Abstract:
A metrology device, such as an ellipsometer, includes a light source that produces a pulsed electromagnetic beam, such as a flash bulb or pulsed laser, and a spatially dependent polarizing element that introduces a spatially dependent retardation in the light beam. The use of a pulsed light source is advantageous over a continuous light source, as a pulsed light source generates less heat, is stronger, lasts longer, and does not need the use of a mechanical shutter. The use of a spatially dependent polarizing element advantageously eliminates the use of temporally dependent moving polarization modulation elements, thereby allowing the use of a pulsed light source. Downstream of the spatially dependent polarizing element are the analyzer and a multi-element detector that may be synchronized with the pulsed electromagnetic beam to detect after one or several pulses of light have been emitted from the pulsed light source.

Description:
FIELD OF THE INVENTION 
   The present invention is related to optical metrology and in particular to a metrology device and technique that uses a pulsed light source and a spatially dependent polarizing element as a component of an ellipsometer or a polarized spectrometer. 
   BACKGROUND 
   There is always a need for precise and reliable metrology to monitor the properties of thin films, especially in the semiconductor and magnetic head industries. Thin film properties of interest include the thickness of one or more layers, the surface roughness, the interface roughness between different layers, the optical properties of the different layers, the compositional properties of the different layers and the compositional uniformity of the film stack. Ellipsometry is particularly well suited to this task when the thickness is less than 100 nm, when there are more than two layers present or when there are compositional variations. Additionally, dimensional measurements such as linewidth, sidewall angle, and height can be extracted using ellipsometry and/or reflectometry. 
   An ellipsometer is a measurement tool used to determine the change in polarization state of an electromagnetic wave after interaction with a sample. The determination of this polarization state can yield information about the thin film properties such as those listed above. In general, an ellipsometer is a polarization-state-in, polarization-state-out device.  FIG. 1  shows a simple block diagram of a typical ellipsometer  10 , which includes a Polarization State Generator (PSG)  12  that generates an electromagnetic wave of a known polarization state and a Polarization State Detector (PSD)  16  that determines the polarization state of the electromagnetic wave after interaction with a sample  14 . In  FIG. 1  the interaction is shown in reflection mode, but it should be understood that the interaction may be in transmission mode, i.e., the PSD determines the polarization state of the electromagnetic wave after transmission through a sample. Different kinds of PSG/PSD configurations have been proposed and developed for ellipsometers. The advantages of each configuration are specific to the kind of extracted information that is desired. 
   The use of pulsed light source in metrology devices offers many advantages over conventional continuous light sources, as discussed in U.S. Pat. No. 6,002,477 to Hammer. A pulsed light source enables energization of the light source to be confined to the time over which a measurement is to be made, thereby reducing power consumption and very significantly extending the life of the light source. 
   In the thin film metrology field, the most popular ellipsometry configurations include a rotating polarizing element in the PSG and/or the PSD. Unfortunately, rotating polarizing elements cannot be used with pulsed light sources such as flash bulbs or pulsed lasers. When a rotating polarizing element is used with a pulsed light source, synchronization problems occur, resulting in inaccurate information being extracted. Furthermore, the light source intensity must be very constant over a whole optical rotation when using rotating elements, which is not possible with a pulsed light source, where the light source intensity varies significantly from pulse to pulse. This problem is aggravated for spectroscopic ellipsometry, where usually a multi-channel detector is utilized to record the whole spectrum. Such a photodiode array generally has a minimum reading time, which makes the use of a pulsed source in conjunction with a rotating element impossible. 
   A different kind of ellipsometer that has been extensively developed and used for thin film metrology and that does not have a rotating element is the photoelastic modulator ellipsometer (PME). This instrument employs a photoelastic modulator (PM) to change the polarization state of the light as a function of time either before or after reflection from the sample surface.  FIG. 2  is a block diagram of a conventional PME  20 . The PSG portion  21  of the PME  20  includes a light source  22  and a linear polarizer  24 . The light source  22  generates a collimated beam (monochromatic or broadband radiation) that is transmitted through the linear polarizer  24 . The linearly polarized beam is reflected from the sample surface  26  thereby modifying the polarization state of the electromagnetic beam. The PSD portion  27  of the PME  20  includes a PM (or Pockels cell)  28 , another linear polarizer  30 , and a detector  32 . 
   Unfortunately, photoelastic modulators and Pockels cells introduce a time dependent phase that creates synchronization problems when used with a pulsed light source, similar to those seen in ellipsometers utilizing a rotating polarizing element. Thus, a pulsed light source is impractical in conjunction with an ellipsometer configuration that utilizes a photoelastic modulator or Pockels cell. 
   What is needed is an ellipsometer configuration (monochromatic or spectroscopic) that does not use moving parts or a phase modulator, i.e., a configuration that is time-independent so that it can be used with a pulsed light source, with the advantage over a continuous light source being that a pulsed light source generates less heat, is more intense and has a longer lifetime. Moreover, such an ellipsometer can be compact and robust, minimizing cost and maintenance. Such a configuration will be particularly suitable for integration into existing process tools due to its reduced size. 
   SUMMARY 
   A metrology device, such as an ellipsometer, in accordance with one embodiment of the present invention, includes a variable retarder that introduces a spatially dependent phase shift to an electromagnetic beam. A polarizer and multi-element detector then spatially sample the phase-shifted beam. This is in contrast to the analysis of a time dependent phase shifted beam used in conventional systems. A metrology device, in accordance with one embodiment of the present invention, advantageously, has no moving parts, is compact and utilizes inexpensive components. 
   Accordingly, in one aspect of the present invention, a metrology device that detects the polarization state of a pulsed electromagnetic beam that is incident on a sample includes a polarization state generator with an electromagnetic source that turns on and off to produce a pulsed electromagnetic beam, wherein the polarization state generator produces a pulsed electromagnetic beam of known polarization state that is incident on the sample. The metrology device includes a spatially dependent polarizing element in the path of the expanded electromagnetic beam and a multi-element detector within the path of the pulsed electromagnetic beam after the spatially dependent polarizing element, wherein the multi-element detector measures the intensity of the pulsed electromagnetic beam as a function of position. The metrology device may include a beam expander that spatially expands the electromagnetic beam to a desired size. The metrology device may operate in a spectroscopic mode, in which case the device includes a monochromator or spectrograph. The metrology device may include a synchronizer coupled to the electromagnetic source and the multi-element detector, wherein the synchronizer causes the multi-element detector to measure the intensity of the pulsed electromagnetic beam as a function of position and is time correlated to read out data when the pulsed electromagnetic source is off, i.e., not producing light. The electromagnetic source may also produce a plurality of pulsed electromagnetic beams after the synchronizer causes the multi-element detector to stop reading out the data, i.e., the collected signal can be averaged over multiple consecutive light pulses. 
   Another aspect of the present invention includes a method of ellipsometrically measuring a sample. The method includes turning on and off an electromagnetic beam to produce a pulsed electromagnetic beam to be incident on a sample. The method further includes creating a spatially dependent relative phase difference between the electric field components of the beam. The beam is then polarized and the intensity is sampled at a plurality of locations with the multi-element detector. The method further includes synchronizing the detection of the intensity of the polarized pulsed electromagnetic beam with turning on and off the electromagnetic beam. The method may further comprise turning on and off the electromagnetic beam to produce a plurality of pulsed electromagnetic beams. In one aspect of the invention, the method includes expanding the beam. The method can be used in a monochromatic or spectroscopic mode. In a spectroscopic mode, the method further includes filtering the wavelengths of the beam spatially in a direction orthogonal to the direction of producing the spatially dependent phase difference, wherein the intensity of the polarized light beam is determined as a function of the spatially dependent relative phase shift in one direction and the wavelengths in another direction on a two-dimensional detector. 
   In yet another aspect of the present invention, an interferometer includes an electromagnetic source that turns on and off to produce a pulsed electromagnetic beam, a spatially dependent polarizing element and a multi-element detector. The interferometer may also include a beam expander that spatially expands the electromagnetic beam. The interferometer may also include a synchronizer coupled to the electromagnetic source and the multi-element detector, wherein the synchronizer causes the multi-element detector to measure the intensity of the pulsed electromagnetic beam as a function of position when the electromagnetic source is turned off. The interferometer may further include a wavelength-dispersing component that separates the component wavelengths of the electromagnetic beam. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a simple block diagram view of a typical ellipsometer including a Polarization State Generator (PSG), a sample and a Polarization State Detector (PSD). 
       FIG. 2  is a block diagram of a conventional photoelastic modulator ellipsometer (PME). 
       FIG. 3  is a block diagram of an ellipsometer with no moving parts and no time dependent phase modulator in accordance with an embodiment of the present invention. 
       FIG. 4  shows a perspective view of the PSD from the ellipsometer in  FIG. 3  when used in spectroscopic mode. 
       FIG. 5  shows a reflecting diffraction grating that expands and collimates the beam in the PSD. 
       FIG. 6A  shows a lens system to expand and collimate the beam to cover the entire PSD detector area. 
       FIG. 6B  shows an etalon that is used to spatially expand the reflected beam into several discrete beams to cover the entire PSD detector area. 
       FIG. 6C  shows transmission diffraction grating that is used to spatially expand the reflected beam to cover the entire PSD area. 
       FIGS. 7A ,  7 B, and  7 C show three embodiments of a variable retarder that may be used in the PSD shown in  FIG. 3   
       FIG. 8  shows a perspective view of an ellipsometer with no moving parts indicating the calibration parameters. 
       FIG. 9  shows a representation of the polarization state of an electromagnetic beam in terms of its ellipsometric angles x and Q. 
       FIG. 10A  shows the modulated intensity signal detected by the multi-element detector. 
       FIG. 10B  shows the same modulated intensity detected by three detectors, which collect the partial integrals of the modulated intensity. 
       FIG. 11  is a block diagram of an interferometer in accordance with an embodiment of the present invention. 
       FIG. 12  shows the ellipsometer of  FIG. 3  in accordance with an embodiment of the present invention. 
       FIGS. 13A ,  13 B,  13 C and  13 D show the synchronization of a CCD reading and a pulsed light source in accordance with an embodiment of the present invention. 
   

   DETAILED DESCRIPTION 
   In accordance with an embodiment of the present invention, a metrology device, such as an ellipsometer, is time independent so that it can be used with a pulsed light source, such as a flash bulb or pulsed laser. For example,  FIG. 3  shows a block diagram of an ellipsometer  100  in accordance with an embodiment of the present invention. After the light beam is reflected from the sample  110 , the beam is expanded and passed through a variable retarder  118  to introduce a spatially dependent phase shift. The expanded beam then passes through polarizer  122  and the intensity is measured using multi-element detector  126 . Ellipsometer  100  may be used advantageously for semiconductor thin film applications. Due to its small size, it may be integrated into various semiconductor or other processor tools. 
   As shown in  FIG. 3 , ellipsometer  100  includes an electromagnetic source  102  that generates a collimated beam  104  of monochromatic or broadband radiation that is transmitted through polarizer  106  to produce a polarized beam  108 . The polarized beam  108  is incident on and interacts with the sample surface  110  to produce a reflected beam  112 . Reflected beam  112  has a modified polarization state compared to polarized beam  108 . It should be understood that if desired, ellipsometer  100  may operate in transmission mode in which case the beam passes through the sample. For the sake of simplicity, the present disclosure will describe an ellipsometer operating in reflection mode, with the understanding that a transmitted beam may alternatively be used. 
   After reflection from the sample surface  110 , the reflected beam  112  is expanded in the plane of the drawing (the x direction) by expander  114  to produce an expanded beam  116 . It should be understood, however, that beam expander  114  is used to shape the beam so that it adequately fills the variable retarder  118  and a multi-element detector  126  with the reflected signal. If the reflected beam itself adequately fills the variable retarder  118  and multi-element detector  126 , e.g., if electromagnetic source  102  produces the properly shaped beam, beam expander  114  is unnecessary. 
   The expanded beam  116  is then transmitted through a variable retarder  118  whose geometry is matched to the shape of the expanded beam. The variable retarder  118  has the property of creating a relative phase difference  6  between the electric field components parallel (ordinary or o) and perpendicular (extraordinary or e) to the optical axis of the variable retarder  118  in the x direction. The resulting phase shifted beam  120  is then transmitted through a polarizer (linear polarizer)  122 . A multi-element detector  126  then records the intensity of resulting beam  124 . The detector geometry is chosen to match the geometry of the beam expander  114  and variable retarder  118 . The multi-element detector  126  may be a photodiode array (PDA), a multi-element charge coupled device (CCD), an avalanche photodiode array (APD), a multi-element photomultiplier, or even a multi-element charge injection device (CID) or some similar device. The choice of the appropriate multi-element detector depends on many variables, such as the available light throughput, needed measurement time, and of course, cost. 
   It should be understood that if desired, the expander  114  and variable retarder  118  may be located in the PSG, i.e., before the sample surface  110 . In this embodiment, for example, the expanded beam is focused onto the sample surface  110 . 
   In a spectroscopic embodiment, broadband radiation is emitted from source  102 . Additionally, the light beam must be expanded in the y direction, which will be described below. An additional optical component, such as a band-pass filter array  123 , is required to separate the various wavelengths of the beam. An appropriate band-pass filter  123  has a linear variation of the transmitted wavelength in the y direction. Band-pass filter  123  can also be made up of individual interferometric elements. Interferometric filters are composed of stacks of thin films with different thicknesses chosen such that essentially only one narrow, well-defined wavelength range is transmitted through the filter. It is possible to construct an interferometric filter employing a gradient in thickness of the thin films in one direction such that a continuous spectrum of wavelength filters is obtained. These kind of filters may be custom-manufactured by, e.g., Barr Associates, Inc. located in Westford, Mass. With the gradient oriented in the y direction and a multi-element detector  126  that has elements in the x and y directions, the detector  126  maps the intensity of the resulting beam as a function of retardance δ in the x direction and as a function of wavelength λ in the y direction. The intensities recorded by the detector  126  can then be analyzed to obtain the ellipsometry angles ψ and Δ as a function of wavelength. 
   The relative position of band-pass filter  123  may vary after beam  112  is adequately expanded, i.e., band-pass filter  123  may also be located immediately after beam  112  is expanded and before variable retarder  118 , or even between variable retarder  118  and polarizer  122 . 
     FIG. 4  shows a perspective view of the PSD after beam expansion in ellipsometer  100  in  FIG. 3  where the expanded beam  112  is illustrated as plate  112  for the sake of simplicity. As shown in  FIG. 4 , spatial variable retarder  118  varies the phase δ along the x-axis and interferometric filter  123  varies the wavelength λ along the y-axis. The polarizer  122  creates the sinusoidal modulation of the intensity. Thus, as illustrated in  FIG. 4 , the detector  126  measures the intensity of the light beam as a function of phase δ along the x-axis and wavelength λ along the y-axis. 
   Other hardware configurations can be devised for spectroscopic ellipsometry in accordance with the present invention, as described in U.S. application Ser. No. 09/929,625. For example, as shown in  FIG. 5 , a reflecting diffraction grating  128  is used to collimate the beam in reflection in the x direction as well as separate the wavelengths in the signal by diffraction in the y direction. In this case, the reflecting diffraction grating  128  replaces the interferometric filter  123  shown in  FIGS. 3 and 4  and the collimating components of the beam expanding optics  114  shown in  FIG. 3 . In this configuration, the reflecting diffraction grating  128  operates as part of the expander in the ellipsometer used to expand the reflected beam to fill the variable retarder  118 . Transmission gratings can also be employed to spread the beam in the y direction. See, for example, U.S. Pat. No. 5,392,116, issued Feb. 21, 1995, which is incorporated herein by reference. 
   Numerous techniques can be devised to expand the reflected beam  112  to fill the variable retarder  118  and detector  126 . For example, as shown in  FIG. 3  and in  FIG. 6A , lenses  130  and  132  can be used to expand and collimate the reflected beam  112  to cover the desired PSD area. Alternatively, as shown in  FIG. 6B , an etalon  140  can be used to divide the reflected beam  112  into a plurality of discrete beams to functionally spatially expand the beam. Multiple reflections inside the etalon  140  generate parallel beams of equal intensity from a properly coated etalon. The detector elements in detector  126  should then be aligned to the discrete beams produced by the etalon  140 . Diffractive optics such as a grating  145  can also be used, along with collimating lens  132 , to spatially expand the beam into a plurality of individual beams of equal intensity, as shown in  FIG. 6C . 
   As described in U.S. application Ser. No. 09/929,625, many spatially variable retarders may be designed for use in the present invention. For example,  FIGS. 7A ,  7 B, and  7 C show three illustrative variable retarders that may be used with the present invention. The variable retarder  150  shown in  FIG. 7A  consists of two wedged plates  152  and  154  composed of birefringent material whose outer surfaces are orthogonal to the beam propagation direction. The optical axes of the plates  152  and  154  are perpendicular to each other. An example of variable retarder  150  is manufactured by InRad Inc. located at New Jersey. The effective retardance for variable retarder  150  assuming an orthogonal incident beam is given by: 
                     δ   ⁡     (   x   )       =         4   ⁢   π     λ     ⁢   Δ   ⁢           ⁢   nx   ⁢           ⁢   tan   ⁢           ⁢   Φ       ,           eq   .           ⁢   1               
where x is the distance from the center of the variable retarder  150 , Δn is the birefringence (which is a function of wavelength λ), i.e., the difference between the ordinary and extraordinary refractive indexes assuming both wedges are made of the same material, and Φ is the wedge angle of the internal faces of the two birefringent plates  152  and  154 . The angle Φ is preferably chosen so that the retardance  8  varies over a range of at least 2π radians for the wavelengths of interest. An additional complexity is that the o and e beams start to diverge at the interface of the two wedges and continue to diverge at the exiting air interface. Therefore, Φ should be chosen as small as possible to minimize the separation between the two polarization components. As shown in  FIG. 3 , it is desirable to locate the detector  126  as close as possible to the variable retarder  118 . Alternatively, a lens following the variable retarder  118  may be used to correct this divergence.
 
     FIG. 7B  shows another example of a variable retarder  170  composed of two plates. The first plate  172  has two parallel faces. The second plate  174  has one flat face and a second face with a series of steps of different thicknesses. If desired, the second plate  174  may have a continuously changing thickness rather than a series of steps. The optical axes of the first plate  172  and the second plate  174  are perpendicular to each other similar to the variable retarder  150  described in  FIG. 7A . The relative phase difference  6  is once again a function of position from the center of the plate. The steps in plate  174  could also be varied in thickness in the y direction for spectroscopic applications to maintain a constant phase delay for each wavelength. This configuration of a variable retarder does not result in a divergence of the two o and e components of the polarized beam. The variable retarder shown in  FIG. 7B  is also useful in an interferometer. 
     FIG. 7C  is another example of a variable retarder  180  composed of a single wedge. Variable retarder  180  is a made up of a single plate of birefringent material with non-parallel faces. The optical axis must be at a very small angle (almost parallel) to the beam propagation direction as indicated by arrow  181 . Thus, the optical axis is at an oblique angle with the direction of propagation of the electromagnetic beam. This geometry creates an effective birefringence given by the projection of the ordinary and extraordinary indices of refraction to the plane perpendicular to the direction of propagation. 
   It should be understood that other variable retarders could be used. For example, a liquid crystal array, where it is possible to control the birefringence of individual pixels in the x and y directions may be used, as described in T. Horn and A. Hofmann, “Liquid Crystal Imaging Stokes Polarimeter”, ASP Conference Series Vol. 184, pp. 33–37 (1999), which is incorporated herein by reference. A depolarizer, such as the one fabricated by Karl Lambrecth Co., located in New Jersey, may also be used as a variable retarder. Moreover, a variable retarder that uses artificial dielectrics may be used, such as that described in D. R. S. Cumming and R. J. Blaikie, “A Variable Polarization Compensator Using Artificial Dielectrics”, Opt. Commun. 163, pp. 164–168 (1999), which is incorporated herein by reference. 
   For the system shown in  FIG. 3 , the Mueller formalism can be used to yield the following dependence for the intensity as measured by the multi-element detector  126  as a function of δ(x):
 
 I=Io {1+sin 2( C′−A ′) sin 2( C′−Q ) cos δ( x )cos 2χ+cos 2( C′−A ′)cos 2( C′−Q )cos 2χ−sin2( C′−A ′)sinδ( x )sin2χ}  eq. 2
         where I 0  is the intensity without polarization, C′ is the angle of the optical axis of the variable retarder  118 , and A′ is the angle of the transmission axis of the polarizer  122 . Both the C′ and A′ angles are measured with respect to the plane of incidence, as shown in  FIG. 8 , which shows a perspective view of ellipsometer  100 . The retardance of the variable retarder  118  is represented in equation 2 by δ(x). The ellipticity angle is represented by χ and the tilt angle defining the polarization state of the reflected beam is represented by Q.       

     FIG. 9  is a representation of a polarization state of an electromagnetic beam in terms of its ellipsometry angles χ and Q, with the x-axis parallel to the plane of incidence. When Q is greater than zero, the angle is defined as counter-clockwise for an incoming beam, as shown in  FIG. 9 . The sign of χ determines the handedness of the polarization state, i.e., positive χ indicates left-handed rotation, whereas negative χ indicates right-handed rotation, also shown in  FIG. 9 . 
   The quantities χ and Q are related to the ellipsometry angles χ i and Δ by: 
                   cos   ⁢           ⁢   2   ⁢   ψ     =         cos   ⁢           ⁢   2   ⁢     P   ′       -     cos   ⁢           ⁢   2   ⁢   Q   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   χ         1   -     cos   ⁢           ⁢   2   ⁢   Q   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   χcos2   ⁢           ⁢     P   ′                     eq   .           ⁢   3     ⁢   A                 tan   ⁢           ⁢   Δ     =     -       tan   ⁢           ⁢   2   ⁢   χ       sin   ⁢           ⁢   2   ⁢   Q                   eq   .           ⁢   3     ⁢   B               
where P′ is the angle of the transmission axis of the polarizer  106  with respect to the plane of incidence, as shown in  FIG. 8 . Ellipsometry angles and equations  3 A and  3 B are described in more detail in Joungchel Lee, P. I. Rovira, Ilsin An, and R. W. Collins, “Rotating-Compensator Multichannel Ellipsometry: Applications for Real Time Stokes Vector Spectroscopy of Thin Film Growth”, Rev. Sci. Intrum. 69, pp. 1800–1810 (1998), which is incorporated herein by reference. The ellipsometry angles ψ and Δ can then be modeled using, e.g., the Fresnel formalism to obtain the thin film properties of the sample.
 
   In order to obtain χ and Q, the intensity given by equation 2 may be analyzed, e.g., using regression analysis, once the intensities of the multi-element detector  126  are measured. An additional approach shows the normalized intensity written as:
 
 I′ =1+α cos δ+βsinδ  eq. 4
 
Where α and β are described by the following equations:
 
   
     
       
         
           
             
               
                 α 
                 = 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           C 
                           ′ 
                         
                         - 
                         
                           A 
                           ′ 
                         
                       
                       ) 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           C 
                           ′ 
                         
                         - 
                         Q 
                       
                       ) 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     χ 
                   
                   
                     1 
                     + 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                         ( 
                         
                           
                             C 
                             ′ 
                           
                           - 
                           
                             A 
                             ′ 
                           
                         
                         ) 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                         ( 
                         
                           
                             C 
                             ′ 
                           
                           - 
                           Q 
                         
                         ) 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       χ 
                     
                   
                 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   5A 
                 
               
             
           
         
       
     
   
                 β   =           -   sin     ⁢           ⁢   2   ⁢     (       C   ′     -     A   ′       )     ⁢   sin   ⁢           ⁢   2   ⁢   χ       1   +     cos   ⁢           ⁢   2   ⁢     (       C   ′     -     A   ′       )     ⁢   cos   ⁢           ⁢   2   ⁢     (       C   ′     -   Q     )     ⁢   cos   ⁢           ⁢   2   ⁢   χ         .             eq   .           ⁢     5B                 
One advantageous configuration of angles is P′=45°, C′=0°, and A′=−45°, but other configurations may be used.
 
     FIG. 10A  shows the modulated intensity signal in arbitrary units detected by the multi-element detector  126 . If the intensity is modulated by 2π radians and the photodetector array contains N detectors, as shown in  FIG. 10A , the Fourier coefficients can be obtained from the following relations: 
   
     
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       1 
                       
                         I 
                         sum 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           q 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           I 
                           
                             exp 
                             , 
                             q 
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           δ 
                           q 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   6A 
                 
               
             
           
           
             
               
                 
                   β 
                   = 
                   
                     
                       1 
                       
                         I 
                         sum 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           q 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           I 
                           
                             exp 
                             , 
                             q 
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           δ 
                           q 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   6B 
                 
               
             
           
           
             
               
                 
                   I 
                   sum 
                 
                 = 
                 
                   
                     ∑ 
                     
                       q 
                       = 
                       1 
                     
                     N 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       I 
                       
                         exp 
                         , 
                         q 
                       
                     
                     . 
                   
                 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   6C 
                 
               
             
           
         
       
     
   
   In an alternative approach, using a multi-element detector with a limited number of elements, the output of each element is proportional to the area of the intensity curve, as shown in  FIG. 10B  for the case of a three-element detector. This technique has the potential to improve the data collection throughput. In  FIG. 10B , each element covers one third of the total modulation. Each detector will collect an intensity that is proportional to the partial integrals of I(x). The integrals of the intensity S j (j=1,2,3, . . . ) are referred to in the literature as Hadamard sums. Therefore, for the case of three detectors and a complete modulation period, the following can be written: 
                     S   m     =       ∫     2   ⁢       π   ⁡     (     m   -   I     )       /   3         2   ⁢   π   ⁢           ⁢     m   /   3         ⁢         I   0     ⁡     [     I   +     αcos   ⁡     (     δ   ⁡     (   x   )       )       +     αcos   ⁡     (     δ   ⁡     (   x   )       )         ]       ⁢           ⁢     ⅆ     δ   ⁡     (   x   )               ,           eq   .           ⁢   7               
where:m=1,2,3.
 
Thus:
 
   
     
       
         
           
             
               
                 
                   
                     S 
                     I 
                   
                   = 
                   
                     
                       I 
                       0 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             2 
                             3 
                           
                           ⁢ 
                           π 
                         
                         + 
                         
                           
                             
                               3 
                             
                             2 
                           
                           ⁢ 
                           α 
                         
                         + 
                         
                           
                             3 
                             2 
                           
                           ⁢ 
                           β 
                         
                       
                       ) 
                     
                   
                 
                 , 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   8A 
                 
               
             
           
         
       
     
   
                     S   2     =       I   0     ⁡     (         2   3     ⁢   π     -       3     ⁢   α       )         ,           eq   .           ⁢     8B                   S   3     =         I   0     ⁡     (         2   3     ⁢   π     +         3     2     ⁢   α     -       3   2     ⁢   β       )       .             eq   .           ⁢     8C                 
Inverting these equations, the normalized Fourier coefficients will be given by:
 
   
     
       
         
           
             
               
                 
                   α 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         π 
                       
                       
                         3 
                         ⁢ 
                         
                           3 
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             - 
                             
                               S 
                               1 
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               S 
                               2 
                             
                           
                           - 
                           
                             S 
                             3 
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             S 
                             1 
                           
                           + 
                           
                             S 
                             2 
                           
                           + 
                           
                             S 
                             3 
                           
                         
                         ) 
                       
                     
                   
                 
                 , 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   9A 
                 
               
             
           
           
             
               
                 β 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                     3 
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             S 
                             1 
                           
                           - 
                           
                             S 
                             3 
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             S 
                             1 
                           
                           + 
                           
                             S 
                             2 
                           
                           + 
                           
                             S 
                             3 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
             
             
               
                 eq 
                 . 
                 
                     
                 
                 ⁢ 
                 
                   9B 
                 
               
             
           
         
       
     
   
   Summarizing, in order to obtain the ellipsometry angles ψ and Δ associated with a thin film stack on a sample, the intensity as a function of detector position is first measured. The quantities α and β are calculated either from equations  6 A– 6 C, or equations  9 A– 9 B. Next, the angles χ and Q are calculated from equations  5 A– 5 B after inversion. Finally, the ellipsometry angles ψ and Δ are obtained from equations  3 A– 3 B. 
   In addition, it should be understood that PSD shown in  FIG. 4  may be used with metrology instruments other than the ellipsometer shown in  FIG. 3 . For example, if desired, the PSD with or without a beam expander may be used in an interferometer  300 , shown in  FIG. 11 . Interferometer  300  includes an electromagnetic source  302  followed by a half-wave plate  303  and a polarizer  304 . A beam splitter  305  directs the electromagnetic beam towards the sample  310 . A Wollaston prism  306  splits the light beam into two light beams, which are focused on the sample by lens  308 . The two beams are reflected off sample  310  and travel back through lens  308  and prism  306 , where the two beams are recombined into a single superimposed beam before passing through beam splitter  305 . The beam is then expanded by beam expander  312  and passes through a spatial variable retarder  314 . If the beam does not need expanding, as discussed above, beam expander  312  need not be used. The beam passes through a polarizer  316  and an interferometric filter  318  (if desired) prior to being received by multi-element detector  320 . Thus, the multi-element detector  320  receives a single superimposed electromagnetic beam. The single beam received by detector  320  is appropriately shaped to fill the detector  320  by beam expander  312  (if beam expansion is necessary) or by other optical elements, e.g., lens  308 , prism  306 , beam splitter  305 , or the light source  302  itself, (if beam expansion is not used). In addition, if desired, spatial variable retarder  314  may be a single plate of birefringent material with non-parallel faces, with the optical axis at a small angle (almost parallel) to the beam propagation direction, as discussed in  FIG. 7C . 
   In accordance with other embodiments of the present invention,  FIG. 12  shows ellipsometer  400 , which is similar to ellipsometer  100  shown in  FIG. 3 , except ellipsometer  400  uses a pulsed light source  402 . Pulsed light source  402  may be, for example, a Xenon flash bulb, such as that manufactured by PerkinElmer, located in Santa Clara, Calif. Such flash bulb is a Xenon arc lamp, with a typical wavelength range from 160 nm to 4 μm, stability of better than 3%, and intensity flux up to 120 μJ/cm 2 . This lamp can be pulsed at frequencies as high as 530 Hz, depending upon the choice of electronics. The decay time of the intensity of one pulse can be as long as 150 μs, again, depending upon the choice of electronics. 
   For a monochromatic system, a pulsed laser may be used as pulsed light source  402 , such as that manufactured by Melles Griot, located Carlsbad, Calif. Laser source parameters vary depending on the desired wavelength and intensity. Furthermore, multiple pulsed discrete laser lines may be used in a pseudo-spectroscopic arrangement. Alternatively, a single pulsed laser, e.g., a NdYAG laser, that includes the fundamentals and overtones may be used to create a pseudo-spectroscopic arrangement. 
   The use of a pulsed light source offers many advantages over conventional continuous light sources. A pulsed light source enables energization of the light source to be confined to the time over which a measurement is to be made. This reduces the time over which the sample is illuminated, thereby reducing possible negative effects on the sample that may occur when it is illuminated with a focused, highly radiant light beam. Also, power consumption is reduced and the life of the light source is significantly extended. Furthermore, unlike a conventional detector readout used in conjunction with a continuous light source, no mechanical shutter is needed to block the electromagnetic beam if a background measurement is needed because the pulsed light source is dark between pulses. 
   However, conventional metrology devices cannot utilize a pulsed light source such as a flash bulb or a pulsed laser, as conventional metrology devices are time dependent, causing synchronization problems, and thus, accuracy problems when used with a pulsed light source. The use of a pulsed light source is possible according to the present invention because ellipsometer  400  of  FIG. 12  has no moving parts, e.g., no time dependent phase modulators, as the spatially dependent polarizing element consisting of variable retarder  118  and linear polarizer  122  in  FIG. 12  produces a spatially dependent polarization state. Another advantage of ellipsometer  400  is that it can simultaneously collect a matrix of phase shift range and spectral wavelength range. It is hence insensitive to intensity fluctuations in the illuminating beam and in pulse-to-pulse intensity variations. It should also be understood that interferometer  300  of  FIG. 111  may similarly use a pulsed light source in place of light source  302 . 
   As shown in  FIG. 12 , ellipsometer  400  may also include synchronizer  427  coupled to pulsed light source  402  and multi-element detector  126 . Synchronizer  427  causes multi-element detector  126  to measure the intensity of the electromagnetic beam as a function of position after pulsed light source  402  emits a short pulse of light, e.g., for 150 μs. Consequently, multi-element detector  126  is synchronized with pulsed light source  402  such that multi-element detector  126  is reading out the accumulated charge only when pulsed light source  402  is off, i.e., when no light is produced. 
   It should also be understood that interferometer  300  of  FIG. 11  may similarly include a synchronizer coupled to a pulsed light source and the multi-element detector, causing the multi-element detector to detect the intensity of the electromagnetic beam when the pulsed light source is off. 
   Synchronizer  427  may operate in several ways. For example, synchronizer  427  may receive a signal from multi-element detector  126  indicating when multi-element detector  126  is beginning to measure the amount of light detected, i.e., read out the charge. Based on the signal from multi-element detector  126 , synchronizer  427  can turn pulsed light source  402  on or off so that multi-element detector  126  is reading only when pulsed light source  402  is off and is not reading when pulsed light source  402  is on, i.e., producing light. In another example, synchronizer  427  may receive a signal from pulsed light source  402  indicating when pulsed light source  402  is on or off. Synchronizer  427  can then cause multi-element detector  126  to measure the accumulated charge during the appropriate period, i.e., when pulsed light source  402  is off. In yet another example, an independent frequency source may turn multi-element detector  126  and pulsed light source  402  on and off in a complimentary fashion so that multi-element detector  126  reads out the accumulated charge when pulsed light source  402  is off. There are also many other ways in which synchronizer  427  can work that are well within the understanding of a person of ordinary skill in the art. 
   In one embodiment, synchronizer  427  permits a user to specify the number of pulses of light during data collection, thereby averaging the signal over several light pulses. Thus, data may be read after a plurality of pulses, e.g., after every ten pulses of light or after every fifty pulses of light, when pulsed light source  402  is on. 
     FIGS. 13A ,  13 B,  13 C, and  13 D show the schematics of the time synchronization of a multi-element detector  126  reading and a pulsed light source  402 . In  FIG. 13A , the first signal indicates the Start pulse  510 ( a ), which occurs at time t 0 . Another Start pulse  510 ( b ) occurs at time t 3 . Start pulses  510 ( a ) and  510 ( b ) also define the Stop pulse for the previous scans. The time between two consecutive Start pulses  510 ( a ) and  510 ( b ) is the total cycle time. 
   After Start pulse  510 ( a ), the multi-element detector  126 , which may be a CCD, begins reading out the accumulated charge from each element from a previous scan as indicated by the number of pulses  530  in  FIG. 13B . The total time for reading all the elements depends on the internal clock of the electronics, the number of elements and the mode of operation. Typically, there a large number of elements, and thus a correspondingly large number of pulses are required, indicated by the pulses  530  with broken lines. At time t 1  the multi-element detector  126  is finished reading out the accumulated charge. 
   As shown in  FIG. 13C , after reading out all the data, the multi-element detector  126  produces an End of Scan (EOS) pulse  540  at time t 1 , that can be used for synchronization of external equipment, such as the pulsed light source  402 . After EOS pulse  540  is produced, the pulsed light source  402  produces one or more light pulses depending on how much energy is needed, as shown in pulse train  560  of  FIG. 13D . As the intensity of the flash lamp has an exponential temporal decay, the integration time of the multi-element detector  126  must be a function of the number of flash pulses, so that it is guaranteed that a complete train of pulses fits into the cycle ending at time t 3 . 
   Thus, as can be seen in  FIGS. 13A ,  13 B,  13 C, and  13 D, multi-element detector  126  reads out data, i.e., accumulated charge, while the pulsed light source  402  is off, i.e., not producing light pulses, and while the pulsed light source  402  is on, i.e., producing light pulses, multi-element detector  126  is off, i.e., not reading out data. 
   In another embodiment, the pulsed light source  402  remains on, i.e., the light source  402  continuously produces pulses of light even while multi-element detector  126  is reading out the accumulated charge. However, difficulties arise from such an embodiment. For example, because the accumulated charge in a multi-element detector  126  is typically read out in a serial fashion, i.e., element by element (or in some cases, row by row), if continuously pulsed light is used, the last element in the multi-element detector  402  will be exposed to pulses of light after a previous, i.e., the first, element has been read out. By using multiple cycles, i.e., exposing and reading out, it may be possible to average the accumulated charge over multiple cycles thereby compensating for the difference in the accumulated charge from one element to the next in any one cycle. 
   Although the present invention is illustrated in connection with specific embodiments for instructional purposes, the present invention is not limited thereto. Various adaptations and modifications may be made without departing from the scope of the invention. For example, the ellipsometer in accordance with the present invention may operate in either reflection or transmission mode. Moreover, a single wavelength or multiple wavelengths may be used. Various expanders may be used to expand the reflected (or transmitted) beam to cover the variable retarder. Additionally, various variable retarders may be used in accordance with the present invention. Therefore, the spirit and scope of the appended claims should not be limited to the foregoing description.