Patent Publication Number: US-2022221802-A1

Title: Self-referencing interferometer and dual self-referencing interferometer devices

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority of U.S. Provisional Patent Application No. 62/854,398, which was filed on May 30, 2019, and which is incorporated herein in its entirety by reference. 
    
    
     FIELD 
     The present disclosure relates to interferometric apparatuses and systems, for example, self-referencing interferometer apparatuses for lithographic apparatuses and systems. 
     BACKGROUND 
     A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus may be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., comprising part of, one, or several dies) on a substrate (e.g., a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the target portions parallel or anti-parallel to this scanning direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate. 
     Another lithographic system is an interferometric lithographic system where there is no patterning device, but rather a light beam is split into two beams, and the two beams are caused to interfere at a target portion of the substrate through the use of a reflection system. The interference causes lines to be formed at the target portion of the substrate. 
     During lithographic operation, different processing steps may require different layers to be sequentially formed on the substrate. Accordingly, it may be necessary to position the substrate relative to prior patterns formed thereon with a high degree of accuracy. Generally, alignment marks are placed on the substrate to be aligned and are located with reference to a second object. A lithographic apparatus may use an alignment apparatus for detecting positions of the alignment marks and for aligning the substrate using the alignment marks to ensure accurate exposure from a mask. Misalignment between the alignment marks at two different layers is measured as overlay error. 
     In order to monitor the lithographic process, parameters of the patterned substrate are measured. Parameters may include, for example, the overlay error between successive layers formed in or on the patterned substrate and critical linewidth of developed photosensitive resist. This measurement may be performed on a product substrate and/or on a dedicated metrology target. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of scanning electron microscopes and various specialized tools. A fast and non-invasive form of a specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a target on the surface of the substrate and properties of the scattered or reflected beam are measured. By comparing the properties of the beam before and after it has been reflected or scattered by the substrate, the properties of the substrate can be determined. This can be done, for example, by comparing the reflected beam with data stored in a library of known measurements associated with known substrate properties. Spectroscopic scatterometers direct a broadband radiation beam onto the substrate and measure the spectrum (intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. By contrast, angularly resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle. 
     Such optical scatterometers may be used to measure parameters, such as critical dimensions of developed photosensitive resist or overlay error (OVE) between two layers formed in or on the patterned substrate. Properties of the substrate can be determined by comparing the properties of an illumination beam before and after the beam has been reflected or scattered by the substrate. 
     Fabrication tolerances continue to tighten as semiconductor devices become ever smaller and more elaborate. Hence, there is a need to continue to improve metrology measurements. One exemplary use of scatterometers is for critical dimension (CD) metrology, which is particularly useful for measuring in patterned structures, such as semiconductor wafers. Optical CD metrology techniques include on dome scatterometry, spectral reflectometry, and spectral ellipsometry. All these techniques are based on measuring the reflected intensity of differently polarized light for different incident directions. Such techniques require a high extinction ratio, or purity of polarization. A polarizing beamsplitter (PBS) divides light by polarization state to transmit p-polarized light while reflecting s-polarized light. Though a perfect PBS transmits 100% of the p-polarization and reflects 100% s-polarization, a real PBS transmits and reflects mixtures of s-polarized light and p-polarized light. The ratio between the p-polarized light and s-polarized light is called the extinction ratio. Optical CD requires a high extinction ratio. 
     Another exemplary use of scatterometers is for overlay (OV) metrology, which is useful for measuring alignment of a stack of layers on a wafer. In order to control the lithographic process to place device features accurately on the substrate, alignment marks, or targets, are generally provided on the substrate, and the lithographic apparatus includes one or more alignment apparatuses by which positions of marks on a substrate must be measured accurately. In one known technique, the scatterometer measures diffracted light from targets on the wafer. 
     Ideally, overlay error is solely a product of positioning of the substrate within the lithographic system. In practice, however, overlay error originates from the interaction between the alignment apparatus and the substrate. Variations in the alignment apparatus and substrate can produce errors in assessing the true location of the alignment marks. Such errors are known as “on-process” accuracy errors. Alignment apparatus optics contain manufacturing aberrations and, thus, cannot be made identical. Substrates (e.g., wafer stacks), likewise, have properties variations stemming from manufacturing and post-manufacturing processes. This “on-process” accuracy problem limits the robustness of the alignment apparatus. 
     Accordingly, there is a need to compensate for variations in an alignment apparatus and system, and reduce the overall footprint of optical parts used in the alignment apparatus and system. 
     SUMMARY 
     In some embodiments, a self-referencing interferometer (SRI) system for an alignment sensor apparatus includes a first prism and a second prism. The first prism includes an input surface for an incident beam. The second prism is coupled to the first prism and includes an output surface for a recombined beam. The recombined beam includes a first image and a second image rotated by 180 degrees with respect to the first image. The first and second prisms are identical in shape. 
     In some embodiments, the first and second prisms are adjoined along a beamsplitter interface. In some embodiments, the beamsplitter interface includes a polarizing coating. In some embodiments, first reflection planes in the first prism are either perpendicular or parallel to a first polarization plane of the incident beam, and second reflection planes in the second prism are either perpendicular or parallel to a second polarization plane of the incident beam. In some embodiments, the first and second prisms include an absence of any phase compensating coatings. 
     In some embodiments, the first prism or the second prism includes one or more phase compensating coatings. In some embodiments, the SRI system further includes a rectangular beamsplitter prism adjoined to the first and second prisms and having a beamsplitter interface. In some embodiments, the beamsplitter interface of the rectangular beamsplitter prism includes a polarizing coating. In some embodiments, the SRI system further includes a plate upon which first and second prisms are supported. In some embodiments, a number of contact reflection points of a first optical path and a second optical path within the first and second prisms is no greater than ten. In some embodiments, the number of contact reflection points is no greater than eight. 
     In some embodiments, a dual self-referencing interferometer (DSRI) system for an alignment sensor apparatus includes a first prism assembly and a second prism assembly. The first prism assembly includes an input surface for a first incident beam and a second incident beam. The second prism assembly is coupled to the first prism assembly and includes an output surface for a first recombined beam and a second recombined beam. The first recombined beam includes a first image and a second image rotated by 180 degrees with respect to the first image. The second recombined beam includes a third image and a fourth image rotated by 180 degrees with respect to the third image. The first and second prism assemblies are identical in shape. 
     In some embodiments, the first and second prism assemblies are disposed on a plate. In some embodiments, the first and second prism assemblies are adjoined along a beamsplitter interface. In some embodiments, the beamsplitter interface includes a polarizing coating. In some embodiments, first reflection planes in the first prism assembly are either perpendicular or parallel to a first polarization plane of the first and second incident beams, and second reflection planes in the second prism assembly are either perpendicular or parallel to a second polarization plane of the first and second incident beams. 
     In some embodiments, the DSRI system further includes a rectangular beamsplitter prism adjoined to the first and second prism assemblies and having a beamsplitter interface. In some embodiments, the beamsplitter interface of the rectangular beamsplitter prism includes a polarizing coating. In some embodiments, first reflection planes in the first prism assembly are either perpendicular or parallel to a first polarization plane of the first and second incident beams, and second reflection planes in the second prism assembly are either perpendicular or parallel to a second polarization plane of the first and second incident beams. In some embodiments, the first and second prism assemblies include an absence of any phase compensating coatings. In some embodiments, the first prism assembly or the second prism assembly includes one or more phase compensating coatings. 
     In some embodiments, a lithographic apparatus includes a first illumination optical system configured to illuminate a diffraction pattern, a projection optical system configured to project an image of the diffraction pattern onto a substrate, and an alignment sensor apparatus configured to correct an alignment position error of the lithographic apparatus. The alignment sensor apparatus includes a second illumination optical system configured to transmit at least one illumination beam of radiation along an illumination path, a first optical system including a first optic and a second optic, and configured to transmit the illumination beam toward the diffraction pattern on the substrate and transmit a signal beam including diffraction order sub-beams reflected from the diffraction pattern along a signal path, a second optical system including a first polarizing optic configured to separate and transmit the signal beam into a first polarization optical branch and a second polarization optical branch based on the polarization of the signal beam, a detector system including one or more detectors, and configured to measure an alignment position of the diffraction pattern based on the signal beam outputted from the first polarization branch and the second polarization branch, and a processor coupled to the detector system, and configured to measure a change in the alignment position of the diffraction pattern. The first polarization optical branch includes a first prism assembly and the second polarization optical branch includes a second prism assembly. The first and second prism assemblies are identical in shape. 
     In some embodiments, the first and second prism assemblies are adjoined to a rectangular beamsplitter prism. In some embodiments, the first and second prism assemblies include an absence of any phase compensating coatings. 
     Further features and advantages of the invention, as well as the structure and operation of various embodiments of the invention, are described in detail below with reference to the accompanying drawings. It is noted that the invention is not limited to the specific embodiments described herein. Such embodiments are presented herein for illustrative purposes only. Additional embodiments will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES 
       The accompanying drawings, which are incorporated herein and form part of the specification, illustrate the present invention and, together with the description, further serve to explain the principles of the invention and to enable a person skilled in the relevant art(s) to make and use the invention. 
         FIG. 1A  is a schematic illustration of a reflective lithographic apparatus, according to an exemplary embodiment. 
         FIG. 1B  is a schematic illustration of a transmissive lithographic apparatus, according to an exemplary embodiment. 
         FIG. 2  is a more detailed schematic illustration of the reflective lithographic apparatus, according to an exemplary embodiment. 
         FIG. 3  is a schematic illustration of a lithographic cell, according to an exemplary embodiment. 
         FIGS. 4A and 4B  are schematic illustrations of enhanced alignment apparatuses, according to various exemplary embodiments. 
         FIG. 5  is a schematic illustration of an alignment sensor apparatus, according to an exemplary embodiment. 
         FIG. 6  illustrates a flow diagram for detecting an alignment position using a self-referencing interferometer, according to an exemplary embodiment. 
         FIGS. 7A-7C  illustrate perspective transparent views of an exemplary self-referencing interferometer prism system and paths of radiation beams, according to an exemplary embodiment. 
         FIG. 7D  illustrates a cross-sectional transparent view of the exemplary self-referencing interferometer prism system and paths of radiation beams of  FIG. 7C . 
         FIG. 7E  illustrates a top plan transparent view of the exemplary self-referencing interferometer prism system and paths of radiation beams of  FIG. 7C . 
         FIG. 7F  illustrates a front plan transparent view of the exemplary self-referencing interferometer prism system and paths of radiation beams of  FIG. 7C . 
         FIGS. 8A-8C, 9A-9C, and 10A-10C  illustrate perspective transparent views of exemplary self-referencing interferometer prism systems and paths of radiation beams, according to various exemplary embodiments. 
         FIGS. 8D, 9D, and 10D  illustrate cross-sectional transparent views of the exemplary self-referencing interferometer prism systems and paths of radiation beams of  FIG. 8C ,  FIG. 9C , and  FIG. 10C , respectively. 
         FIGS. 8E, 9E, and 10E  illustrate top plan transparent views of the exemplary self-referencing interferometer prism systems and paths of radiation beams of  FIG. 8C ,  FIG. 9C , and  FIG. 10C , respectively. 
         FIGS. 8F, 9F, and 10F  illustrate front plan transparent views of the exemplary self-referencing interferometer prism systems and paths of radiation beams of  FIG. 8C ,  FIG. 9C , and  FIG. 10C , respectively. 
         FIG. 11  illustrates a perspective transparent view of an exemplary dual self-referencing interferometer prism system, according to an exemplary embodiment. 
         FIGS. 12A-12C, 13A-13C, 14A-14C, 15A-15C, 16A-16C, 17A-17C, 18A-18C, 19A and 19B, and 20A-20C  illustrate perspective transparent views of exemplary dual self-referencing interferometer prism systems and paths of radiation beams, according to various exemplary embodiments. 
         FIGS. 12D, 13D, 14D, 15D, 16D, 17D, 18D, 19C, and 20D  illustrate cross-sectional transparent views of the exemplary dual self-referencing interferometer prism systems and paths of radiation beams of  FIG. 12C ,  FIG. 13C ,  FIG. 14C ,  FIG. 15C ,  FIG. 16C ,  FIG. 17C , 
         FIG. 18C ,  FIG. 19B , and  FIG. 20C , respectively. 
         FIGS. 12E, 13E, 14E, 15E, 16E, 17E, 18E, 19D, and 20E  illustrate top plan transparent views of the exemplary dual self-referencing interferometer prism systems and paths of radiation beams of  FIG. 12C ,  FIG. 13C ,  FIG. 14C ,  FIG. 15C ,  FIG. 16C ,  FIG. 17C ,  FIG. 18C ,  FIG. 19B , and  FIG. 20C , respectively. 
         FIGS. 12F, 13F, 14F, 15F, 16F, 17F, 18F, 19E, and 20F  illustrate front plan transparent views of the exemplary dual self-referencing interferometer prism systems and paths of radiation beams of  FIG. 12C ,  FIG. 13C ,  FIG. 14C ,  FIG. 15C ,  FIG. 16C ,  FIG. 17C ,  FIG. 18C ,  FIG. 19B , and  FIG. 20C , respectively. 
     
    
    
     The features and advantages of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings, in which like reference characters identify corresponding elements throughout. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. Additionally, generally, the left-most digit(s) of a reference number identifies the drawing in which the reference number first appears. Unless otherwise indicated, the drawings provided throughout the disclosure should not be interpreted as to-scale drawings. 
     DETAILED DESCRIPTION 
     This specification discloses one or more embodiments that incorporate the features of this invention. The disclosed embodiment(s) merely exemplify the invention. The scope of the invention is not limited to the disclosed embodiment(s). The invention is defined by the claims appended hereto. 
     The embodiment(s) described, and references in the specification to “one embodiment,” “an embodiment,” “an example embodiment,” etc., indicate that the embodiment(s) described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is understood that it is within the knowledge of one skilled in the art to effect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described. 
     Spatially relative terms, such as “beneath,” “below,” “lower,” “above,” “on,” “upper,” “top,” “bottom,” “front,” “back” and the like, may be used herein for ease of description to describe one element or feature&#39;s relationship to another element(s) or feature(s) as illustrated in the figures. The spatially relative terms are intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. The apparatus may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein may likewise be interpreted accordingly. 
     The term “about” or “substantially” as used herein indicates the value of a given quantity that can vary based on a particular technology. Based on the particular technology, the term “about” or “substantially” can indicate a value of a given quantity that varies within, for example, 10-30% of the value (e.g., ±10%, ±20%, or ±30% of the value). 
     Embodiments of the disclosure may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the disclosure may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.), and others. Further, firmware, software, routines, and/or instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, instructions, etc. 
     Before describing such embodiments in more detail, however, it is instructive to present an example environment in which embodiments of the present disclosure may be implemented. 
     Example Lithographic Systems 
       FIGS. 1A and 1B  are schematic illustrations of a lithographic apparatus  100  and lithographic apparatus  100 ′, respectively, in which embodiments of the present invention may be implemented. Lithographic apparatus  100  and lithographic apparatus  100 ′ each include the following: an illumination system (illuminator) IL configured to condition a radiation beam B (for example, deep ultra violet or extreme ultra violet radiation); a support structure (for example, a mask table) MT configured to support a patterning device (for example, a mask, a reticle, or a dynamic patterning device) MA and connected to a first positioner PM configured to accurately position the patterning device MA; and, a substrate table (for example, a wafer table) WT configured to hold a substrate (for example, a resist coated wafer) W and connected to a second positioner PW configured to accurately position the substrate W. Lithographic apparatus  100  and  100 ′ also have a projection system PS configured to project a pattern imparted to the radiation beam B by patterning device MA onto a target portion (for example, comprising one or more dies) C of the substrate W. In lithographic apparatus  100 , the patterning device MA and the projection system PS are reflective. In lithographic apparatus  100 ′, the patterning device MA and the projection system PS are transmissive. 
     The illumination system IL may include various types of optical components, such as refractive, reflective, catadioptric, magnetic, electromagnetic, electrostatic, or other types of optical components, or any combination thereof, for directing, shaping, or controlling the radiation beam B. 
     The support structure MT holds the patterning device MA in a manner that depends on the orientation of the patterning device MA with respect to a reference frame, the design of at least one of the lithographic apparatus  100  and  100 ′, and other conditions, such as whether or not the patterning device MA is held in a vacuum environment. The support structure MT may use mechanical, vacuum, electrostatic, or other clamping techniques to hold the patterning device MA. The support structure MT can be a frame or a table, for example, which can be fixed or movable, as required. By using sensors, the support structure MT can ensure that the patterning device MA is at a desired position, for example, with respect to the projection system PS. 
     The term “patterning device” MA should be broadly interpreted as referring to any device that can be used to impart a radiation beam B with a pattern in its cross-section, such as to create a pattern in the target portion C of the substrate W. The pattern imparted to the radiation beam B can correspond to a particular functional layer in a device being created in the target portion C to form an integrated circuit. 
     The patterning device MA may be transmissive (as in lithographic apparatus  100 ′ of  FIG. 1B ) or reflective (as in lithographic apparatus  100  of  FIG. 1A ). Examples of patterning devices MA include reticles, masks, programmable mirror arrays, or programmable LCD panels. Masks are well known in lithography, and include mask types such as binary, alternating phase shift, or attenuated phase shift, as well as various hybrid mask types. An example of a programmable mirror array employs a matrix arrangement of small mirrors, each of which can be individually tilted so as to reflect an incoming radiation beam in different directions. The tilted mirrors impart a pattern in the radiation beam B, which is reflected by a matrix of small mirrors. 
     The term “projection system” PS can encompass any type of projection system, including refractive, reflective, catadioptric, magnetic, electromagnetic and electrostatic optical systems, or any combination thereof, as appropriate for the exposure radiation being used, or for other factors, such as the use of an immersion liquid on the substrate W or the use of a vacuum. A vacuum environment can be used for EUV or electron beam radiation since other gases can absorb too much radiation or electrons. A vacuum environment can therefore be provided to the whole beam path with the aid of a vacuum wall and vacuum pumps. 
     Lithographic apparatus  100  and/or lithographic apparatus  100 ′ can be of a type having two (dual stage) or more substrate tables WT (and/or two or more mask tables). In such “multiple stage” machines, the additional substrate tables WT can be used in parallel, or preparatory steps can be carried out on one or more tables while one or more other substrate tables WT are being used for exposure. In some situations, the additional table may not be a substrate table WT. 
     The lithographic apparatus may also be of a type wherein at least a portion of the substrate may be covered by a liquid having a relatively high refractive index, e.g., water, so as to fill a space between the projection system and the substrate. An immersion liquid may also be applied to other spaces in the lithographic apparatus, for example, between the mask and the projection system. Immersion techniques are well known in the art for increasing the numerical aperture of projection systems. The term “immersion” as used herein does not mean that a structure, such as a substrate, must be submerged in liquid, but rather only means that liquid is located between the projection system and the substrate during exposure. 
     Referring to  FIGS. 1A and 1B , the illuminator IL receives a radiation beam from a radiation source SO. The source SO and the lithographic apparatus  100 ,  100 ′ can be separate physical entities, for example, when the source SO is an excimer laser. In such cases, the source SO is not considered to form part of the lithographic apparatus  100  or  100 ′, and the radiation beam B passes from the source SO to the illuminator IL with the aid of a beam delivery system BD (in  FIG. 1B ) including, for example, suitable directing mirrors and/or a beam expander. In other cases, the source SO can be an integral part of the lithographic apparatus  100 ,  100 ′, for example, when the source SO is a mercury lamp. The source SO and the illuminator IL, together with the beam delivery system BD, if required, can be referred to as a radiation system. 
     The illuminator IL can include an adjuster AD (in  FIG. 1B ) for adjusting the angular intensity distribution of the radiation beam. Generally, at least the outer and/or inner radial extent (commonly referred to as “σ-outer” and “σ-inner,” respectively) of the intensity distribution in a pupil plane of the illuminator can be adjusted. In addition, the illuminator IL can comprise various other components (in  FIG. 1B ), such as an integrator IN and a condenser CO. The illuminator IL can be used to condition the radiation beam B to have a desired uniformity and intensity distribution in its cross section. 
     Referring to  FIG. 1A , the radiation beam B is incident on the patterning device (for example, mask) MA, which is held on the support structure (for example, mask table) MT, and is patterned by the patterning device MA. In lithographic apparatus  100 , the radiation beam B is reflected from the patterning device (for example, mask) MA. After being reflected from the patterning device (for example, mask) MA, the radiation beam B passes through the projection system PS, which focuses the radiation beam B onto a target portion C of the substrate W. With the aid of the second positioner PW and position sensor IF 2  (for example, an interferometric device, linear encoder, or capacitive sensor), the substrate table WT can be moved accurately (for example, so as to position different target portions C in the path of the radiation beam B). Similarly, the first positioner PM and another position sensor IF 1  can be used to accurately position the patterning device (for example, mask) MA with respect to the path of the radiation beam B. Patterning device (for example, mask) MA and substrate W can be aligned using mask alignment marks M 1 , M 2  and substrate alignment marks P 1 , P 2 . 
     Referring to  FIG. 1B , the radiation beam B is incident on the patterning device (for example, mask MA), which is held on the support structure (for example, mask table MT), and is patterned by the patterning device. Having traversed the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. The projection system has a pupil conjugate PPU to an illumination system pupil IPU. Portions of radiation emanate from the intensity distribution at the illumination system pupil IPU and traverse a mask pattern without being affected by diffraction at the mask pattern and create an image of the intensity distribution at the illumination system pupil IPU. 
     The projection system PS projects an image MP′ of the mask pattern MP, where image MP′ is formed by diffracted beams produced from the mark pattern MP by radiation from the intensity distribution, onto a photoresist layer coated on the substrate W. For example, the mask pattern MP may include an array of lines and spaces. A diffraction of radiation at the array and different from zeroth order diffraction generates diverted diffracted beams with a change of direction in a direction perpendicular to the lines. Undiffracted beams (i.e., so-called zeroth order diffracted beams) traverse the pattern without any change in propagation direction. The zeroth order diffracted beams traverse an upper lens or upper lens group of the projection system PS, upstream of the pupil conjugate PPU of the projection system PS, to reach the pupil conjugate PPU. The portion of the intensity distribution in the plane of the pupil conjugate PPU and associated with the zeroth order diffracted beams is an image of the intensity distribution in the illumination system pupil IPU of the illumination system IL. The aperture device PD, for example, is disposed at or substantially at a plane that includes the pupil conjugate PPU of the projection system PS. 
     The projection system PS is arranged to capture, by means of a lens or lens group L, not only the zeroth order diffracted beams, but also first-order or first- and higher-order diffracted beams (not shown). In some embodiments, dipole illumination for imaging line patterns extending in a direction perpendicular to a line may be used to utilize the resolution enhancement effect of dipole illumination. For example, first-order diffracted beams interfere with corresponding zeroth-order diffracted beams at the level of the wafer W to create an image of the line pattern MP at highest possible resolution and process window (i.e., usable depth of focus in combination with tolerable exposure dose deviations). In some embodiments, astigmatism aberration may be reduced by providing radiation poles (not shown) in opposite quadrants of the illumination system pupil IPU. Further, in some embodiments, astigmatism aberration may be reduced by blocking the zeroth order beams in the pupil conjugate PPU of the projection system associated with radiation poles in opposite quadrants. This is described in more detail in U.S. Pat. No. 7,511,799 B2, issued Mar. 31, 2009, which is incorporated by reference herein in its entirety. 
     With the aid of the second positioner PW and position sensor IF (for example, an interferometric device, linear encoder, or capacitive sensor), the substrate table WT can be moved accurately (for example, so as to position different target portions C in the path of the radiation beam B). Similarly, the first positioner PM and another position sensor (not shown in  FIG. 1B ) can be used to accurately position the mask MA with respect to the path of the radiation beam B (for example, after mechanical retrieval from a mask library or during a scan). 
     In general, movement of the mask table MT can be realized with the aid of a long-stroke module (coarse positioning) and a short-stroke module (fine positioning), which form part of the first positioner PM. Similarly, movement of the substrate table WT can be realized using a long-stroke module and a short-stroke module, which form part of the second positioner PW. In the case of a stepper (as opposed to a scanner), the mask table MT can be connected to a short-stroke actuator only or can be fixed. Mask MA and substrate W can be aligned using mask alignment marks M 1 , M 2 , and substrate alignment marks P 1 , P 2 . Although the substrate alignment marks (as illustrated) occupy dedicated target portions, they can be located in spaces between target portions (known as scribe-lane alignment marks). Similarly, in situations in which more than one die is provided on the mask MA, the mask alignment marks can be located between the dies. 
     Mask table MT and patterning device MA can be in a vacuum chamber V, where an in-vacuum robot IVR can be used to move patterning devices such as a mask in and out of vacuum chamber. Alternatively, when mask table MT and patterning device MA are outside of the vacuum chamber, an out-of-vacuum robot can be used for various transportation operations, similar to the in-vacuum robot IVR. Both the in-vacuum and out-of-vacuum robots need to be calibrated for a smooth transfer of any payload (e.g., mask) to a fixed kinematic mount of a transfer station. 
     The lithographic apparatus  100  and  100 ′ can be used in at least one of the following modes: 
     1. In step mode, the support structure (for example, mask table) MT and the substrate table WT are kept essentially stationary, while an entire pattern imparted to the radiation beam B is projected onto a target portion C at one time (i.e., a single static exposure). The substrate table WT is then shifted in the X and/or Y direction so that a different target portion C can be exposed. 
     2. In scan mode, the support structure (for example, mask table) MT and the substrate table WT are scanned synchronously while a pattern imparted to the radiation beam B is projected onto a target portion C (i.e., a single dynamic exposure). The velocity and direction of the substrate table WT relative to the support structure (for example, mask table) MT can be determined by the (de-)magnification and image reversal characteristics of the projection system PS. 
     3. In another mode, the support structure (for example, mask table) MT is kept substantially stationary holding a programmable patterning device, and the substrate table WT is moved or scanned while a pattern imparted to the radiation beam B is projected onto a target portion C. A pulsed radiation source SO can be employed and the programmable patterning device is updated as required after each movement of the substrate table WT or in between successive radiation pulses during a scan. This mode of operation can be readily applied to maskless lithography that utilizes a programmable patterning device, such as a programmable mirror array. 
     Combinations and/or variations on the described modes of use or entirely different modes of use can also be employed. 
     In a further embodiment, lithographic apparatus  100  includes an extreme ultraviolet (EUV) source, which is configured to generate a beam of EUV radiation for EUV lithography. In general, the EUV source is configured in a radiation system, and a corresponding illumination system is configured to condition the EUV radiation beam of the EUV source. 
       FIG. 2  shows the lithographic apparatus  100  in more detail, including the source collector apparatus SO, the illumination system IL, and the projection system PS. The source collector apparatus SO is constructed and arranged such that a vacuum environment can be maintained in an enclosing structure  220  of the source collector apparatus SO. An EUV radiation emitting plasma  210  may be formed by a discharge produced plasma source. EUV radiation may be produced by a gas or vapor, for example Xe gas, Li vapor, or Sn vapor in which the very hot plasma  210  is created to emit radiation in the EUV range of the electromagnetic spectrum. The very hot plasma  210  is created by, for example, an electrical discharge causing at least a partially ionized plasma. Partial pressures of, for example, 10 Pa of Xe, Li, Sn vapor, or any other suitable gas or vapor may be required for efficient generation of the radiation. In an embodiment, a plasma of excited tin (Sn) is provided to produce EUV radiation. 
     The radiation emitted by the hot plasma  210  is passed from a source chamber  211  into a collector chamber  212  via an optional gas barrier or contaminant trap  230  (in some cases also referred to as contaminant barrier or foil trap), which is positioned in or behind an opening in source chamber  211 . The contaminant trap  230  may include a channel structure. Contamination trap  230  may also include a gas barrier or a combination of a gas barrier and a channel structure. The contaminant trap or contaminant barrier  230  further indicated herein at least includes a channel structure. 
     The collector chamber  212  may include a radiation collector CO, which can be a so-called grazing incidence collector. Radiation collector CO has an upstream radiation collector side  251  and a downstream radiation collector side  252 . Radiation that traverses collector CO can be reflected off a grating spectral filter  240  to be focused in a virtual source point IF. The virtual source point IF is commonly referred to as the intermediate focus, and the source collector apparatus is arranged such that the intermediate focus IF is located at or near an opening  219  in the enclosing structure  220 . The virtual source point IF is an image of the radiation emitting plasma  210 . Grating spectral filter  240  is used in particular for suppressing infra-red (IR) radiation. 
     Subsequently the radiation traverses the illumination system IL, which may include a faceted field mirror device  222  and a faceted pupil mirror device  224  arranged to provide a desired angular distribution of the radiation beam  221 , at the patterning device MA, as well as a desired uniformity of radiation intensity at the patterning device MA. Upon reflection of the beam of radiation  221  at the patterning device MA, held by the support structure MT, a patterned beam  226  is formed and the patterned beam  226  is imaged by the projection system PS via reflective elements  228 ,  229  onto a substrate W held by the wafer stage or substrate table WT. 
     More elements than shown may generally be present in illumination optics unit IL and projection system PS. The grating spectral filter  240  may optionally be present, depending upon the type of lithographic apparatus. Further, there may be more mirrors present than those shown in the  FIG. 2 , for example there may be one to six additional reflective elements present in the projection system PS than shown in  FIG. 2 . 
     Collector optic CO, as illustrated in  FIG. 2 , is depicted as a nested collector with grazing incidence reflectors  253 ,  254 , and  255 , just as an example of a collector (or collector mirror). The grazing incidence reflectors  253 ,  254 , and  255  are disposed axially symmetric around an optical axis O and a collector optic CO of this type is preferably used in combination with a discharge produced plasma source, often called a DPP source. 
     Exemplary Lithographic Cell 
       FIG. 3  shows a lithographic cell  300 , also sometimes referred to a lithocell or cluster. Lithographic apparatus  100  or  100 ′ may form part of lithographic cell  300 . Lithographic cell  300  may also include one or more apparatuses to perform pre- and post-exposure processes on a substrate. Conventionally these include spin coaters SC to deposit resist layers, developers DE to develop exposed resist, chill plates CH, and bake plates BK. A substrate handler, or robot, RO picks up substrates from input/output ports I/O1, I/O2, moves them between the different process apparatuses and delivers them to the loading bay LB of the lithographic apparatus  100  or  100 ′. These devices, which are often collectively referred to as the track, are under the control of a track control unit TCU, which is itself controlled by a supervisory control system SCS, which also controls the lithographic apparatus via lithography control unit LACU. Thus, the different apparatuses can be operated to maximize throughput and processing efficiency. 
     Exemplary Alignment Apparatus 
     In order to control the lithographic process to place device features accurately on the substrate, alignment marks are generally provided on the substrate, and the lithographic apparatus includes one or more alignment apparatuses and/or systems by which positions of marks on a substrate must be measured accurately. These alignment apparatuses are effectively position measuring apparatuses. Different types of marks and different types of alignment apparatuses and/or systems are known from different times and different manufacturers. A type of system widely used in current lithographic apparatus is based on a self-referencing interferometer as described in U.S. Pat. No. 6,961,116 (den Boef et al.). Generally marks are measured separately to obtain X- and Y-positions. A combined X- and Y-measurement can be performed using the techniques described in U.S. Publication No. 2009/195768 A (Bijnen et al.), however. The full contents of both of these disclosures are incorporated herein by reference. 
       FIG. 4A  illustrates a schematic of a cross-sectional view of an alignment apparatus  400  that can be implemented as a part of lithographic apparatus  100  or  100 ′, according to an embodiment. In an example of this embodiment, alignment apparatus  400  may be configured to align a substrate (e.g., substrate W) with respect to a patterning device (e.g., patterning device MA). Alignment apparatus  400  may be further configured to detect positions of alignment marks on the substrate and to align the substrate with respect to the patterning device or other components of lithographic apparatus  100  or  100 ′ using the detected positions of the alignment marks. Such alignment of the substrate may ensure accurate exposure of one or more patterns on the substrate. 
     According to an embodiment, alignment apparatus  400  may include an illumination system  412 , a beamsplitter  414 , an interferometer  426 , a detector  428 , a beam analyzer  430 , and an overlay calculation processor  432 , according to an example of this embodiment. Illumination system  412  may be configured to provide an electromagnetic narrow band radiation beam  413  having one or more passbands. In an example, the one or more passbands may be within a spectrum of wavelengths between about 500 nm to about 900 nm. In another example, the one or more passbands may be discrete narrow passbands within a spectrum of wavelengths between about 500 nm to about 900 nm. Illumination system  412  may be further configured to provide one or more passbands having substantially constant center wavelength (CWL) values over a long period of time (e.g., over a lifetime of illumination system  412 ). Such configuration of illumination system  412  may help to prevent the shift of the actual CWL values from the desired CWL values, as discussed above, in current alignment systems. And, as a result, the use of constant CWL values may improve long-term stability and accuracy of alignment systems (e.g., alignment apparatus  400 ) compared to the current alignment apparatuses. 
     Beamsplitter  414  may be configured to receive radiation beam  413  and split radiation beam  413  into at least two radiation sub-beams, according an embodiment. In an example, radiation beam  413  may be split into radiation sub-beams  415  and  417 , as shown in  FIG. 4A . Beamsplitter  414  may be further configured to direct radiation sub-beam  415  onto a substrate  420  placed on a stage  422 . In one example, the stage  422  is movable along direction  424 . Radiation sub-beam  415  may be configured to illuminate an alignment mark or a target  418  located on substrate  420 . Alignment mark or target  418  may be coated with a radiation sensitive film in an example of this embodiment. In another example, alignment mark or target  418  may have one hundred and eighty degrees (i.e., 180°) symmetry. That is, when alignment mark or target  418  is rotated 180° about an axis of symmetry perpendicular to a plane of alignment mark or target  418 , rotated alignment mark or target  418  may be substantially identical to an unrotated alignment mark or target  418 . The target  418  on substrate  420  may be (a) a resist layer grating comprising bars that are formed of solid resist lines, or (b) a product layer grating, or (c) a composite grating stack in an overlay target structure comprising a resist grating overlaid or interleaved on a product layer grating. The bars may alternatively be etched into the substrate. This pattern is sensitive to chromatic aberrations in the lithographic projection apparatus, particularly the projection system PL, and illumination symmetry and the presence of such aberrations will manifest themselves in a variation in the printed grating. One in-line method used in device manufacturing for measurements of line width, pitch, and critical dimension makes use of a technique known as “scatterometry”. Methods of scatterometry are described in Raymond et al., “Multiparameter Grating Metrology Using Optical Scatterometry”, J. Vac. Sci. Tech. B, Vol. 15, no. 2, pp. 361-368 (1997) and Niu et al., “Specular Spectroscopic Scatterometry in DUV Lithography”, SPIE, Vol. 3677 (1999), which are both incorporated by reference herein in their entireties. In scatterometry, light is reflected by periodic structures in the target, and the resulting reflection spectrum at a given angle is detected. The structure giving rise to the reflection spectrum is reconstructed, e.g. using Rigorous Coupled-Wave Analysis (RCWA) or by comparison to a library of patterns derived by simulation. Accordingly, the scatterometry data of the printed gratings is used to reconstruct the gratings. The parameters of the grating, such as line widths and shapes, may be input to the reconstruction process, performed by processing unit PU, from knowledge of the printing step and/or other scatterometry processes. 
     Beamsplitter  414  may be further configured to receive diffraction radiation beam  419  and split diffraction radiation beam  419  into at least two radiation sub-beams, according to an embodiment. In an example, diffraction radiation beam  419  may be split into diffraction radiation sub-beams  429  and  439 , as shown in  FIG. 4A . 
     It should be noted that even though beamsplitter  414  is shown to direct radiation sub-beam  415  towards alignment mark or target  418  and to direct diffracted radiation sub-beam  429  towards interferometer  426 , the disclosure is not so limiting. It would be apparent to a person skilled in the relevant art that other optical arrangements may be used to obtain the similar result of illuminating alignment mark or target  418  on substrate  420  and detecting an image of alignment mark or target  418 . 
     As illustrated in  FIG. 4A , interferometer  426  may be configured to receive radiation sub-beam  417  and diffracted radiation sub-beam  429  through beamsplitter  414 . In an example embodiment, diffracted radiation sub-beam  429  may be at least a portion of radiation sub-beam  415  that may be reflected from alignment mark or target  418 . In an example of this embodiment, interferometer  426  comprises any appropriate set of optical-elements, for example, a combination of prisms that may be configured to form two images of alignment mark or target  418  based on the received diffracted radiation sub-beam  429 . It should be appreciated that a good quality image need not be formed, but that the features of alignment mark  418  should be resolved. Interferometer  426  may be further configured to rotate one of the two images with respect to the other of the two images 180° and recombine the rotated and unrotated images interferometrically. 
     In an embodiment, detector  428  may be configured to receive the recombined image via interferometer signal  427  and detect interference as a result of the recombined image when alignment axis  421  of alignment apparatus  400  passes through a center of symmetry (not shown) of alignment mark or target  418 . Such interference may be due to alignment mark or target  418  being 180° symmetrical, and the recombined image interfering constructively or destructively, according to an example embodiment. Based on the detected interference, detector  428  may be further configured to determine a position of the center of symmetry of alignment mark or target  418  and consequently, detect a position of substrate  420 . According to an example, alignment axis  421  may be aligned with an optical beam perpendicular to substrate  420  and passing through a center of image rotation interferometer  426 . Detector  428  may be further configured to estimate the positions of alignment mark or target  418  by implementing sensor characteristics and interacting with wafer mark process variations. 
     In a further embodiment, detector  428  determines the position of the center of symmetry of alignment mark or target  418  by performing one or more of the following measurements: 
     1. measuring position variations for various wavelengths (position shift between colors);
 
2. measuring position variations for various orders (position shift between diffraction orders); and
 
3. measuring position variations for various polarizations (position shift between polarizations). This data can for example be obtained with any type of alignment sensor, for example a SMASH (SMart Alignment Sensor Hybrid) sensor, as described in U.S. Pat. No. 6,961,116 that employs a self-referencing interferometer with a single detector and four different wavelengths, and extracts the alignment signal in software, or Athena (Advanced Technology using High order ENhancement of Alignment), as described in U.S. Pat. No. 6,297,876, which directs each of seven diffraction orders to a dedicated detector, which are both incorporated by reference herein in their entireties.
 
     In an embodiment, beam analyzer  430  may be configured to receive and determine an optical state of diffracted radiation sub-beam  439 . The optical state may be a measure of beam wavelength, polarization, or beam profile. Beam analyzer  430  may be further configured to determine a position of stage  422  and correlate the position of stage  422  with the position of the center of symmetry of alignment mark or target  418 . As such, the position of alignment mark or target  418  and, consequently, the position of substrate  420  may be accurately known with reference to stage  422 . Alternatively, beam analyzer  430  may be configured to determine a position of alignment apparatus  400  or any other reference element such that the center of symmetry of alignment mark or target  418  may be known with reference to alignment apparatus  400  or any other reference element. Beam analyzer  430  can be a point or an imaging polarimeter with some form of wavelength-band selectivity. According to an embodiment, beam analyzer  430  may be directly integrated into alignment apparatus  400 , or connected via fiber optics of several types: polarization preserving single mode, multimode, or imaging, according to other embodiments. 
     In an embodiment, beam analyzer  430  may be further configured to determine the overlay data between two patterns on substrate  420 . One of these patterns may be a reference pattern on a reference layer. The other pattern may be an exposed pattern on an exposed layer. The reference layer may be an etched layer already present on substrate  420 . The reference layer may be generated by a reference pattern exposed on the substrate by lithographic apparatus  100  and/or  100 ′. The exposed layer may be a resist layer exposed adjacent to the reference layer. The exposed layer may be generated by an exposure pattern exposed on substrate  420  by lithographic apparatus  100  or  100 ′. The exposed pattern on substrate  420  may correspond to a movement of substrate  420  by stage  422 . In an embodiment, the measured overlay data may also indicate an offset between the reference pattern and the exposure pattern. The measured overlay data may be used as calibration data to calibrate the exposure pattern exposed by lithographic apparatus  100  or  100 ′, such that after the calibration, the offset between the exposed layer and the reference layer may be minimized. 
     In an embodiment, beam analyzer  430  may be further configured to determine a model of the product stack profile of substrate  420 , and may be configured to measure overlay, critical dimension, and focus of target  418  in a single measurement. The product stack profile contains information on the stacked product such as alignment mark, target  418 , or substrate  420 , and may include mark process variation-induced optical signature metrology that is a function of illumination variation. The product stack profile may also include product grating profile, mark stack profile, and mark asymmetry information. An example of beam analyzer  430  is Yieldstar™ manufactured by ASML, Veldhoven, The Netherlands, as described in U.S. Pat. No. 8,706,442, which is incorporated by reference herein in its entirety. Beam analyzer  430  may be further configured to process information related to a particular property of an exposed pattern in that layer. For example, beam analyzer  430  may process an overlay parameter (an indication of the positioning accuracy of the layer with respect to a previous layer on the substrate or the positioning accuracy of the first layer with respective to marks on the substrate), a focus parameter, and/or a critical dimension parameter (e.g., line width and its variations) of the depicted image in the layer. Other parameters are image parameters relating to the quality of the depicted image of the exposed pattern. 
     In some embodiments, an array of detectors (not shown) may be connected to beam analyzer  430 , and allows the possibility of accurate stack profile detection as discussed below. For example, detector  428  can be an array of detectors. For the detector array, a number of options are possible: a bundle of multimode fibers, discrete pin detectors per channel, or CCD or CMOS (linear) arrays. The use of a bundle of multimode fibers enables any dissipating elements to be remotely located for stability reasons. Discrete PIN detectors offer a large dynamic range but each need separate pre-amps. The number of elements is therefore limited. CCD linear arrays offer many elements that can be read-out at high speed and are especially of interest if phase-stepping detection is used. 
     In an embodiment, a second beam analyzer  430 ′ may be configured to receive and determine an optical state of diffracted radiation sub-beam  429 , as shown in  FIG. 4B . The optical state may be a measure of beam wavelength, polarization, or beam profile. Second beam analyzer  430 ′ may be identical to beam analyzer  430 . Alternatively, second beam analyzer  430 ′ may be configured to perform at least all the functions of beam analyzer  430 , such as determining a position of stage  422  and correlating the position of stage  422  with the position of the center of symmetry of alignment mark or target  418 . As such, the position of alignment mark or target  418  and, consequently, the position of substrate  420 , may be accurately known with reference to stage  422 . Second beam analyzer  430 ′ may also be configured to determine a position of alignment apparatus  400 , or any other reference element, such that the center of symmetry of alignment mark or target  418  may be known with reference to alignment apparatus  400 , or any other reference element. Second beam analyzer  430 ′ may be further configured to determine the overlay data between two patterns and a model of the product stack profile of substrate  420 . Second beam analyzer  430 ′ may also be configured to measure overlay, critical dimension, and focus of target  418  in a single measurement. 
     In an embodiment, second beam analyzer  430 ′ may be directly integrated into alignment apparatus  400 , or it can be connected via fiber optics of several types: polarization preserving single mode, multimode, or imaging, according to other embodiments. Alternatively, second beam analyzer  430 ′ and beam analyzer  430  may be combined to form a single analyzer (not shown) configured to receive and determine the optical states of both diffracted radiation sub-beams  429  and  439 . 
     In an embodiment, processor  432  receives information from detector  428  and beam analyzer  430 . For example, processor  432  may be an overlay calculation processor. The information may comprise a model of the product stack profile constructed by beam analyzer  430 . Alternatively, processor  432  may construct a model of the product mark profile using the received information about the product mark. In either case, processor  432  constructs a model of the stacked product and overlay mark profile using or incorporating a model of the product mark profile. The stack model is then used to determine the overlay offset and minimizes the spectral effect on the overlay offset measurement. Processor  432  may create a basic correction algorithm based on the information received from detector  428  and beam analyzer  430 , including but not limited to the optical state of the illumination beam, the alignment signals, associated position estimates, and the optical state in the pupil, image, and additional planes. The pupil plane is the plane in which the radial position of radiation defines the angle of incidence and the angular position defines the azimuth angle of the radiation. Processor  432  may utilize the basic correction algorithm to characterize the alignment apparatus  400  with reference to wafer marks and/or alignment marks  418 . 
     In an embodiment, processor  432  may be further configured to determine printed pattern position offset error with respect to the sensor estimate for each mark based on the information received from detector  428  and beam analyzer  430 . The information includes but is not limited to the product stack profile, measurements of overlay, critical dimension, and focus of each alignment marks or target  418  on substrate  420 . Processor  432  may utilize a clustering algorithm to group the marks into sets of similar constant offset error, and create an alignment error offset correction table based on the information. The clustering algorithm may be based on overlay measurement, the position estimates, and additional optical stack process information associated with each set of offset errors. The overlay is calculated for a number of different marks, for example, overlay targets having a positive and a negative bias around a programmed overlay offset. The target that measures the smallest overlay is taken as reference (as it is measured with the best accuracy). From this measured small overlay, and the known programmed overlay of its corresponding target, the overlay error can be deduced. Table 1 illustrates how this may be performed. The smallest measured overlay in the example shown is −1 nm. However this is in relation to a target with a programmed overlay of −30 nm. Consequently the process must have introduced an overlay error of 29 nm. 
                                                 TABLE 1                  Programmed overlay   −70   −50   −30   −10   10   30   50       Measured overlay   −38   −19   −1   21   43   66   90       Difference between measured   32   31   29   31   33   36   40       and programmed overlay       Overlay error   3   2   —   2   4   7   11                    
The smallest value can be taken to be the reference point and, relative to this, the offset can be calculated between measured overlay and that expected due to the programmed overlay. This offset determines the overlay error for each mark or the sets of marks with similar offsets. Therefore, in the Table 1 example, the smallest measured overlay was −1 nm, at the target position with programmed overlay of 30 nm. The difference between the expected and measured overlay at the other targets is compared to this reference. A table such as Table 1 can also be obtained from marks and target  418  under different illumination settings, the illumination setting, which results in the smallest overlay error, and its corresponding calibration factor, may be determined and selected. Following this, processor  432  may group marks into sets of similar overlay error. The criteria for grouping marks may be adjusted based on different process controls, for example, different error tolerances for different processes.
 
     In an embodiment, processor  432  may confirm that all or most members of the group have similar offset errors, and apply an individual offset correction from the clustering algorithm to each mark, based on its additional optical stack metrology. Processor  432  may determine corrections for each mark and feed the corrections back to lithographic apparatus  100  or  100 ′ for correcting errors in the overlay, for example, by feeding corrections into the alignment apparatus  400 . 
     Exemplary Alignment Sensor Apparatus 
     As discussed above, on-process accuracy errors (OPAEs) are caused by varying stack thicknesses, materials, and/or processes on each wafer (i.e., process variations) and overlay errors due to the interaction between alignment sensors. Process variations change the optical properties of reflected light from an alignment mark on a substrate, which causes OPAEs. Despite various techniques, such as mark asymmetry reconstruction (MAR), which corrects for asymmetries in an alignment mark, improved sensors (e.g., SMASH), and predictive modeling, wafer stack properties variations (i.e., process variations) cause a lower limit for OPAEs and cannot be reduced further using current techniques and systems. Process variations interact with an alignment sensor and create an alignment position error (APE) that cannot be calibrated. 
     APE is a change or shift in alignment position from a reference alignment position (e.g., calibrated alignment mark on a substrate). However, APE is a function of various physical parameters, for example, beam wavelength, spectral bandwidth, numerical aperture, beam intensity, beam spot size, beam shape, beam pattern, and/or polarization. For example, APE may be modeled as a linear function for one or more physical parameters. When physical parameters are varied in an alignment and/or lithographic apparatus, a change or shift in the reference alignment position due to unknown process variations can be measured and a correction can be determined and applied in order to reduce OPAEs. 
       FIG. 5  illustrates alignment sensor apparatus  500 , according to an exemplary embodiment. Alignment sensor apparatus  500  is configured to correct APE and improve overlay, for example, in lithographic apparatus  100  or  100 ′. Alignment sensor apparatus  500  may include illumination system  502 , spot mirror  516 , focusing lens  518 , polarizing beamsplitter  550 , detector controller  584 , one or more optical filters  506 ,  508 ,  510 ,  512 ,  514 ,  538 ,  548 ,  560 ,  576 , and APE processor  590 . Although alignment sensor apparatus  500  is shown in  FIG. 5  as a stand-alone apparatus, the embodiments of this disclosure are not limited to this example, and alignment sensor apparatus  500  embodiments of this disclosure can be used with other optical systems, such as, but not limited to, lithographic apparatus  100  and/or  100 ′, lithocell  300 , alignment apparatus  400 , and/or other optical systems. 
     Illumination system  502  is configured to transmit illumination beam  504  along an illumination path toward spot mirror  516 . Illumination system  502  is similar to illumination system  412  described in  FIGS. 4A and 4B . For example, illumination system  502  can include an extreme ultraviolet (EUV) source, which is configured to generate a beam of EUV radiation. Illumination system  502  may include a EUV source in a radiation system and a corresponding conditioning system configured to condition the EUV radiation beam of the EUV source. 
     Illumination system  502  may be configured to provide an electromagnetic narrow band illumination beam  504  having one or more passbands. In an example, the one or more passbands may be within a spectrum of wavelengths between about 500 nm to about 900 nm. In an example, the one or more passbands may be within a spectrum of wavelengths between about 10 nm to about 500 nm. In another example, the one or more passbands may be discrete narrow passbands within a spectrum of wavelengths between about 500 nm to about 900 nm. In another example, the one or more passbands may be discrete narrow passbands within a spectrum of wavelengths between about 10 nm to about 500 nm. Illumination system  502  may be further configured to provide one or more passbands having substantially constant center wavelength (CWL) values over a long period of time (e.g., over a lifetime of illumination system  502 ). Such configuration of illumination system  502  may help to prevent the shift of the actual CWL values from the desired CWL values, as discussed above, in current alignment systems. And, as a result, the use of constant CWL values may improve long-term stability and accuracy of alignment systems (e.g., alignment sensor apparatus  500 ) compared to the current alignment apparatuses. 
     In some embodiments, illumination system  502  may use a broadband light source (i.e., one with a wide range of light frequencies or wavelengths—and, thus, of colors) for a radiation source that may give a large etendue (i.e., spread of light, e.g., the product of the area (A) of the source and the solid angle (Ω) that the system&#39;s entrance pupil subtends as seen from the source), allowing the mixing of multiple wavelengths. In some embodiments, illumination beam  504  may include a plurality of wavelengths in the broadband preferably may each have a bandwidth of Δλ and a spacing of at least 2Δλ (i.e., twice the bandwidth). In some embodiments, illumination system  502  may include several “sources” of radiation for different portions of an extended radiation source that have been split using fiber bundles. In this way, angle resolved scatter spectra can be measured at multiple wavelengths in parallel. For example, a 3-D spectrum (wavelength and two different angles) can be measured, which contains more information than a 2-D spectrum. This allows more information to be measured, which increases metrology process robustness. This is described in more detail in EP 1628164 A2, which is incorporated by reference herein in its entirety. 
     Spot mirror  516  is a transmissive cube with a reflective metal layer disposed in the center of the cube. As shown in  FIG. 5 , spot mirror  516  may form a 45° angle with the illumination path in order to direct illumination beam  504  toward substrate  522 . In an embodiment, spot mirror  516  may be a beamsplitter configured to reflect a first portion (e.g., 50%) of illumination beam  504  toward substrate  522  and transmit a second portion (e.g., 50%) of illumination beam  504  toward beam analyzer  536 . Beam analyzer  536  is similar to beam analyzer  430  described in  FIGS. 4A and 4B , and is configured to analyze various properties of illumination beam  504 , for example, intensity, beam shape, alignment position, and/or polarization. 
     As shown in  FIG. 5 , spot mirror  516  may transmit illumination beam  504  toward focusing lens  518 , which focuses illumination beam  504  on diffraction target  520  on substrate  522 . In an embodiment, diffraction target  520  may be an alignment mark. In an embodiment, substrate  522  is supported by stage  524  and centered along alignment axis  526 . In some embodiments, diffraction target  520  on substrate  522  may be a 1-D grating, which is printed such that after development, bars are formed of solid resist lines. In some embodiments, diffraction target  520  may be a 2-D array or grating, which is printed such that, after development, a grating is formed of solid resist pillars or vias in the resist. For example, bars, pillars, or vias may alternatively be etched into substrate  522 . 
     Focused illumination beam  504  on diffraction target  520  creates a signal beam along signal path  535  comprising diffraction order sub-beams  528 ,  530 ,  532  reflected from diffraction target  520 . As shown in  FIG. 5 , first diffraction order sub-beam  528 , second diffraction order sub-beam  530 , and third diffraction order sub-beam  532  reflect off diffraction target  520  back toward focusing lens  518  and create signal path  535 . In some embodiments, focusing lens  518  may be positioned at the pupil plane. The pupil plane is the plane in which the radial position of radiation defines the angle of incidence and the angular position defines the azimuth angle of the radiation. 
     In some embodiments, first diffraction order sub-beam  528  may be a zeroth diffraction order sub-beam, second diffraction order sub-beam  530  may be a first diffraction order sub-beam (e.g., −1), and third diffraction order sub-beam  532  may be a first diffraction order sub-beam (e.g., +1). As shown in  FIG. 5 , spot mirror  516  blocks and/or reflects first diffraction order sub-beam  528  and transmits second and third diffraction order sub-beams  530 ,  532  along signal path  535 . In some embodiments, spot mirror  516  reflects first diffraction order sub-beam  528  toward beam analyzer  536 , which is configured to analyze various properties of first diffraction order sub-beam  528 , for example, intensity, beam shape, alignment position, and/or polarization. 
     Second and third diffraction order sub-beams  530 ,  532  transmit along signal path  535  to polarizing beamsplitter  550  configured to separate and transmit second and third diffraction order sub-beams  530 ,  532  into a first polarization optical branch (e.g. “X” direction, s-polarized) and a second polarization optical branch (e.g., “Y” direction, p-polarized) based on the polarization of sub-beams  530 ,  532 . Polarized radiation with its electric field along the plane of incidence is deemed p-polarized (i.e., transverse-magnetic (TM), while polarized radiation with its electric field normal to the plane of incidence is deemed s-polarized (i.e., transverse-electric (TE)). Polarizing beamsplitter  550  splits signal path  535  into orthogonal polarization components (i.e., first and second polarization optical branches), and transmits s-polarized sub-beams  530 ,  532  into first polarization optical branch (“X” direction, s-polarized) and transmits p-polarized sub-beams  530 ,  532  into second polarization optical branch (“Y” direction, p-polarized). 
     First polarization optical branch is configured to transmit s-polarized sub-beams  530 ,  532  and measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a horizontal or “X” direction, with reference to alignment axis  526 . As shown in  FIG. 5 , first polarization optical branch may include first polarizing filter  552 , “X” direction self-referencing interferometer (SRI-X)  554 , second polarizing filter  556 , and polarizing beamsplitter  558 . S-polarized sub-beams  530 ,  532  transmit through first polarizing filter  552 , SRI-X  554 , and second polarizing filter  556  in order. In some embodiments, first and second polarizing filters  552 ,  556  may each be a waveplate, for example, a half-wave plate at 22.5° (π/8). In some embodiments, first and second polarizing filters  552 ,  556  may each be a waveplate, for example, a quarter-wave plate at 45° (π/4). 
     A half-wave plate induces a phase shift of 180° (π) and rotates a polarization angle θ, formed between the polarization vector and the fast axis vector, to −θ. For linearly polarized light, a half-wave plate rotates θ to 2θ, while for elliptically (e.g., circularly) polarized light, a half-wave plate inverts the chirality (e.g., from right-circular to left-circular). A quarter-wave plate induces a phase shift of 90° (π/2) and the output depends upon an input polarization angle φ, formed between the fast and slow axis vectors. For linearly polarized light, φ=0° produces no change in the linear polarization, φ=45° produces circular polarization, and 0°&lt;φ&lt;45° produces elliptical polarization. 
     First polarizing filter  552 , SRI-X  554 , and second polarizing filter  556  are configured to rotate an image of s-polarized sub-beams  530 ,  532  by 180° and recombine the two images, which are 180° out of phase with the other. The two recombined images are transmitted to polarizing beamsplitter  558 . Polarizing beamsplitter  558  is configured to separate and transmit the difference between the two recombined images into first position detector  566  and the sum of the two recombined images into second position detector  564 . As shown in  FIG. 5 , focusing lens  562  may be included in first polarization optical branch in order to focus the sum of the two recombined images onto second position detector  564 . In some embodiments, an additional focusing lens, similar to focusing lens  562 , may be included between first position detector  566  and polarizing beamsplitter  558  in order to focus the difference of the two recombined images. 
     Second polarization optical branch is similar to first polarization optical branch, and is configured to transmit p-polarized sub-beams  530 ,  532  and measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a vertical or “Y” direction, with reference to alignment axis  526 . As shown in  FIG. 5 , first polarization optical branch may include first polarizing filter  568 , “Y” direction self-referencing interferometer (SRI-Y)  570 , second polarizing filter  572 , and polarizing beamsplitter  574 . P-polarized sub-beams  530 ,  532  transmit through first polarizing filter  568 , SRI-Y  570 , and second polarizing filter  572  in order. In some embodiments, first and second polarizing filters  568 ,  572  may each be a waveplate, for example, a half-wave plate at 22.5° (π/8). In some embodiments, first and second polarizing filters  568 ,  572  may each be a waveplate, for example, a quarter-wave plate at 45° (π/4). 
     First polarizing filter  568 , SRI-Y  570 , and second polarizing filter  572  are configured to rotate an image of p-polarized sub-beams  530 ,  532  by 180° and recombine the two images, which are 180° out of phase with the other. The two recombined images are transmitted to polarizing beamsplitter  574 . Polarizing beamsplitter  574  is configured to separate and transmit the difference between the two recombined images into third position detector  582  and the sum of the two recombined images into fourth position detector  580 . As shown in  FIG. 5 , focusing lens  578  may be included in second polarization optical branch in order to focus the sum of the two recombined images onto fourth position detector  580 . In some embodiments, an additional focusing lens, similar to focusing lens  578 , may be included between third position detector  582  and polarizing beamsplitter  574  in order to focus the difference of the two recombined images. 
     As shown in  FIG. 5 , detector controller  584  may be connected to first position detector  566 , second position detector  564 , third position detector  582 , and fourth position detector  580  via first, second, third, and fourth control signals  586 ,  585 ,  588 ,  587 , respectively. Detector controller  584  is configured to measure and detect an alignment position of diffraction target  520 , with reference to alignment axis  526 , based on the signal beams (e.g., difference and sum) outputted from the first and second polarization branches. In some embodiments, detector controller  584  is configured to measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a horizontal or “X” direction and/or in a vertical or “Y” direction. In some embodiments, detector controller  584  may combine second and third diffraction order sub-beams  530 ,  532  to generate a sinusoidal phase. 
     One or more optical filters may be disposed along an illumination path of illumination beam  504  and/or signal path  535  of second and third diffraction order sub-beams  530 ,  532 . As discussed above, when an optical filter is disposed along the illumination path and/or signal path  535 , one or more physical parameters of illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535  is adjusted, and a change or shift in alignment position of diffraction target  520  from a reference alignment position occurs (e.g., calibrated diffraction target  520  on substrate  522 ). 
     As shown in  FIG. 5 , one or more optical filters may include spectral filter  506 , numerical aperture (NA) filter  508 , neutral density (ND) filter  510 , patterned filter  512 , and/or polarizing filter  514  disposed along an illumination path of illumination beam  504  and/or signal path  535 . In some embodiments, spectral filter  506  may include a bandpass filter, a bandpass interference filter, a notch filter, a shortpass filter, a longpass filter, a step filter, and/or a dichroic filter. In some embodiments, NA filter  508  may include a lens, an objective, and/or a prism configured to change an optical power and/or beam shape of illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . In some embodiments, ND filter  510  may be configured to change an intensity and/or spot size of illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . In some embodiments, patterned filter  512  may include a patterned reticle and/or reference pattern disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . In some embodiments, polarizing filter  514  may include a waveplate disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . For example, polarizing filter  514  may be a half-wave plate or a quarter-wave plate. 
     In some embodiments, as shown in  FIG. 5 , one or more optical filters  506 ,  508 ,  510 ,  512 ,  514  may be disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535  at various positions, including but not limited to optical filter  538 , optical filter  544 , optical filter  548 , optical filter  560 , and/or optical filter  576 . For example, optical filter  538  may be a waveplate. For example, optical filter  560  may be a patterned reticle and/or reference pattern. For example, optical filter  544  may be a patterned reticle and/or reference pattern. 
     APE processor  590  is coupled to detector controller  584  via control signal  592 . APE processor  590  is configured to measure a change and/or shift in an alignment position of diffraction target  520  caused by one or more optical filters  506 ,  508 ,  510 ,  512 ,  514 ,  538 ,  544 ,  548 ,  560 ,  576 . APE processor  590  receives measured alignment position values from detector controller  584  to calculate a reference (i.e., calibrated) alignment position for diffraction target  520 . When one or more optical filters  506 ,  508 ,  510 ,  512 ,  514 ,  538 ,  544 ,  548 ,  560 ,  576  is disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 , APE processor  590  receives a subsequent (i.e., modified) alignment position for diffraction target  520  and calculates any change between the two measured alignment positions. Based on the change, APE processor  590  determines a sensor response function of alignment sensor apparatus  500 . APE processor  590  is configured to correct an APE of alignment sensor apparatus  500  based on the sensor response function. In some embodiments, APE processor  590  is configured to correct an APE by calculating a derivative and/or a minimum value of the sensor response function for one or more physical parameters. In some embodiments, the sensor response function is calculated by APE processor  590  based on a linear model. In some embodiments, the sensor response function is calculated by APE processor  590  based on a non-linear model. 
     In an embodiment, after detector controller  584  and/or APE processor  590  determines a reference (i.e., calibrated) alignment position for diffraction target  520 , spectral filter  506  is disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535  to adjust one or more physical parameters. APE processor  590  is configured to determine the sensor response function based on the change between different wavelengths of illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . For example, illumination beam  504  may have an initial wavelength (λ) of 700 nm. A first alignment position (e.g., reference alignment position) of diffraction target  520  is measured at λ 0 =700 nm to be x 0 =0 nm. Spectral filter  506 , for example, a notch filter, adjusts the wavelength of illumination beam  504  from 700 nm to 710 nm, and a second alignment position of diffraction target  520  is measured at λ 1 =710 nm to be λ 1 =4 nm. Assuming APE is a linear function of wavelength, a sensor response function based on the change between different wavelengths is calculated by APE processor  590 , such that the sensor response function is Δx/Δλ=(4 nm−0 nm)/(710 nm−700 nm)=0.4 or APE=(0.4)·Δλ. 
     In an embodiment, after detector controller  584  and/or APE processor  590  determines a reference (i.e., calibrated) alignment position for diffraction target  520 , NA filter  508  is disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535  to adjust one or more physical parameters. APE processor  590  is configured to determine the sensor response function based on the change between one or more different diffraction order sub-beams  528 ,  530 ,  532  along signal path  535 . For example, illumination beam  504  may have an initial NA of 1.35. A first alignment position (e.g., reference alignment position) of, for example, third diffraction order sub-beam  532  is measured at NA 0 =1.35 to be x 0 =0 nm. NA filter  508  adjusts the NA of illumination beam  504  from 1.35 to 1.20, and a second alignment position of third diffraction order sub-beam  532  is measured at NA 1 =1.20 to be x 1 =3 nm. Assuming APE is a linear function of diffraction order sub-beams, a sensor response function based on the change between different diffraction order-sub-beams is calculated by APE processor  590 , such that the sensor response function is Δx/ΔNA=(3 nm−0 nm)/(1.35−1.20)=20 or APE=(20)·ΔNA (nm). 
     In an embodiment, after detector controller  584  and/or APE processor  590  determines a reference (i.e., calibrated) alignment position for diffraction target  520 , polarizing filter  514  is disposed in illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535  to adjust one or more physical parameters. APE processor  590  is configured to determine the sensor response function based on the change between different polarizations of illumination beam  504  and/or sub-beams  530 ,  532  along signal path  535 . For example, illumination beam  504  may have an initial linear polarization (θ) of 30°. A first alignment position (e.g., reference alignment position) of diffraction target  520  is measured at θ 0 =30° to be x 0 =5 nm. Polarizing filter  514  adjusts the polarization of illumination beam  504  from 30° to 45°, and a second alignment position of diffraction target  520  is measured at θ 1 =45° to be x 1 =8 nm. Assuming APE is a linear function of polarization, a sensor response function based on the change between different polarizations is calculated by APE processor  590 , such that the sensor response function is λx/Δθ=(8 nm−5 nm)/(45°−30°=0.2 or APE=(0.2)·Δθ (nm/°). 
     In some embodiments, alignment sensor apparatus  500  may include beam analyzer  536  and/or mark asymmetry reconstruction (MAR) optical branch  540 . In some embodiments, as shown in  FIG. 5 , MAR optical branch  540  may be disposed between spot mirror  516  and polarizing beamsplitter  550 . MAR optical branch  540  is configured measure and determine asymmetries in diffraction target  520 . MAR optical branch  540  may include beamsplitter  542  and MAR detector  546 . Beamsplitter  542  reflects a portion of sub-beams  530 ,  532  along signal path  535  and transmits a remaining portion of sub-beams  530 ,  532  along signal path  535  toward polarizing beamsplitter  550 . In some embodiments, as shown in  FIG. 5 , MAR optical branch  540  may include optical filter  544 . For example, optical filter  544  may be a patterned reticle and/or reference pattern. In some embodiments, MAR detector  546  is coupled to beam analyzer  536  via control signal  598 . For example, MAR detector  546  can receive and incorporate reference values for various parameters of illumination beam  504 ,  534  and/or first diffraction order sub-beam  528  measured by beam analyzer  536 , and optimize the asymmetries detected for diffraction target  520  based on these reference values 
     In some embodiments, APE processor  590  is coupled to beam analyzer  536  via control signal  596 . For example, APE processor  590  can receive and incorporate reference values for various parameters of illumination beam  504 ,  534  and/or first diffraction order sub-beam  528  measured by beam analyzer  536 , and optimize the alignment position and/or sensor response function based on these reference values. In some embodiments, APE processor  590  is coupled to MAR detector  546  via control signal  594 . For example, APE processor  590  can receive and incorporate asymmetry values of diffraction target  520  measured by MAR detector  546 , and optimize the alignment position and/or sensor response function based on these asymmetry values. 
     Exemplary Optical Systems 
     A prism is a wedge shaped transparent optical element that separates electromagnetic (EM) radiation based on refraction due to a difference in refractive indices. Generally, a prism has a flat, polished surface. The cross-section of a prism is a polygon, and the sides of the prism are anti-parallel. A prism can include a plurality of surfaces and the angles between surfaces of a prism can vary, but there must be an angle between at least two surfaces. A beam-splitting prism is a type of reflective prism configured to split a beam into two or more beams. A polarizing prism is a type of prism configured to split a beam into varying polarization components based on non-linear optics. 
     Non-linear optics (NLO) involves EM radiation in non-linear media, meaning a polarization of the media (i.e., electric dipole moment) interacts non-linearly with the electric field of the EM radiation. The normal linear relationship between an electric field and the dielectric field breaks down in non-linear media. The non-linear interaction can manifest itself as a change in polarization, frequency, phase, and/or beam path. 
     A non-linear prismatic optic can have non-linear refractive index changes. For example, a birefringent material has a refractive index that depends on the polarization and propagation direction of the EM radiation. The birefringent non-linear media causes double refraction, wherein unpolarized EM radiation is split into two beam paths of parallel and perpendicular polarization. The birefringent non-linear media consists of two polarization wave components corresponding to different refractive indices. The ordinary ray (o-ray) has polarization in a direction perpendicular to the optical axis, while the extraordinary ray (e-ray), which does not follow Snell&#39;s law, has polarization in a direction of the optical axis of the medium. 
     Optical interference corresponds to the interaction of two or more light waves yielding a resultant irradiance that deviates from the sum of the component irradiances. If two beams are to interfere to produce a stable pattern, they must have nearly the same frequency (i.e., coherent beams). Interferometry is a field of study based on the superposition of waves or beams to cause interference in order to extract information. An interferometer is a tool or device that combines two or more sources of light to create an interference pattern which can be measured and analyzed. Generally, light from a single source is split into two beams (i.e., coherent beams) that travel in different optical paths and are later combined again to produce an interference pattern. Amplitude-splitting interferometers use a partial reflector to divide the amplitude of the incident wave into separate beams which are later recombined. 
     A self-referencing interferometer (SRI) creates an interference pattern by combining an aberrated beam with a reference beam created by filtering a sample of the aberrated beam. For example, an SRI (i.e., SRI-X  554  and SRI-Y  570  as shown in  FIG. 5 ) can be a plurality of prisms combined to effectively provide two images of an alignment target, rotate one image 180° with respect to the other, and interferometrically recombine the images. The SRI can be polarization based. Generally, s-polarized and p-polarized beams can be separated by a polarizing SRI and recombined to measure an alignment position of an alignment mark. A type of system widely used in current lithographic apparatus is based on a self-referencing interferometer (e.g., rotation interferometer) as described in U.S. Publication No. 2005/0041256 A1 (Kreuzer), the full contents of which are incorporated herein by reference. 
     Exemplary Flow Diagram 
       FIG. 6  illustrates flow diagram  600  for detecting an alignment position using a self-referencing interferometer (SRI), according to an embodiment. It is to be appreciated that not all steps in  FIG. 6  may be needed to perform the disclosure provided herein. Further, some of the steps may be performed simultaneously, or in a different order than shown in  FIG. 6 . Flow diagram  600  shall be described with reference to SRI-X  554  and SRI-Y  570  as shown in  FIG. 5 . However, flow diagram  600  is not limited to those example embodiments. 
     In step  602 , as shown in the example of  FIG. 5 , an initial (e.g., reference) alignment mark position of diffraction target  520  is measured. For example, diffraction target  520  can be a symmetrical alignment mark (e.g., 180° symmetric). Illumination system  502  is configured to transmit illumination beam  504  along an illumination path toward spot mirror  516 . Spot mirror  516  may transmit illumination beam  504  toward focusing lens  518 , which focuses illumination beam  504  on diffraction target  520  on substrate  522 . Focused illumination beam  504  on diffraction target  520  creates a signal beam along signal path  535  comprising diffraction order sub-beams  528 ,  530 ,  532  reflected from diffraction target  520 . Second and third diffraction order sub-beams  530 ,  532  transmit along signal path  535  to polarizing beamsplitter  550 . Polarizing beamsplitter  550  splits signal path  535  into orthogonal polarization components (i.e., first and second polarization optical branches), and transmits s-polarized sub-beams  530 ,  532  into first polarization optical branch (“X” direction, s-polarized) and transmits p-polarized sub-beams  530 ,  532  into second polarization optical branch (“Y” direction, p-polarized). First polarization optical branch is configured to transmit s-polarized sub-beams  530 ,  532  and measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a horizontal or “X” direction, with reference to alignment axis  526 . Second polarization optical branch is similar to first polarization optical branch, and is configured to transmit p-polarized sub-beams  530 ,  532  and measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a vertical or “Y” direction, with reference to alignment axis  526 . The SRI (e.g., SRI-X  554  or SRI-Y  570  as shown in  FIG. 5 ) receives sub-beams  530 ,  532  (e.g., s-polarized sub-beams  530 ,  532  or p-polarized sub-beams  530 ,  532 ) and forms two images of diffraction target  520 . For example, the first image can be based on sub-beam  530  and the second image can be based on sub-beam  532 . 
     In step  604 , one of the images is rotated 180° with respect to the other image. For example, considering SRI-X  554  as shown in  FIG. 5 , s-polarized sub-beam  530  can be rotated 180° with respect to s-polarized sub-beam  532 . This can be accomplished with an SRI comprising two prisms whose respective optical paths are rotated in opposite directions, such that an orientation of the first image reflections (e.g., sub-beam  530 ) are reversed with respect to an orientation of the second image reflections (e.g., sub-beam  532 ). 
     In step  606 , the first and second images are recombined in the SRI. This can be accomplished by arranging the two prisms of the SRI such that the two prisms are combined (i.e., secured, bonded) at a beamsplitter interface. 
     In step  608 , recombined image can be detected by a detector. For example, considering SRI-X  554  as shown in  FIG. 5 , the recombined image of s-polarized sub-beam  530  (i.e., rotated 180°) and s-polarized sub-beam  532  can be detected by first position detector  566  and/or second position detector  564 . 
     In step  610 , the center of the alignment mark position of diffraction target  520  can be measured. For example, detector controller  584  can be configured to measure the center of the alignment mark position of diffraction target  520  based on the image beams (e.g., difference and sum) from sub-beams  530 ,  532  outputted from the SRI (e.g., SRI-X  554  and/or SRI-Y  570  as shown in  FIG. 5 ). In some embodiments, detector controller  584  is configured to measure any change, shift, and/or deviation in an alignment position of diffraction target  520  in a horizontal or “X” direction and/or in a vertical or “Y” direction. 
     Exemplary Self-Referencing Interferometers 
       FIGS. 7A-7C  illustrate perspective transparent views of an exemplary self-referencing interferometer (SRI) system, according to an exemplary embodiment.  FIG. 7D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 7E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 7F  illustrates a front plan transparent view of the exemplary embodiment. SRI system  700  can include a first prism  710  and a second prism  720  adjoined (i.e., combined) to each other along a beamsplitter interface  730 . In some embodiments, SRI system  700  can be SRI-X  554  or SRI-Y  570  as shown in  FIG. 5 . In some embodiments, SRI system  700  can be configured to receive sub-beams  530 ,  532  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ) as shown in  FIG. 5 . In some embodiments, first and second prisms  710 ,  720  can include any suitable materials as known to a person having ordinary skill in the art for making optical elements or prisms. For example, first and second prisms  710 ,  720  can be made from glass. 
     As shown in  FIGS. 7A and 7B , first prism  710  and second prism  720  can be adjoined (i.e., combined) to each other along beamsplitter interface  730 . In some embodiments, beamsplitter interface  730  can include a polarizing coating. In some embodiments, first prism  710  and/or second prism  720  can include one or more phase compensating coatings. 
     As shown in  FIGS. 7C-7F , SRI system  700  can receive an incident beam  740  and output a recombined beam  750 . For example, incident beam  740  can be sub-beam  530  and/or sub-beam  532  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ) as shown in  FIG. 5 . As incident beam  740  enters first prism  710  through input surface  712 , beamsplitter interface  730  splits incident beam  740  into a first optical path  732  and a second optical path  734 . First optical path  732  and second optical path  734  are rotated in opposite directions in SRI system  700 . An orientation of optical reflections along first optical path  732  are reversed with respect to an orientation of optical reflections along second optical path  734 . First and second optical paths  732 ,  734  are recombined at beamsplitter interface  730  to produce recombined beam  750 . Recombined beam  750  exits second prism  720  through output surface  722 . 
     In some embodiments, incident beam  740  can be a non-polarized beam. In some embodiments, incident beam  740  can be a polarized beam. For example, as shown in  FIG. 5 , incident beam  740  can be sub-beams  530 ,  532  polarized by polarizing beamsplitter  550  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ). First and second prisms  710 ,  720  are identical in shape. It is noted that input surface  712  and output surface  722  are interchangeable since first and second prisms  710 ,  720  are identical in shape and first and second optical paths  732 ,  734  are symmetric about beamsplitter interface  730 . In some embodiments, first and second prisms  710 ,  720  can be identical pieces of glass. In some embodiments, one of first and second prisms  710 ,  720  can include one or more phase compensating coatings. For example, first prism  710  can include one or more phase compensating coatings while second prism  720  omits any phase compensating coatings. In some embodiments, beamsplitter interface  730  of first and second prisms  710 ,  720  can include a polarizing coating configured to separate incident beam  740  into a first polarization plane (e.g., first optical path  732 ) and a second polarization plane (e.g., second optical path  734 ). 
     In some embodiments, SRI system  700  splits incident beam  740  into a first image along first optical path  732  and a second image along second optical path  734 , rotates the second image along second optical path  734  by 180° with respect to the first image along first optical path  732 , recombines the first image along first optical path  732  with the rotated second image along second optical path  734 , and outputs recombined beam  750 . 
     In some embodiments, SRI system  700  can include only two identical components. In some embodiments, SRI system  700  can further include a rectangular beamsplitter prism adjoined to first and second prisms  710 ,  720  and having a beamsplitter interface. For example, rectangular beamsplitter prism can include input surface  712  for incident beam  740 , beamsplitter interface  730  to direct first and second optical paths  732 ,  734  to first and second prisms  710 ,  720 , respectively, and output surface  722  for recombined beam  750 . In some embodiments, SRI system  700  can include a rectangular beamsplitter prism coupled to first and second prisms  710 ,  720  and an additional identical set of first and second prisms  710 ,  720 . 
       FIGS. 8A-8C  illustrate perspective transparent views of an exemplary self-referencing interferometer (SRI) system, according to an exemplary embodiment.  FIG. 8D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 8E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 8F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  800  shown in  FIGS. 8A-8F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the similar features of the embodiments of SRI system  800  shown in  FIGS. 8A-8F . Description of SRI system  800  is omitted in the interest of brevity. The main difference between the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  800  shown in  FIGS. 8A-8F  is that first and second prisms  810 ,  820  are not identical in shape, have a different shape than first and second prisms  710 ,  720  of SRI system  700 , and SRI system  800  omits phase compensating coatings since first reflection planes (e.g., first optical path  832 ) in first prism  810  are either perpendicular or parallel to a first polarization plane of incident beam  840  in first optical path  832  and second reflection planes (e.g., second optical path  834 ) in second prism  820  are either perpendicular or parallel to a second polarization plane of incident beam  840  in second optical path  834  (i.e., SRI system  800  eliminates out of plane folds). 
       FIGS. 9A-9C  illustrate perspective transparent views of an exemplary self-referencing interferometer (SRI) system, according to an exemplary embodiment.  FIG. 9D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 9E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 9F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  900  shown in  FIGS. 9A-9F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the similar features of the embodiments of SRI system  900  shown in  FIGS. 9A-9F . Description of SRI system  900  is omitted in the interest of brevity. The main difference between the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  900  shown in  FIGS. 9A-9F  is that first and second prisms  910 ,  920  are identical in shape but have a different shape than first and second prisms  710 ,  720  of SRI system  700 , and SRI system  900  omits phase compensating coatings since reflection planes (e.g., first optical path  932 ) in first prism  910  are either perpendicular or parallel to a first polarization plane of incident beam  940  in first optical path  932  and second reflection planes (e.g., second optical path  934 ) in second prism  920  are either perpendicular or parallel to a second polarization plane of incident beam  940  in second optical path  934  (i.e., SRI system  900  is phase compensated). 
       FIGS. 10A-10C  illustrate perspective transparent views of an exemplary self-referencing interferometer (SRI) system, according to an exemplary embodiment.  FIG. 10D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 10E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 10F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  1000  shown in  FIGS. 10A-10F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the similar features of the embodiments of SRI system  1000  shown in  FIGS. 10A-10F . Description of SRI system  1000  is omitted in the interest of brevity. The main difference between the embodiments of SRI system  700  shown in  FIGS. 7A-7F  and the embodiments of SRI system  1000  shown in  FIGS. 10A-10F  is that first and second prisms  1010 ,  1020  are identical in shape but have a different shape than first and second prisms  710 ,  720  of SRI system  700 , and SRI system  1000  omits phase compensating coatings since reflection planes (e.g., first optical path  1032 ) in first prism  1010  are either perpendicular or parallel to a first polarization plane of incident beam  1040  in first optical path  1032  and second reflection planes (e.g., second optical path  1034 ) in second prism  1020  are either perpendicular or parallel to a second polarization plane of incident beam  1040  in second optical path  1034  (i.e., SRI system  1000  is phase compensated). 
     Exemplary Dual Self-Referencing Interferometers 
     Similar to a self-referencing interferometer (SRI), a dual self-referencing interferometer (DSRI) combines two prism assemblies into a single system. For example, two prism assemblies can be combined on a common optic (e.g., plate), along a beamsplitter interface, into adjacent prisms (e.g., side by side), or into a rectangular beamsplitter prism with prism assemblies on the faces of the rectangular beamsplitter prism to form a DSRI. 
       FIG. 11  illustrates a perspective transparent view of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment. DSRI system  1100  can include a first prism assembly  1110  and a second prism assembly  120  disposed on a plate  1102 . In some embodiments, first and second prism assemblies  1110 ,  1120  can be SRI systems  700 ,  800 ,  900 ,  1000  described above and shown in  FIGS. 7A-7F, 8A-8F, 9A-9F, and 10A-10F . In some embodiments, DSRI system  1100  can be SRI-X  554  or SRI-Y  570  as shown in  FIG. 5 . In some embodiments, DSRI system  1100  can be configured to receive sub-beams  530 ,  532  (e.g., s-polarized sub-beams  530 ,  532  and p-polarized sub-beams  530 ,  532 ) as shown in  FIG. 5 . In some embodiments, first and second prism assemblies  1110 ,  1120  can include any suitable materials as known to a person having ordinary skill in the art for making optical elements or prisms. For example, first and second prism assemblies  1110 ,  1120  can be made from glass. 
     As shown in  FIG. 11 , first prism assembly  1110  and second prism assembly  1120  can be adjacent to each other. In some embodiments, DSRI system  1100  can include a plate  1102  upon which first and second prism assemblies  1110 ,  1120  can be supported (i.e., combined). In some embodiments, first prism assembly  1110  and/or second prism assembly  1120  can be SRI systems  700 ,  800 ,  900 ,  1000  as shown in  FIGS. 7A-7F, 8A-8F, 9A-9F, and 10A-10F , or some combination thereof. 
       FIGS. 12A-12C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 12D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 12E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 12F  illustrates a front plan transparent view of the exemplary embodiment. DSRI system  1200  can include a first prism assembly  1210  and a second prism assembly  1220  coupled to each other by rectangular beamsplitter prism  1204  with beamsplitter interface  1230  (i.e., prism assemblies  1210 ,  1220  disposed on faces of rectangular beamsplitter prism  1204 ). In some embodiments, first and second prism assemblies  1210 ,  1220  are similar to SRI systems  700 ,  800 ,  900 ,  1000  described above and shown in  FIGS. 7A-7F, 8A-8F, 9A-9F, and 10A-10F . In some embodiments, DSRI system  1200  can be SRI-X  554  or SRI-Y  570  as shown in  FIG. 5 . In some embodiments, DSRI system  1200  can be configured to receive sub-beams  530 ,  532  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ) as shown in  FIG. 5 . In some embodiments, first and second prism assemblies  1210 ,  1220  can include any suitable materials as known to a person having ordinary skill in the art for making optical elements or prisms. For example, first and second prism assemblies  1210 ,  1220  can be made from glass. 
     As shown in  FIGS. 12A and 12B , first prism assembly  1210  and second prism assembly  1220  can be coupled to each other by rectangular beamsplitter prism  1204  with beamsplitter interface  1230  (i.e., prism assemblies  1210 ,  1220  disposed on faces of rectangular beamsplitter prism  1204 ). In some embodiments, first prism assembly  1210  and/or second prism assembly  1220  can be SRI systems  700 ,  800 ,  900 ,  1000  as shown in  FIGS. 7A-7F, 8A-8F, 9A-9F , and  10 A- 10 F, or some combination thereof. In some embodiments, beamsplitter interface  1230  of rectangular beamsplitter prism  1204  can include a polarizing coating. In some embodiments, first prism assembly  1210  and/or second prism assembly  1220  can include one or more phase compensating coatings. 
     As shown in  FIGS. 12C-12F , DSRI system  1200  can receive a first incident beam  1240  and a second incident beam  1242 , and output a first recombined beam  1250  and a second recombined beam  1252 , respectively. For example, first incident beam  1240  can be sub-beam  530  (e.g., s-polarized sub-beam  530 , p-polarized sub-beam  530 ) and second incident beam  1242  can be sub-beam  532  (e.g., s-polarized sub-beam  532 , p-polarized sub-beam  532 ) as shown in  FIG. 5 . As first and second incident beams  1240 ,  1242  enter rectangular beamsplitter prism  1204  through input surface  1212 , beamsplitter interface  1230  splits first incident beam  1240  into a first optical path  1232  and a second optical path  1234  and splits second incident beam  1242  into a third optical path  1236  and a fourth optical path  1238 . In DSRI system  1200 , first optical path  1232  and second optical path  1234  are rotated in opposite directions in first prism assembly  1210 , and third optical path  1236  and fourth optical path  1238  are rotated in opposite directions in second prism assembly  1220 . An orientation of optical reflections along first optical path  1232  are reversed with respect to an orientation of optical reflections along second optical path  1234 , and an orientation of optical reflections along third optical path  1236  are reversed with respect to an orientation of optical reflections along fourth optical path  1238 . First and second optical paths  1232 ,  1234  are recombined at beamsplitter interface  1230  to produce first recombined beam  1250 , and third and fourth optical paths  1236 ,  1238  are recombined at beamsplitter interface  1230  to produce second recombined beam  1252 . First and second recombined beams  1250 ,  1252  exit rectangular beamsplitter prism  1204  through output surface  1222 . 
     In some embodiments, first and second incident beams  1240 ,  1242  can be non-polarized beams. In some embodiments, first and second incident beams  1240 ,  1242  can be polarized beams. For example, as shown in  FIG. 5 , first incident beam  1240  can be sub-beam  530  and second incident beam  1242  can be sub-beam  532 , each polarized by polarizing beamsplitter  550  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ). First and second prism assemblies  1210 ,  1220  are identical in shape. For example, as shown in  FIGS. 12A-12F , DSRI system  1200  is constructed from four identical prisms forming first and second prism assemblies  1210 ,  1220  disposed on rectangular beamsplitter prism  1204 . It is noted that input surface  1212  and output surface  1222  are interchangeable since first and second prism assemblies  1210 ,  1220  are identical in shape, and first and second optical paths  1232 ,  1234  are symmetric about beamsplitter interface  1230  and third and fourth optical paths  1236 ,  1238  are symmetric about beamsplitter interface  1230 . In some embodiments, first and second prism assemblies  1210 ,  1220  can be identical pieces of glass. In some embodiments, one of first and second prism assemblies  1210 ,  1220  can include one or more phase compensating coatings. For example, first prism assembly  1210  can include one or more phase compensating coatings while second prism assembly  1220  omits any phase compensating coatings. In some embodiments, beamsplitter interface  1230  of rectangular beamsplitter prism  1204  can include a polarizing coating configured to separate first incident beam  1240  into a first polarization plane (e.g., first optical path  1232 ) and a second polarization plane (e.g., second optical path  1234 ) and separate second incident beam  1242  into the first polarization plane (e.g., third optical path  1236 ) and the second polarization plane (e.g., fourth optical path  1238 ). 
     In some embodiments, DSRI system  1200  splits first incident beam  1240  into a first image along first optical path  1232  and a second image along second optical path  1234  and splits second incident beam  1242  into a third image along third optical path  1236  and a fourth image along fourth optical path  1238 , rotates the second image along second optical path  1234  by 180° with respect to the first image along first optical path  1232  and rotates the fourth image along fourth optical path  1238  by 180° with respect to the third image along third optical path  1236 , recombines the first image along first optical path  1232  with the rotated second image along second optical path  1234  and recombines the third image along third optical path  1236  with the rotated fourth image along fourth optical path  1238 , and outputs first recombined beam  1250  and second recombined beam  1252 . 
     In some embodiments, DSRI system  1200  can include rectangular beamsplitter prism  1204  and four identical components (e.g., first prism assembly  1210  and second prism assembly  1220 ). 
       FIGS. 13A-13C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 13D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 13E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 13F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1300  shown in  FIGS. 13A-13F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the similar features of the embodiments of DSRI system  1300  shown in  FIGS. 13A-13F . Description of DSRI system  1300  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1300  shown in  FIGS. 13A-13F  is that first and second prism assemblies  1310 ,  1320  are identical in shape but have a different shape than first and second prism assemblies  1210 ,  1220  of DSRI system  1200 . 
       FIGS. 14A-14C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 14D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 14E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 14F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1400  shown in  FIGS. 14A-14F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the similar features of the embodiments of DSRI system  1400  shown in  FIGS. 14A-14F . Description of DSRI system  1400  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1400  shown in  FIGS. 14A-14F  is that first and second prism assemblies  1410 ,  1420  are identical in shape but have a different shape than first and second prism assemblies  1210 ,  1220  of DSRI system  1200 , and DSRI system  1400  omits phase compensating coatings since first reflection planes (e.g., first optical path  1432 ) in first prism assembly  1410  are either perpendicular or parallel to a first polarization plane of first incident beam  1440  in first optical path  1432  and second reflection planes (e.g., second optical path  1434 ) in second prism assembly  1420  are either perpendicular or parallel to a second polarization plane of first incident beam  1440  in second optical path  1434 , and third reflection planes (e.g., third optical path  1436 ) in first prism assembly  1410  are either perpendicular or parallel to a first polarization plane of second incident beam  1442  in third optical path  1436  and fourth reflection planes (e.g., fourth optical path  1438 ) in second prism assembly  1420  are either perpendicular or parallel to a second polarization plane of second incident beam  1442  in fourth optical path  1438  (i.e., DSRI system  1400  eliminates out of plane folds). 
       FIGS. 15A-15C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 15D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 15E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 15F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1500  shown in  FIGS. 15A-15F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the similar features of the embodiments of DSRI system  1500  shown in  FIGS. 15A-15F . Description of DSRI system  1500  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1500  shown in  FIGS. 15A-15F  is that first and second prism assemblies  1510 ,  1520  are identical in shape but have a different shape than first and second prism assemblies  1210 ,  1220  of DSRI system  1200 , and DSRI system  1500  omits phase compensating coatings since first reflection planes (e.g., first optical path  1532 ) in first prism assembly  1510  are either perpendicular or parallel to a first polarization plane of first incident beam  1540  in first optical path  1532  and second reflection planes (e.g., second optical path  1534 ) in second prism assembly  1520  are either perpendicular or parallel to a second polarization plane of first incident beam  1540  in second optical path  1534 , and third reflection planes (e.g., third optical path  1536 ) in first prism assembly  1510  are either perpendicular or parallel to a first polarization plane of second incident beam  1542  in third optical path  1536  and fourth reflection planes (e.g., fourth optical path  1538 ) in second prism assembly  1520  are either perpendicular or parallel to a second polarization plane of second incident beam  1542  in fourth optical path  1538  (i.e., DSRI system  1500  eliminates out of plane folds). 
       FIGS. 16A-16C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 16D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 16E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 16F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1600  shown in  FIGS. 16A-16F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the similar features of the embodiments of DSRI system  1600  shown in  FIGS. 16A-16F . Description of DSRI system  1600  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1600  shown in  FIGS. 16A-16F  is that first and second prism assemblies  1610 ,  1620  are identical in shape but have a different shape than first and second prism assemblies  1210 ,  1220  of DSRI system  1200 . 
       FIGS. 17A-17C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 17D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 17E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 17F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1700  shown in  FIGS. 17A-17F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the similar features of the embodiments of DSRI system  1700  shown in  FIGS. 17A-17F . Description of DSRI system  1700  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1200  shown in  FIGS. 12A-12F  and the embodiments of DSRI system  1700  shown in  FIGS. 17A-17F  is that first and second prism assemblies  1710 ,  1720  are identical in shape but have a different shape than first and second prism assemblies  1210 ,  1220  of DSRI system  1200 . 
       FIGS. 18A-18C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 18D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 18E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 18F  illustrates a front plan transparent view of the exemplary embodiment. DSRI system  1800  can include a first prism assembly  1810  and a second prism assembly  1820  coupled to each other along a beamsplitter interface  1830 . In some embodiments, DSRI system  1800  can be SRI-X  554  or SRI-Y  570  as shown in  FIG. 5 . In some embodiments, DSRI system  1800  can be configured to receive sub-beams  530 ,  532  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ) as shown in  FIG. 5 . In some embodiments, first and second prism assemblies  1810 ,  1820  can include any suitable materials as known to a person having ordinary skill in the art for making optical elements or prisms. For example, first and second prism assemblies  1810 ,  1820  can be made from glass. 
     As shown in  FIGS. 18A and 18B , first prism assembly  1810  and second prism assembly  1820  can be adjoined (i.e., combined) to each other along beamsplitter interface  1830 . In some embodiments, beamsplitter interface  1830  can include a polarizing coating. In some embodiments, first prism assembly  1810  and/or second prism assembly  1820  can include one or more phase compensating coatings. 
     As shown in  FIGS. 18C-18F , DSRI system  1800  can receive a first incident beam  1840  and a second incident beam  1842 , and output a first recombined beam  1850  and a second recombined beam  1852 , respectively. For example, first incident beam  1840  can be sub-beam  530  (e.g., s-polarized sub-beam  530 , p-polarized sub-beam  530 ) and second incident beam  1842  can be sub-beam  532  (e.g., s-polarized sub-beam  532 , p-polarized sub-beam  532 ) as shown in  FIG. 5 . As first and second incident beams  1840 ,  1842  enter first prism assembly  1810  through input surface  1812 , beamsplitter interface  1830  splits first incident beam  1840  into a first optical path  1832  and a second optical path  1834  and splits second incident beam  1842  into a third optical path  1836  and a fourth optical path  1838 . In DSRI system  1800 , first optical path  1832  and second optical path  1834  are rotated in opposite directions in first and second prism assemblies  1810 ,  1820 , and third optical path  1836  and fourth optical path  1838  are rotated in opposite directions in first and second prism assemblies  1810 ,  1820 . An orientation of optical reflections along first optical path  1832  are reversed with respect to an orientation of optical reflections along second optical path  1834 , and an orientation of optical reflections along third optical path  1836  are reversed with respect to an orientation of optical reflections along fourth optical path  1838 . First and second optical paths  1832 ,  1834  are recombined at beamsplitter interface  1830  to produce first recombined beam  1850 , and third and fourth optical paths  1836 ,  1838  are recombined at beamsplitter interface  1830  to produce second recombined beam  1852 . First and second recombined beams  1850 ,  1852  exit second prism assembly  1820  through output surface  1822 . 
     In some embodiments, first and second incident beams  1840 ,  1842  can be non-polarized beams. In some embodiments, first and second incident beams  1840 ,  1842  can be polarized beams. For example, as shown in  FIG. 5 , first incident beam  1840  can be sub-beam  530  and second incident beam  1842  can be sub-beam  532 , each polarized by polarizing beamsplitter  550  (e.g., s-polarized sub-beams  530 ,  532 , p-polarized sub-beams  530 ,  532 ). It is noted that input surface  1812  and output surface  1822  are interchangeable. In some embodiments, one of first and second prism assemblies  1810 ,  1820  can include one or more phase compensating coatings. For example, first prism assembly  1810  can include one or more phase compensating coatings while second prism assembly  1820  omits any phase compensating coatings. In some embodiments, beamsplitter interface  1830  can include a polarizing coating configured to separate first incident beam  1840  into a first polarization plane (e.g., first optical path  1832 ) and a second polarization plane (e.g., second optical path  1834 ) and separate second incident beam  1842  into the first polarization plane (e.g., third optical path  1836 ) and the second polarization plane (e.g., fourth optical path  1838 ). 
     In some embodiments, DSRI system  1800  splits first incident beam  1840  into a first image along first optical path  1832  and a second image along second optical path  1834  and splits second incident beam  1842  into a third image along third optical path  1836  and a fourth image along fourth optical path  1838 , rotates the second image along second optical path  1834  by 180° with respect to the first image along first optical path  1832  and rotates the fourth image along fourth optical path  1838  by 180° with respect to the third image along third optical path  1836 , recombines the first image along first optical path  1832  with the rotated second image along second optical path  1834  and recombines the third image along third optical path  1836  with the rotated fourth image along fourth optical path  1838 , and outputs first recombined beam  1850  and second recombined beam  1852 . 
     In some embodiments, DSRI system  1800  can further include a rectangular beamsplitter prism adjoined to first and second prism assemblies  1810 ,  1820  and having a beamsplitter interface. For example, rectangular beamsplitter prism can include input surface  1812  for first and second incident beams  1840 ,  1842 , beamsplitter interface  1830  to direct first and third optical paths  1832 ,  1836  to first prism assembly  1810  and second and fourth optical paths  1834 ,  1838  to second prism assembly  1820 , and output surface  1822  for first and second recombined beams  1850 ,  1852 . In some embodiments, DSRI system  1800  can include a rectangular beamsplitter prism coupled to first and second prism assemblies  1810 ,  1820  and an additional identical set of first and second prisms  1810 ,  1820 . 
       FIGS. 19A and 19B  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 19C  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 19D  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 19E  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the embodiments of DSRI system  1900  shown in  FIGS. 19A-19E  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the similar features of the embodiments of DSRI system  1900  shown in  FIGS. 19A-19E . Description of DSRI system  1900  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the embodiments of DSRI system  1900  shown in  FIGS. 19A-19E  is that first and second prism assemblies  1910 ,  1920  are identical in shape. 
       FIGS. 20A-20C  illustrate perspective transparent views of an exemplary dual self-referencing interferometer (DSRI) system, according to an exemplary embodiment.  FIG. 20D  illustrates a cross-sectional transparent view of the exemplary embodiment.  FIG. 20E  illustrates a top plan transparent view of the exemplary embodiment.  FIG. 20F  illustrates a front plan transparent view of the exemplary embodiment. The embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the embodiments of DSRI system  2000  shown in  FIGS. 20A-20F  are similar. Similar reference numbers are used to indicate similar features of the embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the similar features of the embodiments of DSRI system  2000  shown in  FIGS. 20A-20F . Description of DSRI system  2000  is omitted in the interest of brevity. The main difference between the embodiments of DSRI system  1800  shown in  FIGS. 18A-18F  and the embodiments of DSRI system  2000  shown in  FIGS. 20A-20F  is that first and second prism assemblies  2010 ,  2020  are identical in shape. 
     The embodiments may further be described using the following clauses: 
     1. A self-referencing interferometer (SRI) system for an alignment sensor apparatus comprising: 
     a first prism having an input surface for an incident beam; and 
     a second prism coupled to the first prism and having an output surface for a recombined beam, 
     wherein the recombined beam comprises a first image and a second image rotated by 180 degrees with respect to the first image, wherein the first and second prisms are identical in shape. 
     2. The SRI system of clause 1, wherein the first and second prisms are adjoined along a beamsplitter interface, the beamsplitter interface comprising a polarizing coating.
 
3. The SRI system of clause 2, wherein:
 
     first reflection planes in the first prism are either perpendicular or parallel to a first polarization plane of the incident beam; and 
     second reflection planes in the second prism are either perpendicular or parallel to a second polarization plane of the incident beam. 
     4. The SRI system of clause 1, wherein the first and second prisms comprise an absence of any phase compensating coatings.
 
5. The SRI system of clause 1, wherein the first prism or the second prism comprises one or more phase compensating coatings.
 
6. The SRI system of clause 1, further comprising a rectangular beamsplitter prism adjoined to the first and second prisms and comprising a beamsplitter interface, the beamsplitter interface comprising a polarizing coating.
 
7. The SRI system of clause 1, further comprising a plate upon which first and second prisms are supported.
 
8. The SRI system of clause 1, wherein a number of contact reflection points of a first optical path and a second optical path within the first and second prisms is no greater than ten.
 
9. The SRI system of clause 8, wherein the number of contact reflection points is no greater than eight.
 
10. A dual self-referencing interferometer (DSRI) system for an alignment sensor apparatus comprising:
 
     a first prism assembly having an input surface for a first incident beam and a second incident beam; and 
     a second prism assembly coupled to the first prism assembly and having an output surface for a first recombined beam and a second recombined beam, 
     wherein the first recombined beam comprises a first image and a second image rotated by 180 degrees with respect to the first image, wherein the second recombined beam comprises a third image and a fourth image rotated by 180 degrees with respect to the third image, wherein the first and second prism assemblies are identical in shape. 
     11. The DSRI system of clause 10, wherein the first and second prism assemblies are disposed on a plate.
 
12. The DSRI system of clause 10, wherein the first and second prism assemblies are adjoined along a beamsplitter interface, the beamsplitter interface comprising a polarizing coating.
 
13. The DSRI system of clause 12, wherein:
 
     first reflection planes in the first prism assembly are either perpendicular or parallel to a first polarization plane of the first and second incident beams; and 
     second reflection planes in the second prism assembly are either perpendicular or parallel to a second polarization plane of the first and second incident beams. 
     14. The DSRI system of clause 10, further comprising a rectangular beamsplitter prism adjoined to the first and second prism assemblies and comprising a beamsplitter interface, the beamsplitter interface comprising a polarizing coating.
 
15. The DSRI system of clause 14, wherein:
         first reflection planes in the first prism assembly are either perpendicular or parallel to a first polarization plane of the first and second incident beams; and       

     second reflection planes in the second prism assembly are either perpendicular or parallel to a second polarization plane of the first and second incident beams. 
     16. The DSRI system of clause 10, wherein the first and second prism assemblies comprise an absence of any phase compensating coatings.
 
17. The DSRI system of clause 10, wherein the first prism assembly or the second prism assembly comprises one or more phase compensating coatings.
 
18. A lithographic apparatus comprising:
 
     a first illumination optical system configured to illuminate a diffraction pattern; 
     a projection optical system configured to project an image of the diffraction pattern onto a substrate; and 
     an alignment sensor apparatus configured to correct an alignment position error of the lithographic apparatus, the alignment sensor apparatus comprising:
         a second illumination optical system configured to transmit at least one illumination beam of radiation along an illumination path;   a first optical system comprising a first optic and a second optic, and configured to transmit the illumination beam toward the diffraction pattern on the substrate and transmit a signal beam comprising diffraction order sub-beams reflected from the diffraction pattern along a signal path;   a second optical system comprising a first polarizing optic configured to separate and transmit the signal beam into a first polarization optical branch and a second polarization optical branch based on the polarization of the signal beam;   a detector system comprising one or more detectors, and configured to measure an alignment position of the diffraction pattern based on the signal beam outputted from the first polarization branch and the second polarization branch; and   a processor coupled to the detector system, and configured to measure a change in the alignment position of the diffraction pattern,   wherein the first polarization optical branch comprises a first prism assembly and the second polarization optical branch comprises a second prism assembly, and   wherein the first and second prism assemblies are identical in shape.
 
19. The lithographic apparatus of clause 18, wherein the first and second prism assemblies are adjoined to a rectangular beamsplitter prism.
 
20. The lithographic apparatus of clause 18, wherein the first and second prism assemblies comprise an absence of any phase compensating coatings.
       

     Although specific reference may be made in this text to the use of lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications, such as the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat-panel displays, LCDs, thin-film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “wafer” or “die” herein may be considered as synonymous with the more general terms “substrate” or “target portion”, respectively. The substrate referred to herein may be processed, before or after exposure, in for example a track unit (a tool that typically applies a layer of resist to a substrate and develops the exposed resist), a metrology unit and/or an inspection unit. Where applicable, the disclosure herein may be applied to such and other substrate processing tools. Further, the substrate may be processed more than once, for example in order to create a multi-layer IC, so that the term substrate used herein may also refer to a substrate that already contains multiple processed layers. 
     Although specific reference may have been made above to the use of embodiments of the invention in the context of optical lithography, it will be appreciated that the invention may be used in other applications, for example imprint lithography, and where the context allows, is not limited to optical lithography. In imprint lithography a topography in a patterning device defines the pattern created on a substrate. The topography of the patterning device may be pressed into a layer of resist supplied to the substrate whereupon the resist is cured by applying electromagnetic radiation, heat, pressure or a combination thereof. The patterning device is moved out of the resist leaving a pattern in it after the resist is cured. 
     It is to be understood that the phraseology or terminology herein is for the purpose of description and not of limitation, such that the terminology or phraseology of the present specification is to be interpreted by those skilled in relevant art(s) in light of the teachings herein. 
     The term “substrate” as used herein describes a material onto which material layers are added. In some embodiments, the substrate itself may be patterned and materials added on top of it may also be patterned, or may remain without patterning. 
     Embodiments of the invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the invention may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, a machine-readable medium may include read only memory (ROM); random access memory (RAM); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical, or other forms of propagated signals, and others. Further, firmware, software, routines, and/or instructions may be described herein as performing certain actions. However, it should be appreciated that such descriptions are merely for convenience and that such actions in fact result from computing devices, processors, controllers, or other devices executing the firmware, software, routines, and/or instructions. 
     The following examples are illustrative, but not limiting, of the embodiments of this disclosure. Other suitable modifications and adaptations of the variety of conditions and parameters normally encountered in the field, and which would be apparent to those skilled in the relevant art(s), are within the spirit and scope of the disclosure. 
     Although specific reference may be made in this text to the use of the apparatus and/or system according to the invention in the manufacture of ICs, it should be explicitly understood that such an apparatus and/or system has many other possible applications. For example, it can be employed in the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, LCD panels, thin-film magnetic heads, etc. The skilled artisan will appreciate that, in the context of such alternative applications, any use of the terms “reticle,” “wafer,” or “die” in this text should be considered as being replaced by the more general terms “mask,” “substrate,” and “target portion,” respectively. 
     While specific embodiments of the invention have been described above, it will be appreciated that the invention may be practiced otherwise than as described. The description is not intended to limit the invention. 
     It is to be appreciated that the Detailed Description section, and not the Summary and Abstract sections, is intended to be used to interpret the claims. The Summary and Abstract sections may set forth one or more but not all exemplary embodiments of the present invention as contemplated by the inventor(s), and thus, are not intended to limit the present invention and the appended claims in any way. 
     The present invention has been described above with the aid of functional building blocks illustrating the implementation of specified functions and relationships thereof. The boundaries of these functional building blocks have been arbitrarily defined herein for the convenience of the description. Alternate boundaries can be defined so long as the specified functions and relationships thereof are appropriately performed. 
     The foregoing description of the specific embodiments will so fully reveal the general nature of the invention that others can, by applying knowledge within the skill of the art, readily modify and/or adapt for various applications such specific embodiments, without undue experimentation, without departing from the general concept of the present invention. Therefore, such adaptations and modifications are intended to be within the meaning and range of equivalents of the disclosed embodiments, based on the teaching and guidance presented herein. 
     The breadth and scope of the present invention should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.