Photonic crystal technology offers the potential oflossless control of the propagation of light at micro-electronic and nanoelectronic size scales [1]. Thistechnology may be instrumental in producing thefirst truly dense integrated photonic circuits and sys-tems. Numerous important physical characteristicshave already been demonstrated. These phenomenainclude the photonic bandgap [1], the superprismeffect [2–4], negative refraction [5,6], and negativediffraction [7–9]. Individual components that havebeen demonstrated include waveguides [10,11], reso-nators [12–15], filters [15–18], waveguide couplers[19–22], directional couplers [23], demultiplexers[24], antennas [25], switches [26], and sensors [27].There are a number of techniques currently usedto fabricate photonic crystal structures and devices.Some of these are two-photon polymerization [28,29],focused-ion-beam etching [30], self-assembly [31–33],optical lithography [34–36], and e-beam lithography[35,37–39]. A relatively new technique for definingthe photoresist mask and polymer templates used
to construct photonic crystal structures is multi-beam-interference lithography (MBIL) [40–42]. Byinterfering two or more coherent beams of light, per-iodic interference patterns can be created and areused to expose and define photosensitive materialsto be used as etching masks or templates.
Previously, specific configurations of wave vectorswere presented that produce interference patternswith the symmetry of all 14 three-dimensional Bra-vais lattices through the interference of four nonco-planar beams [43,44]. To produce a lithographicallyuseful pattern, Cai et al.[44,45] introduced a conceptof uniform contrast defined by two requirements:(1) contrast of the interference terms should be thesame for multiple primitive lattice directions and(2) the absolute contrast should be as large as possi-ble. Within four-beam interference, constraints onthe intensities and polarizations of the individualbeams have been given to satisfy the uniform con-trast condition. However, in the literature there is nocomplete treatment of uniform contrast itself.
Here we introduce four conditions for primitive-lattice-vector-direction equal contrasts to describe thecontrast in four-beam interference for producinglithographically useful interference patterns. A briefdescription of multibeam interference is given as a
basis for the subsequent discussion of four-beam in-terference. The concept of uniform contrast is alsodescribed. The new nomenclature, conditions forprimitive-lattice-vector-direction equal contrasts, anda symbolic designationCðmÞ
n are introduced to providea definitive description of the resulting interferencepatterns. For each condition for primitive-lattice-vector-direction equal contrasts, a thorough mathe-matical description is given including the requiredconstraints on the plane wave parameters, the re-sulting locations of the maxima, minima, and saddlepoints within the unit cell of the periodic inter-ference pattern, and subordinate conditions formaximum absolute contrast. Within each conditionfor primitive-lattice-vector-direction equal contrasts,three lattices are treated (simple cubic, face-centeredcubic, and body-centered cubic) with tables present-ing the plane wave parameters and the maximumabsolute contrast for each case.
2. Multibeam Interference
The time-average electric field intensity that is dueto the interference ofN linearly polarized, monochro-matic plane waves can be expressed as
ITðrÞ ¼ I0
�1þ
XNi¼1
XNj>i
Vij cosðGji · rþ ϕi − ϕjÞ�; ð1Þ
where the DC intensity term (I0), interference coeffi-cients (Vij), polarization efficiency factors (eij), andspatial cosine wave vectors (Gji) are given by
I0 ¼ 12
XNk¼1
E2k; Vij ¼
EiEjeijI0
;
eij ¼ ei · ej; Gji ¼ kj − ki; ð2Þ
respectively. The terms Ei, ei, ki, and ϕi are the elec-tric field amplitude, polarization vector, wave vector,and initial phase, respectively, of the ith plane wave.For this analysis, the initial phases of the interferingbeams are set to zero (ϕi ¼ 0). This constraint guar-antees that a set of global intensity extrema (maximaor minima) are located at the lattice points whose ori-gin is at r ¼ 0. Nonzero initial phases merely trans-late the locations of these intensity extrema.
3. Four-Beam Interference
In general, the interference of four linearly polar-ized, monochromatic planes waves will produce athree-dimensional, periodic interference pattern.The primitive basis vectors a, b, and c are used to de-fine the translational symmetry of the desired inter-ference pattern. The corresponding reciprocal latticevectors A, B, and C are expressed as
A ¼ 2π b × ca · b × c
; B ¼ 2π c × aa · b × c
;
C ¼ 2π a × ba · b × c
: ð3Þ
A set of four wave vectors that will produce an inter-ference pattern with the translational symmetrygiven by a, b, and c is found by locating the circum-center (P) of a pyramid (tetrahedron) defined interms of the three reciprocal lattice vectors A, B,and C as
P ¼ 12jAj2ðB × CÞ þ jBj2ðC × AÞ þ jCj2ðA × BÞ
A · B × C: ð4Þ
The four recording wave vectors are then given by
k1 ¼ P; k2 ¼ P − A;
k3 ¼ P − B; k4 ¼ P − C: ð5Þ
Unlike the three-beam case [46], the choice of the pri-mitive basis vectors determines the wavelength re-quired to produce the desired interference pattern.The recording wavelength needed has a magnitudeof λ ¼ 2π=jPj. Given this methodology, it is clear thatdifferent sets of primitive basis vectors that definethe same translational symmetry will produce differ-ent sets of recording wave vectors. While the transla-tion symmetry of each will be identical, the locationsof symmetry elements and stationary points withinthe interference pattern will differ.
Equation (1) can be written in the case of four-beam interference as
The expression results in a DC intensity term and sixspatial cosines corresponding to the six interfering-beam pairs. It should be noted that there is a funda-mental relationship between the desired reciprocallattice vectors and the spatial cosine wave vectors.G21, G31, and G41 are equal to A, B, and C, respec-tively, while G32, G42, and G43 are equal to ðB − AÞ,ðC − AÞ, and ðC − BÞ, respectively.
Here we treat three lattices: simple cubic, face-centered cubic, and body-centered cubic. Using theabove methodology, the required recording wave vec-tors are calculated for the desired translational sym-metry given by its primitive lattice vectors. Table 1summarizes the primitive lattice vectors and thecalculated wave vectors. The recording wave vectorterm ko is defined as 2π=λ ¼ jPj.
4. Contrast
Proper selection of the recording wave vectors, ki, hasbeen shown to produce interference patterns withthe translational symmetry of all 14 Bravais lattices[43]. The usefulness of an interference pattern forlithography can be improved by systematically se-lecting the plane wave parameters of the recordingbeams. This is done most often through nonlinear op-timization by maximizing the absolute contrast of
the interference pattern for which the absolute con-trast is defined as
Vabs ¼Imax − Imin
Imax þ Imin: ð7Þ
The nonlinear constraints applied during the optimi-zation determine the locations of symmetry elementsand saddle points within the unit cell of the peri-odic interference pattern. Lithographically speaking,these determine the shape of the intensity contoursthat will define the final structures. Within four-beam interference, there is one set of nonlinear con-straints that has been applied during the optimiza-tion process and is referred to in the literature as theuniform contrast condition [44,47]. These lithogra-phically useful interference patterns are producedby choosing the plane wave parameters such that allsix interference coefficients Vij are equal. Mathema-tically, the resulting interference patterns have equalcontrasts in the three primitive lattice vector direc-tions (a, b, c) from each lattice point. If the assump-tion about including all six interference coefficients isrelaxed, a more fundamental and complete under-standing of multibeam interference is possible whilestill achieving equal contrasts in more than oneprimitive lattice vector direction. That is, all six in-terference coefficients do not have to be equal andpositive. Avoiding the previously ambiguous termi-nology of uniform contrast [44–46], a more de-scriptive nomenclature is introduced here, namely,condition for primitive-lattice-vector-direction equalcontrasts.In actuality, multiple conditions for primitive-
lattice-vector-direction equal contrasts exist for MBILincluding the four-beam interference case treatedhere. For three-dimensional periodic interferencepatterns produced through four-beam interference,there are six spatial cosines as described byEq. (6). However, only three are required to pro-duce three-dimensional periodicity. Consequently,four conditions for primitive-lattice-vector-directionequal contrasts exist. Based on the current research,a symbolic designation CðmÞ
n is introduced to differ-entiate between the various conditions for primitive-lattice-vector-direction equal contrasts. Quantity n re-presents the total number of interfering beams andm the number of nonzero interference coefficients(number of interfering beam pairs). Consequently,the resulting nonzero interference coefficients Vijof Eq. (6) are denoted by V ðmÞ
n similarly. In our pre-viously published work for three-beam interference
[46], a second uniform contrast condition was intro-duced along with the original uniform contrast con-dition. However, given the new nomenclature andsymbolic designation introduced in this work, a morecomplete description is denoted by Cð3Þ
3 and Cð2Þ3 for
the first and second uniform contrast conditions, re-spectively, for three-beam interference [46].
Considering our previous work [46] and work thatis presented here, it is important to note the relation-ships between the interference coefficient V ðmÞ
n andthe absolute contrast Vabs. This suggests that config-urations of wave vectors and polarizations can resultin interference coefficients that are positive or nega-tive. The physical meaning of this statement is asfollows: If a configuration satisfies a condition forprimitive-lattice-vector-direction equal contrasts andresults in a positive interference coefficient, VðmÞ
n >0, volumes of high intensity surround the latticepoints. Similarly, if a configuration satisfies a con-dition for primitive-lattice-vector-direction equal con-trasts and results in a negative interference coeffi-cient, VðmÞ
n < 0, volumes of low intensity surroundthe lattice points. However, regardless of the signor magnitude of the interference coefficient, V ðmÞ
n , de-termined by the polarization states and amplitudesof the recording beams, the intensity contours will beidentical (but differ in intensity value) provided therecording wave vectors remain unchanged. Conse-quently, another superscript can be added to the gi-ven symbolic designation to describe more accuratelythe interference pattern resulting in �CðmÞ
n where �denotes either a positive or a negative interferencecoefficient. This concept of positive and negativeinterference coefficients enables the pairing of the in-terference pattern with positive and negative photo-resists in a manner analogous to pairing positive andnegative photoresist with light- and dark-field masksin conventional lithography. Given a particular litho-graphic process chemistry, a designer has the abilityto choose between these light-field and dark-field in-terference patterns.
For each lattice, a nonlinear optimization for max-imizing absolute contrast is used here to deter-mine the plane wave parameters while satisfyingthe nonlinear constraints given by each conditionfor primitive-lattice-vector-direction equal contrasts.While subordinate conditions for unity absolute con-trast (Vabs ¼ 1 with Imin ¼ 0) are discussed, thesesubordinate conditions cannot always be satisfied.However, the nonlinear optimization neverthelessproduces solutions that maximize the absolute
Table 1. Primitive Basis Vectors and their Corresponding Recording Wave Vectors
contrast given a particular configuration of recordingwave vectors. More detailed information regardingthe approach used to determine these optimizedparameters is described in the Appendix.
5. Condition for Primitive-Lattice-Vector-DirectionEqual Contrasts Cð6Þ
4
The previously discussed uniform contrast conditionwithin four-beam interference [44,47] corresponds toa symbolic designation of Cð6Þ
4 . Each of the six inter-ference coefficients (corresponding to an interferingbeam pair) contributes to the modulation of intensitywithin the interference pattern as
Vð6Þ4 ¼ V12 ¼ V13 ¼ V14 ¼ V23 ¼ V24
¼ V34½Condition Cð6Þ4 �: ð8Þ
Applying this condition leads to the following set ofconstraints on the polarizations and amplitudes ofthe recording beams [44]:
e12e34 ¼ e13e24 ¼ e14e23; ð9Þ
E2
E1¼ e13
e23;
E3
E1¼ e12
e23;
E4
E1¼ e12
e24: ð10Þ
Once these constraints are satisfied, the interferencecoefficient can be written in terms of the polarizationefficiency factors as [44]
V ð6Þ4 ¼ 2e12e13e23
e212 þ e213 þ e223 þ e213e223=e
234
: ð11Þ
When Cð6Þ4 is satisfied, one set of global intensity
extrema (maxima or minima) is located at the latticepoints (vertices of the primitive unit cell with the ori-gin at r ¼ 0) and the other set of global intensity ex-trema (minima or maxima) is located at the facecenters of the primitive unit cell (r ¼ a=2þ b=2,r ¼ a=2þ c=2, r ¼ b=2þ c=2). Additional stationarypoints (saddle points) occur at the body center (r ¼a=2þ b=2þ c=2) and edge centers (r ¼ a=2, r ¼ b=2,r ¼ c=2) of the primitive unit cell. Given the locationsof the intensity extrema, the absolute contrast can bewritten in terms of the interference coefficient as
Vabs ¼���� 4
1=V ð6Þ4 þ 2
����: ð12Þ
A. Subordinate Condition for Unity Absolute Contrast for−Cð6Þ
4
Two subordinate conditions also exist for unity abso-lute contrast (Vabs ¼ 1 with Imin ¼ 0) for Cð6Þ
Fig. 1. Orientation of polarizations for the subordinate conditionfor unity absolute contrast [or any combination of the polarizationvectors (ei), where one, multiple, or all are inverted (−ei)] for −Cð6Þ
4 ,where V ð6Þ
4 ¼ −1=3. α ¼ cos−1ð−1=3Þ ≈ 109:47°.
Table 2. Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for −Cð6Þ4
(109:47° between polarization vectors) or one of anadditional 15 orientations of the polarization vectorsthat are obtained by inverting (�ei) any single, anypair, any triplet, or all of the polarization vectors. Theorientation of the resulting polarization vectors de-scribed in Eq. (13) (which define the vertices of a reg-ular tetrahedron) is illustrated in Fig. 1. For Fig. 1and subsequent illustrations of the subordinate con-ditions for unity absolute contrast, the orientationsof the polarizations are fixed relative to each other.However, the set of polarization vectors, as a whole,may have any arbitrary rotational orientation rela-tive to the origin. In these illustrations, the specificorientation of the set of polarizations is chosen sym-metrically with respect to the Cartesian axes for easeof visualization.
B. Subordinate Condition for Unity Absolute Contrast forþCð6Þ
4
For V ð6Þ4 ¼ 1=2, an example of þCð6Þ
4 , the additionalconstraints on the polarizations are
eij ¼ e12 ¼ e13 ¼ e14 ¼ e23 ¼ e24 ¼ e34 ¼ 1 ð14Þ(parallel polarization vectors) or one of an additional15 orientations of the polarization vectors that areobtained by inverting (�ei) any single, any pair, anytriplet, or all the polarization vectors. This constraintindicates that the recording wave vectors are copla-nar and polarizations are collinear. In this case, theresulting interference pattern will be invariant in atleast one direction, i.e., exhibit only two-dimensionalperiodicity. Thus, optimizing the orientations ofpolarizations for wave vectors that exhibit three-dimensional periodicity will result in an absolutecontrast Vabs < 1 while satisfying þCð6Þ
4 .A nonlinear optimization to maximize absolute
contrast is used to determine the plane wave para-meters. Solutions for both −Cð6Þ
4 and þCð6Þ4 are sum-
marized in Tables 2 and 3, respectively. Intensitycontours for each solution are also illustrated in Fig. 2with volumes of higher intensity being enclosed.
6. Condition for Primitive-Lattice-Vector-DirectionEqual Contrasts Cð5Þ
4
A second condition for primitive-lattice-vector-direction equal contrasts [Cð5Þ
4 ] occurs when five of thesix interference coefficients (corresponding to inter-fering beam pairs) contribute to the modulation of in-tensity within the interference pattern as
V ð5Þ4 ¼ V12 ¼ V13 ¼ V14 ¼ V23 ¼ V24;
V34 ¼ 0½Condition Cð5Þ4 �: ð15Þ
Applying this condition leads to the following set ofconstraints on the polarizations and amplitudes ofthe recording beams:
aElectric field amplitude ratios and polarization vectors with resulting interference coefficient and absolute contrast.
Fig. 2. Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubicperiodicity that satisfy the þCð6Þ
4 (upper) and −Cð6Þ4 (lower) condi-
tions for primitive-lattice-vector-direction equal contrasts.
When Cð5Þ4 is satisfied, one set of global intensity ex-
trema (maxima or minima) is located at the latticepoints (vertices of the primitive unit cell with the ori-gin at r ¼ 0) and the other set of global intensity ex-trema (minima or maxima) is located at a face centerof the primitive unit cell (r ¼ b=2þ c=2). Additionalstationary points (saddle points) occur at r ¼ a=2þb=4þ 3c=4, r ¼ a=2þ 3b=4þ c=4, the edge centers(r ¼ a=2, r ¼ b=2, r ¼ c=2), the body center (r ¼a=2þ b=2þ c=2), and the rest of the face centers(r ¼ a=2þ b=2, r ¼ a=2þ c=2) of the primitive unitcell. Given the locations of the intensity extrema,the absolute contrast can be written in terms ofthe interference coefficient as
Vabs ¼���� 4
1=V ð5Þ4 þ 1
����: ð19Þ
A. Subordinate Condition for Unity Absolute Contrast for−Cð5Þ
4
Two subordinate conditions also exist for unity abso-lute contrast (Vabs ¼ 1 with Imin ¼ 0) for Cð5Þ
(109:47° or 114:09° between polarization vectors) orone of an additional 15 orientations of the polariza-tion vectors that are obtained by inverting (�ei) anysingle, any pair, any triplet, or all the polarizationvectors. The orientation of the resulting polarizationvectors described in Eq. (20) is illustrated in Fig. 3(a).
Fig. 3. Orientation of polarizations for the subordinate conditions for unity absolute contrast [or any combination of the polarizationvectors (ei), where one, multiple, or all are inverted (−ei)] for (a) þCð5Þ
4 , where V ð5Þ4 ¼ 1=3 and (b) −Cð5Þ
4 , where V ð5Þ4 ¼ −1=3 and
α ¼ cos−1ð−1=3Þ ≈ 109:47°.
Table 4. Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for −Cð5Þ4
B. Subordinate Condition for Unity Absolute Contrast forþCð5Þ
4
For V ð5Þ4 ¼ 1=3, an example of þCð5Þ
4 , the additionalconstraints on the polarizations are
e12 ¼ 1; e13 ¼ e14 ¼ e23 ¼ e24 ¼ 1=ffiffiffi2
pð21Þ
(0° or 45° between polarization vectors) or one of anadditional 15 orientations of the polarization vectorsthat are obtained by inverting (�ei) any single, anypair, any triplet, or all the polarization vectors. Thisconstraint indicates that all six polarizations are co-planar with e3 and e4 orthogonal and e1 and e2 col-linear, bisecting e3 and e4. The orientation of the re-sulting polarization vectors described in Eq. (21) isillustrated in Fig. 3(b).A nonlinear optimization to maximize absolute
contrast is used to determine the plane wave para-meters. Solutions for both −Cð5Þ
4 and þCð5Þ4 are sum-
marized in Tables 4 and 5, respectively. Intensitycontours for each solution are also illustrated in Fig. 4with volumes of higher intensity being enclosed.
7. Condition for Primitive-Lattice-Vector-DirectionEqual Contrasts Cð4Þ
4
A third condition for primitive-lattice-vector-directionequal contrasts [Cð4Þ
4 ] occurs when four of the six in-terference coefficients (corresponding to interferingbeam pairs) contribute to the modulation of intensitywithin the interference pattern as
V ð4Þ4 ¼ V12 ¼ V13 ¼ V14 ¼ V23;
V24 ¼ V34 ¼ 0½Condition Cð4Þ4 �: ð22Þ
Applying this condition leads to the following set ofconstraints on the polarizations and amplitudes ofthe recording beams:
e24 ¼ e34 ¼ 0 ð23Þ
(polarization vectors are orthogonal),
E2
E1¼ e13
e23;
E3
E1¼ e12
e23;
E4
E1¼ e12e13
e14e23: ð24Þ
Once these constraints are satisfied, the interferencecoefficient can be written in terms of the polarizationefficiency factors as
V ð4Þ4 ¼ 2e12e13e23
e212 þ e213 þ e223 þ e212e213=e
214
: ð25Þ
When Cð4Þ4 is satisfied, one set of global intensity ex-
trema (maxima or minima) is located at the latticepoints (vertices of the primitive unit cell with the ori-gin at r ¼ 0) and the other set of global intensity ex-trema (minima or maxima) is located at r ¼ a=3þ2b=3þ c=2 and r ¼ 2a=3þ b=3þ c=2. Additionalstationary points (saddle points) occur at the edgecenters (r ¼ a=2, r ¼ b=2, r ¼ c=2), body center (r ¼a=2þ b=2þ c=2), and face centers (r ¼ a=2þ b=2,r ¼ a=2þ c=2, r ¼ b=2þ c=2) of the primitive unitcell, and also at r ¼ a=3þ 2b=3, and r ¼ 2a=3þb=3. Given the locations of the intensity extrema,the absolute contrast is written in terms of theinterference coefficient as
Table 5. Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for þCð5Þ4
Lattice E2=E1 E3=E1 E4=E1 e1 e2 e3 e4 V Vabs
Simple cubic 1 2 2 0:0−0:70710:7071
0@
1A 0:0
−0:70710:7071
0@
1A −0:7071
−0:70710:0
0@
1A 0:7071
−0:70710:0
0@
1A 1=5 2=3
Face-centered cubic 1ffiffiffi2
p−
ffiffiffi2
p0:7071−0:7071
0:0
0@
1A 0:7071
−0:70710:0
0@
1A 0:0969
−0:90310:4184
0@
1A −0:9031
0:09690:4184
0@
1A 1=3 1.0
Body-centered cubic 1 2 −2 0:0−0:70710:7071
0@
1A 0:0
−0:70710:7071
0@
1A −0:7071
−0:70710:0
0@
1A −0:7071
0:70710:0
0@
1A 1=5 2=3
aElectric field amplitude ratios and polarization vectors with resulting interference coefficient and absolute contrast.
Fig. 4. Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubicperiodicity that satisfy the þCð5Þ
4 (upper) and −Cð5Þ4 (lower) condi-
tions for primitive-lattice-vector-direction equal contrasts.
(114:09°, 125:26°, or 120° between polarization vec-tors) or one of an additional 15 orientations of the po-
larization vectors that are obtained by inverting (−ei)any single, any pair, any triplet, or all the polariza-tion vectors. The orientation of the resulting polari-zation vectors described in Eq. (27) is illustrated inFig. 5(a).
B. Subordinate Condition for Unity Absolute Contrast forþCð4Þ
4
For V ð4Þ4 ¼ 2=5, an example of þCð4Þ
4 , the additionalconstraints on the polarizations are
e12 ¼ e13 ¼ e14 ¼ 1=ffiffiffi2
p; e23 ¼ 1 ð28Þ
(45° or 0° between polarization vectors) or one of anadditional 15 orientations of the polarization vectorsthat are obtained by inverting (−ei) any single, anypair, any triplet, or all the polarization vectors. Thisconstraint indicates that all six polarizations arecoplanar with e4 orthogonal to e2 and e3 (which arecollinear) and e1 bisecting e2 and e4. The orientation
Fig. 5. Orientation of polarizations for the subordinate conditions for unity absolute contrast [or any combination of the polarizationvectors (ei), where one, multiple, or all are inverted (−ei)] for (a) þCð4Þ
4 , where V ð4Þ4 ¼ 2=5 and (b) −Cð4Þ
4 where V ð4Þ4 ¼ −1=4 and
α ¼ cos−1ð1= ffiffiffi3
p Þ ≈ 125:26°.
Table 6. Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for −Cð4Þ4
of the resulting polarization vectors described inEq. (28) is illustrated in Fig. 5(b).A nonlinear optimization to maximize absolute
contrast is used to determine the plane wave para-meters. Solutions for both −Cð4Þ
4 and þCð4Þ4 are sum-
marized in Tables 6 and 7, respectively. Intensitycontours for each solution are also illustrated in Fig. 6with volumes of higher intensity being enclosed.
8. Condition for Primitive-Lattice-Vector-DirectionEqual Contrasts Cð3Þ
4
The fourth, and final, condition for primitive-lattice-vector-direction equal contrasts [Cð3Þ
4 ] occurswhen three of the six interference coefficients (corre-sponding to interfering beam pairs) contribute to themodulation of intensity within the interference pat-tern as
Vð3Þ4 ¼ V12 ¼ V13 ¼ V14;
V23 ¼ V24 ¼ V34 ¼ 0½Condition Cð3Þ4 �: ð29Þ
Applying this condition leads to the following set ofconstraints on the polarizations and amplitudes ofthe recording beams:
e23 ¼ e24 ¼ e34 ¼ 0 ð30Þ
(polarization vectors are orthogonal),
E3
E2¼ e12
e13;
E4
E2¼ e12
e14: ð31Þ
Once these constraints are satisfied, the interferencecoefficient can be written in terms of the polarizationefficiency factors as
V ð3Þ4 ¼ 2E1E2e12
E21 þ E2
2ð1þ e212=e213 þ e212=e
214Þ
: ð32Þ
When Cð3Þ4 is satisfied, one set of global intensity ex-
trema (maxima or minima) is located at the latticepoints (vertices of the primitive unit cell with theorigin at r ¼ 0) and the other set of global intensityextrema (minima or maxima) is located at the bodycenter of the primitive unit cell (r ¼ a=2þ b=2þ c=2).Additional stationary points (saddle points) occur atthe edge centers (r ¼ a=2, r ¼ b=2, r ¼ c=2) and theface centers (r ¼ a=2þ b=2, r ¼ a=2þ c=2, r ¼ b=2þ c=2) of the primitive unit cell. Given the locationsof the intensity extrema, the absolute contrast can bewritten in terms of the interference coefficient as
Vabs ¼ j3Vð3Þ4 j: ð33Þ
Given a configuration of polarizations that satisfyEqs. (30) and (31), the absolute contrast [or interfer-ence coefficient in Eq. (32)] can be maximized by set-ting the electric field ratio E2=E1 equal to
aElectric field amplitude ratios and polarization vectors with resulting interference coefficient and absolute contrast.
Fig. 6. Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubicperiodicity that satisfy the þCð4Þ
4 (upper) and −Cð4Þ4 (lower) condi-
tions for primitive-lattice-vector-direction equal contrasts.
V ð3Þ4 ¼ e12e13e14ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e213e214 þ e212e
214 þ e212e
213
q : ð35Þ
Generally speaking, there is no difference in inter-ference patterns satisfying −Cð3Þ
4 and þCð3Þ4 in that the
contours around intensity maxima and minima areidentical (differing only in intensity value). This par-ticular condition for primitive-lattice-vector-directionequal contrasts in three dimensions is similar to thecondition for primitive-lattice-vector-direction equalcontrasts in two dimensions designated as Cð2Þ
3 (pre-viously referred to as the second uniform contrastcondition in two dimensions) in this regard [46]. Gi-ven a bias intensity value of Ib (where Imin ≤ Ib ≤
Imax), these two identical surfaces are described as
While the interference coefficient Vð3Þ4 of a specific so-
lution that satisfies Cð3Þ4 will be positive or negative,
this metric simply describes whether interferencemaxima or minima are located at lattice points (r ¼0 and all equivalent points in the periodic interfer-ence pattern). Two interference patterns with coeffi-cients of V ð3Þ
4 and −Vð3Þ4 are identical when either is
translated by a=2þ b=2þ c=2.
A. Subordinate Condition for Unity Absolute Contrast forCð3Þ
4
One subordinate condition exists for unity absolutecontrast (Vabs ¼ 1with Imin ¼ 0) for Cð3Þ
4 . ConsideringEq. (33), unity absolute contrast occurs when the in-terference coefficient Vð3Þ
4 equals −1=3 or 1=3. The ad-ditional constraints on the polarizations are
e12 ¼ e13 ¼ e14 ¼ 1=ffiffiffi3
pð37Þ
(54:74° between polarization vectors) or one of an ad-ditional 15 orientations of the polarization vectorsthat are obtained by inverting (−ei) any single, anypair, any triplet, or all the polarization vectors. Theorientation of the resulting polarization vectors de-scribed in Eq. (37) is illustrated in Fig. 7.
A nonlinear optimization to maximize absolutecontrast is used to determine the plane wave para-meters. Solutions for Cð3Þ
4 are summarized in Table 8.Intensity contours for each solution are also illu-strated in Fig. 8 with volumes of higher intensitybeing enclosed.
9. Summary and Discussion
Optimized solutions for three different latticeswith all four different conditions for primitive-lattice-vector-direction equal contrasts in three dimen-sions exhibiting the maximum possible absolutecontrast have been given in Tables 3 through 8.These tables provide guidelines for determininglithographic process parameters and enable choos-ing between processes based on positive-tone ornegative-tone photoresists. The lithographic process
Fig. 7. Orientation of polarizations for the subordinate conditionsfor unity absolute contrast [or any combination of the polarizationvectors (ei), where one, multiple, or all are inverted (−ei)] for Cð3Þ
4 ,where V ð3Þ
4 ¼ �1=3 and α ¼ cos−1ð ffiffiffi3
p=3Þ ≈ 54:74°.
Table 8. Optimized Plane Wave Parameters a for Lattices Maximizing Absolute Contrast for þCð3Þ4
will be easier to implement and will provide greaterprocessing latitude when an interference patternwith the largest absolute contrast is used.From the quantitative descriptions of the
conditions for primitive-lattice-vector-direction equalcontrasts and the illustrations of the intensity con-tours for the three lattices, a more qualitative de-scription of the resultant interference patterns canbe given. For the Cð6Þ
4 and Cð3Þ4 , there is equal contrast
in each of the primitive lattice directions (a, b, and c)from each lattice point. For Cð5Þ
4 , there is equal con-trast in two of the primitive lattice directions (aand c) from each lattice point. For Cð4Þ
4 , there is equalcontrast in two of the primitive lattice directions (aand b) from each lattice point. For the three latticestreated, there are significant differences in the abso-lute contrast of the optimized solutions between eachcondition for primitive-lattice-vector-direction equalcontrasts and between the distinct þCðmÞ
n and −CðmÞn
solutions. For the simple cubic lattice, the absolutecontrast ranges from 0.5 when satisfying þCð6Þ
4 tounity absolute contrast when satisfying −Cð6Þ
4 . Forthe face-centered cubic lattice, the absolute contrastranges from 0.2599 when satisfying −Cð6Þ
4 to unity ab-solute contrast when satisfying þCð5Þ
4 . For the body-centered cubic lattice, the absolute contrast rangesfrom 0.5 when satisfying þCð6Þ
4 to unity absolute con-trast when satisfying −Cð6Þ
4 . We have presented fourdifferent conditions in the case of four-beam interfer-ence that will result in lithographically useful inter-ference patterns suitable for lithographic processing.Four-beam interference also serves as a basis for de-termining and describing these interference patternsin terms of the crystallographic space groups that de-termine the symmetry of the patterns.
Appendix: Optimization Approaches
To maximize the absolute contrast, an objective func-tion must be maximized (or minimized) under con-straints given by the physical problem and by eachcondition for primitive-lattice-vector-direction equalcontrasts. In the most general of terms, this problemwill have 12 variables and either 10 or 11 constraintequations (depending on the condition for primitive-
lattice-vector-direction equal contrasts). The 12 vari-ables are the 12 Cartesian coordinates that describeeach of the four polarization vectors as
ei · ki ¼ ei;xki;x þ ei;yki;y þ ei;zki;z ¼ 0; ðA3Þ
where the wave vector is
ki ¼ ðki;x; ki;y; ki;zÞ: ðA4Þ
Equations (A2) and (A3) constitute eight of the con-straint equations (two for each recording wavevector). The final two or three constraint equationsare determined by the condition for primitive-lattice-vector-direction equal contrasts in Eq. (9) forCð6Þ
4 , in Eq. (16) for Cð5Þ4 , in Eq. (23) for Cð4Þ
4 , and inEq. (30) for Cð3Þ
4 . The objective function, absolute con-trast, is calculated in terms of these 12 variables andmaximized under the 10 or 11 constraint equations.
A commonly used strategy for solving constrainedoptimization problems is the method of Lagrangemultipliers. Given the function to be minimized ormaximized,
with the introduction of m Lagrange multipliers(λi). A subset of the stationary points of this uncon-strained function, given by
∇Λ ¼ 0; ðA8Þ
is the solution to the constrained optimizationproblem. As described above, the constrained optimi-zation problem of 12 variables and 10 or 11 con-straint equations yields a Lagrangian function of22 or 23 variables (constituting 10 or 11 Lagrangemultipliers).
Fig. 8. Intensity contours for interference patterns with body-centered cubic (BCC), face-centered cubic (FCC), and simple cubicperiodicity that satisfy the Cð3Þ
4 condition for primitive-lattice-vector-direction equal contrasts.
For this work, however, we utilized a differentapproach that incorporates only four variables andtwo or three constraint equations. The four variablesare the counterclockwise angular rotations (whenlooking antiparallel to the wave vector ki), ψ i, ofthe vector
ei;0 ¼ z × ki ðA9Þ
about ki, which corresponds to a polarization vectorei, for wave vector ki. This is accomplished by apply-ing five rotational transformations to ei;0. Given therotational matrices for the counterclockwise angularrotation (when looking toward the origin) of the x, y,and z axes as RxðαÞ, RyðβÞ, and RzðγÞ, respectively, ψ idescribes the polarization vector ei as
ei ¼ Rzð−ϕÞRyð−θÞRzð−ψÞRyðθÞRzðϕÞei;0; ðA10Þ
where θ and ϕ are the spherical coordinates of ki.By definition, ei;0 and ei satisfy the orthogonalityand normalized conditions, eliminating the need toincorporate the eight constraint equations given byEqs. (A2) and (A3). The objective function, abso-lute contrast, is then calculated in terms of thesefour variables and maximized under the two orthree constraint equations given by each conditionfor primitive-lattice-vector-direction equal contrasts.This results in a Lagrangian function of onlysix or seven variables (constituting two or threeLagrange multipliers). This approach resulted in aquicker and more stable implementation of the non-linear optimization.
This research was supported in part by the Na-tional Science Foundation (NSF) under grantECCS-0925119. .
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