## Abstract

By use of a superresolution pupil filtering technique to achieve a lateral optical superresolution and a differential confocal microscopy technique to achieve an axial resolution at the nanometer level, we propose a high spatial resolution bipolar absolute differential confocal approach for the ultraprecision measurement of three-dimensional microstructures. The feasibility of the proposed approach has been proved by use of a shaped annular beam differential confocal microscopy system. The experimental results indicate that the lateral and axial resolutions of the shaped annular beam differential confocal system are better than 0.2 μm and 2 nm, respectively, when λ=632.8 nm, *ε*=0.5, *u _{M}*=6.95, and with a 0.85 numerical aperture.

©2004 Optical Society of America

## 1. Introduction

It is of great significance to develop a measurement system with a higher spatial resolution to satisfy the stringent requirement for ultraprecision measurement of three-dimensional (3-D) microstructures; micromachining technology is now being used for 3-D machining at deep submicrometer and nanometer levels. A spatial resolution at the nanometer level has been achieved with scanning probe measurement instruments based on the principles of scanning tunneling microscopy, atomic force microscopy, and near-field optics, and it is difficult to use the instruments on a shop floor for the measurement of workpieces with large sudden skipping positions. A confocal microscope (CM) can be easily fitted with a superresolution pupil filter to achieve optical superresolution. In addition to a wider measurement range and stronger antiinterference capability, a CM has a lateral resolution that is 1.4 times better than a conventional microscope under the same conditions. The capability of a CM makes it a useful tool with great potential for its use in the optical measurement field, and it can be readily used for the measurement of 3-D microstructures and 3-D surface contours.[1–4] Lee *et al*., for example, proposed the concept of differential confocal microscopy (DCM) and achieved an axial resolution of 2 nm when DCM was used for surface profiling.[1] Udupa *et al*. proposed a confocal scanning optical microscope based on DCM for the measurement of two-dimensional (2-D) surface roughness, 3-D surface topography, and form errors.[2–3] A fiber-optic confocal sensor has been proposed for the measurement of grooves in CDs.[4] Because the slopes of CM axial response curves are used for measurements in the methods mentioned above, It is obvious that their measurement accuracies are limited by the nonlinearity of response curves in confocal microscopy systems. The measurement accuracies are susceptible to the disturbance in intensity of the light source and ambient lighting, the reflection characteristics of the measured surface, and, above all, there is no absolute measurement zero required for absolute measurement and tracing in the confocal microscopy system response curves.

The CM methods currently used for superresolution measurements can be classified into two categories[5]: (1) methods based on extrapolation of the information contained in the bandpass of the system and estimation of the information,[5–7] and (2) methods based on pupil filtering superresolution techniques. Unless prior information is added, the gain in resolution is modest. Confocal microscopy system resolution is usually improved by the proper use of a superresolution pupil filtering technique in a CM, and there are quite a few reports about the use of such techniques. [8–18] A rectangular pupil filter was designed to make the lateral straight-edge response sharper by as much as 2.36 times[8]; two-zone or multizone phase-only pupil filters were designed by use of diffractive optical technology[9]; resolution improved by increasing the spatial frequency of a partial beam in an optical system[10]; and superresolution pupil filters with continuous phase variation were designed to further improve the superresolution capability of a confocal microscopy system.[11] To improve the 3-D superresolution imaging capability of a confocal microscopy system, Liu *et al*. designed a three-zone annular amplitude pupil filter to enable the point-spread function (PSF) to be compressed in the lateral direction and sharpened in the axial direction.[12] Two types of three-zone phase-only pupil filter were designed by use of axial and lateral half-width at half-maximum (HWHM) of the PSF as the optimized object function to satisfy the superresolution requirements.[13] Two pupil filters were combined to improve the spatial resolution of the CM. When existing 3-D superresolution filters are used in confocal microscopy systems for 3-D superresolution measurements, the improvement of spatial resolution is usually not obvious because the superresolution must be achieved simultaneously in both the lateral and the axial directions. Moreover, ultraprecision 3-D microstructural workpieces feature high-dimensional accuracy and discontinuity, and this necessitates the use of a measurement approach with both higher spatial resolution and absolute tracing capability. Therefore, a new absolute confocal approach to higher spatial resolution is proposed to achieve high accuracy measurement of 3-D microstructures, such as microsteps and microgrooves.

## 2. Measurement principle

An optimized pupil filter is used in confocal microscopy systems to modify the mask of pupil function and to sharpen the mainlobe of the Airy spot, thereby improving the lateral resolution of confocal microsopy systems. A dual-receiving confocal light path arrangement with an axial offset of a pinhole from the focal plane of a collective lens and normalized differential subtraction of two signals received from detectors are used to improve the axial resolution of confocal microscopy systems, thereby achieving high spatial resolution. The unique dual-receiving light path arrangement enables the high spatial resolution of confocal microscopy systems to be used for bipolar absolute and high accuracy tracing and aiming measurements. None of the existing 3-D superresolution pupil confocal approaches can be used to fulfill this function.

As shown in Fig. 1, the measurement system is based on a reflection superresolution pupil filtering confocal microscopy system and the axial offset of a pinhole that changes the axial offset of an axial curve in confocal microscopy systems only. Measurement system A in the dash-dotted area is a superresolution pupil filtering confocal microscopy system with a pinhole in the axial near-focal plane of receiving lens 1. Measurement system B in the dashed area is a superresolution pupil filtering confocal microscopy system with a pinhole in the axial far-focal plane of receiving lens 2. In the measurement system as a whole, that part of the laser beam that passes through the extender and pupil filter is used as the measuring beam. It passes through the polarized beam splitter (PBS), and the measuring beam is reflected by the object and returns to the PBS. The measuring beam reflected by the PBS is divided by a beam splitter (BS) into two components that are focused separately by collecting lenses 1 and 2. Pinhole 1 is placed before the focal plane of collecting lens 1, the distance between the focal plane and pinhole 1 is *M*; detector 1 is near pinhole 1; pinhole 2 is placed behind the focal plane of collecting lens 2, and the distance between them is also *M*; detector 2 is near pinhole 2, and the optical coordinate corresponding *M* is expressed as *u _{M}*. When the parameters of a micro objective, collecting lenses 1 and 2, are the same, and the object is scanned in both the axial and the lateral directions, the signal received by detector 1 in measuring system A is

*I*

_{1}(

*v*,

*u*,+

*u*) and the signal received by detector 2 in measuring system B is

_{M}*I*

_{2}(

*v*,

*u*,-

*u*). Signal

_{M}*I*

_{3}(

*v*,

*u*,

*u*) obtained through the differential subtraction of

_{M}*I*

_{2}(

*v*,

*u*,-

*u*) and

_{M}*I*

_{1}(

*v*,

*u*,+

*u*) is

_{M}$$\left({\mid \left[2{\int}_{0}^{1}P\left(\rho \right)\xb7{e}^{j{\rho}^{2}\frac{\left(u-{u}_{M}\right)}{2}}{J}_{0}\left(\mathit{\rho v}\right)\mathit{\rho d\rho}\right]\mid}^{2}-{\mid \left[2{\int}_{0}^{1}P\left(\rho \right)\xb7{e}^{j{\rho}^{2}\frac{\left(u+{u}_{M}\right)}{2}}{J}_{0}\left(\mathit{\rho v}\right)\mathit{\rho d\rho}\right]\mid}^{2}\right)$$

where *J*
_{0} is a zero-order Bessel function, *ρ* is a radial normalized radius, *u* and *v* are axial and lateral normalized coordinates, respectively.

The axial response curves with an axial offset of a pinhole when the object is scanned in the axial direction only are shown in Fig. 2(a). The 3-D intensity response curves of a DCM system, while the object is scanned in axial and lateral directions, are shown in Fig. 2(b).

It can be seen from Fig. 2(a) that curves *I*
_{1}(0,*u*,+*u _{M}*) and

*I*(0,

*u*,-

*u*) are the same shape as curve

_{M}*I*(0,

*u*,0) of a CM and have an offset difference of

*u*from curve

_{M}*I*(0,

*u*,0) only in the axial direction. There is a zero measurement in curve

*I*

_{3}(0,

*u*,

*u*) obtained through differential subtraction of

_{M}*I*

_{2}(0,

*u*,-

*u*) and

_{M}*I*

_{1}(0,

*u*,+

*u*) without an absolute zero with both linearity and sensitivity in measurement interval

_{M}*ab*of curve

*I*

_{3}(

*v*,

*u*,

*u*), which is obviously better than in measurement interval

_{M}*cd*of the CM curve

*I*(0,

*u*,0). The improvement in the axial resolution of a confocal microscope system is achieved by optimization of the location of pinhole

*u*.

_{M}The lateral resolution of a confocal microscope system was improved by optimization of pupil filter parameters. The pupil filters used to achieve lateral superresolution include phase-only pupil filters, amplitude pupil filters, and a hybrid of amplitude and phase pupil filters. The pupil filtering function *P*(*ρ*) is

where *t*
_{j} is the transmission function of zone *j*, *φ _{j}* is the phase difference between zones 1 and

*j*, and

*N*is the total number of zones.

When a pupil filter is an amplitude pupil filter, *φ _{j}*=constant C, and

*t*is a variant in Eq. (2), where

_{j}*j*=1,2,3…,

*N*. When a pupil filter is a phase-only pupil filter,

*t*= constant C, and

_{j}*φ*is a variant in Eq. (2), where

_{j}*j*= 1,2,3…,

*N*. When a pupil filter is a hybrid pupil filter,

*φ*and

_{j}*t*are variants in Eq. (2), where

_{j}*j*= 1,2,3…,

*N*.

Three characteristic parameters used for evaluation of the superresolution property of a confocal microscope system are as follows: *G* is the ratio between the HWHM of PSF lateral intensity response curves with or without a pupil filter, Strehl ratio *S* is the ratio between the focal intensities with or without a pupil filter, and *M* is the ratio of the maximum intensity of the sidelobe with a pupil filter to the intensity of the mainlobe without a pupil filter. In the measurement system as a whole, the optimum of parameters *t _{j}*,

*φ*, and

_{j}*N*make

*G*and

*M*as small as possible, whereas S is as large as possible to satisfy the lateral superresolution requirements for a confocal microscopy system.

## 3. Lateral resolution improvement

It can be concluded from a comparison of the pupil filters used to achieve the same *G* that *S* and *M* of an amplitude-type pupil filter are smaller than those of a phase-only pupil filter, and the decrease in *S* can be compensated by increasing the intensity of the light source and using a diffractive pupil filter. To take advantage of an amplitude pupil filter with smaller *M* and to compensate for the deficiency of its smaller *S*, we used a shaped annular incident beam to improve the lateral resolution of a confocal microscopy system.

A binary optical element (BOE) with sixteen phase levels was used to shape a Gaussian beam into an annular beam. *N*=2, *φ _{j}*=0,

*t*

_{1}=0, and

*t*

_{2}≈1 when the shaped annular beam pupil filter is expressed by pupil function (2). Fig. 3 is the equivalent diagram of the light path shown in Fig. 1.

If the amplitude of the annular beam annulus is *A*, the intensity distribution of the shaped beam can be expressed as

According to the energy conservation law,

so

For a shaped annular beam differential confocal measurement, Eq. (1) can be rewritten as

$$=\frac{1}{{\left(1-{\epsilon}^{2}\right)}^{2}}\xb7{\mid 2{\int}_{\epsilon}^{1}{e}^{\frac{\left(\mathit{j}{\mathit{u\rho}}^{2}\right)}{2}}{J}_{0}\left(\mathit{\rho v}\right)\mathit{\rho d\rho}\mid}^{2}\times \left({\mid 2{\int}_{\epsilon}^{1}{e}^{\frac{\mathit{j}{\mathit{\rho}}^{2}\left(u-{u}_{M}\right)}{2}}{J}_{0}\left(\mathit{\rho v}\right)\mathit{\rho d\rho}\mid}^{2}-{\mid 2{\int}_{\epsilon}^{1}{e}^{\frac{\mathit{j}{\mathit{\rho}}^{2}\left(u+{u}_{M}\right)}{2}}{J}_{0}\left(\mathit{\rho v}\right)\mathit{\rho d\rho}\mid}^{2}\right)$$

where *ε* is the normalized radius of an annular beam inner aperture.

It can be seen from Fig. 4, obtained by use of Eq. (6), that, as *ε* increases, there is an obvious improvement in the lateral resolution and a decrease in the intensity of lateral response curves because of the shift of the annular beam from an inner ring to an outer ring while a BOE with a small energy loss is used. In a laser confocal microscopy system, the decrease of energy mentioned above is compensated by increasing the intensity of an incident beam.

## 4. Axial resolution improvement

It can be seen from Fig. 5 obtained by use of Eq. (6) that *u _{M}* has an optimum value that can be used to optimize the resolution property of axial response curves. Gradient

*k*(0,

*u*,

*u*) obtained by use of differentiation of differential signal

_{M}*I*

_{3}(0,

*u*,

*u*) in Fig. 5 on

_{M}*u*is

$${\left(1-{\epsilon}^{2}\right)}^{2}\xb7\mathrm{sin}c\left[\frac{{u}_{M}\left(1-{\epsilon}^{2}\right)}{4\pi}\right]\times \left\{\frac{\left\{\frac{\left(2u+{u}_{M}\right)\left(1-{\epsilon}^{2}\right)}{4}\right\}\xb7\mathrm{cos}\left\{\frac{\left(2u+{u}_{M}\right)\left(1-{\epsilon}^{2}\right)}{4}\right\}-\mathrm{sin}\frac{\left(2u+{u}_{M}\right)\left(1-{\epsilon}^{2}\right)}{4}}{{\left\{\frac{\left(2u+{u}_{M}\right)\left(1-{\epsilon}^{2}\right)}{4}\right\}}^{2}}\right\}$$

where *k*(0,0,*u _{M}*) and

*k*(0,

*u*,

*u*) are equal in the linear range, and the gradient in the linear measurement range of

_{M}*I*

_{3}(0,

*u*,

*u*) can be expressed in

_{M}*k*(0,0,

*u*) at

_{M}*u*=0 as shown below.

The variations of *k*(0,0,*u _{M}*) with pinhole offset

*u*for different

_{M}*ε*are shown in Fig. 6, and optimum offset

*u*of a pinhole and corresponding values of

_{M}*k*(0,0,

*u*) obtained by use of Eq. (8) for different e are as follows:

_{M}*u _{M}* =5.21 and

*k*(0,0,

*u*)=0.54, when

_{M}*ε*=0.00;

*u*=5.56 and

_{M}*k*(0,0,

*u*)= 0.4748, when

_{M}*ε*=0.25;

*u*=6.95 and

_{M}*k*(0,0,

*u*)=0.3038, when

_{M}*ε*=0.50;

*u*=11.91 and

_{M}*k*(0,0,

*u*)=0.1034, when

_{M}*ε*=0.75.

It can be concluded from the above that the sensitivity of a differential curve is the best when *u _{M}* is one of the above values. In addition to

*u*, the pinhole size and lateral offset are two factors that have their effect on the measurement sensitivity of a differential curve, but their effect can be easily eliminated by readjustment of the optical path. Usually the optimum diameter of a pinhole is between 5 and 10 μm.

_{M}## 5. Experiments

As shown in Fig. 7, a BOE with 16 phase levels is placed into input plane *P*
_{1} to shape a Gaussian beam into an annular beam. The distance between *P*
_{1} and output plane *P*
_{2} is *L*=2 mm, the input wavelength is λ=632.8 nm, the maximum diameter of input beam *d _{i}*=4.3 mm, the inner diameter of annular beam

*d*

_{0}=0.87 mm, and the outer diameter of annular beam

*D*

_{0}≈3.45 mm. The BOE is fitted with an adjustable diaphragm to shape a laser beam into an annular beam with an inner diameter of 0.87 mm and an outer diameter of <3.45 mm, as required for superresolution measurements. The normalized radius of the annular beam is

*ε*, its outer diameter is changed by the adjustable diaphragm, and

*ε*≈0.50 when

*d*

_{0}≈1.74 mm.

When *ε*=0.50 and the numerical aperture is 0.85, the actual axial intensity response curves of a DCM system are shown in Fig. 8. The signal received from detector 1 corresponds to the maximum of curve *e* and the differential signal corresponds to point *b* of curve *s* when the object is at point C in the near-focal plane of the micro objective. The signal received from detector 1 is in the middle of the descending range of curve *e*, and the signal received from detector 2 is in the middle of the ascending range of curve *f* The differential signal corresponds to absolute zero A of curve s when the object is at the focal point of the micro objective. The signal received from detector 2 corresponds to the maximum of curve *f*, and the differential signal corresponds to point *a* of curve *s* when the object is at point D in the far-focal plane of the micro objective. When the object is moved in the CD range near the focus, the differential signal varies in the *ab* range of curve s and its magnitude corresponds to the convex-concave variation of the object with respect to the focal plane, to which the surface contour and microdimensions of the object are reconstructed.

It can be seen from Fig. 8 that the sensitivity of response curve *s* depends on *u _{M}* to be given. In the

*ab*interval of an actual curve, its measurement range is 4 μm, its sensitivity is 1.25 V/μm, and its axial resolution is 2 nm. The slope in bevel interval

*ab*of differential confocal curve

*s*is larger than that in bevel interval

*hi*of confocal curve

*e*, the sensitivity and axial resolution of curve

*s*are improved, the measurement range in linear measurement interval

*ab*is larger than that in linear measurement interval

*hi*, and the linearity of measurement interval

*ab*is thus improved.

We further verified the improved resolution of the system by measuring the standard step supplied with the Dimension 3100 atomic force microscope (AFM) produced by Digital Instruments Company. The measurements of the standard step obtained with the AFM are shown in Fig. 9, where the step height between two identification points is approximately 118.23 nm and the horizontal skip distance of the step between the two identification points is 0.1367 μm.

We used a standard step as the object, a micro objective of 60×0.85 was used as the measuring lens, and the step placed on the workbench was moved in the axial direction by a microadjusting setup to keep the light probe focused on its surface and was then moved in the lateral direction perpendicular to the light probe. The micromotion system, which has a resolution of 2 nm and a movement range of 13 μm, can drive the step at a feed rate of 2 nm per driving pulse. The movement of the step is measured by use of a dual-frequency laser interferometer with a resolution of 0.01 μm.

As shown in Fig. 10, when *ε*=0.0, the lateral scanning curve of the standard step is measured by the system mentioned above without a BOE, and the system shown in Fig. 3 corresponds to a conventional differential confocal microscopy system at this time, where the height of the measured step is approximately 120 nm and the skip distance of the step is 0.403 μm, including the skip width of the step itself. When *ε*=0.50, the lateral scanning curve of the standard step is measured by the system mentioned above with a BOE, the height of the measured step is approximately 120 nm, and the skip distance of the step is 0.268 μm. The lateral resolution with superresolution is better than 0.2 μm if the slope of the step itself is 0.1367μm, including the lateral resolution of the AFM. Comparison of the step cross sections above indicates that the step cross section measured with superresolution by use of a BOE has a steeper change in slope in the skip range of the step.

## 6. Conclusions

We proved that a differential confocal scanning approach based on confocal microscopy and superresolution pupil filtering techniques by use of a shaped annular beam confocal microscopy system is feasible to achieve the high spatial resolution required for bipolar absolute tracing measurements. Preliminary experimental results indicate that confocal microscopy systems based on this approach have axial and lateral resolutions of better than 2 nm and 0.2 μm, respectively. The method has the following characteristics:

- improvements in the measurement range and lateral resolution of a confocal microscopy system; and
- suppression of common-mode noise that results from environmental differences, disturbance in intensity of the light source, and electrical drift of the detector, all leading to improvements in sensitivity, linearity, and signal-to-noise ratio of the measurement system.

We can therefore conclude that the proposed approach can be used for noncontract optical measurement of 3-D microstructures and 3-D surface contours.

## Acknowledgments

Thanks to National Science Foundation of China (No.50475035) for the support.

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