Abstract:
A method for designing a base includes (1) selecting a location of a first center of expansion of a child part (CE child ) relative to a parent part; (2) determining a location of a second center of expansion of a bond joint (CE bond ) bonding the child part to the base; and (3) determining a location of a third center of expansion of the base (CE base ) on a centerline, which is defined by the CE child  and the CE bond , so the CE child  does not substantially move relative to the parent part under a temperature change. To determine the location of the CE base , the method further includes (a) determining a length change to the child part from the CE bond  to the CE child  under the temperature change; (b) determining a length of the base that produces the same length change under the temperature change; and (c) locating the CE base  at the length away from the CE bond .

Description:
DESCRIPTION OF RELATED ART 
   For a child part that is bolted on to a parent structure, maintaining an invariant relationship is one of the cardinal issues for precision engineers. There are typically two states that impose different athermalization requirements: operation and shipment/storage. During operation, a distance measuring interferometer (DMI) that is bolted onto a metrology frame must maintain a constant position relative to the metrology frame despite thermal cycling. During shipment/storage, the DMI is not being used but it will see environmental excursions orders of magnitudes higher than it will operationally; which is more likely to cause a permanent misalignment of the system. Thus, a method and an apparatus are needed to athermalize a child part relative to the parent structure. 
   SUMMARY 
   In one embodiment of the invention, a method for designing a base includes (1) selecting a location of a first center of expansion of a child part (CE child ) relative to a parent part; (2) determining a location of a second center of expansion of a bond joint (CE bond ) bonding the child part to the base; and (3) determining a location of a third center of expansion of the base (CE base ) on a centerline, which is defined by the CE child  and the CE bond , so that the CE child  does not substantially move relative to the parent part under a temperature change. To determine the location of the CE base , the method further includes (a) determining a length change to the child part from the CE bond  to the CE child  under the temperature change; (b) determining a length of the base that produces the same length change under the temperature change; and (c) locating the CE base  away from the CE bond  at the length determined from step (b). 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  illustrates an assembly of a child part, a base part, and a parent structure in one embodiment of the invention. 
       FIGS. 2A and 2B  illustrate a concept for setting a center of expansion of the base part that causes a center of expansion of the child part to remain substantially motionless relative to the parent structure in one embodiment of the invention. 
       FIGS. 3A and 3B  illustrate a refined concept for setting a center of expansion of the base part that causes a center of expansion of the child part to remain substantially motionless relative to the parent structure in one embodiment of the invention. 
       FIGS. 4A and 4B  illustrate flexures of the base part in one embodiment of the invention. 
       FIGS. 5A ,  5 B,  5 C, and  5 D illustrate a spring force balance analysis of a base part with asymmetric flexure placement in one embodiment of the invention. 
       FIGS. 6A ,  6 B,  6 C,  6 D, and  6 E are schematics of a child part bonded atop a base part in one embodiment of the invention. 
       FIG. 7  is a flowchart of a process for designing a base part that causes a center of expansion of a child part to remain substantially motionless relative to the parent structure in one embodiment of the invention. 
   

   Use of the same reference numbers in different figures indicates similar or identical elements. 
   DETAILED DESCRIPTION 
     FIG. 1  illustrates an assembly  10  in one embodiment of the invention. A child part  13  is bonded atop a base part  14  by a bond joint  15 . Base part  14  has flexures  16  (only one is labeled for clarity) for mounting child part  13  atop a parent structure  18 . The flexure feet have small contact surfaces designed in conjunction with the flexure heights so that the shear force from differential thermal expansion between child part  13  and parent structure  18  is less than the static friction force between the flexure feet and parent structure  18 . 
     FIG. 2A  illustrates a concept for designing base part  14  in one embodiment of the invention. A position for the center of expansion of child part  13  (hereafter “CE child ”) is selected. CE child  is a point on child part  13  that is desired to remain substantially motionless relative to parent structure  18  when the temperature changes. 
   In one embodiment, child part  13  is a distance measuring interferometer and parent structure  18  is a metrology frame. Interferometer  13  includes a polarizing beam splitter  20  having a measurement path&#39;s quarter-wave plate  23 . Typically, a measurement beam exits quarter-wave plate  23 , bounces off a measurement mirror mounted to a stage that is being measured, and returns to quarter-wave plate  23 . CE child  is selected to be located on the outer face of quarter-wave plate  23 , which must remain substantially motionless relative to metrology frame  18  for accurate distance measurements of the stage. In one embodiment, a representative requirement is for quarter-wave plate  23  to move less than 10 nm/° C. relative to metrology frame  18 . 
   Initially, the center of expansion of base part  14  (hereafter “CE base ”) is thought to produce the desired result if it coincides with CE child .  FIG. 2B  illustrates a simplified cross-section view used to determine any change in the position of CE child  relative to metrology frame  18  under a temperature change. As interferometer  13  is bonded to base part  14  with bond joint  15 , expansions of interferometer  13  and base part  14  are determined relative to the center of expansion of bond joint  15  (hereafter “CE bond ”), which is shown as axis  28  in  FIG. 2B . Depending on the materials used, the coefficient of thermal expansion of base part  14  (hereafter “CTE base ”) can be greater or smaller than the coefficients of thermal expansion of polarizing beam splitter  20  and quarter-wave plate  23  (hereafter respectively as “CTE PBS ” and “CTE QWP ”). The change in the position of CE child  is calculated as follows:
 
δ CE     child     =CTE   base    ΔTl   base   −CTE   PBS   ΔTl   PBS   −CTE   QWP   ΔTl   QWP ,  (1)
 
where δ CE     child    is the change in the position of CE child , ΔT is the temperature change, l base  is the length of base part  14  from CE bond  to CE base  along a centerline  30  defined by CE bond  and CE child , l PBS  is ½ length of polarizing beam splitter  20  (i.e., the length of polarizing beam splitter  20  from CE bond  to quarter-wave plate  23 ), and l QWP  is the length of quarter-wave plate  23 . As l base  is equal to the sum of l PBS  and l QWP , equation (1) can be rewritten as:
 
δ CE     child     =CTE   base   ΔT ( l   PBS   +l   QWP )− CTE   PBS   ΔTl   PBS   −CTE   QWP   ΔTl   QWP             δ CE     child   =( CTE   base   ΔTl   PBS   −CTE   PBS   ΔTl   PBS )+( CTE   base   Δl   QWP   −CTE   QWP   ΔTl   QWP )         δ CE     child     =ΔT[l   PBS ( CTE   base   −CTE   PBS )+ l   QWP ( CTE   base   −CTE   QWP )].  (2)
 
In a typical case, the parameters for equation (2) are:
       CTE base =9.9×10 −6  1/° C. based on 416 stainless steel;   CTE PBS =7.1×10 −6  1/° C. based on BK-7;   CTE QWP =13.2×10 −6  1/° C. based on quartz;   l PBS =15.25×10 −3  m; and   l QWP =0.75×10 −3  m.
 
Typical precision photolithography environment parameters are:
       

                   ⅆ   T       ⅆ   t       ≤     28   ×     10     -   6       ⁢           ⁢     °   ⁢   C     ⁢     /     ⁢     s   ⁡     (     0.1   ⁢           ⁢     °   ⁢   C     ⁢     /     ⁢   hr     )           ;       and   ⁢           ⁢     t   process       =     300   ⁢           ⁢     s   .     
     ⁢   Thus           ,       Δ   ⁢           ⁢   T     =     8.4   ×     10     -   3       ⁢           ⁢   °C             
So the change in the position of CE child  would be 0.34 nm. Thus, δ CE     child    is marginally allowable at the current state of technology, but the technology requirements will soon surpass this and a minor reorientation of features is already required due to the shipment/storage requirements for a potentially zero δ CE     child   .
 
   To ensure that CE child  remains substantially motionless relative to metrology frame  18 , l base  is set as a variable in equation (1) and δ CE     child    is set to 0 so that interferometer  13  and base part  14  would expand to the same length when subjected to a temperature change. 
                   δ     CE   child       =             CTE   base     ⁢   Δ   ⁢           ⁢     Tl   base       -       CTE   PBS     ⁢   Δ   ⁢           ⁢     Tl   PBS       -       CTE   QWP     ⁢   Δ   ⁢           ⁢     Tl   QWP         ⇒     
     ⁢   0     =             CTE   base     ⁢   Δ   ⁢           ⁢     Tl   base       -       CTE   PBS     ⁢   Δ   ⁢           ⁢     Tl   PBS       -       CTE   QWP     ⁢   Δ   ⁢           ⁢     Tl   QWP         ⇒     
     ⁢     l   base       =           CTE   PBS       CTE   base       ·     l   PBS       +         CTE   QWP       CTE   base       ·     l   QWP                     (   3   )               
With the values listed above, l base  is determined to be 11.94×10 −3  m. The calculations described above form part of a step  110  ( FIG. 7 ) in method  100  described later. Referring to  FIGS. 3A and 3B , CE base  is placed l base  away from CE bond  on centerline  30 . The desired location of CE base  can be set by placing flexures  16  so their lines of action  31  intersect at the desired location of CE base . See  FIG. 7 , step  112 . The line of action of a flexure  16  is defined by its cross-section. For a flexure  16  having a rectangular cross-section with a high aspect ratio (e.g., 6), the line of action is defined by the minor axis of the cross-section. Note that CE base  and CE child  are not coincident in  FIG. 3A .
 
   In one embodiment, metrology frame  18  includes B datum pins  33  and  34 , and C datum pin  36  for initially positioning interferometer  13  on metrology frame  18 . Accordingly, base part  14  includes a B datum feature  38  (e.g., a recessed plane) for receiving B datum pins  33  and  34 , and a C datum feature  40  (e.g., a recessed plane) for receiving C datum pin  36 . Datum features  38  and  40  are placed so the directions of their planes run through CE base , see  FIG. 7 , step  114 . This prevents base part  14  from thermally expanding against datum pins  33 ,  34 , and  36 . This is important during the large temperature excursions of shipment/storage. If the base expansion is restricted by the pins, this would cause the flexure feet to slip, thus loosing the system alignment. 
   When base part  14  differentially expands or contracts, it becomes convex or concave due to the tip rotation of flexures  16 . Typically, it is preferred to not have an induced curvature on the surface of base part  14 .  FIGS. 4A and 4B  illustrate flexure pairs  16 A that each consists of two parallel flexures (e.g., a simple leaf linear spring or parallel plate flexures) in one embodiment of the invention. A flexure pair  16 A, base part  14 , and parent structure  18  form a four bar mechanism that mitigates curvature on the surface of base part  14 . Mounting holes  42  are formed in base part  14  between the flexures of flexure pairs  16 A. Fasteners (e.g., machine screws) are passed through mounting holes  42  to secure base part  14  to metrology frame  18 . Hereafter, a flexure pair and its corresponding machine screw are collectively referred to as a “flexure-machine screw set.” 
   The length of the flexure is determined so that the shear force experienced by the flexure, which is caused by flexure deflection from the differential thermal expansion between base part  14  and parent structure  18 , does not cause the flexure foot to slip on parent structure  18 . In one embodiment, the length of the flexure is calculated using a bending model as follows. 
                     l   f     =       [           E   f     ⁡     (       CTE   base     -     CTE   parent       )       ⁢   Δ   ⁢           ⁢     Tbh   3     ⁢     l   B         f   zx       ]       1   /   3         ,           (   4   )               
where l f  ( FIG. 4A ) is the length of the flexure, E f  is Young&#39;s modulus of the flexure, CTE base  is the coefficient of thermal expansion of base part  14 , CTE parent  is the coefficient of thermal expansion of parent structure  18 , ΔT is the temperature change, b and h ( FIG. 4B ) are the base and the height of the flexure cross-section, l B  ( FIG. 4B ) is the distance from CE base  to the center of the flexure, and f zx  is the shear force experienced by the flexure under deflection, see  FIG. 7 , step  116 . To determine the minimum flexure length l f  that would prevent the flexure foot from slipping on parent structure  18 , shear force f zx  is set equal to the static friction force between the flexure foot and parent structure  18 . Only a bending model is used to determine flexure length l f  because the use of flexure pairs and the use of high aspect ratio cross-section (e.g., 6) essentially ensure that the flexures are in pure bending without shearing. To determine the minimum flexure height, flexure distance l B  is set to the distance of the farthest flexure. The conditions that constrain the flexure geometry are,
 τ zx ≦τ μ     s   , which prevents slippage,  (4.1) σ vonMises &lt;σ YC     parent   , which prevents yielding in the parent structure, and  (4.2) σ vonMises &lt;σ YC     child   , which prevents yielding in the child part,  (4.3) 
where τ zx  is the shear stress of the flexure foot under thermal cycling, τ μ     s    is the static shear stress of the flexure foot under friction, σ vonMises  is the vonMises stress of the flexure to parent contact under thermal cycling, σ YC     parent    is the yield stress of the parent, and σ YC     child    is the yield stress of the child. See  FIG. 7 , step  116 .
 
   A spring force balance analysis may be necessary to determine the movement of CE base  relative to parent structure  18  due to temperature change. This is because flexure pairs  16 A may be placed asymmetrically about CE base . For example, flexure pairs  16 A may be spaced apart at different angles from each other, or flexure pairs  16 A may be spaced at different radial lengths from CE base . The asymmetric placement of flexure pairs  16 A causes them to exert asymmetric forces that cause CE base  to move with temperature. 
   To prevent CE base  from moving with temperature, the thermal forces from differential thermal expansion of base part  14  and parent structure  18  must be balanced with the spring forces from flexure pairs  16 A. Note that the spring forces due to the machine screws that secure base part  14  to parent structure  18  must also be taken into account. Thus, there are three sets of springs (i.e., three flexure-machine screw sets), with each set containing an outer and inner flexure and a machine screw centered between the flexures. The spring force balance can be determined as follows in one embodiment of the invention. 
   The radial and tangential spring constants of a flexure are: 
                     k   r     =         R   bh     ⁢     E   f     ⁢     h   4         l   f   3         ,   and           (   5   )                   k   t     =         R   bh   3     ⁢     E   f     ⁢     h   4         l   f   3         ,           (   6   )               
where k r  is the radial spring constant of the flexure, R bh  is the aspect ratio of the flexure cross-section (i.e., R bh ≡b/h), and k t  is the tangential spring constant of the flexure. Note that tangential spring constant k t  is R bh   2  times larger than radial spring constant k r . Since a typical aspect ratio is six, then the tangential stiffness is typically 36 times larger than the radial stiffness and thus a line of action of the flexure is defined along the radial direction.
 
   The spring constant of a machine screw is: 
                     k   ms     =       3   ⁢   π   ⁢           ⁢     E   ms     ⁢     r   4         l   ms   3         ,           (   7   )               
where k ms  is the spring constant of the machine screw, E ms  is the Young&#39;s modulus of the machine screw, r is the minimum radius of the machine screw, and l ms  is the distance from CE base  to the machine screw.
 
     FIGS. 5A and 5B  illustrate a one-dimensional simple spring force balance model of base part  14  mounted on parent structure  18  by flexures  16 L and  16 R in one embodiment of the invention.  FIG. 5A  shows the effect of a net contraction of base part  14  relative to parent structure  18  under a temperature change. A notional pin  52  is used to convey this concept by restraining the movement of base part  14  relative to parent structure  18 . Specifically, base part  14  contracts a length δ th,1  at one end and a length δ th,2  at another end relative to CE base . These length changes cause (1) a flexure  16 L (represented by a spring having a spring constant k 1 ) to exert a force on parent structure  18  and the parent structure  18  to exert an equal but opposite force F th,1 , and (2) a right flexure  16 R (represented by a spring having a spring constant k 2 ) to exert a force on parent structure  18  and the parent structure  18  to exert an equal but opposite force F th,2 . 
     FIG. 5B  shows a translation of base part  14  caused by an imbalance of forces due to the contraction of base part  14  after the removal of notional pin  52  that restrained the location of CE base . Specifically, CE base  has moved a distance δ ε . The translation of base part  14  causes (1) flexure  16 L to exert a force on parent structure  18  and the parent structure  18  to exert an equal but opposite force F δε,1 , and (2) flexure  16 R to exert a force on parent structure  18  and the parent structure  18  to exert an equal but opposite force F δε,2 . When base part  14  and parent structure  18  are in equilibrium, then the sum of all the external forces due to thermal contraction/expansion and the translation of base part  14  must sum to zero.
 Σ F= 0 =ΣF   th   +ΣF   δ     ε   , or Σ F   δ     ε     =−ΣF   th ,  (8) 
where ΣF th  (subscript should not be bold and not italized) is the sum of the external forces due to thermal contraction/expansion relative to CE base , and ΣF δ     ε    is the sum of the external forces due to the translation of base part  14 . The analytical model is constructed in this form so that δ ε  can be calculated and then the model variables can be iteratively modified until δ ε  is within the design specification.
 
   In the two dimensional design model, the F th &#39;s are radial relative to CE base  as everything expands/contracts radially from CE base . Referring to  FIG. 5C , the radial forces of the three flexure-machine screw sets due to thermal contraction/expansion are: 
                     Σ   ⁢           ⁢     F     th   ,   x         =         (       F     th   ,     11   ⁢   r         +     F     th   ,     12   ⁢   r         +     F     th   ,     1   ⁢   ms           )     ⁢   cos   ⁢           ⁢     θ   1       +       (       F     th   ,     21   ⁢   r         +     F     th   ,     22   ⁢   r         +     F     th   ,     2   ⁢   ms           )     ⁢   cos   ⁢           ⁢     θ   2       +       (       F     th   ,     31   ⁢   r         +     F     th   ,     32   ⁢   r         +     F     th   ,     3   ⁢   ms           )     ⁢   cos   ⁢           ⁢     θ   3           ,     
     ⁢   and           (   9   )                   Σ   ⁢           ⁢     F     th   ,   y         =         (       F     th   ,     11   ⁢   r         +     F     th   ,     12   ⁢   r         +     F     th   ,     1   ⁢   ms           )     ⁢   sin   ⁢           ⁢     θ   1       +       (       F     th   ,     21   ⁢   r         +     F     th   ,     22   ⁢   r         +     F     th   ,     2   ⁢   ms           )     ⁢   sin   ⁢           ⁢     θ   2       +       (       F     th   ,     31   ⁢   r         +     F     th   ,     32   ⁢   r         +     F     th   ,     3   ⁢   ms           )     ⁢   sin   ⁢           ⁢     θ   3           ,           (   10   )               
where F th,11r  is the radial force of flexure  11 , F th,12r  is the radial force of flexure  12 , F th,1ms  is the force of machine screw  1 , θ 1  is the angle of the line of action through the flexure pair formed by flexures  11  and  12 , and so forth for the other two set of forces.
 
   Only the forces for one flexure-machine screw set will be described. The forces on the flexure-machine screw set formed by flexures  31  and  32  and machine screw  3  are: 
                     F     th   ,     31   ⁢   r         =       -   Δ     ⁢           ⁢     CTE   ·   Δ     ⁢           ⁢     T   ·     l   31     ·         R     bh   ,   3       ⁢     E   f     ⁢     h   3   4           (     l     f   ⁢           ⁢   3       )     3             ,           (   11   )                   F     th   ,     32   ⁢   r         =       -   Δ     ⁢           ⁢     CTE   ·   Δ     ⁢           ⁢     T   ·       l   31     ⁡     (     1   -       (       g   3     +     h   3       )       l   31         )       ·         R     bh   ,   3       ⁢     E   f     ⁢     h   3   4           (     l     f   ⁢           ⁢   3       )     3             ,   and           (   12   )                   F     th   ,     3   ⁢   ms         =       -   Δ     ⁢           ⁢     CTE   ·   Δ     ⁢           ⁢     T   ·       l   31     ⁡     (     1   -       (       g   3     +     h   3       )       2   ⁢     l   31           )       ·   3     ⁢   π   ⁢           ⁢     E   ms     ⁢     r   3   4     ⁢     1       (       l     f   ⁢           ⁢   3       +     l     CH   ⁢           ⁢   3         )     3           ,           (   13   )               
where ΔCTE is the relative coefficient of thermal expansion between base part  14  and parent structure  18  (i.e., ΔCTE≡CTE base −CTE parent ), ΔT is the temperature change, l 31  is the distance from CE base  to the center of flexure  31 ,
 
               R     bh   ,   3       ⁢     E   f     ⁢     h   3   4           (     l     f   ⁢           ⁢   3       )     3           
is the radial spring constant of flexures  31  and  32 , R bh,3  is the aspect ratio of flexures  31  and  32  (i.e., b 3  to h 3 ), l f3  is the flexure length of flexures  31  and  32 , l 31   
           (     1   -       (       g   3     +     h   3       )       l   31         )         
is the distance from CE base  to the to center of flexure  32 , g 3  is the distance between flexures  31  and  32 ,
 
           3   ⁢   π   ⁢           ⁢     E   ms     ⁢     r   3   4     ⁢     1       (       l     f   ⁢           ⁢   3       +     l     CH   ⁢           ⁢   3         )     3             
is the spring constant of machine screw  3 ,
 
             l   31     ⁡     (     1   -       (     g   +   h     )       2   ⁢     l   31           )           
is the distance from CE base  to the center of machine screw  3 , and l CH3  is clearance hole depth. The formulas for the other flexure-machine screw sets are the same but for the substitution of the corresponding parameters.
 
   For the left hand side of equation 8, and referring to  FIG. 5D , the forces of a flexure-machine screw set i due to the translation of base part  14  is: 
                       F       δ   ɛ     ,   i       ≡       [           F       δ   ɛ     ⁢   x                 F       δ   ɛ     ⁢   y             ]     i       ,   where     ⁢          ⁢                 [           F       δ   ɛ     ⁢   x                 F       δ   ɛ     ⁢   y             ]     i     =       [           cos   ⁡     (     -     θ   i       )             sin   ⁡     (     -     θ   i       )                 -     sin   ⁡     (     -     θ   i       )               cos   ⁡     (     -     θ   i       )             ]     ⁡     [             2   ⁢     k     r   ,   i         +     k     ms   ,   i             0           0           2   ⁢     k     t   ,   i         +     k     ms   ,   i               ]         ⁢                               ⁢         [           cos   ⁢           ⁢     θ   i             sin   ⁢           ⁢     θ   i                   -   sin     ⁢           ⁢     θ   i             cos   ⁢           ⁢     θ   i             ]     ⁡     [           -     δ     ɛ   ⁢           ⁢   x                   -     δ     ɛ   ⁢           ⁢   y               ]       ,                     (   14   )               
where F δ     ε     ,i  is the force vector of set i caused by the translation of base part  14 , F δ     ε     x  and F δ     ε     y  are the x and y components of force vector F δ     ε     ,i , θ i  is the angle of the line of action through set i, k r,i  is the radial spring constant of a flexure in set i, k ms,i  is the spring constant of a machine screw in set i.
 
   When multiplied out, equation (14) becomes: 
   
     
       
         
           
             
               
                 
                   
                     
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                                 i 
                               
                             
                             
                               
                                 b 
                                 i 
                               
                             
                           
                           
                             
                               
                                 b 
                                 i 
                               
                             
                             
                               
                                 d 
                                 i 
                               
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 - 
                                 
                                   δ 
                                   
                                     ɛ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     x 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   δ 
                                   
                                     ɛ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     y 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       a 
                       i 
                     
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                           
                             k 
                             r 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 
                                   θ 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   R 
                                   bh 
                                   2 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 
                                   θ 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         k 
                         ms 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       b 
                       i 
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                         
                           k 
                           r 
                         
                         ⁡ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               R 
                               bh 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         i 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         i 
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     d 
                     i 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           k 
                           r 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               
                                 θ 
                                 i 
                               
                             
                             + 
                             
                               
                                 R 
                                 bh 
                                 2 
                               
                               ⁢ 
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               
                                 θ 
                                 i 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         k 
                         ms 
                       
                       . 
                     
                   
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   The sum of all the forces from the three flexure-machine screw sets can be written as: 
   
     
       
         
           
             
               
                 
                   ∑ 
                   
                     F 
                     
                       δ 
                       ɛ 
                     
                   
                 
                 ≡ 
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 a 
                                 1 
                               
                               + 
                               
                                 a 
                                 2 
                               
                               + 
                               
                                 a 
                                 3 
                               
                             
                           
                           
                             
                               
                                 b 
                                 1 
                               
                               + 
                               
                                 b 
                                 2 
                               
                               + 
                               
                                 b 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               
                                 b 
                                 1 
                               
                               + 
                               
                                 b 
                                 2 
                               
                               + 
                               
                                 b 
                                 3 
                               
                             
                           
                           
                             
                               
                                 d 
                                 1 
                               
                               + 
                               
                                 d 
                                 2 
                               
                               + 
                               
                                 d 
                                 3 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               
                                 δ 
                                 
                                   ɛ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   x 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 δ 
                                 
                                   ɛ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   y 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   Equation (8) can now be rewritten as: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   A 
                                 
                                 
                                   B 
                                 
                               
                               
                                 
                                   B 
                                 
                                 
                                   D 
                                 
                               
                             
                             ] 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     - 
                                     
                                       δ 
                                       
                                         ɛ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         x 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       δ 
                                       
                                         ɛ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         y 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   
                                     F 
                                     thx 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     F 
                                     thy 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       , 
                       where 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     A 
                     = 
                     
                       ( 
                       
                         
                           a 
                           1 
                         
                         + 
                         
                           a 
                           2 
                         
                         + 
                         
                           a 
                           3 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     B 
                     = 
                     
                       ( 
                       
                         
                           b 
                           1 
                         
                         + 
                         
                           b 
                           2 
                         
                         + 
                         
                           b 
                           3 
                         
                       
                       ) 
                     
                   
                   , 
                   and 
                 
                 ⁢ 
                 
                    
                 
                 ⁢ 
                 
                   D 
                   = 
                   
                     
                       ( 
                       
                         
                           d 
                           1 
                         
                         + 
                         
                           d 
                           2 
                         
                         + 
                         
                           d 
                           3 
                         
                       
                       ) 
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   Thus, the movement of CE base  is: 
                   [           δ     ɛ   ⁢           ⁢   x                 δ     ɛ   ⁢           ⁢   y             ]     =           [         A       B           B       D         ]       -   1       ⁡     [           F   thx               F   thy           ]       .             (   19   )               
The design of base part  14  can be modified until the movement of CE base  is acceptable. See  FIG. 7 , step  118 .
 
     FIGS. 6A ,  6 B,  6 C,  6 D,  6 E, and  6 F illustrate base  14  for mounting interferometer  13  atop metrology frame  18  in one embodiment of the invention. As can be seen in  FIGS. 6B and 6D , flexure pairs  16 A can have different lengths depending on their placement. 
     FIG. 7  illustrates a method  100  for designing base part  14  for mounting a child part  13  to a parent structure  18  in one embodiment of the invention. 
   In step  102 , the materials of child part  13  and parent structure  18  are determined. The materials of child part  13  and parent structure  18  are application specific. Typical materials for a precision photolithography application were provided above. 
   In step  104 , the material of base part  14  is selected. Typically, the material of base part  14  is selected so its coefficient of thermal expansion (CTE) matches that of either child part  13  or parent structure  18 , or its CTE is a compromise between those of child part  13  and parent structure  18 . 
   In step  106 , the desired position for the center of expansion (CE) of child part  13  is selected. As described above, this CE is a point on child part  13  that is desired to remain substantially motionless relative to parent structure  18 . 
   In step  108 , the CE of bond joint  15  is determined. Typically, bond joint  15  is symmetrical so this CE is typically located at its geometric center. If bond joint  15  is not symmetrical, then finite element analyses or experimental tests can be performed to determine the CE of bond joint  15 . 
   In step  110 , the CE of base part  14  is determined. In one embodiment, the position of this CE is placed along a centerline defined by CE child  and CE bond  at a distance l base  away from CE bond . As described above, distance l base  can be determined using equation (3). 
   In step  112 , flexures  16  or flexure pairs  16 A are positioned on base part  14  so that their lines of action intersect at the desired location of CE base . 
   In step  114 , datum features  38  and  40  are positioned on base part  14  so the directions of their planes run through CE base . 
   In step  116 , additional parameters of assembly  10  are set (or modified in subsequent loops through step  116 ). For example, aspect ratio R bh  and flexure length l f  of flexure  16  or flexure pairs  16 A are set. With the parameters set, the design is checked to make sure that flexures  16  or flexure pairs  16 A do not slip with temperature change and they, along with parent structure  18 , do not yield. Furthermore, in subsequent loops through  116 , the location of CE base  can be changed to minimize the movement of CE child . 
   In step  118 , a spring force balance analysis is performed to determine the movement of CE base  with temperature in the current design of base part  14 . As described above, the spring force balance may be necessary when flexures  16  or flexure pairs  16 A are asymmetrically placed about CE base . 
   In step  120 , the movement of CE base  relative to parent structure  18  is compared with the desired tolerance. If the movement is less than the tolerance, then step  120  is followed by step  122 . Otherwise step  120  is followed by step  116  where the parameters such as aspect ratio R bh , flexure length l f , and the location of CE base  are adjusted to reduce the movement of CE base  relative to parent structure  18 . 
   In step  122 , a numeral analysis is performed to determine the movement of CE child  with temperature in the current design of base part  14 . In one embodiment, a finite element analysis is performed on the current design of base part  14 . 
   In step  124 , the movement of CE child  relative to parent structure  18  is compared with the desired tolerance. If the movement is less than the tolerance, then step  124  is followed by step  126  that ends method  100 . Otherwise step  124  is followed by step  116  where the parameters such as aspect ratio R bh ; flexure length l f , the location of CE base  are adjusted to reduce the movement of CE child  relative to parent structure  18 . 
   Various other adaptations and combinations of features of the embodiments disclosed are within the scope of the invention. Although examples for designing a base for mounting a distance measuring interferometer to a metrology frame are described, the general design process can be applied for designing a base for mounting any child part to any parent structure. Furthermore, the location of CE base  can be applied to other interface features in addition to flexures between the base part and the parent structure, such as a ball in groove interface. Numerous embodiments are encompassed by the following claims.