## Abstract

The linear systems optical resolution limit is a dense grating pattern at a λ/2 pitch or a critical dimension (resolution) of λ/4. However, conventional microscopy provides a (Rayleigh) resolution of only ~ 0.6λ/*NA*, approaching λ/1.67 as *NA* → 1. A synthetic aperture approach to reaching the λ/4 linear-systems limit, extending previous developments in imaging-interferometric microscopy, is presented. Resolution of non-periodic 180-nm features using 633-nm illumination (λ/3.52) and of a 170-nm grating (λ/3.72) is demonstrated. These results are achieved with a 0.4-*NA* optical system and retain the working distance, field-of-view, and depth-of-field advantages of low-*NA* systems while approaching ultimate resolution limits.

© 2007 Optical Society of America

## 1. Introduction

Optical microscopy is a well-established field with many important applications. Despite a long and rich history, much effort and significant advances continue. Many newly demonstrated techniques take advantage of nonlinearities associated with fluorescence to achieve resolutions well beyond the diffraction limit, for example: stimulated-emission-depletion (STED; λ/50) [1, 2] and structured illumination and saturation (λ/10) [3]. In another direction, efforts have continued to enhance resolution using synthetic aperture approaches that extend the collected spatial frequency content towards the 2/λ limit of the transmission medium [4–13].

The resolution of a traditional microscopy system is given by the well-known Rayleigh criterion, *CD* ~ 0.6×λ/*NA* where, borrowing from lithographic terminology, *CD* is the critical dimension or the smallest resolvable linear dimension, λ is the optical wavelength, and *NA* is the numerical aperture of the optical system. This limit directly arises from the low-pass spatial bandwidth constraints of the optical system, which extend to *NA*/λ for coherent illumination and to 2*NA*/λ with reduced fidelity at high spatial frequencies for incoherent illumination [14].

In contrast, the transmission medium (assuming air or vacuum with a unity refractive index, immersion will be discussed below) has a bandpass of 2/λ corresponding to the highest angle at which diffracted light from an object is available (incident beam at grazing incidence and diffraction in the backward direction). If an optical system could record the scattering from a 1-D structure object for frequencies up to 2/λ, a higher resolution image, extending to *CD*s as small as λ/4 would be achievable. For arbitrary patterns, the resolution is somewhat lower as a result of the need to capture an extended range of frequencies around the center frequency to define the image; in the same spirit as Rayleigh’s criterion, this limit is *CD* ~ λ/3.3.

The goal of imaging interferometric microscopy (IIM) is to use a relatively low-*NA* microscope objective but, in multiple partial images with off-axis illumination and interferometric optics, to assemble a composite image corresponding to a total frequency-space coverage extending out to 2/λ. This is a synthetic aperture approach in which different regions of frequency space are recorded in separate partial images. A related concept, imaging interferometric lithography or IIL, has been introduced for lithographic image formation [6]. A major advantage for microscopy is that the partial images can be electronically manipulated, whereas in the lithography case the image information is chemically stored in the photoresist and is not directly accessible. Because each partial image is obtained with a low-*NA* optical system, the depth-of-field, field-of view and working distance associated with the low-*NA* system are retained, but the final composite image has a resolution roughly a factor of two better than that possible with any high-*NA* lens using conventional imaging approaches as a result of the large (~ 1) modulation transfer coefficients extending out to the edges of frequency space.

We have previously presented a simple case with only two offset partial images, one each in orthogonal directions, and demonstrated the possibility of resolving 0.5-μm features using a 0.4 *NA* objective at a 633-nm wavelength (λ/1.2)[4]. Alexandrov *et al*. [8, 9] introduced a related concept wherein the images were recorded directly in the Fourier plane, which can increase the system information capacity for a given camera, but also can lead to ambiguity of phase determination and possible information loss for patterns with high periodicity due to a restricted detection dynamic range. In a recent series of papers, the use of a vertical cavity surface emitting laser (VCSEL) array to provide the off-axis illumination has been explored, providing a simple optical system that is reconfigured by either turning on individual VCSELs or by relying on their mutual incoherence to record all of the offset images in a single step [10–13].

In this paper, we first describe a straightforward extension of the frequency space coverage by using multiple offset partial images in each direction and introduce a signal-processing algorithm to deal with the resulting overlaps in the frequency space coverage. With the same *NA* = 0.4 optical system, the resolution is extended to ~ λ/2.76 (230 nm at λ = 633 nm), well beyond the capability of a traditional microscopy system with the same objective lens, and better than the resolution accessible even using a high-*NA* objective with conventional illumination as a result of the decreasing modulation transfer function at high frequencies (the Rayleigh resolution for a 0.9-*NA* objective at a 633-nm wavelength is 422 nm). Access to additional frequency space coverage by tilting the object plane is then demonstrated. Image distortion is associated with this non-collinear, multi-axis optical configuration; additional image processing is introduced to compensate this distortion. Using this approach a resolution for non-periodic objects to 180 nm (λ/3.52), with some limitations due to the loss of the highest frequencies of the image, and to 170 nm for simple gratings (λ/3.72) is demonstrated, extending optical microscopy close to the linear systems limits of the optical transmission medium. Importantly these techniques are applicable to standard transmission/reflection microscopy configurations and do not rely on fluorescent transitions or nonlinear response (except for the square-law intensity detection of the camera). These results are achieved with modest optics and retain the depth-of-field, field of view and working distance advantages of low-*NA* microscopy.

## 2. Imaging interferometric microscopy

The key to IIM is oblique (off-axis) illumination, which shifts the higher spatial frequencies diffracted from the object in one spatial direction into the bandpass of the objective lens as shown in Fig. 1. A 0-order beam is brought around the objective lens with an auxiliary optical system and interferometrically reintroduced on the low NA side. The interference between this 0-order beam and the diffracted beams transmitted through the objective shifts the collected diffracted information back to high frequency. As a result of the square-law intensity response, the resulting frequency coverage is represented by a pair of circles of radius *NA*/λ shifted away from zero frequency as shown in Fig. 2(b). A collimated illumination beam is incident on the object at an angle of incidence *β*. In the illustrated case, *β* is beyond the collection angle of the objective lens and an auxiliary optical system is used to collect the zero-order transmission, appropriately adjust its divergence, direction, and phase and re-inject it onto the image plane where it interferes with the diffracted beams from the object to reconstruct a partial image. Alternatively, instead of using the zero-order transmission, which might be blocked by the objective-lens mount, a portion of the illumination beam is split off before the object and directed around the objective lens. For convenience of presentation, Fig. 1 is drawn for a ~ 2× magnification, although a 20× objective is used in the present experiments. As an aside, IIM is not directly applicable to fluorescence microscopy because of the need for interferometric introduction of the coherent 0-order beam.

The frequency-space conceptualization of IIM is shown in Fig. 2 which presents the frequency space coverage of our partial images using a *NA* = 0.4 objective mapped against the Fourier amplitude coefficients of a 240-nm *CD* Manhattan (*x*, *y*) geometry structure [shown in Fig. 2(a)]. At the He-Ne laser wavelength of 633 nm, the resolution limit of this objective with conventional illumination is ~ 0.6λ/*NA* (~ 950 nm). The two circles at radii of *NA*/λ (0.4/λ) and 2*NA*/λ (0.8/λ) correspond to the approximate frequency space limits for coherent and incoherent illumination, respectively, and reflect the low-pass transmission characteristic of the objective lens. The inner sets of small shifted circles (radius *NA*/λ) in Fig. 2, that extend from -3*NA*/λ to 3*NA*/λ (±1.2/λ) in the *x*- and *y*-directions, show the frequency space coverage added with two offset partial images, one in each direction. The imaging is single side-band, only the diffracted plane waves to one side of the object are collected (opposite to the tilt of the illumination beam), the square law (intensity) response of the image formation and detection process restores the conjugate frequency space components, resulting in the two symmetrically displaced circles in Fig. 2(b) for each partial image. The offset (off-axis tilt) for these images was chosen to ensure that there was no overlap between the spectral coverage of the low-frequency partial image (extending out to *NA*/λ) and the offset images. As discussed previously, improved images can be obtained by subtracting the dark-field components of the image (with the zero-order transmission blocked) [4]. In the present experiments, this provided a cosmetic, not a dramatic, improvement to the images.

Additional frequency space coverage is available with a second pair of off-axis images, represented by the outer sets of shifted circles, with a larger tilt of the illumination plane wave, approaching grazing incidence (limited to 80° in the present experiment). The maximum frequency coverage in these images extends to [sin(80)+*NA*]/λ = (0.98+*NA*)/λ (1.38/λ). Clearly, the frequency-space coverage of the outer circles is necessary to capture the fundamental frequency components of the line-space portion of this pattern. There is significant overlap between the frequency coverage of the first and second set of partial images as illustrated in Fig 2. To provide a faithful image, it is necessary to exclude the double coverage of frequency space associated with the image spectral overlaps. This can be accomplished by filtering the images either optically (with appropriate apertures in the back focal plane) or electronically once the images are recorded. Importantly, since each of the partial images involves only the *NA* of the objective, this imaging concept retains the working distance, depth-of-field and field-of-view associated with the low-*NA* objective, but has a resolution beyond that achievable with even the highest NA objective using traditional illumination approaches.

## 3. Experimental results

The experimental setup is shown in Fig. 3. A TEM_{00}, 633-nm He-Ne laser illumination source, beam splitters and shutters (not shown) are used to produce and sequence the various illumination and reference beams. The object is illuminated with a 2-mm diameter collimated Gaussian beam. The ~40-μm diameter field (limited by the size of the CCD camera) is imaged by a microscope objective (*NA*=0.4, 20X, ~ 1-cm working distance) along with a tube lens. A set of lenses in a confocal arrangement is used to relay an accessible Fourier plane where the zero-order beam is reintroduced. The image is then magnified with a second microscope objective (*NA*=0.25, 10X). Finally, the image (200X) is recorded with a 640×480 pixel CCD camera with ~ 12×12 μm^{2} pixel size. The coherent reference beam is transmitted around the microscope objective using an optical fiber and reintroduced in the Fourier plane on the low *NA* side of the objective. As a result of the long coherence length of the HeNe laser, balancing the lengths of the various interferometric optical paths is straightforward. Intensities of the beams are adjusted by filters.

The angles of the off-axis illumination and the reference beam are adjusted to capture different ranges of frequency space. For *NA* = 0.4, we first take a picture with on-axis illumination, then illuminate the object at an angle of 53° and then at an angle of 80°.

An adjusting stage controlling the end of the fiber in the Fourier plane is used to mode-match the reference beam to the off-axis characteristics of the illumination beam, such as angle of incidence and phase. A reference grating on the mask is used to determine the correct angle and phase.

Blocking beams 2 and 3 provides the on-axis illumination partial image. The dark field image is taken by blocking the 0-order in the Fourier plane, and the background is taken by moving the object slightly to a nearby unpatterned area. Blocking beam 1 and opening beams 2 and 3 provides the high frequency partial image. The high-frequency dark field is obtained by illuminating the object only with beam 2 and the background of the reference beam is taken by using only beam 3. The corresponding dark field and background images are subtracted from each partial image without changing their amplitudes and, finally, the baseline and contrast are adjusted in correspondence to a reference object. The pattern is rotated to provide additional frequency coverage in the orthogonal spatial direction while keeping the polarization fixed (TE) for optimal contrast.

Experimental results for Manhattan (*x*, *y*) structures (20-nm thick chrome-on-glass) with a *CD* of 240 nm (high-frequency period of 480 nm) are shown in Fig. 4. The coherent low-frequency image [Fig. 4(a)] corresponds to the spatial frequency coverage of the center circle [Fig. 2(b)]. By adding high-frequency images corresponding to the inner pairs of offset circles of Fig. 2(b) in the *x*- and *y*-directions to the low-frequency image and subtracting darkfields and backgrounds the image in Fig. 4(b) is obtained. As expected, there is insufficient resolution to image the pattern, since the frequency space coverage does not include the significant spectral content near the outer edges of the frequency-space circle that correspond to the fundamental frequency components of the periodic line:space structures in the object.

Adding the high-frequency content images corresponding to outer pairs of offset circles of Fig. 2(b) the reconstructed image shown in Fig. 4(c) is obtained. Finally, Fig. 4(d) shows the result of electronically eliminating the overlap in spectral frequencies, by taking Fourier transforms of the partial images, filtering appropriately, and inverse transforming back to real space. Somewhat better images were achieved with electronic filtering than by eliminating the overlap with spatial filters in the back focal plane. This may be due to the difficulty of precisely aligning the pupil-plane filters in the optical system. While these images clearly reflect the major features of the object, there remains significant image modulation associated with an abrupt cut-off in frequency space (Gibbs phenomenon) coupled with a finite signal/noise and the difficulty in precisely matching the experimental partial images in both amplitude and phase. For the present case of a binary image, applying a nonlinear threshold to the image provides significant improvement. This procedure will be discussed in more detail in a subsequent publication.

## 4. Extension to frequency space limits using a tilted object plane

The spatial frequency space coverage can be increased beyond (1+*NA*)/λ by tilting the mask and capturing higher spatial frequencies, up to the transmission medium band pass limit of 2/λ as shown schematically in Fig. 5. The incident beam, the tilt axis, and the reference beam are all in the same (≡ *x*, *z*) plane. In our experimental case, the object tilt was θ_{tilt} = 39° and *NA* = 0.4, so *α* is between -23.6° and +23.6°; the incident beam angle is *β*= 80° and the effective aperture sin(θ_{tilt})+ sin*β* is between 1.25 and 1.87; *β _{ref}*- is the incident angle of the reference beam for the reconstruction of the partial image.

The frequency space coverage using a *NA* = 0.4 objective is shown in Fig. 6(a) superimposed on the frequency space intensity plot for a 180-nm *CD* object of the same pattern as shown in Fig. 2(a). The second pair of off-axis images with the object tilted extends the frequency space coverage out to ~1.87/λ. As a result of the non-collinear, multi-axis conditions of this arrangement, the frequency coverage is distorted from circles to ellipses, and a frequency-mapping algorithm, described below, is required to relate the measured spatial frequencies to the actual image frequencies. As the object linewidth is decreased to 170 nm, the corresponding frequency peak extends beyond the collection limit [Fig. 6(b)] and the image is distorted.

Since this multi-axis optical system is non-paraxial, sin[*θ _{tilt}*±

*α*] ≠ sin(

*θ*)± sin(

_{tilt}*α*), it is necessary to correct the measured frequencies. First, the range of captured frequencies versus mask tilt

*f*

_{min,max}= [sin(

*β*) + sin(

*θ*∓ sin

_{tilt}^{-1}

*NA*)]/

*λ*is shown in Fig. 7 for an incident angle

*β*= 80°. The vertical dashed line corresponds to the experimental conditions with a frequency extent in the plane of the tilt from

*NA*= 1.25 to

_{min}*NA*= 1.87 [the minor axes of the ellipses in Fig. 6(a)]. The effective aperture in the direction along the tilt decreases as the tilt increases. In the orthogonal direction the effective aperture is invariant to the tilt, so the covered frequency region becomes elliptical rather than circular. More generally, the measurement gives offsets from the origin of the tilted coordinate system of normalized frequencies

_{max}*f*’ [≡ sin(

_{x}*α*)/λ] and

_{obs}*f*’ where

_{y}*f*’ is in the plane of the tilt. These frequencies are in the laboratory measurement coordinate system, within the NA of the lens.

_{x}*f*’ is unaffected by the tilt; after some algebra the corrected frequencies are:

_{y}$${f}_{x}={f}_{x}^{\prime}{\mathrm{cos}\theta}_{\mathrm{tilt}}+\sqrt{1-{f}_{x}^{\mathrm{\prime 2}}-{f}_{y}^{\mathrm{\prime 2}}}{\mathrm{sin}\theta}_{\mathrm{tilt}}$$

The conversion between measured frequencies and actual frequencies is shown in Fig. 8 where the measured spatial frequencies are referenced to their position in the pupil of the objective. The calculation is for the experimental situation: θ_{tilt}, *β*, and *NA* as described (39°, 80°, 0.4). Results are shown for different values of *f _{y}*’. As

*f*’ increases, the range of accessible spatial frequencies along

_{y}*f*’ (

_{x}*f*) decreases as a consequence of the circular pupil aperture of the objective. Experimentally, we measure the spatial frequencies in real space by adding in a reference beam and monitoring the interference pattern between the reference beam and the collected diffracted beams. The angle of the reference beam,

_{x}*β*, and its phase are set using a reference grating positioned around the object of interest.

_{ref}The measured image frequencies must be corrected before reconstructing the complete image. This is accomplished in several steps: first, the dark field real space image is subtracted; then a fast Fourier transform (FFT) of the experimental (distorted) real space high-frequency image is taken to provide a frequency space image; and the experimental frequencies are corrected according to Fig. 8. The resulting frequency space information is not immediately suitable for an inverse FFT since the new frequencies are no longer on a regular (integer) grid. The corrected frequencies were assigned to the nearest frequencies on the appropriate grid and the inverse transform was taken to convert back to real space. The procedure is simpler and more robust if the focal point and the spatial origin of the FFT are coincident, since the phase is invariant to image tilt around this point. By comparing the final images with corresponding filtered model images using a mean-square-error metric we find the optimum position, angle and intensity of the partial images. This procedure is necessary as a consequence of experimental uncertainties in defining both the focal position for the tilted object and its relative translation from the optical axis. For an unknown object, a nearby test structure will be necessary to allow the same manipulations. Finally the total image is reconstructed [Fig 9(a)] by adding restored high frequency images with low and middle frequency images and compared with the model [Fig 9(b)].

A crosscut of the reconstructed image [along the red line in Fig. 9(a)] compared with the crosscut of the model is shown in Fig. 10. The crosscut of the reconstructed image has additional peaks on the background due to the lack of accuracy in phase and amplitude matching in conjunction with Gibbs effect and the impact of system noise.

The results of experiments with 170-nm structures are shown in Fig. 11(a). The reconstructed image is not as well resolved as for the 180-nm structures as this image is closer to the limit of resolution [Fig. 6(b)], but the image of a grating with the same period [Fig. 11(b)] is quite good and provides a technique to observe large scale grating defects, which might be useful for grating inspection. As discussed above, both of these structures are beyond the traditional Rayleigh limit (~λ/3.3) for the transmission medium.

Tilting the object plane introduces significant distortions into the optical system. Empirically, the most noticeable is that the in-focus field-of-view is decreased in the direction of the tilt. If the depth of focus of the objective is 6 μm, inclination of the mask to 39° will make the field of view in focus only about 9-μm wide. This is a direct consequence of the non-paraxial effects discussed above that distort the observed spatial frequencies of the diffracted fields in a nonlinear fashion. Correcting this frequency space distortion also restores the field of view.

In light of the precision required, an alternate algorithm to that described above was used. Instead of shifting to the nearest discrete frequencies in the inverse transform, to allow an FFT, the corrected, non-integer frequencies were used for numerical evaluation of the inverse transform. The result is illustrated in Fig. 12 which shows the *x*-offset high frequency partial images of two adjacent test structures (the one on the left is the 180-nm object (Fig. 9) while the one on the right is the 170-nm test structure, each structure is about 20 CD wide (3.6 μm) and their separation is 12 μm. The optical system was adjusted so that for the experimental image the smaller (right hand) structure was approximately in focus while the larger (left hand) structure was behind the focal plane and substantially blurred. The dotted lines and arrows show the significant shifting of the positions of the intense features arising from the transform from the laboratory frame to the image frame.

The final reconstructed images are shown in Fig. 13. The reconstruction procedure restores both images and clearly improves the field of view that was limited by the tilt of the object plane relative to the focal plane. This method requires very precision knowledge of the mask tilt and incident illumination offset in order to obtain high-quality, extended-field images.

## 5. Conclusion

In traditional optical microscopy, the resolution is limited by the optical system bandpass to ~ 0.6λ/*NA*, while the bandpass limitation set by the optical transmission medium extends to 0.25λ independent of *NA*. Imaging interferometric microscopy, closely related to off-axis illumination, but with an interferometric re-introduction of the zero-order transmission on the low-*NA* side of the optical systems extends the resolution to ~0.6λ/(1+*NA*). Tilting the object plane allows collection of the highest spatial frequency diffraction components from the object and further extends the resolution to ~0.3λ, again independent of *NA*. For simple gratings, the resolution is as high as 0.25λ reaching the limit set by optical transmission (assuming a refractive index of unity).

This approach requires only modest optical components. We have demonstrated resolution of a 240-nm *CD* Manhattan geometry (*x*, *y* pattern) object with a 0.4 *NA* objective at a wavelength of 633 nm (*CD* ~ 0.14λ/*NA* ~ λ/2.9) without tilt of the object plane and of a 180-nm *CD* pattern with λ = 633 nm and *NA* = 0.4 (*CD* ~ 0.11λ/*NA* ~ λ/3.52) with tilt of the object plane.

Each of the partial images retains the depth-of-field, working distance and field-of-view advantages of the low-*NA* objective, but the composite image has a resolution superior by almost a factor of two to that achievable with any possible single objective using conventional illumination. For a tilted object plane partial image, an algorithm is described to restore the field-of-view that is restricted by geometric optics with some decrease in quality as the field is extended due to experimental accuracy limitations and noise.

Another advantage of IIM is the possibility of using signal-processing enhancements on the partial images. Examples are the subtraction of dark-field images; the use of Fourier-transform filtering techniques to eliminate double coverage of the same spectral components in two partial images; and the nonlinear frequency transformation for the tilted images. The full apparatus of digital image processing is quite rich and many additional techniques will find application in IIM, although care must be exercised to avoid the introduction of artifacts.

The present examples used a Manhattan geometry structure with the major spectral components along orthogonal axes. For more general objects, additional partial images covering all of frequency space can be added. Phase measurements are readily available since the zero-order beam is independently accessible and can be advanced/retarded independently of the diffracted beams. For example, this will allow detailed microscopy of phase-shift lithography masks, which is difficult with traditional techniques. In comparison to strategies that use the nonlinear response of fluorescence transitions to enhance the resolution, IIM is broadly appliable to all microscopy measurements including transmission and reflection.

Finally, the ultimate resolution can be further extended with immersion techniques. Immersion microscopy is certainly well known. Recently, lithography applications of immersion have attracted much interest for future IC generations. Table I gives the rough limits to resolution corresponding to HeNe laser sources (633 nm) and ArF laser sources (193 nm, near the transmission limits of quartz). The refractive index of the immersion fluid is taken to match that of a silica substrate to minimize Fresnel reflection coefficients; somewhat higher index fluids and substrates are available with corresponding increases in optical resolution.

## Acknowledgments

We are thankful to Felix Jaeckel for making the mask with small features. Support for this work was provided by DARPA as part of the University Photonics Research Program.

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