Optical Microscopy at sub-0.1 µm Resolution: Fiction or Vision?

Until recently, optical microscopes enjoyed approximately a 1.6X resolution advantage over their stepper counterparts due to the microscopes' higher numerical aperture objectives (currently at 0.95) relative to the stepper lens' NA of ~0.65. As stepper optics transition from the violet part of the light spectrum (436 nm, g-line) to ultraviolet (UV, 365 nm, i-line) to deep UV (248 nm, KrF-line and 193 nm, ArF-line), microscopy's resolution advantage in the visible spectrum (400-700 nm) slowly diminishes.

UV/DUV microscopes offer improved resolution, while further increases in NA are impractical due to the physical limitation of a maximum of NA = 1 for non-immersion lenses.

Breaking the classical resolution barrier, a promising new technique potentially improves resolution by a factor of 1.5-2.5X over that of current optical microscopes. At the least, 'super-resolution' technology should deliver improvements in contrast and visualization, similar to benefits offered by confocal microscopy. This article compares super-resolution technology with advances in confocal microscopy, UV/DUV microscopy and photon tunneling microscopy (PTM) techniques for their resolution-enhancement capabilities. It emphasizes production tool requirements, so slow-scan methods such as scanning near-field optical microscopy are not discussed.

Defining resolution

There is a surprising amount of confusion regarding the meaning of 'resolution' in the semiconductor industry. Using the Rayleigh criterion to define the resolving power of image forming systems,

Fig. 1. The composite intensity distribution of the image of two adjacent object points is characterized by an intensity dip at 73.5% of the intensity maximum according to the Rayleigh criterion.

Fig. 2. NA is the product of the refractive index of the medium between the objective and specimen and the sine of the angle of incidence.

There is the underlying assumption that the resolution of two adjacent object points of equal brightness is limited only by diffraction (i.e., a perfect imaging system). The image of each object point is referred to as the Airy pattern or point spread function (PSF). According to the Rayleigh criterion, two overlapping Airy patterns can be perceived as comprising of two distinct Airy patterns, if the intensity maximum of one pattern coincides with the first intensity minimum of the second pattern. In that case, the lateral distance Dx of the intensity maxima of both Airy patterns acts according to Equation 1, where at the wavelength of light, NA is the numerical aperture of the system, and there is a detectable intensity dip at 0.735X the intensity of the maxima at either side (Fig. 1).

Even though the 73.5% dip is a numerically exact result from the Rayleigh criterion, it is nevertheless arbitrary, as its assumptions are arbitrary. It is well stated that, 'This rule is convenient on account of its simplicity, and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution'¹.

There is also a widespread factor of two confusion regarding resolution. For instance, a microscope objective with NA = 0.95 (Fig. 2) and imaging wavelength of 0.55 µm (corresponding with the maximum sensitivity of the human eye), gives a 0.35 µm resolution, often interpreted as the minimum feature size, given the NA and l. This is incorrect! The specified objective can actually resolve ~0.18 µm features. Minimum resolved feature size is about half the value of the Rayleigh resolution, derived by investigating how object gratings instead of object points are imaged by an optical system.

This alternative approach to resolution is based on the Fourier transform relationships between the object/image planes and the exit pupil plane of an optical imaging system². In this theory, the optical system is represented by its optical transfer function, OTF, and its modulus, the modulation transfer function, MTF. The MTF or contrast, C, measures the transfer of the relative peak-to-valley intensity difference for grating-like objects of sinusoidal intensity distribution to the image plane by:

Equation 2

Assuming equal line and space width, a sinusoidal object grating with a spatial frequency of f_o lines and spaces (L&S) per millimeter is imaged by an optical system of magnification M into a sinusoidal image grating with f_I = f_o/M L&S/mm and an image contrast of C_I = MTF(f_I)C_o. Finite resolution corresponds with an MTF of zero above a certain cut-off frequency:

Equation 3

The reciprocal value, w_min = 1/f_max is the width of one line or space for which the image modulation, or contrast, has become zero:

Equation 4

For the above mentioned microscope objective with l = 0.55 µm and NA = 0.95, f_max is at 3455 L&S/mm, and the width of one L&S is 0.29 µm at f_max. Equating feature size with the width of a single line, image contrast becomes zero at 0.145 µm. Allowing for some finite image contrast, practical feature size resolution is ~0.18 µm.

Figure 3 shows the on-axis white light MTF for the Leica Plan Apo 100X/0.95 objective. The dashed curve represents the physical limit due to diffraction. Note that cut-off frequency is different from the 3455 L&S example, because the MTF was calculated for a superposition of multiple wavelengths. As shown, design performance is at the physical borderline. The same holds for production performance in other studies^{3, 4, 5}.

Paths to increased resolution

Most optical textbooks incorrectly define ultimate resolution limit as the resolving power according to Equation 4, 'set' by physical laws. By generalizing the constant factor in front of l/NA by some variable, say k>0.5, one accounts for system noise, detector response functions, etc. From this, it seems the only way to improve resolution is to decrease wavelength and increase NA. This is incorrect, at least in theory, as discussed in detail in 'Resolution: a Survey'¹. Because the classical resolution criteria are taken for granted, it is hard to believe there may be options to break this barrier. First we analyzed benefits offered by confocal microscopy, UV/DUV microscopy and photon tunneling microscopy (PTM) before exploring the limits of resolution.

Confocal microscopes

Fig. 3. Minimum feature size is resolved slightly above the spatial frequency, where the MTF, or contrast, goes to zero.

Fig. 4. Confocal microscopes provide contrast enhancement and pattern z discrimination.

Confocal microscopes essentially provide no resolution improvement compared to standard microscopes. However, because the Airy pattern of a single point object in the image of a confocal microscope is ~73% that of a standard microscope, initial studies concluded a 1.4X resolution gain (1/0.73). This is not true, though it remains true in the minds of many people. Rather than resolution enhancement, the smaller width of the Airy pattern provides better image contrast and therefore better visualization of the object structure. Higher contrast improves the inspection of wafer or mask patterns at the borderline of optical resolution.

Confocal microscopes are scanning microscopes, available as white-light systems with a rotating Nipkow disk in the intermediate plane or as laser confocal microscopes with a laser scanning the object. In both cases, the microscope illuminates a single object point through a pinhole located in the microscope's image plane (or a conjugate plane) (Fig. 4). Reflected light travels back through the same pinhole to the detector. High-intensity light reaches the detector only when object points are within the focal plane of the microscope objective. Out-of-focus images at the pinhole plane blur over a large area, so only light from a small slice (±Dz) of the focal plane contributes to the image. For a three-dimensional structure such as a patterned wafer, one can go through focus and have only one slice sharply imaged at a time. This slicing feature is very different from standard microscopes, where out-of-focus object points are superimposed to the focal plane image, reducing image contrast significantly.

In a white-light confocal microscope, images can be viewed through eyepieces in the same way as with a standard microscope and, at least in principle, with the same color information. However, some residual amount of longitudinal chromatic aberration occurs with even the most sophisticated confocal microscopes, resulting in different z focus positions as a function of wavelength. This color coding at different levels of topography can benefit the user, depending on the application and level of color correction of the objective. One major disadvantage of the white-light confocal microscope is its low light efficiency and low image brightness at high magnifications. By increasing the pinhole size and/or increasing the number of pinholes on the Nipkow disk, image brightness improves but at the expense of resolution, contrast and sectioning sensitivity.

Laser confocal microscopes provide sufficient image brightness, yet offer no color information, can suffer from interference effects due to laser light coherence and cannot be used in real-time without a reduced signal-to-noise ratio.

UV/DUV microscopes

Microscopes operating at UV/DUV wavelengths were introduced later than their stepper counterparts due to:

• 1.6X resolution advantage provided by high NA objectives;
• Lack of availability of suitable light sources
  (i.e., mercury arc lamps are dark, and UV/DUV lasers of feasible size, price and reliability only recently became available);
• Lack of high spatial resolution UV/DUV detectors and
• High cost/price of UV/DUV microscopes.

Depending on pattern type and materials, a UV/DUV image may look unfamiliar compared with a VIS image, making a system that can switch between VIS and UV/DUV desirable. The challenge of manufacturing UV/DUV objectives with similar image performance to VIS objectives requires tighter tolerances on lens elements and new interferometer technology for testing. It is quite likely that the real potential of laser confocal microscopy arises when combined with UV/DUV optics. The UV/DUV resolution gain over VIS objectives is 1.4-2.0, depending on wavelength and detector performance. There may also be a gain in contrast, as UV/DUV light absorption depends more on the material than VIS.

Photon tunneling microscope

Fig. 5. The PTM's tunneling depth (~100 nm) requires that the specimen be brought within 100 nm of the objective's front surface, allowing evanescent light to penetrate the material of the specimen and be removed from internally reflected light.

PTMs provide real-time images and a significant gain in resolution using visible light. Because non-immersion objectives are as close to the theoretical limit of NA = 1.0 as is feasible to design and produce, the only means to increase NA is to utilize immersion.

Near-field optical microscopy is based on frustrated total internal reflection^{6, 7, 8}. The objective is designed so that the focal plane coincides with the planar surface of the front lens. The wavelength of light within the glass of the front lens with refractive index n is only 1/n of that in air, and in this special configuration, the front lens acts like an immersion, a solid immersion! Assuming sins = 0.95, NA for this objective becomes 0.95n. For n = 2, say NA = 1.9 with a factor of two gain in resolution. The question then becomes, 'How can a wafer surface be imaged with such a strange type of zero working distance objective?'

When the angle of incidence, s, at the front surface exceeds sind>1/n, there is total internal reflection. The front surface acts like a mirror, and the viewed image is an empty image of uniform brightness across the field of view. However, part of the light impinging at the front surface penetrates across the glass/air surface into the air with an exponential reduction in intensity. These penetrating waves, called surface or evanescent waves⁹, advance parallel to the glass/air surface (Fig. 5). Tunneling depth, on the order of wavelength (~100 nm), requires that the specimen be brought within 100 nm of the objective's front surface, allowing evanescent light to penetrate the material of the specimen and be removed from the internally reflected light. The closer the specimen is to the evanescent wave field, the darker the image.

It is not clear whether this exotic imaging method will apply to wafer inspection in a production environment, especially with its very small free working distance. However, it is not very different from near-field scanning optical microscopy or atomic force microscopy, which are already used in semiconductor applications. Other issues with PTM include image appearance different from classical bright-field imaging, making image interpretation difficult; potential for wafer damage and the limited range of pattern types to which PTM can be applied, requiring feature heights of 1 µm or less.

Super-resolution

The physical and mathematical models on which the classical resolution criterion are based are quite artificial: assuming the typical assumed objects, e.g., two adjacent points, are mathematical objects, an imaging system free of noise and a perfectly known OTF of the imaging optics. It is also frequently overlooked that within this framework, the mathematical object is always perfectly known! Classical resolution criteria are based on describable mathematical models with detailed a priori information about the object. In the semiconductor industry, because a priori knowledge about wafers and masks exists, there is actually no limit to resolution for calculated images; it is possible to fit a mathematical model of the object to the measured data¹. Because the ideal image of the ideal wafer pattern is known, geometrical parameters of the object model or its calculated image can be fit to the known wafer pattern's detected image using an iterative procedure. Ultimate resolution is not limited by Equations 1 or 4, but by deficiencies in measurement precision (e.g. noise and imperfect knowledge of the OTF), resulting particularly in uniqueness problems¹⁰. An understanding of super-resolution and its options beyond classical resolution limits requires a paradigm shift in thinking.

Tychinsky investigated super-resolution techniques in optical microscopy^{11, 12} using interference microscopy imaging. Up to now, two-point super-resolution has not been demonstrated; however, remarkable preliminary results were demonstrated for the imaging of isolated edges. Under special circumstances, edge inclines of 20 nm were imaged experimentally, implying a 10-fold improvement in resolution compared to classical imaging. Theoretical modeling indicates a need for careful image interpretation, as very steep edges are not actual edge images but a particular interference phenomenon (phase singularities¹³), developing deep but not necessarily steep structured edges. The method is not yet fully understood.

Most importantly, Tychinsky's interference microscopy imaging technique does not take advantage of prior object information, perhaps the most promising path to super-resolution for the semiconductor industry. In contrast to UV/DUV and PTM, concrete estimates of possible resolution improvement cannot be given. Most experts agree that a factor of 1.5X improvement is feasible.

Outlook

The options to higher resolution (see Table) differ significantly in their likelihood of realization. The table compares resolution limits using the Rayleigh equation and k = 0.61, affording some margin. The figures in the super-resolution row refer to bright-field microscopy with an assumed improvement of 1.5X. Due to the restricted availability of optical materials for UV and DUV optics, the figures for PTM are based on different refractive indices for the solid immersion lens: n = 2.0 for VIS light, n = 1.8 for UV and n = 1.6 for DUV.

In summary, optical microscopy techniques can break classical resolution barriers. UV/DUV microscopy will establish itself as a standard tool for wafer inspection and metrology. Laboratory PTMs exist, and though the general technique is feasible, designing a tool for production and its applicability are speculative. Probably the highest degree of uncertainty prevails in super-resolution techniques. However, even in the case where no significant improvement in resolution is realized, a minimal outcome might be improved contrast and visualization.

References

1. A.J. den Dekker, A. van den Bos, 'Resolution: a Survey,' J. Optical Society of America (JOSA), Vol. A14, No. 3, March 1997, p. 547.
2. J.W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1968.
3. R. Blendowske, U. Voigt, W. Vollrath, 'DELTA Optics: Theoretical Aspects of Design and Production,' Leica Scientific and Technical Information, Vol.X, No. 5, June 1993, p. 147.
4. W. Vollrath, 'Microscope Objectives: The State of the Art in Design and Production,' OSA Proc. Of International Optical Design Conf., Vol. 22, 1994, p. 214.
5. R. Blendowske, W.Vollrath, 'Strehl Ratio Split for Production-Limited Optics,' Optics & Photonics News, Vol. 8, No. 5, May 1997.
6. T.R. Corle, G.S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems, Academic Press, 1996.
7. G.S. Kino, S.M. Mansfield, 'Solid Immersion Lens Photon Tunneling Microscopy,' SPIE, 1556, 1991, p. 2.
8. M. Guerra, 'Photon Tunneling Microscope,' Applied Optics, Vol. 29, 1990, p. 3741.
9. E. Hecht, A. Zajac, Optics, Addison-Wesley, 1974.
10. P.J. Sementilli, B.R. Hunt, M.S. Nadar, 'Analysis of the Limit to Superresolution in Incoherent Imaging,' JOSA, Vol. A11, 1993, p. 2265.
11. V.P. Tychinsky, 'Wavefront Dislocations and Registering Images Inside the Airy Disk,' Optics Communications, Vol. 81, 1991, p. 131.
12. V.P. Tychinsky, C.H.F. Velzel, 'Super-resolution in Microscopy,' Current Trends in Optics, Academic Press, 1994.
13. M. Totzeck, H.J. Tiziani, 'Phase-Singularities in 2D Diffraction Fields and Interference Microscopy,' Optics Communications, Vol. 38, 1997, p. 365.