1. Introduction

Through-focus scanning optical microscopy (TSOM) is an optical model-based metrology method that was developed to overcome the classical resolution limit due to diffraction in optical microscopy [1–3]. This bright-field microscopy technique uses an ordinary optical reflection microscope but produces nanometer dimensional measurement sensitivity [4–9]. In comparison to a conventional bright-field microscope, which directly extracts the diffraction-limited optical information from a single in-focus image, TSOM indirectly extracts nanometer scale sensitive information by analyzing stacked through-focus (TF) images as a whole (TSOM data cube) or a cross-sectional image along the scanning direction, which is referred to as a TSOM image. The TSOM data analysis typically includes library-matching [4–9] or machine-learning methods [10,11]. Both methods require reference TSOM data/images that are either experimentally or computationally collected. Therefore, the sensitivity and accuracy of the TSOM method strongly depends on the similarities between the conditions in which the target and reference TSOM data/images are acquired or simulated. Figure 1 provides a schematic diagram of the TSOM process describing four major steps: 1) through-focus scanning along the optical axis, 2) image acquisition at multiple through-focus positions during the scanning process, 3) TSOM data cube/image generation, and 4) TSOM data processing.

 

Fig. 1. Schematic diagram of the TSOM process

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It should be emphasized again that TSOM is a nanodetection technique but does not provide real improved resolving power such as fluorescent stimulated emission depletion microscopy (STED) [12]. The nanometer sensitivity is obtained with great care paid in optimizing the image acquisition conditions such as averaging, noise reduction, and calibration, and in normalizing the images so that they can be compared with each other or with a library of images from a reference library [3,13–14]. In particular, major efforts have been focused on mitigation of the dissimilarity in the through-focus scanning and illumination conditions in which the target and reference TSOM images are taken or simulated [15–23].

Regarding the mitigation efforts for the instability issue, several approaches have been proposed: 1) a z-axis scanning-free strategy utilizing the residual chromatic aberration of an objective lens [15,16]; 2) temporal averaging by the exposure time increment [17]; 3) instability correction with Fourier transform [18,19]; and 4) active compensation by the use of adaptive optics technology [20–22]. The first approach requires precise knowledge of the residual chromatic aberration, which is not typically provided and it prohibits the use of monochromatic light sources such as lasers desired for high volume metrology and inspection applications [24]. The second approach averages out the random position instability by increasing the exposure time of the camera, which sacrifices the efficacy or the throughput. In addition, the slow lateral movement such as drift cannot be removed out even after the temporal averaging. The third approach, i.e. lateral movement correction using Fourier transform, effectively decreases the lateral movement of the pattern image due to the optical and mechanical noise, simultaneously. But this approach requires precise knowledge of the target structure and material properties and the increased TSOM data processing time. The use of adaptive optics [25,26] was initially proposed to compensate the lateral jitter motion with a tip/tilt mirror close-looped with a Shack-Hartmann wavefront sensor in the imaging path of the microscope [20–22]. However, due to the limited angular resolution of the Shack-Hartmann sensor, the lateral position error is small but still non-zero, i.e. 33 nm in peak-to-valley (PV) value [22].

We here propose further improvement from the tip/tilt compensated TSOM, i.e. a motion-free TSOM. The motion-free TSOM reduces the instability down to practically zero by replacing the tip/tilt mirror with a deformable mirror which performs through-focus scanning by deforming its mirror surface.

2. Understanding of through-focus scanning process

2.1 Optical understanding of through-focus scanning

Figure 2 shows a schematic diagram of an infinite-conjugate microscope consisting of an objective lens, a tube lens, and a phase modulator located at the Fourier plane of the microscope, which is a deformable mirror in our study. In terms of geometrical optics, an object plane at the focus of the objective lens is conjugated to the image plane located at the focus of the tube lens with lateral magnification $M $ and longitudinal magnification ${\textrm{M}_z}$ as follows:

 

Fig. 2. Schematic diagram of an infinite-conjugate microscopy

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When the object plane is shifted along the optical axis by an amount ${\varepsilon _z}$, the conjugated image plane is accordingly shifted along the optical axis by an amount $ - {M^2}{\; }{\varepsilon _z}$. Without the corresponding shift of either the object plane or image plane, the imaging is referred to as defocused. In terms of wave optics, the wavefront aberration $\textrm{W}\left( {x,\textrm{y}} \right)$ of the defocused imaging is given as

Therefore, the through-focus scanning can be done by changing either the object or image distances or by generating equivalent optical aberrations by chromatic sweeping or active phase modulation. Attotta thoroughly reviewed over 15 types of through-focus image acquisition methods potentially suitable for TSOM [27]: scanning the sample stage along the focus axis, scanning the objective lens along the focus axis, scanning the image plane along the focus axis, multifocal plane microscopy, wavelength scanning method, flexible-membrane liquid lens, liquid-tunable lens, adaptive optics, multifocus microscopy, aperture-scanning Fourier ptychography, confocal microscopy, light sheet microscopy, light field microscopy, phase retrieval techniques, and digital holography microscopy. Among the reported methods, sample stage scanning is most widely used in the TSOM studies by adopting a manual or motorized z-axis stage [2,4–11,13,14,16,19,23].

2.2 Numerical simulation of through-focus scanning

Many studies numerically simulated the through-focus scanning process by iterative calculation of the light field with movement of the target [2,4–9,11–17]. Recently, a Fourier modal method (FMM) was proposed to improve the computational efficiency by applying a once-solved scattering matrix in the frequency domain to obtain field distributions at any z-position [18], which was used for building up reference TSOM images in this study.

2.3 Necessity of stable through-focus scanning mechanism

The use of the mechanical scanning inevitably incurs positional ambiguity and instability, mostly in the lateral direction across the through-focus scanning direction. First, sub-pixel position ambiguity naturally exists due to the use of a finite sized detector which fundamentally limits the accuracy of the TSOM method. Second, it is very costly to remove relevant random jitter and slow drift during the z-axis stage operation, especially when dealing with targets of nanometer feature size. It was reported that the slow-drift or tilt in the TSOM image can also occur due to non-uniform angular illumination asymmetry (ANILAS) across the field-of-view [28,29]. Figure 3 shows conceptual TSOM image stacks with the lateral positional ambiguity and instability and corresponding exemplary simulated TSOM images.

 

Fig. 3. TSOM image stacks with the lateral positional ambiguity and instability: (a)-(d) conceptual and (e)-(h) exemplary simulated TSOM images

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A quantitative analysis was studied to investigate the effect of the lateral position errors over the TSOM method for critical dimension (CD) measurement on a 60 nm single line fin pattern based on the mean square deference (MSD) as given by

Figure 4 shows the mean square differences (MSD) variations due to CD change and various positional errors of a 60 nm linewidth single line fin pattern. The MSD value of 31.2 × 10⁻⁴ corresponds to the 10% CD difference, i.e. 6 nm. For securing 10% CD measurement accuracy, the sub-pixel position ambiguity, random jitter, and liner drift should be smaller than 15 nm, 12 nm, and 0.4 degree, respectively. Therefore, it is obviously necessary to reduce the positional errors for achieving nm measurement sensitivity in the TSOM method.

 

Fig. 4. MSD variations due to (a) CD change and (b)-(d) individual positional errors of 60 nm linewidth single line fin pattern.

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Table 1. TSOM simulation conditions for investigating the lateral instability and ambiguity.

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3. Implementation

3.1 Adaptive optics supported TSOM

Adaptive optics is an optical technology that removes the wavefront distortion introduced by a turbulent medium by introducing a controllable counter wavefront distortion that both spatially and temporally follows that of the disturbance [25,30]. The use of adaptive optics technology was proposed to actively compensate the lateral instability via two approaches [20–22]. The first approach uses a simple tip/tilt mirror to compensate any optical-axis instability while moving the wafer stage. The second approach uses a deformable mirror to optically perform the through-focus scanning without any mechanical movement. Figure 5 presents schematic diagrams of the two approaches.

 

Fig. 5. Schematic diagram of the adaptive optics supported TSOM

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3.2 Experience with a tip/tilt correction TSOM implementation

The first approach was implemented on an optical bench with commercial off-the-shelf components. The technical details of the implementation are tabulated in Table 2. The motorized z-axis stage experienced center beam movement of 4.1 µm over the 3.0 µm z-axis scanning, as seen in Fig. 6(a). With the help of the active tip/tilt compensation, the beam movement was dramatically reduced down to 33 nm in PV (peak-valley), as presented in Fig. 6(b). However, considering the results of section 2.3, further reduction of the positional error is still desired.

 

Fig. 6. Beam center movement of a tip/tilt corrected TSOM during the through-focus scanning process without and with tip/tilt correction

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Table 2. Main parameters of a tip/tilt corrected TSOM implementation.

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3.3 Motion-free TSOM with a deformable mirror

There are several types of deformable mirrors (DMs) including stacked array, bimorph, voice-coil actuator, micro electro-mechanical system (MEMS), and optically addressed DMs [30,31]. Among them, a bimorph DM is chosen in our study since it provides excellent correction or formation of most common optical aberrations, especially focus [32,33].

The wavefront deformation of the deformable mirrors can be controlled via either zonal or modal control approaches. In the modal control approach, each control channel corresponds to one optical mode, which is often a Zernike polynomial. The wavefront deformation ${\textrm{W}_{DM}}{\; }({x,\textrm{y}} )\; $is then expressed by

Figure 7 shows a motion-free TSOM tool developed with the technical details in Table 3. As expected from Eq. (8), the tool could provide a through-focus range of ± 25 µm with a 25 nm step for a × 50 (NA 0.55) objective lens. The illumination NA and spectrum could be simply changed by changing the aperture and spectral filter in the illumination path, marked as ⑧ in Fig. 7.

 

Fig. 7. Developed motion-free TSOM tool: ① light source ② He-Ne laser ③ z-axis stage ④ objective lens ⑤deformable mirror ⑥ imaging detector ⑦ Shack-Hartmann wavefront sensor ⑧ aperture & spectral filter ⑨ control PC

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Table 3. Main parameters of a motion-free TSOM implementation.

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3.4 Experiments with single line fin patterns

The sub-nanometer scale sensitivity of the motion-free TSOM tool was tested with four single line fin patterns having design width 60, 80, 100, and 120 nm with a height of 50 nm, which were measured to have CDs of 55.2, 78.9, 94.7, and 110.4 nm, respectively, by a scanning electron microscope (SEM), as shown in Fig. 8. Figure 9 shows the experimentally measured and corresponding simulated TSOM images of the four single line fin patterns. Each measured TSOM image was compared with reference TSOM images featuring sub-nanoscale linewidth variations in terms of the MSD. The linewidth (CD) could be estimated at the minimum point of the MSD under a reliable tolerance level of 0.1 nm [18]. The estimated line widths were all within 4 nm from the SEM measured values, as listed in Table 4.

 

Fig. 8. SEM images of the manufactured four single line fin patterns

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Fig. 9. TSOM images experimentally obtained by (a)-(d) the motion-free TSOM tool and numerically by (e)-(h) the Fourier Modal Method for the single fin patterns having width of 55.2, 78.9, 94.7, and 110.4 nm

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Table 4. CD values of the four single line fin patterns in nm.

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4. Conclusion

TSOM is a nanodetection technique that overcomes the resolution limit but the nanometer sensitivity only is obtained with great care paid in optimizing the image acquisition conditions and minimizing the dissimilarity in the through-focus scanning and illumination conditions in which the target and reference TSOM images are taken or simulated. Adaptive optics techniques were introduced to stabilize the through-focus scanning process. The first implementation with a tip/tilt mirror successfully reduced the PV lateral position error from 4 µm to 33 nm. However, it was necessary to further reduce the residual to a few nanometers. A motion-free TSOM by the use of a deformable mirror (DM) was then proposed. A motion-free TSOM tool with a bimorph mirror was developed and it could provide a through-focus range of ± 25 µm with a 25 nm increment for a ×50 (NA 0.55) objective lens. We demonstrated that the motion-free TSOM tool estimated the critical dimension (CD) values of single line fin patterns in the range of 60 ∼ 120 nm with accuracy of < 4 nm based on the minimum MSD approach.