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nanometrology free of unwanted fluorescence labelling. Finally, we end with a short conclusion and outlook( Sect. 6).
2 Enhancement of classical light microscopy by through-focus microscopy
In microscopy, a typical task is to determine the distance between two points on a surface. From this spatial resolution is defined as the ability to distinguish two points like objects on from a single object. Vertical resolution is not uniquely defined in microscopy. Most often we speak about the ability to distinguish two separated neighbouring point like objects on a surface at different vertical positions from a single object. However, in microscopy we can create a 3D image of the object by making a vertical image stack of 2D images. The separation between the 2D images is related to axial resolution of the microscope, which is the ability to separate two points like objects on the axial line from a single object.
Classical bright-field optical microscopy is the simplest, most widely used system for dimensional metrology at the micro- to nanoscale. However, due to diffraction the spatial resolution is limited to about k /( 2NA) according to the Rayleigh limit, where NA is the numerical aperture [ 24 ]. The axial resolution is limited by the depth of field of the microscope, which may be estimated by 2k /( NA 2).
Scanning confocal microscopy offers several advantages over bright-field microscopy, including the ability to control depth of field and reducing the scattered light away from the focal plane, the capability to serial image thick objects, and the ability to achieve high lateral resolution. The key to confocal advantages is the use of spatial filtering with a pinhole to eliminate out-of-focus light from the object and the use of a laser or narrow line-width LED to form a coherent imaging system. A coherent imaging system is described by linear addition of the fields from the object, whereas a normal incoherent bright-field microscope is described by linear addition of intensities. The point-spread function( PSF) is a measure of the resolving power of a microscope and the observed microscope image is the convolution between the true image and the PSF. The true image may in principle be obtained by deconvolving the observed picture with the PSF of the microscope, if the PSF is known. However, in reality we only get an improved image due to the lack of available information. The deconvoluted image is the limit of classical light microscopy with a typical spatial resolution limit of approximately 75 nm for scanning confocal microscopy working in the visible wavelength range [ 25 ].
Through-focus confocal microscopy( TFCM) is a technique, in which the image is obtained using inverse methods, where a simulated image stack is compared to the measured image stack to obtain the best fit [ 25, 26 ]. The phase information in the complex fields usually requires rigorous treatment of the light-matter interaction and the light propagating through the microscope. The first step for a partially resolved image is to improve the measured image by deconvolution with the PSF of the microscope. This image is used to extract the starting values for the inverse modelling.
Here we present a digital twin of an optical microscope focusing on TFCM. The working principle of the digital twin can be described by the following steps:
1. The incident field is a normal incident wave with a normalized k-vector( 0, 0, 1) T in the back focal plane.
2. The light beam coordinate system defined by the unit vectors for TE-amplitude, TM-amplitude, and k-vector is used to describe the optical propagation.
3. The response from the periodic structure is calculated as a Jones matrix using the Fourier modal method.
4. All the points that enter the same pixel in the back focal plane are added coherently. The 2D image formation is done by focusing the far-field angular scattering in the back focal plane for a given vertical position onto the image plane.
5. The 3D image is obtained by performing 2D image formations for all vertical( z) positions and stacking the 2D images( see Fig. 1).
6. For confocal microscopy imaging the 3D image is a set of data points( x, y, z) from the 2D image stack. For each lateral position( x, y), the z value is obtained as the z position in the 2D image stack, for which the intensity is maximal.
We have performed TFCM simulations of silicon line gratings with a pitch of 250 nm, 300 nm and 400 nm of ridge height 138 nm( for the smallest pitches) and a ridge height of 148 nm for the largest pitch. In the simulations, we use light of wavelength 405 nm and an objective with NA = 0.95. We demonstrate that TFCM simulations resolve these lateral pitch structures and that the resolution along the z direction is improved by deconvolution of the PSF of the objective.
We obtain the PSF by Fourier transformation of the optical transfer function, which is defined by a step function in cylindrical coordinates [ 27 ]. The components of the light propagation vector in the lateral directions( k x, k y) are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined by k 2 2 x þ k y 2p NA. The out-of-plane propagation vector is defined by k z ¼ ð2p = kÞ 2 � k 2 2
k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x � k y which
enters the out-of-focus phase factor of the PSF given by e ikz z. The image step size defines the sampling in spatialfrequency space. Figure 2 shows the result of applying this procedure to the silicon line grating with pitch of 300 nm.
In a real microscope measurement, the procedure is slightly different. First, the sample is measured, and we observe the measured profile given by the black line in Figure 2. Secondly, we do a deconvolution to improve the optical profile to find a good starting point for TFCM inverse modelling. Thirdly, we calculate simulated optical profiles from simulated z-stack images to find the simulated optical profile that best match the measured optical profile( black curve). The shape of the measured optical profile is found from knowledge of the shape for the best matching simulated profile.