Patent Publication Number: US-9404872-B1

Title: Selectably configurable multiple mode spectroscopic ellipsometry

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     The present application is related to and claims the benefit of the earliest available effective filing date(s) from the following listed application(s) (the “Related Applications”) (e.g., claims earliest available priority dates for other than provisional patent applications or claims benefits under 35 USC §119(e) for provisional patent applications, for any and all parent, grandparent, great-grandparent, etc. applications of the Related Application(s)). 
     RELATED APPLICATIONS 
     
         
         
           
             For purposes of the USPTO extra-statutory requirements, the present application constitutes a regular (non-provisional) patent application of U.S. Provisional Patent Application entitled APPARATUS AND METHODS OF LIGHT SOURCE TRACKING IN OPTICAL METROLOGY MEASUREMENT SYSTEM, naming Shankar Krishnan, Guorong Vera Zhuang, Klaus Flock, and Johannes D. de Veer, as inventors, filed Jun. 29, 2011, Application Ser. No. 61/502,507. 
             For purposes of the USPTO extra-statutory requirements, the present application constitutes a regular (non-provisional) patent application of U.S. Provisional Patent Application entitled SYSTEMS AND METHODS FOR CONDUCTING MULTIPLE SPECTROSCOPIC ELLIPSOMETRIC MEASUREMENTS IN SEMICONDUCTOR MANUFACTURING PROCESSES, naming Shankar Krishnan and Haiming Wang, as inventors, filed Aug. 26, 2011, Application Ser. No. 61/527,884. 
           
         
       
    
    
    
     TECHNICAL FIELD 
     The present invention generally relates to a method and system for performing spectroscopic ellipsometry, and, in particular, a method and system for performing multiple mode spectroscopic ellipsometry. 
     BACKGROUND 
     Broadband spectrum illumination sources are commonly implemented in spectroscopic based optical metrology tools. Such broadband based metrology tools may be utilized to measure various parameters associated with a given sample, such as a semiconductor wafer or lot of semiconductor wafers. Fabricating semiconductor devices such as logic and memory devices typically includes processing a substrate such as a semiconductor wafer using a large number of semiconductor fabrication processes to form various features and multiple levels of the semiconductor devices. 
     One type of broadband based metrology includes spectroscopic ellipsometry (SE). Spectroscopic ellipsometry (SE) is commonly implemented in order to measure thicknesses of film stacks on semiconductor wafers and critical dimensions (CD) of semiconductor structures in a variety of semiconductor manufacturing processes. One type of SE system includes a rotating polarizer SE (RPSE), which typically includes a broadband light source (e.g., xenon lamp), a rotating polarizer, a fixed analyzer (may be rotated at discrete angles), and a spectroscopic detector. RPSE is an extremely powerful tool, initially used for film thickness measurements, and is now widely applied to measure both film thicknesses and critical dimensions of samples. Another type of SE system includes the rotating compensator spectroscopic ellipsometer (RCSE), which typically includes a broadband light source, a fixed polarizer, a rotating waveplate (i.e., “compensator” in ellipsometry), a fixed analyzer, and a spectroscopic detector. 
     As critical dimensions (e.g., the gate width or gate oxide thickness) continue to decrease, measurement of these ever shrinking dimensions (e.g., a few nanometers or sub-nanometer) requires improved measurement performance. One of the fundamental performance requirements includes measurement precision, or the so called “3-sigma” of the measured CD parameters. An additional challenge includes tool-to-tool matching. Both requirements are related to the sensitivity of the implemented measurement system. As the system parameters decrease, the measurable signal change, corresponding to the fractional change in given parameters, are often not distinguishable from system noise for commonly implemented standard RPSE and RCSE systems. As a result, prior art implementations are insufficient both in terms of precision and tool-to-tool matching capability. 
     One method to improve measurement sensitivity includes the measurement of an increased number of spectral properties associated with geometric and material structures of a given sample. In an RPSE system, a motor is used to drive a polarizer with angular frequency ω. The signal measured by the detector of the system consists of three harmonic terms, known as the DC term, and terms related to cos 2 ωt and sin 2 ωt, respectively. In practice, the DC term is used to normalize the cos 2 ωt and sin 2 ωt harmonics, resulting in the commonly termed [α, β] spectra. As a result, one can measure two sets of combinations of Mueller elements, normalized by the first Mueller element (M 00 ). Similarly, in an RCSE system, one may obtain three normalized harmonics, and is then able to measure three combinations of Mueller elements normalized by the first Mueller element (M 00 ). 
     A common approach to measuring additional Mueller elements includes Mueller SE, also referred to as “complete SE,” which includes the implementation of two rotating polarizing elements. One such complete SE system includes a dual-rotating compensator SE (DRCSE) system, which includes a light source, a fixed polarizer, a first rotating waveplate in the illumination path, a second rotating waveplate in the collection path, a fixed analyzer, and a detector. This complete SE system configuration is capable of collecting up to 25 harmonic coefficients, corresponding to certain linear combinations of the rotation frequencies of these two rotating compensators. As a result, all 15 normalized Mueller elements can be resolved and measured. 
     Enhanced sensitivity of Mueller SE is advantageous for measuring small CD parameter changes. Nevertheless, this enhanced sensitivity will also cause the given measurement system to be more susceptible to system errors, due to the sensitivity of the additional Mueller elements to perturbations in the system. As a result, improved systems and methods for calibration of critical ellipsometry parameters are desirable. 
     SUMMARY 
     An apparatus suitable for multiple mode spectroscopic ellipsometry is disclosed. In one aspect, an apparatus may include, but is not limited to, an illumination source configured to illuminate a surface of a sample disposed on a sample stage; a detector configured to detect at least a portion of light reflected from the surface of the sample; a selectably configurable optical system including an illumination arm and a collection arm, the illumination source and the detector being optically coupled by the optical system, the optical system further comprising: a rotatable polarizing element disposed in the illumination arm of the optical system, the rotatable polarizing element being rotatable about an optical axis of the illumination arm; an analyzing element disposed in the collection arm of the optical system and arranged at a selected analyzer angle; and a rotatable-translatable compensator element disposed in the collection arm of the optical system, the rotatable-translatable compensator element being rotatable about an optical axis of the collection arm, the rotatable-translatable compensator element being linearly translatable along a direction perpendicular to the optical axis of the collection arm; and a control system communicatively coupled to the rotatable polarizing element and the rotatable-translatable compensator element, wherein the control system is configured to selectably configure the optical system in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode by controlling an operation state of at least one of the rotatable polarizing element and the rotatable-translatable compensator element. 
     In a further embodiment, the apparatus may include a multi-axis actuation control system, wherein the illumination source is disposed on a multi-axis actuation stage of the multi-axis actuation control system; an additional control system communicatively coupled to the multi-axis actuation control system and the detector, wherein the control system is configured to: acquire a set of diagnostic parameters from a diagnostic sample disposed on the sample stage, wherein the set of diagnostic parameters are indicative of position drift of the illumination source as measured relative to one or more components of the optical system; determine a magnitude of illumination source position drift by comparing the acquired set of diagnostic parameters to an initial set of parameters obtained from the diagnostic sample at a previously measured alignment condition; determine a direction of illumination source position drift; and provide a set of illumination source position adjustment parameters configured to correct the determined magnitude and direction of the illumination source position drift to the multi-axis actuation control system. 
     An apparatus suitable for calibration of one or more ellipsometric parameters is disclosed. In one aspect, an apparatus may include, but is not limited to, an illumination source configured to illuminate a surface of a calibration sample disposed on a sample stage; a detector configured to collect at least a portion of light reflected from the surface of the calibration sample; a selectably configurable optical system including an illumination arm and a collection arm, the illumination source and the detector being optically coupled by the optical system, the optical system further comprising: a rotatable polarizing element disposed in the illumination arm of the optical system, the rotatable polarizing element being rotatable about an optical axis of the illumination arm; an analyzing element disposed in the collection arm of the optical system and arranged at a selected analyzer angle; and a rotatable-translatable compensator element disposed in the collection arm of the optical system, the rotatable-translatable compensator element being rotatable about an optical axis of the collection arm, the rotatable-translatable compensator element being linearly translatable along a direction perpendicular to the optical axis of the collection arm; and a control system communicatively coupled to the rotatable polarizing element and the rotatable and translatable compensator element, wherein the control system is configured to: configure the optical system in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode; measuring one or more spectroscopic ellipsometry properties of one or more calibration samples in at least one of the RCSE mode, the RPSE mode, and the RPRC mode; and calibrating one or more parameters of the optical system utilizing the measured one or more spectroscopic ellipsometry properties. 
     A method for performing multiple mode spectroscopic ellipsometry is disclosed. In one aspect, a method may include, but is not limited to, selectably configuring an optical system of the spectroscopic ellipsometer in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode; illuminating a surface of a sample disposed on a sample stage via an illumination arm of the optical system; and detecting at least a portion of light reflected from the surface of the sample via a collection arm of the optical system. 
     In a further embodiment, the method includes measuring one or more spectroscopic ellipsometry properties of one or more calibration samples in at least one of the RCSE mode, the RPSE mode, and the RPRC mode; and calibrating the optical system utilizing the measured one or more spectroscopic ellipsometry properties. 
     In another aspect, a method may include, but is not limited to, selectably configuring an optical system of the spectroscopic ellipsometer in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode; illuminating a surface of a diagnostic sample disposed on a sample stage via an illumination arm of the optical system utilizing an illumination source disposed on a multi-axis actuation stage of a multi-axis actuation control system; detecting at least a portion of light reflected from the surface of the diagnostic sample via a collection arm of the optical system, wherein the illumination source and the detector are optically direct-coupled via the optical system; acquiring a set of diagnostic parameters indicative of illumination source position drift from the diagnostic sample; determining a magnitude of the illumination source position drift by comparing the acquired set of diagnostic parameters to an initial set of parameters obtained from the diagnostic sample at a previously measured alignment condition; determining a direction of the illumination source position drift; and providing a set of illumination source position adjustment parameters configured to correct the determined magnitude and direction of the illumination source position drift to the multi-axis actuation control system of the multi-axis sample stage. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not necessarily restrictive of the invention as claimed. The accompanying drawings, which are incorporated in and constitute a part of the specification, illustrate embodiments of the invention and together with the general description, serve to explain the principles of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The numerous advantages of the disclosure may be better understood by those skilled in the art by reference to the accompanying figures in which: 
         FIG. 1A  illustrates a schematic view of a selectably configurable multimode spectroscopic ellipsometry (MMSE) system, in accordance with one embodiment of the present invention. 
         FIG. 1B  illustrates a schematic view of a rotating polarizing spectroscopic ellipsometry (RPSE) mode of the selectably configurable MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 1C  illustrates a schematic view of a rotating compensator spectroscopic ellipsometry (RCSE) mode of the selectably configurable MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 1D  illustrates a schematic view of a rotating polarizer-rotating compensator spectroscopic ellipsometry (RPRC SE) mode of the selectably configurable MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 2A  illustrates a schematic view of a selectably configurable multimode MMSE system equipped with an illumination tracking and adjustment sub-system, in accordance with one embodiment of the present invention. 
         FIG. 2B  illustrates a schematic view of a thin film based diagnostic sample suitable for implementation in the sub-system for tracking and adjusting illumination source position drift in the MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 2C  illustrates a schematic view of a patterned grating based diagnostic sample suitable for implementation in the sub-system for tracking and adjusting illumination source position drift in the MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 3A  illustrates a schematic view of a selectably configurable multimode MMSE system equipped with a direct imaging sub-system for tracking and adjusting illumination source position drift in the MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 3B  illustrates a schematic view of a direct imaging sub-system equipped with two cameras and a translatable multi-axis sample stage suitable implementation in the MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 4A  illustrates a schematic view of a selectably configurable MMSE system equipped with a beam condition module, in accordance with one embodiment of the present invention. 
         FIG. 4B  illustrates a schematic view of a beam condition module suitable for implementation in the selectably configurable MMSE system, in accordance with one embodiment of the present invention. 
         FIG. 5  illustrates a schematic view of a selectably configurable MMSE system equipped with an adjustable polarization slit, in accordance with one embodiment of the present invention. 
         FIG. 6  is a flow diagram illustrating a method for performing multiple mode spectroscopic ellipsometry, in accordance with one embodiment of the present invention. 
         FIG. 7  is a flow diagram illustrating a method for performing multiple mode spectroscopic ellipsometry, in accordance with an alternative embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Reference will now be made in detail to the subject matter disclosed, which is illustrated in the accompanying drawings. 
     Referring generally to  FIGS. 1A through 5 , systems for performing multimode spectroscopic ellipsometry are described in accordance with the present disclosure. The present disclosure is directed toward systems and methods for performing multimode spectroscopic ellipsometry on a single platform, allowing for the selectable configuration of the system in one of multiple spectroscopic ellipsometry modes. In this regard, the system may be configured in one of a rotating polarizer SE (RPSE) mode, a rotating compensator SE (RCSE), or a rotating polarizer-rotating compensator SE (RPRC SE) mode. The present disclosure is also directed to a systematic, self-consistent, and unified calibration methodology, allowing for calibration of the multimode system across all modes. The unique ability of the system to be configured in multiple modes provides enhanced calibration ability, which in turn provides enhanced measurement sensitivity and accuracy. The improved sensitivity and calibrated systematic error provided by the present invention in turn provides for improved tool-to-tool matching. The present disclosure is further directed to the implementation of multimode spectroscopic ellipsometry (MMSE) in a direct coupled optical setting with illumination source tracking and feedback, which may further improve the stability and sensitivity of the presently disclosed selectably configurable MMSE system. 
       FIG. 1A  illustrates a schematic view of a configurable multiple mode spectroscopic ellipsometer  100 , in accordance with one embodiment of the present invention. The system  100  may include an illumination source  102  configured to illuminate a surface of a sample  112  disposed on a sample stage  114 , a detector  104  configured to detect light reflected from the surface of the sample  112 , and a selectably configurable optical system  106 , which acts to optically couple the illumination source  102  and the detector  104 . The system  100  may further include a computer control system  107  suitable for controlling the configuration of the selectably configurable optical system  106 . 
     In one aspect of the present invention, the selectably configurable optical system  106  of the system  100  may include an illumination arm  108  and a collection arm  110 . The illumination source  102  and the detector  104  may be optically coupled via the illumination arm  108  and the collection arm  110  of the optical system  106 . In this manner, light may emanate from the illumination source  102  and travel along the illumination arm  108  to the surface of the sample  112 . Light reflected from the sample  112  may then travel from the surface of the sample  112  to the detector  104  along the collection arm  110 . 
     In another aspect of the present invention, the illumination arm  108  of the optical system  106  may include a rotatable polarizing element  116 . In one aspect, the rotatable polarizing element  116  is configured for selective rotation about the optical axis of the illumination pathway  117  of the illumination arm  108  of the optical system  106 . In a general sense, the rotatable polarizing element  116  may include any polarizer and means for rotating the polarizer known in the art. For example, the rotatable polarizing element  116  may include a polarizer mechanically coupled to a rotatable motor. It is recognized herein that the polarizer of the rotating polarizing element  116  may include any polarizer known in the art. For example, the polarizer may include a rotating polarizer, such as but not limited to, a Rochon prism. In another example, the polarizer may include a beam displacer. The rotatable polarizer element  116  is configured to operate within the system  100  in either a rotationally active or rotationally inactive state. For instance, a rotatable motor of the rotatable polarizing element  116  may be inactive such that the rotating element  116  remains rotationally fixed about the axis  117  of the illumination arm  108 . In another instance, the rotatable motor may act to rotate the polarizer of a rotatable polarizing element  116  at a selected angular frequency, ω p , about the optical axis  117  of the illumination arm  108 . 
     In another aspect of the present invention, the collection arm  110  of the optical system  106  may include a rotatable-translatable compensator element  118 . In one aspect, the rotatable-translatable compensator element  118  is configured for selective rotation about the optical axis of the optical pathway  119  of the collection arm  110  of the optical system  106 . In addition, the rotatable-translatable compensator element  118  is configured for the selective translation in and out of the optical pathway  119  of the collection arm  110 . In a general sense, the rotatable-translatable compensator element  118  may include any compensator (e.g., waveplate) and means for rotating and/or translating the compensator known in the art. For example, the rotatable-translatable compensator element  118  may include a compensator mechanically coupled to a rotatable motor, wherein the rotatable motor is further coupled to a linear translation stage suited for moving the compensator-rotational motor assembly in and out of the collection pathway  119 . It is recognized herein that the compensator of the rotatable-translatable compensator element  118  may include any compensator known in the art suitable for ellipsometry analysis. The rotatable-translatable compensator element  118  is configured to operate within the system  100  in either a rotationally active or rotationally inactive state. For instance, the translation stage coupled to the compensator may be controlled to either place the compensator element  118  within the collection pathway  119  or remove the element  118  from the pathway  119 . Further, the rotatable-translatable compensator element  118  is configured to operate within the system  100  in either a rotationally active or rotationally inactive state. For instance, a rotatable motor of the rotatable-translatable compensator element  118  may be inactive such that the element  118  remains rotationally fixed about the axis  119  of the collection arm  110 . In another instance, the rotatable motor may act to rotate the compensator of the rotatable-translatable compensator element  118  at a selected angular frequency, ω c , about the optical axis  119  of the collection arm  110 . 
     In another aspect of the invention, the collection arm  110  of the optical system  106  may include an analyzer  120 . In one embodiment, the analyzer  120  may be arranged at a fixed analyzer angle. For example, the analyzer  120  may be arranged at a fixed analyzer angle of −25°. 
     In a general sense, the system  100  may provide for multiple configurations of the various polarizing elements of the system (i.e., polarizing element  116 , compensator element  118 , or analyzer  120 ). In this regard, the system  100  may provide for any combination of a rotational condition of the polarizing element  116 , compensator element  118 , or analyzer  120 . For example, the polarizer, compensator and analyzer of the system  100  may each rotate continuously or discretely. Further, the polarizer, compensator, and analyzer may rotate simultaneously or in any combination with one another. For instance, the polarizer and compensator may rotate simultaneously in a continuous or discrete fashion. In another, instance the polarizer and analyzer may rotate simultaneously in a continuous or discrete fashion. Even further, the analyzer and compensator may rotate simultaneously in a continuous or discrete fashion. The system  100  of the present invention provides for complete control of all rotational conditions of the various polarizing elements of the optical system  106  in a single platform environment. 
     In another aspect of the invention, the control system  107  may configure the multimode system  100  into one of multiple modes. For example, the control system  107  may control the rotatable polarizing element  116  and/or the rotatable-translatable compensator element  118  in order to configure the system  100  into one of the RPSE mode, the rotating compensator spectroscopic ellipsometry RCSE mode, or the RPRC SE mode. In this regard, the control system  107  may consist of a computing system configured to control the rotatable polarizing element  116  and/or the rotatable-translatable compensator element  118  in order to achieve one of the above selected modes (discussed in greater detail further herein). 
     It should be recognized that the various control steps associated with configuring of the multimode system  100  described throughout the present disclosure may be carried out by a single computer system  107  or, alternatively, a multiple computer system  107 . Moreover, different subsystems of the system  100  may include a computer system suitable for carrying out at least a portion of the steps described above. Further, the one or more computer systems  107  may be configured to perform any other step(s) of any of the method embodiments described herein. 
     In another embodiment, the computer control system  107  may be communicatively coupled to the rotatable polarizing element  116 , the rotatable-translatable compensator element  118 , the illumination source  102  or detector  104  in any manner known in the art. For example, the computer control system  107  may be communicatively coupled to the various sub-systems of system  100  via a wireline or wireless connection. 
     The computer control system  107  may include, but is not limited to, a personal computer system, mainframe computer system, workstation, image computer, parallel processor, or any other device known in the art. In general, the term “computer system,” “computing system(s),” or “computer control system” may be broadly defined to encompass any device(s) having one or more processors, which execute instructions from a memory medium. 
     Program instructions  128  implementing methods such as those described herein may be transmitted over or stored on carrier medium  126 . The carrier medium  126  may be a transmission medium such as a wire, cable, or wireless transmission link. The carrier medium  126  may also include a storage medium such as a read-only memory, a random access memory, a magnetic or optical disk, or a magnetic tape. 
     In one embodiment, the computer control system  107  may receive user inputted instructions via a user interface (not shown). In response to user inputted instructions, the control system  107  may act to configure the multimode system  100  into one of the RPSE mode, the RCSE mode, or the RPCE SE mode. For example, the control system  107  may display a set of mode selections to a user via a display (not shown). The user may then select one or more of the selections via a user interface device (not shown). In response to this selection, the control system  107  may transmit control signals to the rotatable polarizing element  116  and/or the rotatable-translatable compensator element  118  in order to direct the elements  116  or  118  to adjust in order to correspond with the selected configuration. 
     In another embodiment, the system  100  need not require a centralized control system. In this sense, the rotatable polarizing element  116  and the rotatable-translatable compensator element  118  of the system  100  may be controlled by independent controllers each configured to receive user input independently. 
     In additional embodiments, the illumination arm  108  and/or the collection arm  110  may include, but are not limited to, one or more additional optical elements,  122  and  124 . Those skilled in the art should recognize that numerous optical elements  122 ,  124  may be utilized within the illumination arm  108  or collection arm  110  within the scope of the present invention. For example, the optical elements  122  of the illumination arm  108  may include, but are not limited to, one or more lenses (e.g., focusing lenses), one or more mirrors, one or more filters, or one or more collimators. Similarly, the optical elements  124  of the collection arm  110  may include, but are not limited to, one or more lenses, one or more mirrors, one or more filters, or one or more collimators. 
     In another aspect of the present invention, the Illumination source  102  may include any broadband illumination source known in the art. In one embodiment, the illumination source  102  may include, but is not limited to, a halogen light source (HLS). For instance, the halogen light source may include, but is not limited to, a tungsten based halogen lamp. In another example, the illumination source  102  may include a xenon arc lamp. By yet another example, the illumination source  102  may include a deuterium arc lamp. In a general sense, any illumination source capable of producing illumination in the visible, infrared, and ultraviolet spectral ranges is suitable for implementation in the present invention. For example, a xenon arc lamp is capable of delivering light in a spectral range of 190 nm to 2000 nm, with a gradual radiant intensity decrease below 400 nm. In another embodiment, the illumination source  102  may include, but is not limited to, any discharge plasma source known in the art. In yet another embodiment, the illumination source  102  may include, but is not limited to, a laser-driven plasma source. It should be recognized by those skilled in the art that the above described illumination sources do not represent limitations, but should merely be interpreted as illustrative. 
     It is noted that the above description relating to the various types of illumination sources should not be interpreted as limiting, but rather merely as illustrative. Those skilled in the art should recognize that any broadband illumination source is suitable for implementation in the present invention. Moreover, it is further contemplated herein that two or more broadband illumination source may be combined in order to tailor to a required broadband spectral range. In this manner, a first source emitting illumination in a first spectral range may be combined with a second source emitting illumination in a second spectral range. For example, a first light source may include a xenon lamp, while a second light source may include a deuterium lamp. 
     In another aspect of the present invention, the detector  104  may include any light detection system known in the art suitable for implementation in a broadband spectrometer, ellipsometer, reflectometer, or scatterometer setting. In a general sense, any detector capable of measuring spectra across the visible, infrared, and ultraviolet spectral ranges is suitable for implementation in the present invention. 
     In one embodiment of the present invention, the illumination source  102  of multiple mode SE system  100  may be coupled to the collection optics of the system via one or more optical fibers (not shown). In a fiber-coupled optical setting, the optical fiber of the system  100  may be utilized to present light from a broadband source  102  to the collection optics of the system  100 . The utilization of optical fiber to couple a broadband source to collection optics aids in improving pointing stability and intensity fluctuations created due to beam pointing errors. Moreover, optical fiber may act as a light scrambler, causing the intensity spatial distribution of the transmitted light to become more uniform as it propagates through the optical fiber. Further, the spatial dimension of the illumination source (as it is presented to the collection optics) is defined by the physical diameter of the optical fiber. As such, optical fiber plays a key role in both beam shaping and beam stabilization. 
     In another embodiment of the present invention, the illumination source  102  of the multiple mode SE system  100  may be direct-coupled to the detector  104  of the system  100 . While it is contemplated herein that the system  100  is applicable in settings wherein the illumination source  102  and detector  104  are coupled via an optical fiber, due to the degradation of UV throughput in optical fibers as a result of solarization and UV enhanced photocontamination, in certain instances, it is desirable to operate the multimode SE system  100  in a direct-coupled configuration. 
     As used throughout the present disclosure, the term “direct-coupling” generally refers to a configuration wherein light from an illumination source is delivered to a measurement system (e.g., detector  104 ) through free space (i.e., no intervening transmission medium, such as an optical fiber). The direct free space coupling may take place in vacuum, an ambient atmosphere, or a purged environment (e.g., inert gas purge). As will be discussed in greater detail further herein, the implementation of a direct-coupled illumination-detector system may lead to instabilities in the position of the illumination source  102  relative to the measurement optics of the system  100 . It is generally desirable to maintain the relative position of the illumination source  102  with respect to the measurement optics, allowing for long term measurement system stability. 
     In another embodiment of the present invention, the selectably configurable optical system  106  may include a reflective-based optical system. In an alternative embodiment, the selectably configurable optical system  106  may include a refractive-based optical system. 
       FIG. 1B  illustrates the rotating polarizing spectroscopic mode (RPSE) of the multimode system  100 . In the RPSE mode, the compensator is “flipped out” or removed from the collection path  117  of the collection arm  110 . In this manner, the linear translation stage of the rotatable-translatable element  118  acts to translate the compensator along a direction generally perpendicular to the collection path  119 . While in the RPSE mode, the system  100  may rotate the polarizer of the rotatable polarizer element  116  using a rotation motor at an angular frequency of ω p . Further, the analyzer  120  is set at a fixed analyzer angle (e.g., 25°). Rotating polarizer based spectroscopic ellipsometry is generally described in U.S. Pat. No. 5,581,350 and U.S. Pat. No. 5,608,526, which are incorporated herein in their entirety. 
     In the RPSE mode, the detector signal may be expressed in terms of three harmonics of the frequency of the rotating polarizer, ω p . In this regard, the detector signal is given by:
 
 S   D   =S   F   └f   0   +f   1  cos 2ω p   t+f   2  sin 2ω p   t┘   (Eq. 1)
 
     where the scale factor, S F , is proportional to the intensity of the incident light beam, I i , and the fixed analyzer angle, respectively. The scale factor, intensity, and harmonic coefficients f 0 , f 1 , and f 2  are given by: 
                     S   F     =       I   i     4             (     Eq   .           ⁢   2     )                 f   0   =M   00   +M   10  cos 2 A−M   20  sin 2 A      f   1   =M   01   +M   11  cos 2 A−M   21  sin 2 A      f 2=−( M   02   +M   12  cos 2 A−M   22  sin 2 A )  (Eq. 3)
 
     In this regard, the detector signal S D  consists of three harmonics of the polarizer angle P=ω p t. These second harmonic coefficients are then normalized to produce the standard [α,β] factors in the RPSE mode, given by: 
     
       
         
           
             
               
                 
                   
                     
                       
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                         = 
                         
                           
                             
                               f 
                               2 
                             
                             
                               f 
                               0 
                             
                           
                           = 
                           
                             - 
                             
                               
                                 
                                   M 
                                   02 
                                 
                                 + 
                                 
                                   
                                     M 
                                     12 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                                 - 
                                 
                                   
                                     M 
                                     22 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 
                                   M 
                                   00 
                                 
                                 + 
                                 
                                   
                                     M 
                                     10 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                                 - 
                                 
                                   
                                     M 
                                     20 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     Utilizing these factors and their relationship to the associated Mueller matrix components, the RPSE mode of the system  100  may be fully calibrated. 
       FIG. 1C  illustrates the rotating compensator spectroscopic mode (RCSE) of the multimode system  100 . In the RCSE mode, the compensator is “flipped in” the collection path  119  of the collection arm  110 . In this manner, the linear translation stage of the rotatable-translatable element  118  acts to translate the compensator along a direction generally perpendicular to the collection path  119  until the compensator is positioned adequately in the optical pathway  119  of the collection arm  110 . In the RCSE mode, the system  100  rotates the compensator of the rotatable-translatable polarizer element  118  using a rotation motor at an angular frequency of ω c . Further, the polarizer of the rotatable polarizing element  116  is fixed at a selected polarizer angle, while the analyzer  120  is set at a fixed analyzer angle. Rotating compensator based spectroscopic ellipsometry is generally described in U.S. Pat. No. 5,798,837, which is incorporated herein in its entirety. 
     In the RCSE mode, the detector signal may be expressed in terms of five harmonics (with only four of them being independent) of the rotating compensator angle, C=ω e t. In this regard, the detector signal is given by:
 
 S   D   =S   F   └f   0   +f   1  sin 2( A−ω   c   t )+ f   3  cos 4ω p   t+f   4  sin 4ω p   t┘   (Eq. 5)
 
     where the four independent harmonic coefficients are given by: 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       0 
                     
                     = 
                     
                       
                         
                           
                             1 
                             - 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Γ 
                             
                           
                           2 
                         
                         ⁢ 
                         
                           f 
                           a0 
                         
                       
                       + 
                       
                         
                           
                             1 
                             + 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Γ 
                             
                           
                           2 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               f 
                               
                                 p 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                             + 
                             
                               
                                 f 
                                 
                                   p 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               P 
                             
                             + 
                             
                               
                                 f 
                                 
                                   p 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               P 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       1 
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               M 
                               30 
                             
                             + 
                             
                               
                                 M 
                                 31 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               P 
                             
                             - 
                             
                               
                                 M 
                                 32 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               P 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Γ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       3 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Γ 
                           
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               f 
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                           + 
                           
                             
                               f 
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       4 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Γ 
                           
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               f 
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                           + 
                           
                             
                               f 
                               
                                 a 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     where the retardation of the compensator (i.e., waveplate), Γ, is related to the polarizing transmission coefficient (t p , t s ), e jΓ =τ=t p /t s . The coefficients of Eq. 6 are given by:
 
 f   p0   =M   00   +M   10  cos 2 A−M   20  sin 2 A  
 
 f   p1   =M   01   +M   11  cos 2 A−M   21  sin 2 A  
 
 f   p2 =−( M   02   +M   12  cos 2 A−M   22  sin 2 A )
 
 f   a0   =M   00   +M   01  cos 2 P−M   02  sin 2 P  
 
 f   a1   =M   10   +M   11  cos 2 P−M   12  sin 2 P  
 
 f   a2 =—( M   20   +M   21  cos 2 P−M   22  sin 2 P )  (Eq. 7)
 
     Utilizing these coefficients and their relationship to the associated Mueller matrix components, the RCSE mode of the system  100  may be fully calibrated. 
       FIG. 1D  illustrates the rotating polarizer-rotating compensator spectroscopic ellipsometry (RPRC SE) mode of the system  100 . In the RPRC SE mode, the compensator is “flipped in” the collection pathway  119  of the collection arm  110 , with the polarizer positioned along the illumination pathway  117  of the illumination arm  108 . In addition, both the compensator and polarizer are rotating at constant angular frequency. The compensator is rotating at constant angular frequency, ω c , while the polarizer is rotating at constant angular frequency, ω p . 
     It is recognized herein that in addition to measuring certain off-diagonal Mueller elements, the RPRC SE Mueller configuration is also capable of improved system calibration using additional system parameters. One group of the critical system parameters includes the polarization properties of the compensator (i.e., waveplate). Traditionally, the waveplate compensator is considered “ideal,” whereby only the polarizing phase (the retardation) is used to characterize the waveplate. In practice, however, proper characterization of the compensator waveplate requires quantification of the so called “diattenuation” or “dichroism.” In this regard, retardation of the compensator (i.e., waveplate), Γ, is related to the polarizing transmission coefficient by: e jΓ =τ=t p /t s . Equivalently, the compensator waveplate may be described using the following three parameters: 
     
       
         
           
             
               
                 
                   
                     s 
                     = 
                     
                       
                         
                           1 
                           - 
                           
                             
                                
                               τ 
                                
                             
                             2 
                           
                         
                         
                           1 
                           + 
                           
                             
                                
                               τ 
                                
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                         &lt;&lt; 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     p 
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                            
                           τ 
                            
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Γ 
                       
                       
                         1 
                         + 
                         
                           
                              
                             τ 
                              
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     r 
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                            
                           τ 
                            
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Γ 
                       
                       
                         1 
                         + 
                         
                           
                              
                             τ 
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     Utilizing the parameters of equation 8, the detector signal in the RPRC SE mode may be expressed in terms of a linear combination of harmonics of both the polarizer and compensator angular frequencies. The detector signal is provided by: 
     
       
         
           
             
               
                 
                   
                     S 
                     D 
                   
                   = 
                   
                     
                       S 
                       F 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 f 
                                 0 
                               
                               + 
                               
                                 
                                   f 
                                   1 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   ω 
                                   p 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 
                                   f 
                                   2 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   ω 
                                   p 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 
                                   f 
                                   3 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   c 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 
                                   f 
                                   4 
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   ω 
                                   c 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   f 
                                   5 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       p 
                                     
                                     - 
                                     
                                       ω 
                                       c 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 
                                   f 
                                   6 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           ω 
                                           p 
                                         
                                       
                                       - 
                                       
                                         2 
                                         ⁢ 
                                         
                                           ω 
                                           c 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 t 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         f 
                                         7 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             p 
                                           
                                           + 
                                           
                                             ω 
                                             c 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                     
                                       
                                         f 
                                         8 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             p 
                                           
                                           + 
                                           
                                             ω 
                                             c 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     t 
                                     + 
                                     
                                       
                                         f 
                                         9 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                       ⁢ 
                                       
                                         ω 
                                         c 
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       f 
                                       10 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     4 
                                     ⁢ 
                                     
                                       ω 
                                       c 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         f 
                                         11 
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ω 
                                                 a 
                                               
                                             
                                             - 
                                             
                                               4 
                                               ⁢ 
                                               
                                                 ω 
                                                 c 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                     
                                       
                                         f 
                                         12 
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ω 
                                                 a 
                                               
                                             
                                             - 
                                             
                                               4 
                                               ⁢ 
                                               
                                                 ω 
                                                 c 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           f 
                                           13 
                                         
                                       
                                       ⁢ 
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ω 
                                                 a 
                                               
                                             
                                             + 
                                             
                                               4 
                                               ⁢ 
                                               
                                                 ω 
                                                 c 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       t 
                                     
                                     + 
                                     
                                       
                                         f 
                                         14 
                                       
                                       ⁢ 
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 ω 
                                                 a 
                                               
                                             
                                             - 
                                             
                                               4 
                                               ⁢ 
                                               
                                                 ω 
                                                 c 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       t 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     where the harmonic coefficients of equation 9 may be related to the system and sample parameters by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         0 
                       
                       = 
                       
                         
                           
                             
                               1 
                               - 
                               p 
                             
                             2 
                           
                           ⁢ 
                           
                             M 
                             00 
                           
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             2 
                           
                           ⁢ 
                           
                             f 
                             
                               p 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         1 
                       
                       = 
                       
                         
                           
                             
                               1 
                               - 
                               p 
                             
                             2 
                           
                           ⁢ 
                           
                             M 
                             01 
                           
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             2 
                           
                           ⁢ 
                           
                             f 
                             
                               p 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         2 
                       
                       = 
                       
                         
                           
                             - 
                             
                               
                                 1 
                                 - 
                                 p 
                               
                               2 
                             
                           
                           ⁢ 
                           
                             M 
                             02 
                           
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             2 
                           
                           ⁢ 
                           
                             f 
                             
                               p 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       f 
                       3 
                     
                     = 
                     
                       
                         
                           - 
                           r 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           M 
                           30 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                       + 
                       
                         s 
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               10 
                             
                             + 
                             
                               
                                 M 
                                 00 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               A 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       4 
                     
                     = 
                     
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           M 
                           30 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                       - 
                       
                         s 
                         ⁡ 
                         
                           ( 
                           
                             
                               M 
                               20 
                             
                             - 
                             
                               
                                 M 
                                 00 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       5 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   r 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         M 
                                         31 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                     + 
                                     
                                       
                                         M 
                                         32 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     M 
                                     11 
                                   
                                   + 
                                   
                                     M 
                                     22 
                                   
                                   + 
                                   
                                     
                                       M 
                                       01 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     A 
                                   
                                   - 
                                   
                                     
                                       M 
                                       02 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       6 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 r 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         M 
                                         32 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                     - 
                                     
                                       
                                         M 
                                         31 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     M 
                                     12 
                                   
                                   - 
                                   
                                     M 
                                     21 
                                   
                                   + 
                                   
                                     
                                       M 
                                       01 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                   + 
                                   
                                     
                                       M 
                                       02 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       7 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   - 
                                   r 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         M 
                                         31 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       A 
                                     
                                     - 
                                     
                                       
                                         M 
                                         32 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     M 
                                     11 
                                   
                                   - 
                                   
                                     M 
                                     22 
                                   
                                   + 
                                   
                                     
                                       M 
                                       01 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                   + 
                                   
                                     
                                       M 
                                       02 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       8 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 r 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         M 
                                         32 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                     + 
                                     
                                       
                                         M 
                                         31 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       A 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               s 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     M 
                                     12 
                                   
                                   + 
                                   
                                     M 
                                     21 
                                   
                                   - 
                                   
                                     
                                       M 
                                       01 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                   + 
                                   
                                     
                                       M 
                                       02 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     A 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       9 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               M 
                               10 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                           + 
                           
                             
                               M 
                               20 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       10 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               M 
                               10 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                           - 
                           
                             
                               M 
                               20 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       11 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         4 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   M 
                                   11 
                                 
                                 + 
                                 
                                   M 
                                   22 
                                 
                               
                               ) 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   M 
                                   12 
                                 
                                 - 
                                 
                                   M 
                                   21 
                                 
                               
                               ) 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       12 
                     
                     = 
                     
                       - 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     12 
                                   
                                   - 
                                   
                                     M 
                                     21 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     11 
                                   
                                   + 
                                   
                                     M 
                                     22 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ] 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       13 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         4 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   M 
                                   11 
                                 
                                 - 
                                 
                                   M 
                                   22 
                                 
                               
                               ) 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   M 
                                   12 
                                 
                                 + 
                                 
                                   M 
                                   21 
                                 
                               
                               ) 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             A 
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       14 
                     
                     = 
                     
                       - 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     12 
                                   
                                   + 
                                   
                                     M 
                                     21 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     M 
                                     11 
                                   
                                   - 
                                   
                                     M 
                                     22 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     In addition to providing multiple selectable modes of spectroscopic ellipsometry, the system  100  of the present disclosure further provides enhanced calibration capabilities. In the case of the RPSE mode, the calibration of the system  100  is relatively straightforward due to the relative simplicity of RPSE. The most critical parameters, such as wavelength as a function of detector pixel, angle of incidence (AOI) of illumination on the sample  112 , the polarizer home position, the analyzer angle, and the azimuth angle of the plane of incidence relative to a sample of grating lines, may be systematically calibrated. 
     In the case of the RCSE mode, calibration becomes more difficult. Difficulty in calibration in the RCSE mode arises from the fact that the retardation of the compensator waveplate  118  is function of illumination wavelength. The difficulty is further compounded by the fact that in the RCSE mode the compensator waveplate serves as a critical component of the optical system  106 , with the compensator rotating at a constant frequency, ω c . As such, errors in calibration may significantly impact measurement performance of the system  100 . 
     In one aspect of the present invention, the multimode system  100  may be used to calibrate compensator retardation at each wavelength of illumination. In this regard, a calibration sample (wafer), consisting of a single-layer film stack, may be implemented for the purposes of calibrating the system  100 . In one embodiment, the calibration wafer may include a single-layer film stack of silicon dioxide disposed on a silicon substrate. Since the calibration is isotropic, the Mueller matrix associated with the calibration sample may be simplified as: 
     
       
         
           
             
               
                 
                   
                     M 
                     sample 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             M 
                             00 
                           
                         
                         
                           
                             M 
                             01 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             M 
                             10 
                           
                         
                         
                           
                             M 
                             11 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             M 
                             22 
                           
                         
                         
                           
                             M 
                             23 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             M 
                             32 
                           
                         
                         
                           
                             M 
                             33 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     where the components of the simple matrix are given by: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       11 
                     
                     = 
                     
                       
                         M 
                         00 
                       
                       = 
                       
                         
                           
                             R 
                             p 
                           
                           + 
                           
                             R 
                             s 
                           
                         
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       M 
                       10 
                     
                     = 
                     
                       
                         M 
                         01 
                       
                       = 
                       
                         
                           
                             R 
                             p 
                           
                           - 
                           
                             R 
                             s 
                           
                         
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       M 
                       22 
                     
                     = 
                     
                       
                         M 
                         33 
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             
                               R 
                               s 
                             
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       M 
                       23 
                     
                     = 
                     
                       
                         - 
                         
                           M 
                           32 
                         
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             
                               R 
                               s 
                             
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
     As such, the harmonics associated with the RPRC Mueller SE mode of the system  100  may then be written as: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         f 
                         0 
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             + 
                             
                               R 
                               s 
                             
                           
                           2 
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             4 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               - 
                               
                                 R 
                                 s 
                               
                             
                             ) 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           A 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         1 
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             - 
                             
                               R 
                               s 
                             
                           
                           2 
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             4 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               + 
                               
                                 R 
                                 s 
                               
                             
                             ) 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           A 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         2 
                       
                       = 
                       
                         
                           
                             1 
                             + 
                             p 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             
                               R 
                               s 
                             
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δsin2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         3 
                       
                       = 
                       
                         s 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 - 
                                 
                                   R 
                                   s 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   + 
                                   
                                     R 
                                     s 
                                   
                                 
                                 2 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         4 
                       
                       = 
                       
                         s 
                         ⁢ 
                         
                           
                             
                               R 
                               p 
                             
                             + 
                             
                               R 
                               s 
                             
                           
                           2 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         5 
                       
                       = 
                       
                         
                           
                             
                               r 
                               ⁢ 
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   ⁢ 
                                   
                                     R 
                                     s 
                                   
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δ 
                             
                             2 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           A 
                         
                         + 
                         
                           s 
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               4 
                             
                             + 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   - 
                                   
                                     R 
                                     s 
                                   
                                 
                                 4 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         6 
                       
                       = 
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ⁢ 
                                   
                                     
                                       
                                         R 
                                         p 
                                       
                                       ⁢ 
                                       
                                         R 
                                         s 
                                       
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                 
                                 2 
                               
                               + 
                               
                                 s 
                                 ⁢ 
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     - 
                                     
                                       R 
                                       s 
                                     
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         7 
                       
                       = 
                       
                         
                           
                             - 
                             
                               
                                 r 
                                 ⁢ 
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           A 
                         
                         + 
                         
                           s 
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               4 
                             
                             - 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   - 
                                   
                                     R 
                                     s 
                                   
                                 
                                 4 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               A 
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         8 
                       
                       = 
                       
                         
                           - 
                           
                             ( 
                             
                               
                                 
                                   r 
                                   ⁢ 
                                   
                                     
                                       
                                         R 
                                         p 
                                       
                                       ⁢ 
                                       
                                         R 
                                         s 
                                       
                                     
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                 
                                 2 
                               
                               - 
                               
                                 s 
                                 ⁢ 
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     - 
                                     
                                       R 
                                       s 
                                     
                                   
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         9 
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               R 
                               p 
                             
                             - 
                             
                               R 
                               s 
                             
                           
                           ) 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         10 
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               R 
                               p 
                             
                             - 
                             
                               R 
                               s 
                             
                           
                           ) 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         11 
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         12 
                       
                       = 
                       
                         
                           - 
                           
                             
                               1 
                               - 
                               p 
                             
                             4 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         13 
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         f 
                         14 
                       
                       = 
                       
                         
                           
                             1 
                             - 
                             p 
                           
                           4 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   R 
                                   p 
                                 
                                 + 
                                 
                                   R 
                                   s 
                                 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                               2 
                             
                           
                           ) 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                         ⁢ 
                         A 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
     It is noted herein that utilizing an analyzer  120  angle of 0° or 45° results in the removal of some dependent harmonics. As such, equation 13 may reduce to: 
     
       
         
           
             
               
                 
                   
                     
                       
                         f 
                         0 
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             + 
                             
                               R 
                               s 
                             
                           
                           2 
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             4 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               - 
                               
                                 R 
                                 s 
                               
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         1 
                       
                       = 
                       
                         
                           
                             
                               R 
                               p 
                             
                             - 
                             
                               R 
                               s 
                             
                           
                           2 
                         
                         + 
                         
                           
                             
                               1 
                               + 
                               p 
                             
                             4 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               + 
                               
                                 R 
                                 s 
                               
                             
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         2 
                       
                       = 
                       
                         
                           ( 
                           
                             1 
                             + 
                             p 
                           
                           ) 
                         
                         ⁢ 
                         
                           δ 
                           a 
                         
                         ⁢ 
                         
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             
                               R 
                               s 
                             
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         f 
                         3 
                       
                       = 
                       
                         
                           R 
                           p 
                         
                         ⁢ 
                         s 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       f 
                       4 
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               R 
                               p 
                             
                             + 
                             
                               R 
                               s 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             s 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               δ 
                               a 
                             
                           
                           ) 
                         
                       
                       → 
                       0 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       5 
                     
                     = 
                     
                       
                         
                           r 
                           ⁢ 
                           
                             
                               
                                 R 
                                 p 
                               
                               ⁢ 
                               
                                 R 
                                 s 
                               
                             
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                         
                         + 
                         
                           s 
                           ⁡ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               + 
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       6 
                     
                     = 
                     
                       
                         - 
                         r 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         δ 
                         a 
                       
                       ⁢ 
                       
                         
                           
                             R 
                             p 
                           
                           ⁢ 
                           
                             R 
                             s 
                           
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       7 
                     
                     = 
                     
                       
                         
                           
                             - 
                             r 
                           
                           ⁢ 
                           
                             
                               
                                 R 
                                 p 
                               
                               ⁢ 
                               
                                 R 
                                 s 
                               
                             
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                         
                         + 
                         
                           s 
                           ⁡ 
                           
                             ( 
                             
                               
                                 R 
                                 p 
                               
                               - 
                               
                                 
                                   
                                     
                                       R 
                                       p 
                                     
                                     ⁢ 
                                     
                                       R 
                                       s 
                                     
                                   
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                               
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       8 
                     
                     = 
                     
                       f 
                       6 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       9 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         4 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             R 
                             p 
                           
                           - 
                           
                             R 
                             s 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       10 
                     
                     = 
                     
                       
                         f 
                         9 
                       
                       ⁢ 
                       
                         δ 
                         a 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       11 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         4 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 R 
                                 p 
                               
                               + 
                               
                                 R 
                                 s 
                               
                             
                             2 
                           
                           + 
                           
                             
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   ⁢ 
                                   
                                     R 
                                     s 
                                   
                                 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δ 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       12 
                     
                     = 
                     
                       
                         - 
                         
                           f 
                           11 
                         
                       
                       ⁢ 
                       
                         δ 
                         a 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       13 
                     
                     = 
                     
                       
                         
                           1 
                           - 
                           p 
                         
                         4 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 R 
                                 p 
                               
                               + 
                               
                                 R 
                                 s 
                               
                             
                             2 
                           
                           - 
                           
                             
                               
                                 
                                   
                                     R 
                                     p 
                                   
                                   ⁢ 
                                   
                                     R 
                                     s 
                                   
                                 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δ 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       f 
                       14 
                     
                     = 
                     
                       
                         f 
                         13 
                       
                       ⁢ 
                       
                         δ 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     In turn, a user or the control system  107  may calibrate one or more of the parameters of the system  100 . For example, parameters suitable for calibration may include, but are not limited to, deviation of the analyzer angle, δ a  (measured from zero), the compensator dichroism-related parameter s, and the parameter related to the real part of the compensator retardation, p. Each of these parameters may be expressed in terms of the coefficients from Eq. 14 as follows: 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       a 
                     
                     = 
                     
                       
                         1 
                         6 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               f 
                               10 
                             
                             
                               f 
                               9 
                             
                           
                           - 
                           
                             
                               f 
                               12 
                             
                             
                               f 
                               11 
                             
                           
                           + 
                           
                             
                               f 
                               14 
                             
                             
                               f 
                               13 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     p 
                     = 
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             0 
                           
                         
                         - 
                         
                           f 
                           1 
                         
                         - 
                         
                           3 
                           ⁢ 
                           
                             ( 
                             
                               
                                 f 
                                 11 
                               
                               + 
                               
                                 f 
                                 13 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           f 
                           1 
                         
                         + 
                         
                           f 
                           11 
                         
                         + 
                         
                           f 
                           13 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     s 
                     = 
                     
                       
                         
                           3 
                           + 
                           p 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           f 
                           3 
                         
                         
                           
                             f 
                             0 
                           
                           + 
                           
                             f 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     It is noted herein that the parameter, r, related to the imaginary part of the compensator retardation cannot be directly solved using equation 14 as it is always combined with the imaginary part of the sample ellipsometric parameter, sin Δ. However, the r-parameter may be solved indirectly using the s- and p-parameters of Eq. 15. In addition, the r-parameter may be determined independently utilizing the RCSE mode of the multimode system  100 . 
     In one aspect, the r-parameter may be determined utilizing information from a calibration sample, described previously herein. For example, single layer film stacks, such as a single layer of silicon dioxide film, having thicknesses of approximately 200 Å, have been widely used for calibration purposes. Based on measurements from such a calibration sample, it is possible to eliminate the 180° uncertainly in the polarizing phase term. In this regard, it is known at each wavelength whether the phase term Δ of ellipsometric parameter (tan Δ=|r p /r s |, where Δ=Arg[r p /r s ]) of the sample is in the [0, 180] region, or the [180, 360] region. Based on this information, it is possible to eliminate the 180° uncertainty in the compensator retardation. As a result, one may use these equations to over-determine sample ellipsometric parameters: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           p 
                         
                         - 
                         
                           R 
                           s 
                         
                       
                       
                         
                           R 
                           p 
                         
                         + 
                         
                           R 
                           s 
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             3 
                             + 
                             p 
                           
                           ) 
                         
                         ⁢ 
                         
                           f 
                           9 
                         
                       
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             0 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               1 
                               + 
                               p 
                             
                             ) 
                           
                           ⁢ 
                           
                             f 
                             1 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         s 
                       
                       
                         
                           R 
                           p 
                         
                         + 
                         
                           R 
                           s 
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             3 
                             + 
                             p 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               f 
                               11 
                             
                             + 
                             
                               f 
                               13 
                             
                             - 
                             
                               f 
                               9 
                             
                           
                           ) 
                         
                       
                       
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             f 
                             0 
                           
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               p 
                             
                             ) 
                           
                           ⁢ 
                           
                             f 
                             1 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         R 
                         p 
                       
                       
                         
                           R 
                           p 
                         
                         + 
                         
                           R 
                           s 
                         
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             p 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               f 
                               0 
                             
                             + 
                             
                               f 
                               1 
                             
                           
                           ) 
                         
                       
                       
                         
                           4 
                           ⁢ 
                           
                             f 
                             0 
                           
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               p 
                             
                             ) 
                           
                           ⁢ 
                           
                             f 
                             1 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               R 
                               p 
                             
                             ⁢ 
                             
                               R 
                               s 
                             
                           
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Δ 
                       
                       
                         
                           R 
                           p 
                         
                         + 
                         
                           R 
                           s 
                         
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             f 
                             11 
                           
                           - 
                           
                             f 
                             13 
                           
                         
                         
                           
                             f 
                             11 
                           
                           + 
                           
                             f 
                             13 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ) 
                 
               
             
           
         
       
     
     where equation (16) acts to over-determine the ellipsometric parameters, R P , R s , cos Δ, and sin Δ. As a result, utilizing the known ellipsometric parameters the compensator spectral retardation r is provided by: 
     
       
         
           
             
                 
             
             ⁢ 
             
               ( 
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 17 
               
               ) 
             
           
         
       
       
         
           
             
               r 
               ⁢ 
               
                   
               
               ⁢ 
               tan 
               ⁢ 
               
                   
               
               ⁢ 
               Δ 
             
             = 
             
               
                 
                   
                     ( 
                     
                       1 
                       - 
                       p 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         f 
                         5 
                       
                       - 
                       
                         f 
                         7 
                       
                     
                     ) 
                   
                 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         f 
                         11 
                       
                       - 
                       
                         f 
                         13 
                       
                     
                     ) 
                   
                 
               
               - 
               s 
             
           
         
       
       
         
           
             
               r 
               ⁢ 
               
                 
                   
                     
                       
                         R 
                         p 
                       
                       ⁢ 
                       
                         R 
                         s 
                       
                     
                   
                   ⁢ 
                   sin 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Δ 
                 
                 
                   
                     R 
                     p 
                   
                   + 
                   
                     R 
                     s 
                   
                 
               
             
             = 
             
               
                 
                   
                     
                       ( 
                       
                         3 
                         + 
                         p 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         p 
                       
                       ) 
                     
                   
                   4 
                 
                 ⁢ 
                 
                   
                     
                       f 
                       5 
                     
                     - 
                     
                       f 
                       7 
                     
                   
                   
                     
                       2 
                       ⁢ 
                       
                         f 
                         0 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           1 
                           + 
                           p 
                         
                         ) 
                       
                       ⁢ 
                       
                         f 
                         1 
                       
                     
                   
                 
               
               - 
               
                 
                   s 
                   2 
                 
                 ⁢ 
                 
                   
                     
                       f 
                       11 
                     
                     - 
                     
                       f 
                       13 
                     
                   
                   
                     
                       f 
                       11 
                     
                     + 
                     
                       f 
                       13 
                     
                   
                 
               
             
           
         
       
     
     An overview of the relationship between various ellipsometric parameters as well as the principles of ellipsometry is provided generally in Harland G. Tompkins and Eugene A. Irene,  Handbook of Ellipsometry,  1st ed, William Andrew, Inc., 2005, which is incorporated herein by reference in the entirety. In addition, Mueller matrix ellipsometry in the context of imperfect compensators is discussed in detail in P. S. Hauge,  Mueller Matrix Ellipsometry with Imperfect Compensators ”, J. of the Optical Soc. of Am. A 68(11), 1519-1528, 1978, which is incorporated herein in its entirety. Mueller matrix ellipsometry in the context of rotating polarizer and analyzer is discussed in detail in R. M. A Azzam,  A Simple Fourier Photopolarimeter with Rotating Polarizer and Analyzer for Measuring Jones and Mueller Matrices , Opt Comm 25(2), 137-140, 1978, which is incorporated herein in its entirety. Further, the concept of “complete” ellipsometry is discussed in M. L. Aleksandrov, et. al. “ Methods and Apparatus for Complete Ellipsometry  ( review )”, J. Appl. Spectroscopy 44(6), 559-578, 1986, which is incorporated herein in its entirety. 
     In another aspect of the present invention, upon calibration of the various system  100  parameters, the multimode system  100  may then be utilized to carry out spectroscopic ellipsometry measurements on one or more test samples. The determination of the ellipsometric parameters, such as p, s, δ a , and r provides for improved precision, accuracy, stability, tool-to-tool matching. In addition, the multimode system  100  provides for improved baseline matching. The ability to configure system  100  in multiple ellipsometric modes allows for unique measurement capabilities, which are further amplified due to the improved calibration abilities of the system  100 . 
       FIG. 2A  illustrates the multimode spectroscopic ellipsometer equipped with beam tracking capabilities in a direct-coupled illumination setting, in accordance with an alternative embodiment of the present invention. 
     It is recognized herein that long term stability of the multimode SE system  100  may be measured using measurement results acquired over a period of extended time, such as weeks or months. In this manner, the stability of the measurement system may be acquired and monitored by analyzing measurement results obtained under identical measurement conditions (e.g., identical recipes) as a function of time. Those skilled in the art will further recognize that changes in the measurement results as a function of time may be influenced by numerous factors. The factors may include, but are not limited to, illumination source position relative to the optical system, slow decay of illumination source intensity, and changes in sample conditions, such as changes in optical properties of the measured sample (e.g., optical changes due to accumulated UV exposure). 
     In one embodiment of the present invention, the system  100  may include a means for active tracking of the illumination source  102 . Application of an active illumination source tracking function may enable the system  100  to adjust the spatial positioning of the illumination source  102  relative to the various optical components of the measurement system  100 . It is contemplated herein that the active illumination source tracking function may be executed via a closed-loop feedback operation. 
     As shown in  FIG. 2A , the multimode SE system  100  may further include a multi-axis actuation stage  202  of a multi-axis actuation control system  204  configured to actuate the illumination source  102  in response to determined illumination source  102  drift. The system  100  may further include a computer control system  206  communicatively coupled to the actuation control system  204  and the detector  104 . The computer control system  206  may be configured to receive one or more signals from the detector  104  indicative of intensity measurement data obtained from a diagnostic sample loaded on the sample stage  114 . The computer control system  206  may further be configured to transmit one or more instruction signals to the multi-axis actuation control system  204  based on a source position diagnostic algorithm  212 , as will be discussed in greater detail further herein. In the embodiment of  FIG. 2A , the optical system  106  is configured to direct-couple the illumination source  102  and the detector  104 . 
     In one embodiment of the present invention, the multi-axis actuation stage  202  may include any actuation stage known in the art. For example, the multi-axis actuation stage  202  may include a translational stage or rotational stage. For instance, the multi-axis actuation stage  202  may include, but is not limited to, a motorized translation stage. In another instance, the multi-axis actuation stage  202  may include, but is not limited to, a motorized rotational stage. It is further contemplated herein that the multi-axis actuation stage  202  may consist of a stage having both translational and rotational capabilities. 
     In a further embodiment, the multi-axis actuation stage  202  may include any stage known in the art capable of responding to a control system from one or more computer systems  206 . In this sense, the multi-axis actuation stage  202  may be configured to receive command signals either directly or indirectly from the one or more computer systems  206 . For example, the multi-axis actuation stage  202  may be communicatively coupled to the computer control system  206 , allowing the computer control system  206  to directly control the multi-axis actuation stage  202 . In another example, the multi-axis actuation control system  204  may control the multi-axis actuation stage  202  using information transmitted from the one or more computer systems  206  to the multi-axis actuation control system  204 . 
     In another aspect of the present invention, the computer control system  206  may be configured to execute a source position diagnostic algorithm  212  utilizing information received from a diagnostic sample by the detector  104 . In this manner, the detector  104  may acquire a set of intensity measurement data associated with light reflected from a diagnostic sample loaded onto the sample stage  114 . 
     Those skilled in the art will recognized that the measured intensity of the light reflected from the sample  112  (e.g., diagnostic sample) may vary with respect to a variety of parameters. For example, in the case of the RPSE mode of the multimode system  100 , the measured intensity of illumination is expressed by:
 
 I=I   0 (α·cos 2 P +β·sin 2 P )  (Eq. 18)
 
 P=ω   P   ·t   (Eq. 19)
 
     where I represents the measured intensity by the detector  104 , I 0  represents the intensity of the light emanating from the illumination source  102 , P represents the polarizer angle, and ω p  again represents the polarizer angular frequency. 
     As shown in  FIGS. 2B and 2C , the diagnostic sample utilized to acquire the diagnostic parameters may include a sample engineered to have a selected set of properties. In one embodiment, the diagnostic sample may include, but is not limited to, a thin film sample  214 , having known properties such as, but not limited to, thickness and index of refraction(s). For example, the thin film sample  214  may include a simple film  216  deposited on a substrate  218 . For instance, the thin film  216  may consist of a first layer of material and a second layer, each with known properties. 
     In a further embodiment, it is contemplated that the diagnostic sample utilized to perform the illumination tracking function of the illumination source  102  of the multimode SE system  100  may be the same sample as the calibration sample utilized to perform the calibration function of one or more ellipsometric parameters of the multimode SE system  100 . In this regard, a simple film sample may be fabricated to have a selected set of properties (e.g., thickness, index of refraction, and etc.), while also being suitable for calibrating one or more of the ellipsometric parameters of the system  100 . 
     In another embodiment, the diagnostic sample may include, but is not limited to, a patterned grating sample  220  deposited on a substrate. For example, the patterned grating sample  220  may be constructed to possess a set of selected characteristics. For instance, the selected characteristics of the patterned grating sample  220  may include, but are not limited to, a selected critical dimension (CD)  222 , a selected side-wall angle (SWA)  224 , a selected feature height  226 , or a selected feature periodicity (d)  228 . 
     Upon receiving the measurement results associated with the chosen diagnostic sample, the computer control system  206  may acquire a set of diagnostic parameters utilizing the source position diagnostic algorithm  212 . The measured set of diagnostic parameters may then be compared to an initial set of diagnostic parameters obtained from the diagnostic sample at an initial set of alignment conditions (e.g., optimal or near-optimal alignment conditions) in order to determine the magnitude of the illumination source  102  position change. In this sense, the computer control system  206  may compare diagnostic parameters extracted from intensity data of the diagnostic sample to archived diagnostic sample imagery data taken at an earlier time. For instance, the archived imagery data of the diagnostic sample may be saved in the memory of the computer. Further, it is recognized herein that the archived imaged data of the diagnostic sample may represent imagery data of the diagnostic sample obtained at optimum alignment conditions. 
     It is recognized herein that the illumination source  102  position drift, which occurs over time, may manifest itself in the measured light intensity data as well as the measured diagnostic parameters. While the source intensity is a convolution of normal illumination source  102  intensity decay and illumination source  102  position drift relative to the measurement optics of system  100 , it is recognized herein that the diagnostic parameters are a gauge of the illumination source  102  position drift. As such, the diagnostic parameters (when compared to diagnostic parameters obtained at initial alignment conditions) may be utilized to monitor and compensate for the illumination source  102  position drift and change in illumination source  102  intensity spectral-spatial distribution. It is noted herein that source position drift may also induce spectral distribution changes on the sample. 
     In the case of the thin film sample  214 , the one or more diagnostic parameters that may be extracted may include, but are not limited to, total intensity integrated over a selected spectral range (e.g., intensity integrated over a broad spectrum range), the polarizer asymmetry parameter, or the percentage of higher harmonic components (commonly referred to as “4P”), or relative intensity ratio of DUV to NIR. Each of these diagnostics parameters may be computationally extracted from the spectra obtained from the thin film sample  214  utilizing the computer control system  206 . After these diagnostic parameters are extracted they may be compared to the diagnostic parameters extracted from the diagnostic sample at an alignment condition (e.g., optimal alignment condition) in order to determine the magnitude of the illumination source  102  drift relative to the initial alignment condition. 
     In the case of the patterned grating sample  220 , diagnostic parameters may be extracted from the acquired spectrum by comparing the experimental data to an accepted theoretical model. In this manner, the experimental data may be regressed against the theoretical model, allowing for the extraction of the diagnostic parameters by the computer control system  206 . It is recognized herein that, in the event the position of the illumination source shifts relative to an initial measurement instance, the regressed diagnostic parameters, such as SWA, CD, and periodicity of the grated sample  220 , may deviate from the diagnostic parameters measured during the initial measurement, allowing the system  100  to determine the magnitude of the illumination source  102  drift. It is noted herein that the pattern of sample  220  may include a three-dimensional pattern. Further, it is recognized that the pattern need not be symmetric in nature and may include an asymmetric pattern. 
     Upon determining the magnitude of the illumination source  102  drift, the one or more computer systems  206  may then determine the direction of the illumination source position drift. The computer control system  206  may transmit one or more instruction signals to the multi-axis actuation control system  204  in order to actuate (e.g., translate) the illumination source  102  (via the actuation stage  202 ) by the determined magnitude of illumination source position drift in a first direction within the optical plane and in a second direction perpendicular to the optical plane, as well as tip and tilt. In this sense, the detector  104  may measure intensity changes as a function of the directed actuation from the computer control system  206  and transmit those changes to the computer control system  206 . The computer control system  206  may then deduce the direction of the illumination source  102  position drift utilizing the measurement results from the detector  104 . 
     Upon determining both the magnitude and direction of illumination source  102  position drift, the computer control system  206  may transmit a set of illumination source position adjustment parameters to the multi-axis actuation control system  204  via input  203  of the multi-axis actuation control system  204 . In turn, the multi-axis actuation control system  204  may adjust the position of the multi-axis actuation stage  202  in accordance with the adjustment parameters provided by the computer control system  206 . In this regard, the adjustment parameters may be used by the multi-axis actuation control system  204  and the multi-axis stage  202  to compensate for illumination source drift and/or changes in illumination spectral-spatial distribution. 
     It is further contemplated herein that upon compensating for the illumination source position drift and/or changes in illumination spatial distribution, the system  100  may repeat the diagnostic sequence described above. In this manner, the computer control system  206  may repeat the source position diagnostic algorithm  212  in an effort to refine the compensation of the illumination source position drift until a selected level of compensation is attained. 
     It should be recognized that the various steps described throughout the present disclosure with respect to  FIGS. 2A-2C  may be carried out by a single computer control system  206  or, alternatively, a multiple computer control system  206 . Moreover, different subsystems of the system  100 , such as the multi-axis actuation control system  204 , may include a computer system suitable for carrying out at least a portion of the illumination source tracking/correction steps described above. Therefore, the above description should not be interpreted as a limitation on the present invention but merely an illustration. Further, the one or more computer systems  206  may be configured to perform any other step(s) of any of the method embodiments described herein. 
     It is further recognized that computer control system  206  and the computer control system  107  described previously herein may consist of the same computational unit. While described independently above, it is contemplated that the computer control system  206  and control system  107  may function as a single computational system. In another embodiment, the control system  206  and the control system  107  may be communicatively coupled to one another. In yet another embodiment, the control system  206  and control system  107  may be themselves controlled by an additional control system (not shown). In this regard, control system  206  and control system  107  may be embodied as computational modules existing in a higher-level computational and operational infrastructure (i.e., system including multiple computers and multiple functional capabilities). 
     In another embodiment, the computer control system  206  may be communicatively coupled to the multi-axis actuation control system  204  or detector  104  in any manner known in the art. For example, the computer control system  206  may be communicatively coupled to the multi-axis actuation control system  204  or the detector via a wireline or wireless connection. 
     For example, the one or more control systems  206  may be coupled to a computer system of the multi-axis actuation control system  204 . Moreover, the computer control system  206  of the system  100  may be configured to receive and/or acquire data or information from other systems by a transmission medium that may include wireline and/or wireless portions. In this manner, the transmission medium may serve as a data link between the computer control system  206  and other subsystems of the system  100 . Moreover, the computer control system  206  may send data to external systems via a transmission medium. 
     The computer control system  206  may include, but is not limited to, a personal computer system, mainframe computer system, workstation, image computer, parallel processor, or any other device known in the art. Program instructions  210  for implementing methods illumination drift tracking and adjustment may be transmitted over or stored on carrier medium  208 . The carrier medium  208  may be a transmission medium such as a wire, cable, or wireless transmission link. The carrier medium may also include a storage medium such as a read-only memory, a random access memory, a magnetic or optical disk, or a magnetic tape. 
       FIGS. 3A and 3B  illustrate a schematic view of the multimode SE system  100  equipped with image-based illumination source drift tracking and adjusting functionality, in accordance with an alternative embodiment of the present invention. In one aspect, the system  100  may include a direct imaging system  301  configured to measure the position of an illumination source  102  or the spatial distribution of the light intensity of illumination emanating from the illumination source  102 . 
     In a further aspect, the direct imaging system  301  is communicatively coupled to the computer control system  206  of the system  100 . In this manner, the direct imaging system  301  may be configured to transmit one or more sets of imagery data to the computer control system  206 . The sets of imagery data may include imagery data associated with the position of the illumination source  102  or the spectral-spatial distribution of the illumination intensity of the light emitted by the illumination source. In this manner, the direct imaging system  301  may acquire one or more sets of imagery data of the illumination source  102 . Upon acquiring the imagery data of the illumination source, the direct imaging system  301  may transmit a signal indicative of the imagery data to the computer control system  206 . 
     Upon receiving the imagery data from the direct imaging system  301 , the computer control system  206  may compare the acquired imagery data of the illumination source  102  to imagery data of the illumination source obtained in a preferred alignment condition. In this sense, the imagery data of the illumination source obtained while the illumination source is in a preferred alignment condition may be referred to as “reference imagery data.” For example, the computer control system  206  may compare the acquired imagery data of the illumination source  102  to imagery data (i.e., reference imagery data) of the illumination source  102  as it existed in an optimal or near-optimal alignment condition. The comparison data created by comparing the acquired imagery data and the reference imagery data may then be utilized to determine the magnitude and direction of the position drift of the illumination source  102  relative to the various optical element of the optical system  106 . In a further aspect, the reference imagery data may be stored in the memory of the computer control system  206 . In this regard, the acquired imagery data may be compared to the stored reference imagery data in order to measure the amount of position drift and/or the direction of position drift of the illumination source  102 . It is recognized herein that the various sets of imagery data described throughout the present disclosure may be stored in the memory of the computer control system  206  or a portable memory means (e.g., CD, DVD, flash drive, portable hard drive, communicatively couple server or the like) in any digital image format known in the art. 
     It is contemplated herein in that the computer control system  206  may execute a pre-programmed source position imaging algorithm  305  suitable for comparing the acquired images of the illumination source  102  and the reference images of the illumination source  102 . In turn, the source position imaging algorithm  305  may extract magnitude and direction information of the illumination source  102  drift. 
     Upon determining the magnitude and direction of position drift of the illumination source  102 , the computer control system  206  may determine a set of position adjustment parameters suitable for use by the multi-axis actuation stage  202  of the multi-axis actuation control system  204 . In this sense, the computer control system  206  may calculate the adjustment parameters utilizing the determined position drift of the illumination source  102 . For example, the computer control system  206  may calculate the adjustment parameters required to compensate for the magnitude and direction of the position drift of the illumination source  102 . It is recognized herein that the specific form of the adjustment parameters may depend on the specific type of multi-axis actuation stage  202  employed. For instance, in the event an X-Y translation stage is employed, the adjustment parameters may form a set of displacement values in the X- and Y-directions required to compensate for the position drift. In another instance, in a rotational stage setting, the adjustment parameters may, in part, include a rotational component and radial component required to compensate for the position drift of the illumination source. It is further contemplated herein that the multi-axis actuation stage  202  may include three-dimensional actuation capabilities. In this setting, the adjustment parameters may include displacement components in three dimensions (e.g., X-direction, Y-direction, and Z-direction). In yet another instance, the translational and rotational movement could be coupled to in order to provide increased adjustment flexibility to meet the stringent system measurement requirements. 
     Upon calculating the set of illumination source position adjustment parameters, the computer control system  206  may transmit the set of illumination source position adjustment parameters to the multi-axis actuation control system  204  via the input  203  of the multi-axis actuation control system  204 . In turn, the multi-axis actuation control system  204  system may adjust the position of the multi-axis actuation stage  202  in accordance with the adjustment parameters provided by the computer control system  206 . In this regard, the adjustment parameters may be used by the multi-axis actuation control system  204  and the multi-axis actuation stage  202  to compensate for illumination source drift and/or changes in illumination spectral-spatial distribution. 
     It is further contemplated herein that upon compensating for the illumination source position drift and/or changes in illumination spatial distribution, the system  100  may repeat the source position drift determination sequence described above. In this manner, the computer control system  206  may repeat the source position imaging algorithm  305  in an effort to refine the compensation of the illumination source position drift. 
     In one embodiment, the direct imaging system  301  may include a camera. It is recognized herein that a camera of the direct imaging system  301  may include any camera or camera system known in the art. For example, the camera of the direct imaging system  301  may include a charged coupled device (CCD) base camera. In a further embodiment, a DUV band pass filter may be positioned in front of the camera in order to aid in the determination of source position shift, as the system  100  tends to be most sensitive to the UV spectral region spatial distribution. A DUV band pass filter at this position may provide improvement adjustment sensitivity. 
     In another embodiment, the direct imaging system  301  may include a position sensor. It is recognized herein that the position sensor of the direct imaging system  301  may include any position sensor known in the art. For example, the position sensor of the direct imaging system  301  may include a one-dimensional position sensor. By way of another example, the direct imaging system  301  may include a two-dimensional position sensor. 
     In another embodiment, as depicted in  FIG. 3B , the direct imaging system  301  may include two or more cameras or two o more one-dimensional position sensors. For example, a first camera  302   a  may be arranged perpendicularly with respect to a second camera  302   b . By way of a further example, a first positional sensor may be arranged perpendicularly with respect to a second positional. Such an arrangement may allow for the acquisition of both angular and displacement information of the illumination source  102 . 
     It is further recognized herein that the direct imaging system  301  may include one or more optical elements (e.g., focusing lenses  306   a  and  306   b ) associated with the optical pathway between the illumination source  102  and each camera  302   a,b  of the direct imaging system  301 . 
     In a further embodiment, the direct imagining system  301  may include a camera or position sensor mounted on the optics frame  204  of the system  100  such that camera or position sensor is capable of capturing imagery and/or positional data of the illumination source  102  directly. 
     In a further embodiment, it is contemplated herein that the direct imaging system  301  may acquire imagery or positional data at a pre-determined time. For example, a user may input a selected data acquisition frequency into the computer  206 . At the pre-programmed times, the computer control system  206  may then acquire imagery or positional data using the direct imaging system  301 . By way of another example, the acquisition time may be determined by the computer control system  206  based on an analyzed drift pattern associated with the illumination source  102  and the system  100 . 
       FIGS. 4A and 4B  illustrate a schematic view of the multimode SE system  100  equipped with a beam conditioning module  402 , in accordance with an alternative embodiment of the present invention. The term “beam conditioning module” as used throughout the present disclosure may, in the alternative, may be referred to as a “beam conditioner” or an “illuminator.” 
     It is recognized herein that variations in spectral intensity as function of either illumination source position or illumination wavelength may have a significant adverse effects on system performance. This behavior is compounded in settings where intermediate scramblers or homogenizers (e.g., optical fiber-wiggler or optical fiber and the like) are not arranged between the illumination source  102  and an entrance slit of a downstream optical element. Further, it is noted that illumination source non-uniformity as a function of position or wavelength combined with illumination source movement (e.g., drift) relative to the entrance slit of the polarizer  116  may degrade the performance of the system  100 . For example, Poynting vector error may be introduced due to a circular trajectory, or “wobbling,” of an illumination source  102  relative to a rotating polarizer (e.g., rotating polarizer in RPSE mode of system  100  or rotating polarizer in RPRC SE mode of system  100 ) of the illumination arm  108 . This error may lead to illumination beam displacement at the illumination field stop (e.g., see  410  of  FIG. 4B ). In extreme settings, a plasma based broadband illumination source exhibiting a large intensity gradient may lead to the introduction of harmonics higher than a theoretical  2   w  into the measured signal. It is therefore desirable to introduce a direct-coupled optical system capable of limiting illumination source position drift, discussed in detail above, and illumination source variation. 
     In one aspect, the beam conditioning module  402  is positioned between the illumination source  102  and the entrance slit of the polarizer element  116  (e.g., rotating polarizer in RPSE or RPRC SE mode) of the illumination arm  108 . In a general sense, the beam conditioning module acts to “condition” the light of the illumination path by projecting an image of the illumination source onto a portion of one or more of the optical elements (e.g., slit of polarizing element) such that the projected image has a selected level of uniformity across the projection spot. 
     In one embodiment, the beam conditioning module  402  of system  100  may be configured to project an image of the illumination source onto an optical element of the illumination arm such that the projected image across a selected area of a given optical element has an intensity level above a selected level. For example, the beam conditioning module  402  may be configured to project an image of the illumination source  102  onto the entrance slit of the polarizer  116  (e.g., rotating polarizer) of the illumination arm  108  such that the intensity of the image is above a sufficient level across the entire area of the entrance slit. In a general sense, it is desirable to maintain the source brightness when forming the illumination source image on the entrance slit of the polarizer  116 . As such, the beam conditioning module  402  may be configured to project a source image onto the polarizer slit (or other optical element) such that projected image of the illumination source has an intensity level that is substantially similar to the illumination source intensity level. 
     In another embodiment, the beam conditioning module  402  of system  100  may be configured to project an image of the illumination source  102  onto an optical element of the illumination arm  108  such that the projected image across a selected area of a given optical element has an intensity gradient level below a selected level. In a general sense, it is desirable to project a uniform illumination source image onto the entrance slit of the polarizer  116  across a broad spectral range. For example, the beam conditioning module  402  may be configured to project an image of the illumination source  102  onto the entrance slit of the polarizer  116  (e.g., rotating polarizer) of the illumination arm  108  such that the intensity gradient of the image is below a selected level across the entire area of the entrance slit across a broad spectral range (e.g., significant portion of spectral range of illumination source). 
     In one embodiment, the beam conditioning module  402  may include one or more mirrors  406   a  . . .  406   d  configured to project an image of the illumination source  102  onto an optical element of the illumination arm  108  such that image has an intensity level above a selected level, while maintaining intensity uniformity (i.e., intensity gradient is below an adequate level). In one aspect, the set of mirrors  406   a  . . .  406   d  of the beam conditioning module  402  may include one or more spherical mirrors. The spherical mirrors of the beam conditioning module  402  may be configured to minimize low spatial frequency scattering. In another aspect, the set of mirrors  406   a  . . .  406   d  of the beam conditioning module  402  may include one or more non-spherical mirrors. It is recognized herein that one or more non-spherical mirrors may be implemented within the beam conditioning module  402  in order to limit design complexity of the beam condition module  402 . 
     In another embodiment, one or more internal pupil and/or field stops  408   a ,  408   b  may be included within in the optical pathway of the beam conditioning module  402  in order to reject scattering and stray light from reaching the illumination field stop  410 . 
     The mirrors  406   a  . . .  406   d  of the beam module  402  may be configured to provide magnification greater than unity. In this regard, the mirrors  406   a  . . .  406   d  may be utilized to magnify the image of the illumination source  102  such that only the “sweet spot” (i.e., portion of illumination distribution having highest intensity level) passes through the illumination field stop of the illumination image of the illumination source  102  pass through the illumination field stop  410 . 
     It is recognized herein that several factors may be considered when determining the degree of magnification of the beam condition module  402 . For example, the chosen degree of magnification may depend on the size of the illumination beam sought to be implemented, the illumination uniformity required for the given application (i.e., metrology system specifications), and the polarizer  116  slit size, among other factors. 
     Further, as shown in  FIG. 4B , the projection of the sweet spot onto the polarizer slit may be accomplished with manageable spectral angular distribution by projecting the illumination source  102  image at near normal incidence (e.g., via mirrors  406   a  . . .  406   d ) to downstream optical elements (e.g., polarizer  116 ) in a given oblique incidence metrology tool. 
     In another aspect, the angle of incident of illumination and the relative orientation of the various optical components (e.g., mirrors  406   a  . . .  406   d ) may be selected and configured to provide an illumination beam with residual polarization below a selected level. In this regard, the beam conditioning module  402  may be configured to minimize residual polarization via implementation of a low angle of incidence optical design and selected relative orientations of the various optical components such that birefringence induced by the optical coatings of the various optical components of the beam conditioning module  402  is below a selected level (e.g., birefringence is minimized). Those skilled in the art will recognize that coating induced birefringence is dependent upon both the angle of incidence of illumination relative to a coating surface and the thickness of the coating thickness. 
       FIG. 5  illustrates a schematic view of the multimode SE system  100  equipped with an adjustable polarization slit  502 , in accordance with an alternative embodiment of the present invention. In one aspect, the adjustable polarization slit  502  is positioned along the illumination arm  108  of the optical system  106  between the illumination source  102  and the polarizer  116 . The adjustable polarization slit  502  may be utilized by the system  100  in order to adjust the size of the polarization slit  502  to a size providing adequate illumination beam uniformity. In this sense, the adjustable polarization slit  502  may be utilized to isolate the sweet spot of the illumination beam from the less intense and less uniform portions of the illumination beam. It is recognized herein that the amount of adjustment required may depend on several factors, such as, but not limited to, required illumination beam brightness and required intensity uniformity of the system  100 . 
     The adjustable polarization slit  502  may include any adjustable polarization slit  502  known in the art. For example, the adjustable polarization slit  502  may include an adjustable polarization slit capable of adjustment in one dimension. For instance, the adjustable polarization slit  502  may include an adjustable polarization slit capable of adjusting the slit height or the slit width. By way of another example, the adjustable polarization slit  502  may include an adjustable polarization slit capable of adjustment in two dimensions. For instance, the adjustable polarization slit  502  may include an adjustable polarization slit capable of adjusting the slit height and the slit width or a square or rectangular slit. It is contemplated herein that a four-knife-edge-based adjustable slit is suitable for adjusting the slit size in two dimensions. In other instances, the slit may include any geometric shape known in the art. For example, the slit may include a circular slit, an oval slit, and the like. 
     In one embodiment, the adjustable polarization slit  502  may include an adjustable polarization slit  502  driven by one or motorized micrometers. It is recognized herein that any motorized micrometer known in the art is suitable for implementation in the present invention. In another embodiment, the adjustable polarization slit  502  may include an adjustable polarization slit  502  driven by one or more piezoelectric actuators. It is recognized herein that any piezoelectric actuator known in the art is suitable for implementation in the present invention. 
     In one aspect, the adjustable polarization slit  502  may be communicatively coupled to the computer control system  206 . In this sense, the adjustable polarization slit  502  may be configured to receive instruction signals from the computer control system  206 . In one embodiment, the computer control system  206  may transmit instructions to the adjustable polarization slit  502  based on results from a source position diagnostic algorithm  212 , as discussed in greater detail above. In this manner, the source position diagnostic algorithm  212  may be configured to incorporate the ability to adjust the polarization slit height or width when determining the position adjustment parameters for the multi-axis actuation stage  202 . In this sense, the computer control system  206  may supply adjustment parameters to either the multi-axis actuation stage  202  or the adjustable polarization slit  502  or both based on the source position diagnostic algorithm  212  executed by the computer control system  206 . 
     In another embodiment, the computer control system  206  may transmit instructions to the adjustable polarization slit  502  based on results from a source position imaging algorithm  305 , as discussed in greater detail above. In this manner, the source position imaging algorithm  305  may be configured to incorporate the ability to adjust the polarization slit height or width when determining the position adjustment parameters for the multi-axis actuation stage  202 . In this sense, the computer control system  206  may supply adjustment parameters to either the multi-axis actuation stage  202  or the adjustable polarization slit  502  or both based on the position imaging algorithm executed by the computer control system  206 . 
     In another embodiment, the computer control system  206  may transmit instruction to the adjustable polarization slit  502  based on user input. In this manner, a user may enter instructions into the computer control system  206  via one or more user interfaces (not shown). Then, the computer control system  206  may transmit corresponding instruction signals to the adjustable polarization slit  502 . In a further embodiment, user input may be combined with the results of the source position diagnostic algorithm  305  or the source position imaging algorithm  305  by the computer control system  206  in order to achieve improved illumination characteristics of the system  100 . 
     It is further contemplated herein that the adjustable polarization slit  502  may be implemented in conjunction with the beam condition module  402 , providing for even greater control of illumination source brightness and intensity uniformity. In this sense, although not shown, the beam conditioning module  402  may be positioned between the illumination source  102  and the entrance of the adjustable polarization slit  502 . 
       FIG. 6  illustrates a process flow  600  suitable for implementation by the system  100  of the present invention. In step  602 , an optical system  106  of the spectroscopic ellipsometer may be selectably configured in at least one of in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode. In step  604 , upon being configured in one of the above modes, a surface of a sample disposed on a sample stage may be illuminated by an illumination source  102  via an illumination arm  108  of the optical system  106 . In step  606 , at least a portion of light reflected from the surface of the sample  112  via a collection arm  110  of the optical system  106  may be detected utilizing a detector  104 . In a further step of process flow  600 , one or more spectroscopic ellipsometry properties of one or more calibration samples may be measured in at least one of the RCSE mode, the RPSE mode, and the RPRC mode in at least one of the RCSE mode, the RPSE mode, and the RPRC mode. In a further step of process flow  600 , the optical system  106  may be calibrated utilizing the measured one or more spectroscopic ellipsometry properties. 
       FIG. 7  illustrates an alternative process flow  700  suitable for implementation by the system  100  of the present invention. In step  702 , an optical system  106  of the spectroscopic ellipsometer may be selectably configured in at least one of in at least one of a rotating compensator (RCSE) mode, a rotating polarizer (RPSE) mode, and a rotating polarizer and compensator (RPRC) mode. In step  704 , a surface of a diagnostic sample disposed on a sample stage may be illuminated via an illumination arm of the optical system utilizing an illumination source  102  disposed on a on a multi-axis actuation stage  202  of a multi-axis actuation control system  204 . In step  706 , at least a portion of light reflected from the surface of the diagnostic sample may be detected via a collection arm of the optical system, wherein the illumination source and the detector are optically direct-coupled via the optical system. In step  708 , a set of diagnostic parameters indicative of illumination source position drift may be acquired from the diagnostic sample. In step  710 , a magnitude of the illumination source position drift may be determined by comparing the acquired set of diagnostic parameters to an initial set of parameters obtained from the diagnostic sample at a previously measured alignment condition. In step  712 , a direction of the illumination source position drift may be determined. In step  714 , a set of illumination source position adjustment parameters configured to correct the determined magnitude and direction of the illumination source position drift may be provided to the multi-axis actuation control system  204  of the multi-axis sample stage  202 . In a further step of process flow  700 , a position of the illumination source may be adjusted utilizing the provided set of illumination source position adjustment parameters utilizing the multi-axis sample stage  202  of the multi-axis actuation control system  204 . 
     All of the methods described herein may include storing results of one or more steps of the method embodiments in a storage medium. The results may include any of the results described herein and may be stored in any manner known in the art. The storage medium may include any storage medium described herein or any other suitable storage medium known in the art. After the results have been stored, the results can be accessed in the storage medium and used by any of the method or system embodiments described herein, formatted for display to a user, used by another software module, method, or system, etc. Furthermore, the results may be stored “permanently,” “semi-permanently,” temporarily, or for some period of time. For example, the storage medium may be random access memory (RAM), and the results may not necessarily persist indefinitely in the storage medium. 
     It is further contemplated that each of the embodiments of the method described above may include any other step(s) of any other method(s) described herein. In addition, each of the embodiments of the method described above may be performed by any of the systems described herein. 
     Those having skill in the art will appreciate that there are various vehicles by which processes and/or systems and/or other technologies described herein can be effected (e.g., hardware, software, and/or firmware), and that the preferred vehicle will vary with the context in which the processes and/or systems and/or other technologies are deployed. For example, if an implementer determines that speed and accuracy are paramount, the implementer may opt for a mainly hardware and/or firmware vehicle; alternatively, if flexibility is paramount, the implementer may opt for a mainly software implementation; or, yet again alternatively, the implementer may opt for some combination of hardware, software, and/or firmware. Hence, there are several possible vehicles by which the processes and/or devices and/or other technologies described herein may be effected, none of which is inherently superior to the other in that any vehicle to be utilized is a choice dependent upon the context in which the vehicle will be deployed and the specific concerns (e.g., speed, flexibility, or predictability) of the implementer, any of which may vary. Those skilled in the art will recognize that optical aspects of implementations will typically employ optically-oriented hardware, software, and or firmware. 
     Those skilled in the art will recognize that it is common within the art to describe devices and/or processes in the fashion set forth herein, and thereafter use engineering practices to integrate such described devices and/or processes into data processing systems. That is, at least a portion of the devices and/or processes described herein can be integrated into a data processing system via a reasonable amount of experimentation. Those having skill in the art will recognize that a typical data processing system generally includes one or more of a system unit housing, a video display device, a memory such as volatile and non-volatile memory, processors such as microprocessors and digital signal processors, computational entities such as operating systems, drivers, graphical user interfaces, and applications programs, one or more interaction devices, such as a touch pad or screen, and/or control systems including feedback loops and control motors (e.g., feedback for sensing position and/or velocity; control motors for moving and/or adjusting components and/or quantities). A typical data processing system may be implemented utilizing any suitable commercially available components, such as those typically found in data computing/communication and/or network computing/communication systems. 
     The herein described subject matter sometimes illustrates different components contained within, or connected with, different other components. It is to be understood that such depicted architectures are merely exemplary, and that in fact many other architectures can be implemented which achieve the same functionality. In a conceptual sense, any arrangement of components to achieve the same functionality is effectively “associated” such that the desired functionality is achieved. Hence, any two components herein combined to achieve a particular functionality can be seen as “associated with” each other such that the desired functionality is achieved, irrespective of architectures or intermedial components. Likewise, any two components so associated can also be viewed as being “connected”, or “coupled”, to each other to achieve the desired functionality, and any two components capable of being so associated can also be viewed as being “couplable”, to each other to achieve the desired functionality. Specific examples of couplable include but are not limited to physically mateable and/or physically interacting components and/or wirelessly interactable and/or wirelessly interacting components and/or logically interacting and/or logically interactable components. 
     While particular aspects of the present subject matter described herein have been shown and described, it will be apparent to those skilled in the art that, based upon the teachings herein, changes and modifications may be made without departing from the subject matter described herein and its broader aspects and, therefore, the appended claims are to encompass within their scope all such changes and modifications as are within the true spirit and scope of the subject matter described herein. 
     Furthermore, it is to be understood that the invention is defined by the appended claims. 
     Although particular embodiments of this invention have been illustrated, it is apparent that various modifications and embodiments of the invention may be made by those skilled in the art without departing from the scope and spirit of the foregoing disclosure. Accordingly, the scope of the invention should be limited only by the claims appended hereto. 
     It is believed that the present disclosure and many of its attendant advantages will be understood by the foregoing description, and it will be apparent that various changes may be made in the form, construction and arrangement of the components without departing from the disclosed subject matter or without sacrificing all of its material advantages. The form described is merely explanatory, and it is the intention of the following claims to encompass and include such changes.