters (laser power and writing speed). The linewidths were measured from SEM pictures. Figure 6 shows the linewidth in dependence of the writing depth for focusing with the microscope objective (a) and hybrid optics (b). The width of the lines written with the microscope objective becomes smaller with increasing writing depth. As illustrated in Fig. 3, the writing depth dependent spherical aberration lead to a drop of the peak intensity in the focal plane and consequently to smaller volumes of polymerization giving smaller linewidths. The increase of the linewidth up to a writing depth of 10 μm for the series at 1.5 mW in Fig. 6(a) is caused by the aberration induced broadening of the intensity distribution, before the drop of the intensity becomes dominant, resulting in decreasing linewidths. In contrast, when the hybrid optics is applied for focusing into the polymer, the linewidth remains constant for different writing depths. Due to the correction of the writing depth dependent aberrations, the hybrid optics should theoretically allow for homogeneous structuring with constant writing parameters over the complete working distance range of 577 μm.

 

Fig. 6 Measured linewidth vs. writing depth inside the polymer when focusing with the Zeiss Plan-Apochromat (a) and the hybrid optics (b). Scanning speed for all lines was 10 μm/s. Only (b) shows constant linewidths for increasing writing depths and non-varying writing parameters. The dashed lines are a guide to the eye.

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4. Conclusion

We have demonstrated a hybrid optics specially designed for 2PP with a high numerical aperture of 1.33 and low internal dispersion. The suggested optical concept of an ASIL in combination with a DOE allows for focusing of ultrashort laser pulses into the polymer without introducing any writing depth dependent aberrations over a large working distance range of 577 μm. The improvements for volume structuring of the polymer were experimentally verified in comparison to an oil-immersion microscope objective. The flexible design can be easily adapted to polymers with different refractive index, by adjusting the optical function of the DOE.

Appendix A

This section contains the analytic equations implemented in the software “PSF Lab”, for the calculation of the PSFs in section 2. The following equations and most of the notations are taken from Ref. [8], where the derivation of the equations is described in more detail.

The illumination path is schematically shown in Fig. 7. Linearly polarized monochromatic light with vacuum wavelength λ_ill passes through a Babinet-Soleil compensator and is focused by a microscope objective into the sample after traversing three media of different refractive index (immersion medium, n₁; coverslip, n₂; sample medium, n₃). The design values that were used by the manufacturer for the correction of the objective are denoted by an asterisk. The origin of the right-handed coordinate system is placed in the corrected Gaussian focus, which is the geometrical focus in the presence of stratified media as given by the design case. In this coordinate system, the zenith is denoted by θ (0 ≤ θ ≤ π), the azimuth by ϕ (0 ≤ ϕ < 2π), and the z axis points in the direction of propagation of the incident light. h₁ and h₂ are the coordinates (with increasing values toward the objective) of the interfaces of the coverslip (thickness t = h₁ – h₂) relative to the corrected Gaussian focus. Electric field vectors are represented with a lowercase e, electric strength vectors with an E, and 3 × 3 tensor matrices in bold and underlined (e.g. P).

 

Fig. 7 Schematic of the illumination path of an infinity-corrected microscope objective. BS: Babinet-Soleil compensator; OBJ: microscope objective; P: probe. The origin of the xyz coordinate system is placed in the corrected Gaussian focus.

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The time-averaged electric field in the focal region of an objective lens, after passing through three media, can be expressed in the form of an angular decomposition of interfering plane waves:

(5)$$\begin{array}{lll}{\mathbf{e}}_{\text{ill}}(r_s,{\varphi }_s,z_s)\hfill & =\hfill & \frac{i{k}_1}{2\pi }{\int }_0^{\alpha }{\int }_0^{2\pi }{\mathbf{E}}_3\text{exp}\left[i{k}_0(\mathrm{\Psi }-{\mathrm{\Psi }}^{*})\right]\text{exp}\left[i{k}_1{r}_s\text{sin}{\theta }_1\text{cos}(\varphi -{\varphi }_s)\right]\hfill \\ \hfill & \hfill & \times \text{exp}\left(i{k}_3{z}_s\text{cos}{\theta }_3\right)\mathit{sin}{\theta }_1\text{d}\varphi \text{d}{\theta }_1.\hfill \end{array}$$

The magnitudes of the wave vectors are given by k_j = 2πn_j/λ_ill for the different media (k₀ in vacuo). The half-angle α, subtended by the objective lens, is obtained from the NA and the actual refractive indices. The expression Ψ = h₂n₃ cosθ₃ – h₁n₁ cosθ₁ is the initial aberration function, in which the angles θ₁ and θ₃ are linked by applying Snell’s law across the plane-parallel coverslip interfaces: n₁ sinθ₁ = n₂ sinθ₂ = n₃ sinθ₃. The term ${\mathrm{\Psi }}^{*}=-h_1^{*}{n}_1^{*}\text{cos}{\theta }_1^{*}$ corresponds to the initial aberration function calculated for the objective’s design parameters for refractive index and coverslip thickness. Since the objectives are optimized to focus onto the second interface, Ψ^* corresponds to the two-media case ( $h_2^{*}=0$). This compensation of the phase factor Ψ in case of stratified media, by the phase factor Ψ^* in the design case, allows for the simulation of objectives corrected for use with a certain coverslip and immersion fluid.

The electric strength vector E₃ in the third medium is calculated assuming a plane wave with linear polarization along the x axis: E₀ = (1, 0, 0), which traverses a Babinet-Soleil compensator (BS_ill). The Babinet-Soleil compensator allows to turn the axis of linearly polarized light or to convert E₀ into circularly or elliptically polarized light. The electric strength vector in the immersion medium is consequently given by

(6)$${\mathbf{E}}_1=A_{\text{ill}}({\theta }_1^{*}){\underset{\_}{P}}_{(1)}{\underset{\_}{L}}_{(1)}\underset{\_}{R}{\underset{\_}{BS}}_{\text{ill}}{\mathbf{E}}_0=A_{\text{ill}}\left({\theta }_1^{*}\right)\underset{\_}{R}{\underset{\_}{BS}}_{\text{ill}}{\mathbf{E}}_0.$$

The apodization function for the illumination,

(7)$$A_{\mathit{ill}}\left({\theta }_1^{*}\right)=\text{exp}\left[-{\beta }_G^2\frac{{\text{sin}}^2{\theta }_1^{*}}{{\text{sin}}^2{\alpha }^{*}}\right]{\text{cos}}^{1/2}{\theta }_1^{*},$$

allows the incorporation of a Gaussian intensity profile characterized by the filling parameter β_G, where α^* is the half-angle subtended by the objective in the design case ( $\text{NA}=n_1^{*}\text{sin}{\alpha }^{*}$).

The matrices in Eq. (6) are the generalized Jones matrices given by

(8)$$\begin{array}{cc}{\underset{\_}{L}}_{(1)}=\left(\begin{array}{ccc}\text{cos}{\theta }_1& 0& \text{sin}{\theta }_1\\ 0& 1& 0\\ -\text{sin}{\theta }_1& 0& \text{cos}{\theta }_1\end{array}\right),& {\underset{\_}{P}}_{(j)}=\left(\begin{array}{ccc}\text{cos}{\theta }_j& 0& -\text{sin}{\theta }_j\\ 0& 1& 0\\ \text{sin}{\theta }_j& 0& \text{cos}{\theta }_j\end{array}\right),\\ \underset{\_}{R}=\left(\begin{array}{ccc}\text{cos}\varphi & \text{sin}\varphi & 0\\ -\text{sin}\varphi & \text{cos}\varphi & 0\\ 0& 0& 1\end{array}\right),& {\underset{\_}{BS}}_{\text{ill}}=\left(\begin{array}{ccc}{A}_{\text{ill}}^{+}& B_{\text{ill}}& 0\\ B_{\text{ill}}& A_{\text{ill}}^{-}& 0\\ 0& 0& 1\end{array}\right).\end{array}$$

The index j in P_(j) ranges from 1 to 3, referring to the corresponding media. The matrix BS_ill describes the Babinet-Soleil compensator in the illumination path, defined by the components

(9)$$\begin{array}{l}{A}_{\text{ill}}^{\pm }=\text{cos}({\delta }_{\text{ill}}/2)\pm i\text{cos}\left(2{\varphi }_{\text{BS},\text{ill}}\right)\text{sin}({\delta }_{\text{ill}}/2),\\ B_{\text{ill}}=i\text{sin}(2{\varphi }_{\text{BS},\text{ill}})\text{sin}({\delta }_{\text{ill}}/2).\end{array}$$

The orientation and retardation angles of the Babinet-Soleil compensator are denoted by ϕ_BS,ill and δ_ill, respectively.

The electric strength vector in the third medium can be written as

(10)$${\mathbf{E}}_3={\underset{\_}{R}}^{-1}{\underset{\_}{P}}_{(3)}^{-1}{\underset{\_}{I}}_{\text{ill}}{\mathbf{E}}_1=A_{\text{ill}}\left({\theta }_1^{*}\right){\underset{\_}{R}}^{-1}{\underset{\_}{P}}_{(3)}^{-1}{\underset{\_}{I}}_{\text{ill}}\underset{\_}{R}{\underset{\_}{BS}}_{,\text{ill}}{\mathbf{E}}_0,$$

where the stratified media are included via the diagonal matrix:

(11)$${\underset{\_}{I}}_{\text{ill}}=\left(\begin{array}{ccc}{T}_{\left|\right|\text{ill}}& 0& 0\\ 0& T_{\perp \text{ill}}& 0\\ 0& 0& T_{\left|\right|\text{ill}}\end{array}\right).$$

The transmission coefficients T of all three media for the illumination light are

(12)$$T_{\text{pol},\text{ill}}=\frac{t_{12\text{pol}}{t}_{23\text{pol}}\text{exp}\left[i\left(\beta -{\beta }^{*}\right)\right]}{1+r_{12\text{pol}}{r}_{23\text{pol}}\text{exp}\left[2i\left(\beta -{\beta }^{*}\right)\right]},$$

where the index “pol” stands for either s-polarized (⊥) or p-polarized light (||), β = k₀n₂ |h₂ – h₁| cos θ₂, and ${\beta }^{*}=k_0{n}_2^{*}\left|h_1^{*}\right|\text{cos}{\theta }_2^{*}$. The Fresnel coefficients for transmission and reflection from medium i toward medium j,

(13)$$\begin{array}{cc}{t}_{ij\perp }=\frac{2}{1+a_{ij}{b}_{ij}},& t_{ij\perp }=\frac{2}{a_{ij}+b_{ij}},\\ r_{ij\perp }=\frac{1-a_{ij}{b}_{ij}}{1+a_{ij}{b}_{ij}},& r_{ij\left|\right|}=\frac{a_{ij}-b_{ij}}{a_{ij}+b_{ij}},\end{array}$$

are written as a function of the ratios a_ij = n_j/n_i and b_ij = cosθ_j/cosθ_i. The index “ill” indicates, all parameters appearing in Eq. (12) are those at the illumination wavelength λ_ill.

Evaluating Eq. (10) with Eq. (6), (8), and (11) yields

(14)$$\begin{array}{l}{\text{E}}_{3,x}=A_{\text{ill}}\left({\theta }_1^{*}\right)\left(T_{\left|\right|\text{ill}}\text{cos}{\theta }_3-T_{\perp \text{ill}}\right)\left(A_{\text{ill}}^{+}{\text{cos}}^2\varphi +B_{\text{ill}}\text{sin}\varphi \text{cos}\varphi \right)+T_{\perp \text{ill}}{A}_{\text{ill}}^{+},\\ E_{3,y}=A_{\text{ill}}\left({\theta }_1^{*}\right)\left(T_{\left|\right|\text{ill}}\text{cos}{\theta }_3-T_{\perp \text{ill}}\right)\left(A_{\text{ill}}^{+}\text{sin}\varphi \text{cos}\varphi -B_{\text{ill}}{\text{cos}}^2\varphi \right)+B_{\text{ill}}{T}_{\left|\right|\text{ill}}\text{cos}{\theta }_3,\\ E_{3,z}=A_{\text{ill}}\left({\theta }_1^{*}\right)-T_{\left|\right|\text{ill}}\text{sin}{\theta }_3\left(A_{\text{ill}}^{+}\text{cos}\varphi +B_{\text{ill}}\text{sin}\varphi \right).\end{array}$$

With Eq. (14), the integral over ϕ in Eq. (5) can be evaluated to

(15)$${\int }_0^{2\pi }\begin{array}{c}\text{sin}(n\varphi )\\ \text{cos}(n\varphi )\end{array}\text{exp}\left[i\rho \text{cos}(\varphi -\gamma )\right]\text{d}\varphi =2\pi i^n{J}_n(\rho )\begin{array}{c}\text{sin}(n\gamma )\\ \text{cos}(n\gamma )\end{array}$$

This finally leads to the following set of analytic equations:

(16)$$\begin{array}{l}{\text{e}}_{\text{ill},x}\left(r_s,{\varphi }_s,z_s\right)=i{k}_1\left[A_{\text{ill}}^{+}{I}_{\text{ill}}^{\left(2\right)}\text{cos}(2{\varphi }_s)+B_{\text{ill}}{I}_{\text{ill}}^{\left(2\right)}\text{sin}(2{\varphi }_s)+A_{\text{ill}}^{+}{I}_{\text{ill}}^{\left(0\right)}\right],\\ {\text{e}}_{\text{ill},y}(r_s,{\varphi }_s,z_s)=i{k}_1\left[A_{\text{ill}}^{+}{I}_{\text{ill}}^{\left(2\right)}\text{sin}(2{\varphi }_s)-B_{\text{ill}}{I}_{\text{ill}}^{\left(2\right)}\text{cos}(2{\varphi }_s)+B_{\text{ill}}{I}_{\text{ill}}^{\left(0\right)}\right],\\ {\text{e}}_{\text{ill},z}(r_s,{\varphi }_s,z_s)=2k_1\left[A_{\text{ill}}^{+}{I}_{\text{ill}}^{\left(1\right)}\text{cos}{\varphi }_s+B_{\text{ill}}{I}_{\text{ill}}^{\left(1\right)}\text{sin}{\varphi }_s\right],\end{array}$$

where the integrals $I_{\text{ill}}^{(j)}$ are defined by

(17)$$\begin{array}{lll}{I}_{\text{ill}}^{\left(0\right)}\hfill & =\hfill & {\int }_0^{\alpha }{A}_{\text{ill}}\left({\theta }_1^{*}\right)J_0\left(k_1{r}_s\text{sin}{\theta }_1\right)\left(T_{\perp \text{ill}}+T_{\left|\right|\text{ill}}\text{cos}{\theta }_3\right)\text{exp}\left[i{k}_0\left(\mathrm{\Psi }-{\mathrm{\Psi }}^{*}\right)\right]\hfill \\ \hfill & \hfill & \times \text{exp}\left(i{k}_3{z}_s\text{cos}{\theta }_3\right)\text{sin}{\theta }_1\text{d}{\theta }_1,\hfill \\ I_{\text{ill}}^{\left(1\right)}\hfill & =\hfill & {\int }_0^{\alpha }{A}_{\text{ill}}\left({\theta }_1^{*}\right)J_1\left(k_1{r}_s\text{sin}{\theta }_1\right)T_{\left|\right|\text{ill}}\text{sin}{\theta }_3\text{exp}\left[i{k}_0\left(\mathrm{\Psi }-{\mathrm{\Psi }}^{*}\right)\right]\hfill \\ \hfill & \hfill & \times \text{exp}\left(i{k}_3{z}_s\text{cos}{\theta }_3\right)\text{sin}{\theta }_1\text{d}{\theta }_1,\hfill \\ I_{\text{ill}}^{\left(2\right)}\hfill & =\hfill & {\int }_0^{\alpha }{A}_{\text{ill}}\left({\theta }_1^{*}\right)J_2\left(k_1{r}_s\text{sin}{\theta }_1\right)\left(T_{\perp \text{ill}}-T_{\left|\right|\text{ill}}\text{cos}{\theta }_3\right)\text{exp}\left[i{k}_0\left(\mathrm{\Psi }-{\mathrm{\Psi }}^{*}\right)\right]\hfill \\ \hfill & \hfill & \times \text{exp}\left(i{k}_3{z}_s\text{cos}{\theta }_3\right)\text{sin}{\theta }_{1}d{\theta }_1.\hfill \end{array}$$

Acknowledgments

We acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within priority program SPP1327.

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16. ”PSF Lab” available at onemolecule.chem.uwm.edu/index.php/software.

17. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, 2006).