3.4 Eight-Mirror System
4.1.2 Field Lens and Mask Position
For the Kohler integrator, the position of the mask is located at the back focal length (BFL) of the field lens group. This can be verified by consider a ray parallel to the optics
axis entering the field lens group. In terms of the ABCD matrix, this can be expressed by ray height and angle at the mask, and y is the incident ray height. Here, since the mask is placed at the focal plane of the field lens, rays incident on the field lens parallel to the optical axis by definition must be focused onto the optical axis, therefore
y0= Ay = 0, (101)
or
A= 0 (102)
for non-trivial incident ray height. From here, again in terms of the ABCD matrix, for an incident ray of arbitrary angle and height,
This gives the following relationship
y0= Bu, (104)
which means that at the mask position, the ray height on the mask is dependent only on the ray angle entering the field lens, and is independent of the incident ray height.
Figure 64: The marginal ray angles at the grid source and at the projection tool entrance pupil is related to the magnification of the finite conjugate system formed by the elements in between.
In terms of GGC, this condition can be expressed by
tL2= A(L1, L2)
C(L1, L2), (105)
and GGC expansion gives the following relationship
tL2= 1 − φL1tL1
φL1+ φL2− φL1φL2tL1. (106) 4.1.3 Grid Source NA and Exposure Field Width
Since the grid source and the projector entrance pupil are bound together in a conjugate relationship, the marginal ray angle at the two positions must be bound together by the magnification of the finite conjugate system formed by the elements in between, giving
M(GS, EnP) = tan(αGS)
tan(αEn) (107)
as illustrated in Figure64. Also can be observed in Figure 64is that the exposure field width on the mask wMaskcan now be expressed as
tan(αEn) = wMask
2tEn , (108)
Figure 65: The illumination NA is directly related to the beam size of the collimated LPP source.
and combining the two equations107and108gives the expression for the exposure width
wMask= 2tEntan(αGS)
M(GS, En) . (109)
Here, M(GS, En) is the magnification this finite conjugate system, and can be expressed in terms of GGC as
M(GS, En) = 1
D(GS, En), (110)
which can be expanded to give the following relationship
φF=1 − φL2(tF+ tL+ tL1) − φL1(tF+ tL)(1 − φL2tL1) −M1 tF(1 − φL2(tL+ tL1) − φL1tL(1 − φL2tL1))
(111)
4.1.4 Illumination Optics NA and Collimated Plasma Source Beam Size
As illustrated in Figure65, the illumination NA is directly related to the beam size of the collimated LPP source and the effective focal length (EFL) of the field lens, through
tan(α ) = wLPP
=1
w C(L1, L2) (112)
Figure 66: The number of array elements on the pupil array determines the array pitch and the grid source NA.
in terms of GGC, which can be expanded into
wLPP= tan(αMask)
φL1+ φL2− φL1φL2tL1. (113) 4.1.5 Pupil Pitch Size
By simple geometry (Figure 66), the pupil array pitch is related to the NA of the grid source through
tan(αGS) = DP
2tP, (114)
where
NAGS = sin(αGS). (115)
4.1.6 Number of Array Elements
Likewise, also shown in Figure66, the number of array elements on the mirror arrays can be determined by
NP= 1 +2wLPP
DP . (116)
4.2 Illumination System Design
Combining the equations derived in the previous section, a set of governing equations117 to120relating the positioning and properties of the mirrors can be obtained.
φL1= wLPP+ tan(αMask)tL2
wLPPtL1 (119)
φL2= φL1
φL1tL1− 1+ 1
tL2 (120)
As can be seen from the derived governing equations, the choice of parameter to be made dependent or independent is rather intentional. Mathematically this choice can be arbitrary in principle, however in practice some parameters are more suited to act as in-dependent variables. For example, parameters that must equate to a specific value (e.g.
system specifications), or have a heavy constraint placed upon (e.g. inter-element spac-ings are positive definite), are better left as independent variables. Keeping these parame-ters on hand as independent variables allows for much greater control over the validity of the design during the optimization process. Whereas parameters that can roam free with-out major consequences (e.g. mirror focal lengths) are good candidates for dependent variables.
As a design example, a set of specifications derived from the authors’ previous pro-jection tool design [5] are listed in Table8, along with a set of initial conditions. From the specifications and initial conditions, a complete solution for the illuminator can be
Table 8: A set of specification and initial condition of the illuminator design.
Specifications
Illumination NA (αMask) 0.1 Mask-Pupil Distance (tEn) 1518.73 mm
Exposure Width (wMask) 8 mm Number of Arrays (NP) 51
Table 9: Initial evaluation of the governing equations using specification and initial con-ditions provided in Table8.
EFL (mm) Thickness (mm)
Collimated LPP -
-Pupil Array 250.0 250
Grid Source - 80
Field Array -92.7 400
Field Lens 1 1375.4 300
Field Lens 2 636.9 400
obtained, listed in Table9.
4.2.2 Resolving Obstructions
A paraxial layout of the optical system described by the initial evaluation is shown in Figure 67, resulting in an obstructed system, as would be expected from an arbitrary initial evaluation.
For a single instance between a mirror and a neighboring ray bundle, the obstruction is defined as the angle of the obstructed region as illustrated in Figure68. To resolve the obstruction, a random walk optimizing algorithm as devised in a previous work by the
Figure 67: An initial evaluation of the governing equations.
ε
ObsFigure 68: The amount of obstruction between a ray bundle and mirror is defined as the angle of the overlapping region. [5]
5
o5
o5
o5
oFigure 69: The same initial evaluation with a 5◦tilt introduced.
author [46] is used to mitigate the obstructions, using the governing equations117to120 as the kernel and setting the total amount of obstruction present in the system as the error function of the algorithm.
As fewer number of optical elements are desired in the illuminator, simply adjusting the mirror spacings and focal lengths may not be enough to fully resolve the obstructions.
As such, element tilts and decenters are introduced to assist in the mitigation of the ob-structions. Figure69shows the same result from the initial evaluation, however with a 5◦ tilt applied to each of the mirrors. An obstruction resolved solution of the illuminator is shown in Figure70, with parameters listed in Table10.
Projector Entrance Pupil
Mask
FL
1FL
2FA
Collimated
PA
LPP
Figure 70: Obstruction resolved paraxial layout. The tilt angle required is θtilt= 6.991◦.
Table 10: Obstruction resolved illuminator parameters.
EFL (mm) Thickness (mm)
Collimated LPP -
-Pupil Array 506.4 510.2
Grid Source - 255.1
Field Array 433.6 774.4
Field Lens 1 1274.4 478.5
Field Lens 2 1324.1 502.2
Mask -
-Element Tilt Angle 6.991◦
Figure 71: Direct conversion from the paraxial result. The lower zoomed in part shows that aberration resultant from the plain spherical mirror offsets the illumination profile from different array elements.
4.3 Reflective Illuminator System Embodiment
4.3.1 Illuminator Design Result
After resolving the obstructions, the resultant paraxial results can be converted to actual mirror surface geometries (i.e. radius of curvatures) through the relationship
Rn= −2 · EFLn. (121)
During the conversion, deviation from paraxial calculations is inevitable, due to the pres-ence of optical aberrations. Figure71shows the illuminator design resulting from a direct conversion from the paraxial parameters, and as can be clearly seen in the lower mag-nified portion of the ray trace diagram, at the mask, the illuminated field from different array elements do not overlap as was predicted by the paraxial system.
Figure 72: The aberration corrected illuminator.
the mirror surface of the field lens is changed to a more complex conic profile. A quick optimization sun by setting the chief ray deviation on the mask as the error function yields the corrected result, shown in Figure72. The required compensation parameters are given in Table11.
Table 11: Obstruction resolved illuminator parameters.
Radius Conic Constant
Pupil Array -1012.7
-Field Array -867.1
-Field Lens 1 -2548.7 -19.314
Field Lens 2 -2648.2 13.124
FA Tilt Adjust Array Element Angle Uppermost 0.045◦
Middle 0.013◦
Lowermost 0.035◦
Figure 73: The eight mirror projector design from which the illuminator specifications are derived from.
4.4 Illuminator Projector Integration
At the final step, the illuminated mask is imaged by the projector. Shown in Figure73 is the lens layout of the projection tool for which the specifications of the illuminator are derived from. The result of the illuminator-projector integration is shown in Figure74, and Figure75is the illumination profile at the mask side and the wafer end respectively.
As can be seen in Figure 74, the illuminator matches well with the projector. In this demonstration of the illuminator design, for the explicit purpose of integration with an existing projector design, the lens data of a projector design in a previous work by the authors [5] (Figure 73) is used. As such, important optical properties of the projector (i.e. entrance pupil location and size, and designed mask height) can be known exactly, to ensure that the resultant illuminator designed matches with the projector.
Between projector layout (Figure73) and the combined layout (Figure74), one might notice a difference in the color of the rays shown, which is the result of the color code used by the optics software. In the projector layout, different ray colors represents light originating from different points on the mask. In the combined layout, the different ray colors now represent light originating from the light reflecting off different array elements
Illuminator
Projector
Figure 74: The combined system of the illuminator and the projector.
X-Field (mm)
Figure 75: Upper: The resultant ringfield illumination profile at the mask side. Lower:
The illumination profile at the wafer end.
on the pupil array and field array, which is distributed uniformly onto the mask as can be seen in the illuminator layout (Figure72), and therefore also uniformly onto the wafer as imaged by the projector.
5 Conclusion
A systematic design method for both the EUVL projector and the illuminator has been described. For the projector, an eight mirror design with 0.4 NA has been demonstrated to verify the capability of this design method. This novel application of GGC on EUVL tool design is beneficial in that the relationship between optical properties and requirements can be derived through GGC and implemented into existing optical design software to assist the design process. With GGC, complex optical systems can be analyzed in greater depth, and in relative ease to conventional ABCD matrix.
For the illuminator, using GGC, the optical properties of various parts of the illumi-nator can be derived, and can be combined to form a general governing equation that, if followed, guarantees the desired optical properties. For simple optical system of few elements, the application of GGC analysis may be overly cumbersome and redundant, however for complex optical systems such as the EUVL illuminator and projector tool, GGC greatly eases the analysis and design difficulty.
Using the obtained governing equations as the kernel, an unobstructed illuminator de-sign can be obtained through application of the Monte Carlo random walk algorithm, and can be implemented into existing optical design software to assist in the optical system optimization.
As a design example, the illuminator specifications are derived from the optical prop-erties of an existing projection tool design (author’s previous work [5]). The result of the illuminator design matches well to the projector specs and properties, and is demonstrated to perform well in providing the necessary illumination as required for the projection tool.
6 References References
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7 Appendices
7.1 GGC Implementation into MATLAB
The following is an implementation of the Gaussian bracket in the MATLAB environ-ment, which can be used to assistance with simple analysis of an optical system.
Basic Definition
temp2 = temp; temp2(:,[1 2])=[];
stopflag = 0;
while stopflag == 0;
c1_ = gb_base(temp1(:,1:2)).*c1 + temp2(:,1).*c2;
c2_ = temp1(:,1).*c1 + c2;
out = c1.*gb_base(temp1) + c2.*gb_base(temp2);
end
7.2 GGC Implementation into Code V
The codes demonstrated in this section is an implementation of the Gaussian bracket and GGC into the commercial optics design software CodeV. This can be used to enhance and finetune its already powerful optical design and analysis capability, allowing for more precise and minute control of a complex optical system, where the optical properties of individual subsystem within the entire system is of importance.
Basic Definition
FCT @gb_base(NUM ^gb_base_in(100), NUM ^gb_base_ind) IF ^gb_base_ind = 4;
^gb_base_out ==
^gb_base_in(1)*^gb_base_in(2)*^gb_base_in(3)*^gb_base_in(4) + ^gb_base_in(1)*^gb_base_in(4) + ^gb_base_in(3)*^gb_base_in(4) + ^gb_base_in(1)*^gb_base_in(2) + 1;
ELS IF ^gb_base_ind = 3;
^gb_base_out == ^gb_base_in(1) + ^gb_base_in(3) + ^gb_base_in(1)*^gb_base_in(2)*^gb_base_in(3);
ELS IF ^gb_base_ind = 2;
^gb_base_out == ^gb_base_in(1)*^gb_base_in(2) + 1;
ELS IF ^gb_base_ind = 1;
^gb_base_out == ^gb_base_in(1);
ELS IF ^gb_base_ind = 0;
^gb_base_out == 1;
END IF
END FCT ^gb_base_out
Gaussian Bracket
! =============================================== Begin Code FCT @gbrac(NUM ^gb_in(100), NUM ^gb_ind)
^gb_c1 == ^gb_in(1);
^gb_c2_ == ^gb_in(2*^gb_n)*^gb_c1 + ^gb_c2;
^gb_c1 == ^gb_c1_;
^gb_c2 == ^gb_c2_;
^gb_n == ^gb_n+1;
^c_v(^k*2-1) == ((IND S^k)-(IND S^k-1))/(RDY S^k);
^c_v(^k*2) == -(THI S^k)/(IND S^k);
END FOR
^c_v(^j*2-1) == ((IND S^j)-(IND S^j-1))/(RDY S^j);
^c_out == @gbrac(^c_v, (^j-^i+1)*2-1);
END FCT ^c_out
FCT @a(NUM ^i, NUM ^j) ! --- GGC (A) FOR ^k ^i (^j-1) 1
^a_v(^k*2-1) == ((IND S^k)-(IND S^k-1))/(RDY S^k);
^a_v(^k*2) == -(THI S^k)/(IND S^k);
END FOR
^b_v(^k*2-1) == -(THI S^k)/(IND S^k);
END FOR
^b_out == @gbrac(^b_v, (^j-^i+1)*2-3);
END FCT ^b_out
FCT @d(NUM ^i, NUM ^j) ! --- GGC (D)
^d_v(1) == -(THI S^i)/(IND S^i);
FOR ^k (^i+1) (^j-1) 1
^d_v(^k*2-2) == ((IND S^k)-(IND S^k-1))/(RDY S^k);
^d_v(^k*2-1) == -(THI S^k)/(IND S^k);
END FOR
^d_v(^j*2-2) == ((IND S^j)-(IND S^j-1))/(RDY S^j);
^d_out == @gbrac(^d_v, (^j-^i+1)*2-2);
END FCT ^d_out
!! ============================================== End Code
7.3 GGC Implementation into Zemax
The codes detailed in this section implements the ability to evaluate the GGC of the cur-rent optical system, and any arbitrary subsystems within, as a merit function operand in ZEMAX.
##========================================================##
## Integration of GGC into ZEMAX
##========================================================##
gb_in(ind1) == ( INDX(k)-INDX(k-1) ) * CURV(k) gb_in(ind2) = -THIC(k) / INDX(k)
NEXT
FOR k, s1+1, s2-1, 1 ind1 = 2*(k-s1) ind2 = 2*(k-s1)+1
gb_in(ind1) = ( INDX(k)-INDX(k-1) ) * CURV(k) gb_in(ind2) = -THIC(k) / INDX(k)
NEXT
gb_in(ind1) = ( INDX(k)-INDX(k-1) ) * CURV(k) gb_in(ind2) = -THIC(k) / INDX(k)
NEXT
ind3 = 2*(s2-s1)+1
gb_in(ind3) = ( INDX(s2)-INDX(s2-1) ) * CURV(s2) gb_ind = 2*(s2-s1)+1
gb_in(ind1) = -THIC(k-1) / INDX(k-1)
gb_in(ind2) = ( INDX(k)-INDX(k-1) ) * CURV(k) NEXT
temp = gb_base_out
gb_c1_temp = temp*gb_c1 + gb_in(2*gb_n+1)*gb_c2;
gb_c2_temp = gb_in(2*gb_n)*gb_c1 + gb_c2;
gb_c1 = gb_c1_temp;
gb_out = gb_c1*temp1 + gb_c2*temp2 ENDIF
gb_out = gb_c1*temp1 + gb_c2*temp2 ENDIF
SUB GB_Base
IF gb_base_ind == 0 gb_base_out = 1 ENDIF
IF gb_base_ind == 1
gb_base_out = gb_base_in(1) ENDIF
IF gb_base_ind == 2
gb_base_out = gb_base_in(1)*gb_base_in(2) + 1 ENDIF
IF gb_base_ind == 3
gb_base_out = gb_base_in(1) + gb_base_in(3) +
gb_base_in(1)*gb_base_in(2)*gb_base_in(3) ENDIF
IF gb_base_ind == 4 gb_base_out =
gb_base_in(1)*gb_base_in(2)*gb_base_in(3)*gb_base_in(4) + gb_base_in(1)*gb_base_in(4) + gb_base_in(3)*gb_base_in(4) + gb_base_in(1)*gb_base_in(2) + 1;
ENDIF RETURN