## Abstract

The reduction of wavefront aberrations is essential in a number of fields, including astronomy, microscopy, photography, vision science, lithography, and lasers. Aberrations may be determined either directly with wavefront sensors or indirectly with signal- or image-based optimization algorithms. Here, we introduce a novel wavefront-sensing methodology that employs intensity differences across a beam of light to encode local wavefront slopes via attenuated total internal reflection following surface-plasmon excitation at the surface of a thin gold film. This method excels due to the dense spatial sampling of the wavefront and the fact that the wavefront itself can be determined by straightforward integration of two sets of images captured in orthogonal directions without time-consuming optimization, deconvolution, or spot centroiding.

© 2015 Optical Society of America

Wavefront aberrations (WA) are routinely determined by interferometric techniques such as self-referenced point diffraction [1], shearing and digital methods [2], or by sensing of wavefront slopes. The latter is often realized using a Hartmann–Shack (HS) wavefront sensor (WFS) to sample the wavefront at regular distances set by the lenslet array, with a typical pitch of 100 μm or more, from which WA of the incident light can be determined by numerical reconstruction [3]. Denser sampling is feasible using curvature [4] or pyramidal wavefront sensing [5,6] that allows WA to be determined via intensity differences between distinct pupil images. We have recently shown that dense wavefront sampling is feasible using an array of waveguides that convert local wavefront slopes into power differences in the fraction of wave-guided light without time-consuming spot centroiding [7]. Here, we propose a novel intensity-based wavefront-sensing scheme employing attenuated total internal reflection (ATIR) associated with the excitation of surface-plasmon polaritons (SPP) in near-resonant conditions in the common Kretschmann configuration [8–10]. This method samples the wavefront at a nanometric resolution across the beam and translates any WA into spatial intensity variations from which the wavefront can be reconstructed.

The SPP-WFS is demonstrated experimentally using a 50 nm Au film coated onto a 0.15 mm thick BK7 glass coverslip with a nominal 2 nm Ti binding layer (Phasis, Switzerland). The film roughness is $<1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ and thickness variations are $<1\%$. ATIR measurements for the film mounted with index-matching oil onto a 20 mm rectangular BK7 prism with AR-coated legs are shown in Fig. 1 and compared to the calculated ATIR dependence found using COMSOL Multiphysics. The simulations are based on the dielectric functions ${\epsilon}_{\mathrm{Au}}=-11.740+i1.2611$ and ${\epsilon}_{\mathrm{Ti}}=-6.8655+i20.361$ for Au [11] and Ti, respectively, and the refractive index ${n}_{\mathrm{BK}7}=1.5151$ for BK7 at wavelength $\lambda =632.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. Resonant excitation occurs when the light momentum is best matched to that of the SPP which, for the chosen film, happens at incidence angle ${\theta}_{\mathrm{SPP}}=44.0\xb0$. In the present context, the angles of interest are the ATIR slopes at either side of the resonance where two cases will be considered: configurations I and II. The set angle, around which deviations are measured, has been chosen as 43.7° and 44.5° for configurations I and II, respectively. The steepness of the resonance determines the sensitivity to angular deviations and its width sets the dynamic range. Other film thicknesses or metals will alter these characteristics. The deviations between theory and experiment can be attributed to uncertainties in the dielectric function of Au, scattering by interface roughness, contaminants, humidity caused by the ambient conditions [12], and experimental errors. At the SPP resonance, however, differences are small and the measured width of the ATIR dip matches closely that of its calculated counterpart.

The setup used to verify the SPP-WFS is shown schematically in Fig. 2. A MEMS deformable mirror (DM) with 140 actuators and nominal 3.5 μm stroke (Boston Micromachines, USA) is operated in adaptive optics (AO) closed loop with a HS-WFS (Thorlabs, USA) to generate controllable amounts of Zernike aberration modes up to the 4th radial order across a collimated $4\times $ expanded beam from a He–Ne laser. The prism with the Au film is mounted on a motorized rotation stage with its hypotenuse in the approximate conjugate plane of the DM and HS-WFS. The WA imparts intensity variations by ATIR onto the beam cross-section that is monitored by a CCD camera (Thorlabs, USA). SPP coupling occurs just for $p$-polarized light, and thus only the wavefront derivate in the $x$ direction is measured with the prism as shown in Fig. 2. It can also be analyzed with unpolarized light, albeit the contrast of the induced changes would be correspondingly reduced. To measure the wavefront derivatives in the $y$ direction, a half-wave plate is inserted into the beam and the prism-CCD branch is rotated 90° out of the plane (a setup with two orthogonally mounted prisms and Au films could allow simultaneous detection of both $x$ and $y$ derivatives).

Analytical Zernike modes are shown in Fig. 3 together with their Cartesian derivatives from tip and tilt to spherical aberration: ${Z}_{2}\u2013{Z}_{11}$. The derivatives are shown scaled by a positive and negative Zernike coefficient ${c}_{n}=\pm 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ for $n=2$ to 11, and have been individually adjusted to the full range of the color map.

The CCD camera captures a conjugated pupil image of the ATIR light and shows WA as either an increased or decreased intensity across the beam. In the following, experimental images show the captured difference in intensity between the beam subject to aberrations, ${I}_{\mathrm{WA}}$, and that of a planar reference wave, ${I}_{\text{ref}}$. In this way, the impact of a nonuniform amplitude is reduced. Prior to subtraction, the aberrated and reference images have been low-pass Fourier filtered to reduce high-frequency noise caused by speckle and actuator print-through effects of the DM (see Fig. 2 insets). A different DM or the use of a high-resolution liquid-crystal spatial light modulator could potentially circumvent this complication.

Normalized CCD intensity measurements for the $x$ derivatives of the Zernike polynomials from ${Z}_{2}$ to ${Z}_{11}$ scaled by ${c}_{n}=\pm 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ are shown in Fig. 4. They are plotted on a color scale for enhanced visibility. Systematic brightness variations related to small irregularities of the Au film can potentially be compensated by spatial normalization of the WA across the beam. However, the noise made us refrain from it at this stage. Nonetheless, it is worth noting that addition of the images scaled by $\pm 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ largely cancels the WA as expected.

The corresponding $y$ derivatives have subsequently been measured with the prism-CCD branch rotated 90° out of the plane and with a $\lambda /2$ plate inserted into the beam. The results for this case are shown in Fig. 5 for the same two configurations just below and above the SPP resonance angle.

Figures 4 and 5 show the sensitivity to wavefront slopes of the SPP-WFS. Configuration I appears with inverted contrast when compared to configuration II and the theoretical expectations shown in Fig. 3 due to the opposite slopes at either side of the ATIR resonance. For both the $x$ and $y$ derivatives, configuration I has higher sensitivity and less noise than configuration II.

The ATIR is approximately linear in the region near the bias setting. To explore this, measurements have been made for different Zernike coefficients and modes. Figure 6 shows the outcome for ${Z}_{2}$ and the $x$ derivative in configurations I and II together with a plot of the average image intensity that fits a linear distribution with slopes of $-0.99$ and $+0.93$, respectively (both with an R-squared value of 0.99). A multilayered film for SPP excitation may potentially be designed that would increase linearity further.

An ATIR range from 0.8 to 0.1 equals a 0.79° change in the angle of incidence (see Fig. 1), which translates into a dynamic range of up to $\pm 7.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ for ${c}_{2}$ and ${c}_{3}$ across the incident beam.

For example, a 1% difference in ATIR corresponds to a 0.0022° (i.e., 7.9 arc-sec.) change in the angle of incidence and a Zernike coefficient of 0.040 μm. Even smaller differences may be detected as set by the dynamic range and resolution of the CCD, but ultimately intensity fluctuations and detector noise put a lower limit on the SPP-WFS sensitivity. For the chosen 12-bit CCD camera, the smallest detectable angle equals 0.00011° (i.e., 0.39 arc-sec.) and thus a Zernike coefficient of 0.0020 μm for tip and tilt, provided that the signal is not limited by noise. Operating the SPP-WFS at less than the full ATIR range can potentially be used to extend the sensitivity further.

For other aberrations, an intensity gradient is present that should ideally be resolved with pixel resolution across the CCD camera. Thus, a wide sensor has the largest sensitivity but a reduced dynamic range. The chosen system has $657\times 657$ pixels enclosing the analyzed pupil diameter so that the required smallest range is 0.072° (i.e., 4.3 arc-min.), equal to a Zernike coefficient of 0.25 μm for defocus or 0.36 μm for astigmatism. Binning of adjacent pixels will reduce the range correspondingly and allow detection of smaller-magnitude aberrations. To maintain the pixel-sized resolution for small WA, a higher dynamic range of the CCD used would be required. The finite size of the sampled pupil in the $4\text{-}f$ system induces an angular range of incident wave vectors even in the absence of WA. However, while only analyzing difference images, $\mathrm{\Delta}{I}_{n}={I}_{\mathrm{WA}}\text{-}{I}_{\text{ref}}$, this has no impact on the final estimates.

The last important step is the WA reconstruction (${\mathrm{WA}}^{R}$). This may be accomplished most directly by numerical integration of the wavefront slopes shown in Figs. 4 and 5, assuming linearity. This method is exact for tip, tilt, defocus, and astigmatism but only approximate for higher-order Zernike modes. The results of direct integration are for simplicity not shown here (although an example is included in Fig. 8 for combinations of Zernike modes).

More accurately, the intensity images (proportional to the wavefront derivatives) may be projected onto the analytical Zernike derivatives and the WA calculated from the resulting least-square estimate of the Zernike coefficients. This reduces noise and is the preferred reconstruction method based on wavefront slopes. Thus, the estimate ${c}^{\text{est}}$ for the reconstructed Zernike coefficients can be written as the vector [13]

where $\mathit{A}$ is a tensor of Zernike polynomial derivatives, ${\mathit{A}}^{T}$ is the transpose of $\mathit{A}$, and ${[{\mathit{A}}^{T}\mathit{A}]}^{-1}{\mathit{A}}^{T}$ is the pseudo-inverse of $\mathit{A}$. The vector ${b}^{\text{meas}}$ consists of measured wavefront slopes proportional to the intensity derivatives across the pixels of the CCD images shown in Figs. 4 and 5. Reconstructing the Zernike coefficients for the data from Eq. (1) and calculating the aberrations as a linear combination of Zernike polynomials from ${Z}_{2}$ to ${Z}_{11}$ produces the results shown in Fig. 7.Direct comparison of the SPP-WFS determined wavefront with that of the Zernike polynomials in Fig. 3 shows that the reconstruction is in accord with expectations except for spherical aberrations ${Z}_{11}$, where the reconstruction is partially projected into lower-order Zernike modes (predominantly tilt and astigmatism) due to the truncation of the Zernike series and measurement inaccuracies.

In most situations, combinations of Zernike modes will be present. Figure 8 shows one such case, when the AO subsystem has all odd Zernike coefficients set equal to $-0.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and all even coefficients set equal to $+0.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ for polynomials from ${Z}_{2}$ to ${Z}_{11}$. Good correspondence can be observed between the reconstructed wavefront (in particular the least-square estimate for configuration I) and the AO-applied WA, whereas the straightforward reconstruction by integration is again more prone to errors.

Further insight may be gained by plotting the reconstructed Zernike coefficients determined for the linear combination of Zernike modes.

Figure 9 shows a bar diagram of the applied Zernike terms versus those estimated based on the least-square reconstruction for configurations I and II. The errors, which are largest for tip and tilt, may be reduced with a dual-prism setup, allowing for simultaneous detection of derivatives in two orthogonal directions and the avoidance of DM print-through effects in the generated WA.

In this Letter, a novel SPP-WFS has been introduced and conceptually verified experimentally. The sensor has very dense sampling potential and simplifies numerical reconstruction of aberrations since wavefront derivatives are proportional to the ATIR images recorded by the CCD in the conjugate pupil plane. Although the spatial resolution of the wavefront sampling is at present limited by the pixel spacing, it could easily be further enhanced with optical magnification in the system branch from the Au film to the camera. The main shortcomings of the experimental verification are the coherence of the light and the noise created by print-through effects of the DM used. A light source with a larger spectral bandwidth ($\mathrm{\Delta}\lambda <10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$) would reduce interference artifacts and in a real application, where aberrations originate from elsewhere, the DM would not be needed. Although analyzed for monochromatic light, the SPP-WFS could be used at other wavelengths. In the near-IR it would have higher sensitivity but reduced dynamic range. Most importantly, however, it would allow larger bandwidth of the illumination without significant differences in the ATIR behavior [8].

Refractive errors introduced by the prism are small and can be even further reduced if the prism is cut exactly at the SPP set angle. The calibration of the SPP-WFS is done externally to the prism and the system is therefore in great part immune to any such errors. The film and prism could be vacuum sealed for improved robustness and durability. The fact that a prism is used for the SPP excitation is not a fundamental limit and alternative ATIR schemes may be envisioned such as, for example, the use of gratings and multilayered Bragg reflectors [8], or nanostructured surfaces [14,15] that simultaneously probe various polarization states [16] and may ultimately make the current two-stage process for determination of $x$ and $y$ derivatives of the WA redundant. In this way, the SPP-WFS would allow for real-time AO at a speed set by the CCD frame rate and the multiplication in Eq. (1), as long as the photon flux is above the noise floor.

## Funding

Capes Foundation (Science without Borders); Irish Research Council (New Foundations).

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