Abstract:
A method for determining the surface topography of an object with various surface properties in white light interferometry first generates a set of interferograms, produced by a phase shift driving mechanism supplied with known inputs, obtained from a single reflective surface at the location of the test piece. The sequence of interferograms from the test piece then is used to calibrate a sequence of inputs to the phase shift driving mechanism to compensate for non-linear characteristics of the phase shifting mechanism. The temporal interferometric signal or its transform at each pixel of a set of subsequently acquired interferograms from a test piece then is compared with signals or their transforms modeled with different properties of the measuring surface. The measured surface properties which generate a best match to the modeling surface properties are selected as the desired signal.

Description:
BACKGROUND 
   This invention is related to the field of white light phase shifting interferometry. Phase-shifting interferometry (PSI) has proven to be a highly accurate and efficient method for the measurement of single reflective surfaces in a variety of applications including optical testing, surface profilometry, surface roughness estimation, and surface displacement measurement. PSI was first introduced by Bruning et al. in 1974,  Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses , Applied Optics 13, 2693-26703 (1974). 
   The fundamental concept of PSI is that the phase of an interferogram can be extracted by acquiring a set of a few sequentially phase-shifted interferograms of the original interferogram with a constant phase shift between any two adjacent interferograms (or intensity frames). These phase shifted interferograms are produced by changing the optical path difference (OPD) between a measurement surface and a reference surface, or by changing the wavelength if the OPD is not zero. Thus, all types of PSI measurements rely on some mechanism to shift or change the phase of an interferogram in a regular and predictable manner. 
   PSI also has been successfully applied to measure an object with multiple reflective surfaces, such as a transparent plate, which produces multiple interferograms superimposed on the recording plane of an interferometer. Such systems are disclosed in the U.S. patent to S. Tang U.S. Pat. No. 6,885,461, and the articles to P. DeGroot,  Measurement of Transparent Plates with Wavelength - tuned Phase Shifting Interferometry , Applied Optics, Vol. 39, No. 16, 2658-2663 (2000) and K. Okada et al.  Separate Measurements of Surface Shapes and Reflective Index Inhomogeneity of an Optical Element Using Tunable Source Phase Shifting Interferometry , Applied Optics, Vol. 29, No. 22, 3280-3285 (1990). Each of the interferograms from transparent plates carries topographic information related to its corresponding reflective surface. The phase of each interferogram in the superimposed interferograms shifts at a different speed during the wavelength changes. Consequently, each interferogram is differentiated from the others; and its phase may be extracted from the set of superimposed interferograms, as disclosed in the U.S. patent to S. Tang U.S. Pat. No. 6,856,405. 
   As disclosed in the Tang &#39;405 patent, in order to obtain precise estimates for the various surface phases, phase shift increments between any two adjacent intensity frames must be calibrated to a known constant. While this condition is desirable for the phase measurement of a single surface, it becomes essential when multiple overlapping interferograms are present and the phase contribution from each reflective surface must be segregated from the other surfaces. 
   Since 1990, PSI with a spectrally broad band or white light illumination known as vertical scanning interferometry also has been widely used to profile surfaces. Although white light phase shifting interferometry (WLPSI) is capable of measuring surfaces with nanometer precision and with step height greater than one-fourth of a wavelength, the technique is only able to profile objects with uniform surface properties, and without transparent thin films. This is because the complex surface properties, such as bulk surfaces, single or multi-layer film stacks on a substrate, unresolved micro-structures on a substrate, or as part of a film stack, from the testing object create various phase shifts on reflections. Typically, WLPSI loses its ability to profile such objects. 
   With the introduction of temporal interferometric signal modeling techniques in WLPSI recently, an object with complex surface properties can be measured with improved precision. This signal modeling technique not only obtains the topographic information of the test surface, but also simultaneously or separately determines additional parameters of the test piece, e.g. layer thickness and/or material refractive index for film stacks, or line width and structure depth of micro-structures. Signal modeling techniques may acquire a set of white light phase-shifted interferograms with piezoelectric transducer (PZT) pushers of either the reference flat or the test object. Signal modeling techniques require a constant phase shift between any adjacent interferograms similar to measuring a transparent plate with a tunable laser in PSI. However, the material properties of a PZT cause the phase of interferogram change to be non-linear with respect to the time during the acquisition. Such non-linear phase shifts result in errors in sampling the temporal interferometric signal at each pixel of the interferograms. Such sampling errors greatly deteriorate the measurement accuracy of the signal modeling technique. 
   SUMMARY OF THE INVENTION 
   This invention is directed to WLPSI combining signal modeling techniques and a non-linear phase shift calibration to measure objects with complex surface properties. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic diagram of an interferometer useful in practicing embodiments of the invention; 
       FIG. 2  is a schematic diagram of a different interferometer used in practicing embodiments of the invention; and 
       FIG. 3  is a temporal intensity distribution diagram for a pixel of a measuring surface with multiple transparent thin films on it. 
   

   DETAILED DESCRIPTION 
     FIG. 1  is a schematic diagram of a Mirau interferometer system which may be used in conjunction with the practice of the method of the various embodiments of this invention. As shown in  FIG. 1 , the interferometer is controlled by a processor or computer  20  which coordinates the operation of a phase shifting driving mechanism  50  with other components of the system. The white light from the source  22  is supplied through a collimating lens  24  to a beam splitter  28  which supplies the collimated light to an objective lens  44 . The lens  44  in turn supplies light to a reference flat in the form of a reference mirror  30 , and to a further beam splitter  46  in accordance with the conventional operation of such an interferometer. 
   The reflected light beams from both the reference flat  30  and from both surfaces of a measurement object or test piece  32  are directed by the beam splitter  28  to an imaging lens  38 , which supplies simultaneously multiple interferograms to a CCD camera  40 , or other suitable recording plane. The camera  40  additionally may include a frame grabber (not shown) for storing images detected by the camera; or the processor  20  may be configured to provide this function. In any event, the images obtained by the camera  40  are supplied to the computer  20  for processing to provide the desired profiles, in a suitable form, for immediate display on a TV monitor  42  or for storage for subsequent utilization. 
     FIG. 2  illustrates a Linnick interferometer system which may be used in conjunction with the method of the various embodiments of this invention. As is known, a Linnick interferometer uses identical microscope objective lenses, with the lens  44  being duplicated by another lens  54 , provided with inputs from the beam splitter  28 . The lens  54  then focuses on a reference flat (mirror)  30 , whereas the lens  44  focuses on the object  32 . The reflected images from both the reference flat  30  and the single or multiple surfaces of the test object  32  are gathered and supplied by the beam splitter  28  to the imaging lens  38  for the camera  40 . The circuitry in the processor  20  then processes the information in substantially the same manner as for the Mirau interferometer circuit described previously. It also should be noted that other interferometers which may be used in conjunction with the practice of the methods and embodiments of this invention include Twyman-Green, Fizeau, and Michelson interferometers (none of which are shown) 
   In conjunction with the practice of the method described subsequently, all of the interferometers described above (including those mentioned in addition to the two specifically shown in  FIGS. 1 and 2 ) are operated with a white light source  22 , as described previously. Vertical scanning (parallel) or vertical positioning of the object  32  relative to a reference flat  30  is utilized for measuring the intensity of each pixel in the scan. 
   As shown in both  FIGS. 1 and 2  (and as also would be used in conjunction with the other interferometers mentioned), the step-by-step positioning for each frame of analysis is effected by the processor  20  in synchronization with the operation of the camera  40  by means of a suitable pusher or drive mechanism  50 . Ideally, a piezoelectric (PZT) pusher  50  is employed; but a pneumatic pusher or other suitable mechanical pusher may be employed for this purpose. PZT pushers, however, have been found to be highly effective as a means of mechanically moving the object  32  toward and away from the reference flat  30 . It should be noted that instead of moving the test piece or object  32  with respect to the reference flat  30 , as illustrated in both  FIGS. 1 and 2 , the pusher  50  could be mechanically or otherwise coupled (by coupling means not shown) to the reference flat  30  to move that surface relative to the surfaces of the test piece  32 . The object/test piece  32  and the reference surface  30  are moved in parallel planes relative to one another to produce the repeated measurements for vertical scanning for each of the positions over which the complete scan is made. As is well known, the use of WLPSI allows the entire image field to be captured in one instant, without the need for scanning apertures. This results in profiling with high accuracy over a large range. 
   In the measurement of multiple parallel surfaces or more complex surfaces as described above, a variety of error sources can induce non-linear phase shift to take place between successive interferograms. Consequently, in order to permit a signal modeling technique to be used effectively with a WLPSI system, it has been found that the combination of such a technique with a non-linear phase shift calibration technique allows measurement of an object with complex surface properties using WLPSI. The calibration technique obtains non-linear phase shift information directly from interferograms; so that all error sources that result in the non-linear phase shift are taken into account, including, but not limited to those caused by the use of PZT pushers. Consequently, the calibration method or technique is able to determine a sequence of physical values used as an input during data acquisition to produce an accurate, repeatable, uniform phase shifting speed crucial to signal modeling in order to allow signal modeling to perform properly. This can be accomplished without the addition of any extra hardware or processor circuits beyond that normally used for the different types of interferometers described. By combining non-linear phase shift calibration with signal modeling, significant improvement in the ability of signal modeling to extract desired information from a set of phase-shifted white light interferograms results. 
   The phase shift θ in PSI and in WLPSI is a function of time t. It can be expressed as: 
                   θ   ⁡     (   t   )       =       ∑     m   =   0     M     ⁢           ⁢       c   m     ⁢       t   m     .                 (   1   )               
where C m  is the weight for the sampling intensity. The non-linear phase shift implies M&gt;1 in Equation (1). The phase shift θ can also be expressed as a function of input v(t). That is:
 θ( t )=θ( v ( t )).  (2) 
where the input v(t), related to the physical value such as voltage, is also a function of time t. An embodiment of the invention features a calibration technique that finds an input v(t) such that:
 θ( t )= C   0   +C   1   t,   (3) 
where C O  and C 1  are constants, without adding extra hardware. In other words, an existing non-linear phase shift driving mechanism is made to produce a linear or a constant phase shifting speed during the data acquisition. Such a system and method are disclosed in the above-mentioned patent to Tang U.S. Pat. No. 6,856,405, the disclosure of which is incorporated herein by reference in its entirety.
 
   To determine the input v(t) in a white light interferometer, the phase shift driving mechanism in the interferometer first has to be calibrated with an object  32  having a single reflective surface. A set of interferograms is acquired with a known input v m (t)=kt from the computer  20  to the phase shift driving mechanism  50  in  FIG. 1  and  FIG. 2 . The initial constant k may be approximately determined if the measurement hardware information is known. Otherwise, a small value k arbitrarily is selected. A mean phase shift from the set of interferograms acquired with the input v m (t) can be determined in the frequency domain by means of Fourier transform. Assume the set of interferograms can be expressed as x(t), then its Fourier transform is: 
                   X   ⁡     (   ω   )       =       ∑   i     ⁢           ⁢       x   ⁡     (   t   )       ⁢     ⅇ       -   j     ⁢           ⁢   ax                   (   4   )               
and its energy density spectrum is:
   S (ω)=| X (ω)| 2   (5) 
This is the distribution of phase-shifting energy in the frequency domain. The mean phase shift is equal to the angular frequency {dot over (ω)} m  where the maximum of S({dot over (ω)}), O&lt;{dot over (ω)}&lt;π, occurs. Now k is modified by:
 
                   k   new     =       k   old     ⁢     Θ     ω   m                 (   6   )               
where Θ is the desired phase shift between adjacent interferograms. Θ=π/2 in most applications. The above process is repeated until |Θ−{dot over (ω)} m |&lt;π/180.
 
   After v m  (t) is obtained from the above linear phase shift calibration, the non-linear phase shift calibration is started. A phase shift θ m (t) is achieved from a set of interferograms acquired by the input v m (t). That is, for a given pixel location (x,y) the phase shift θ at the pixel for the k-th frame of the interferograms can be calculated by: 
                       θ   k     ⁡     (     x   ,   y     )       =       cos     -   1       ⁢           I     k   +   2       ⁡     (     x   ,   y     )       -       I     k   -   2       ⁡     (     x   ,   y     )           2   ⁢     (         I     k   +   1       ⁡     (     x   ,   y     )       -       I     k   -   1       ⁡     (     x   ,   y     )         )             ,           (   7   )               
where I k (x,y) is the intensity value at the pixel position (x,y) in the k-th interferogram. The mean phase shift θ m (k) for the k-th frame interferogram is the mean of phase shifts calculated only at pixels where I k+1 (x,y)−I k−1 (x,y)&gt; mean of all I k+1 (x,y)−I k−1 (x,y). If the set of interferograms acquired from an interferometer is noisy, the phase shift θ m (t) calculated from it may need additional processing to smooth the results, such as by using a least-squares fitting technique to approximate θ m (t) with a polynomial P m (t):
 
                       P   n     ⁡     (   t   )       =       ∑     k   =   0     n     ⁢           ⁢       a   k     ⁢     t   k           ,           (   8   )               
where n≧2 and ak is a constant. After θ m (t)v achieved with input v m (t), a new known non-linear input can be obtained by:
 
                   v   ⁡     (   t   )       =         Θ   ⁢           ⁢       tv   m     ⁡     (   t   )             θ   m     ⁡     (   t   )         .             (   9   )               
The above non-linear phase shift calibration process is repeated by using the calculated v(t) as new v m (t) until the linearity of phase shift θ m (t) is satisfied.
 
   The non-linear phase shift calibration also can be done with the derivatives of v m (t) and θ m (t). Similar to Equation (9), a new derivative of the non-linear input can be expressed as: 
                     v   ′     ⁡     (   t   )       =       Θ   ⁢           ⁢       v   m   ′     ⁡     (   t   )             θ   m   ′     ⁡     (   t   )                 (   10   )               
Thus, the new non-linear input becomes:
 
                     v   ⁡     (   t   )       =       ∫         Θ   ⁢           ⁢       v   m   ′     ⁡     (   t   )             θ   m   ′     ⁡     (   t   )         ⁢     ⅆ   t         +     v   0         ,           (   11   )               
where v O  is a constant.
 
   Once the phase shift calibration has taken place, the non-linear input v(t) can be used for the ongoing measurement of an object  32  with complex surfaces. This is done for n frames of data using the calibrated non-linear input steps v(t). The temporal interferometric signal x(z) at a pixel of the recording plane from a white light phase-shifting interferometer can be expressed as:
 
 x ( z )=∫ 0   ∞   F ( k )(1+∫ 0   θ     0    cos(2 k  cos θ( z−h )+φ( k ,θ))sin θ cos θ d θ) dk,   (12)
 
where z is the distance of the reflecting point from the focus, h is the distance of the reference mirror from the focus, k is the wavenumber, θ O  is related to the numerical aperture of the objective or N.A.=sin θ O ,φ((k,θ) is the phase offset resulting from the reflectance phase of the measuring surface and the spectral modulation phase; and F(k) is the modulation amplitude. F(k) can be expressed as:
 
 F ( k )= R ( k ) R   g ( k )= R   f ( k ),  (13)
 
where R g (k) is the amplitude of spectral distribution of the light source, and R(k) is the reflectance amplitude of the measuring surface. Equation 12 is used by way of example, and other expressions of a temporal interferometric signal in either a different or more general form may be substituted for equation 12.
 
   A typical temporal interferometric signal of Equation (12) is depicted in  FIG. 3 . Various associated signals can be derived from this signal, such as its temporal amplitude, its temporal phase, or the spectral amplitude of its Fourier transform. One, two or more signals are selected to compare with a group of corresponding signals modeled with different properties such as film thicknesses, refractive indexes, height, etc, of the measuring surface. Once an optimum or best match is found, the measuring surface properties, such as film thicknesses, refractive indexes and height are determined from the modeling surface properties that generate the optimum or best match signal or signals. 
   In summary, the frames of interferometric signals are gathered from an interferometer, such as those discussed above and two examples of which are shown in  FIGS. 1 and 2 . After the non-linear phase shift calibrations have been effected by the computer  20 , n frames of data are acquired while the white light fringe pattern from the light source  22  is scanning through the field of view, with the frames having a plurality of pixels. Depending upon the specific property which is the object of the interferometric test being conducted, a comparison is made by the computer  20  with the appropriate model signal to determine the best match, as mentioned above. Once a match has been determined, the particular information, based on the model with which the comparison is being made, is selected to provide the output from the computer  20  to the monitor  42 , or other output device, for subsequent utilization. 
   The foregoing description of embodiments of the invention is to be considered as illustrative and not limiting. Various changes and modifications will occur to those skilled in the art for performing substantially the same function, in substantially the same way, to achieve substantially the same result, without departing from the true scope of the invention as defined in the appended claims.