Contrast Analysis of Polarization in Three-Beam Interference Lithography

applied sciences

Article Contrast Analysis of Polarization in Three-Beam Interference Lithography

Fuping Peng 1,2 , Jing Du 2, Jialin Du 1,2, Simo Wang 1,2 and Wei Yan 2,*

1 University of Chinese Academy of Sciences, Beijing 100049, China; [email protected] (F.P.); [email protected] (J.D.); [email protected] (S.W.) 2 Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China; [email protected] * Correspondence: [email protected]

Abstract: This paper analyzes the effect of polarization and the incident angle on the contrasts of interference patterns in three-beam interference lithography. A non-coplanar laser interference system was set up to simulate the relationship between contrast, beam polarization, and the incident angle. Different pattern periods require different incident angles, which means different contrast losses in interference lithography. Two different polarization modes were presented to study the effects of polarization with different incident angles based on theoretical analysis simulations. In the case of the co-directional component TE polarization mode, it was demonstrated that the pattern contrast decreases with the increase in the incident angle and the contrast loss caused by the polarization angle error also grew rapidly. By changing the mode to azimuthal (TE-TE-TE) polarization, the contrast of the interference pattern can be ensured to remain above 0.97 even though the incident angle is large. In addition, TE-TE-TE mode can accept larger polarization angle errors. This conclusion provides a theoretical basis for the generation of high-contrast light ﬁelds at different incident angles, and the conclusion is also applicable to multi-beam interference lithography.

Keywords: pattern contrast; polarization; incident angle; interference lithography

Citation: Peng, F.; Du, J.; Du, J.; Wang, S.; Yan, W. Contrast Analysis of Polarization in Three-Beam 1. Introduction Interference Lithography. Appl. Sci. As a new type of material with a photonic band gap, photonic crystals play an 2021 11 , , 4789. https://doi.org/ important role in modern optical systems, such as photonic crystal ﬁbers, waveguides, low- 10.3390/app11114789 threshold lasers, multifunctional sensors, cavity quantum electrodynamics, and quantum information processing [1–6]. Over the past decades, many techniques have been developed Received: 29 April 2021 for the fabrication of photonic crystals, of which the laser interference lithography (LIL) is Accepted: 18 May 2021 Published: 23 May 2021 a powerful technology due to its low cost, short time consumption, and lack of necessity for precise focusing. LIL was ﬁrst used to fabricate one-dimensional (1D) grating using

Publisher’s Note: MDPI stays neutral double-beam interference. With the addition of a third or further beams, 2D and 3D with regard to jurisdictional claims in lattice interference patterns can be generated, including all 2D and 3D Bravais lattice published maps and institutional affil- structures [7–11]. iations. Since the energy of the interference fringes is sinusoidally distributed, a high-contrast LIL can write sub-diffraction-limited features in photoresist [12,13]. This is possible because of the non-linear response of positive photoresist to the light dose. The feature size (line width) is reduced by overexposing the high-contrast sinusoidal energy distribution light field [14,15]. Therefore, the actual limit of the line width is set by the light field contrast Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. and the sensitivity of the photoresist. In order to obtain the smallest possible feature This article is an open access article size, the contrast of the interference light field needs to be sufficiently high. To ensure distributed under the terms and sufficient interference contrast, the design of the interferometer and the selection of the conditions of the Creative Commons beam parameters are crucial. Attribution (CC BY) license (https:// As shown in the following papers [16–18], the source of the contrast loss of the creativecommons.org/licenses/by/ double-beam interference in the same plane mainly includes the beam pointing deviation, 4.0/). the polarization state, and the intensity ratio of the interference beam. Different from

Appl. Sci. 2021, 11, 4789. https://doi.org/10.3390/app11114789 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 4789 2 of 11

double-beam interference, multi-beam interference lithography (MIL) is the superposition of the intensity of multiple groups of non-coplanar interference with different periods, directions, and extremum [19–21]. The value of the incident angle controls the period of the interference pattern. For non-coplanar interference, the contrast of the interference light field is simultaneously affected by the azimuth angle, incident angle, and polarization state of each beam [22–24]. Therefore, it is necessary to select a suitable combination of beam polarization states for different incident angles to obtain the best contrast and consider the influence of the beam pointing stability on the contrast under that incident combination. In this paper, we discuss the two-beam non-coplanar interference, study the relationship between fringe visibility and the spatial incidence direction of the two beams, and analyze the source of the contrast loss in the non-coplanar interference. We calculate how to select the appropriate combination of beam polarization states to ensure sufficient contrast under different incident angles. Then, according to the non-coplanar interference model, we demonstrate the specific beam polarization combinations that should be used for different pattern periods in three-beam LIL. Although the focus of this paper is three-beam LIL, the same analysis of contrast loss and the beam polarization combination mode is applicable for N-beam (n > 3) interference lithography.

2. Contrast in Double-Beam Non-Coplanar Interference Patterns When two laser beams from the same light source coincide, an interference pattern is produced, which contains all of the information including the intensity and phase of the two interfering beams. The interference of two beams is the superposition of electric ﬁeld vectors. The electric ﬁeld vector of ith beam vector can be written as

Ei = Aiei exp[i(ki · r + ϕi)] (1)

where Ai is the amplitude, ei is the polarization vector, ki is the wave propagation vector, r is the position vector at interference area, and ϕi is the initial phase of the light wave. The intensity of two-beam interference can be calculated as follows:

∗ 2 2 T I2−beam = (E1 + E2)(E1 + E2) = |A1| + |A2| + 2A1 A2 e1e2 cos[(k1 − k2) · r + (ϕ1 − ϕ2)] (2) Interference contrast is deﬁned as I − I K = max min (3) Imax + Imin

Imax and Imin represent the minimum and the maximum intensity of the interference ﬁeld. Substituting Equation (2) into Equation (3) yields

T 2A1 A2 e1e2 K ∝ 2 2 (4) |A1| + |A2|

Equation (4) states that contrast is affected by the intensity mismatch and the polarization states for each beam. We already know that the contrast loss caused by the unequal amplitude of the two beams is very low. Even if the amplitude ratio of the two beams reaches 2:3, the contrast loss is only 0.076. Therefore, the main source of contrast loss for non-coplanar interference is the polarization state of the two beams. In Equation (1), according to the coordinate rotation transformation matrix ei, ki and r can be expressed as

 T  T eix i(cos αi cos θi cos φi − sin αi sin φi) ei =  eiy  =  j(sin αi cos θi cos φi + cos αi sin φi)  (5) eiz −k sin θi cos φi ki = kix kiy kiz = k i cos αi sin θi j sin αi sin θi −k cos θi (6) Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 11

T ()αθφ− αφT e i cosiii cos cos sin ii sin ix  == ()αθφ + αφ eeiiy  jsin i cos i cos i cos ii sin (5)  −k sinθφ cos eiz ii ==[ αθ αθ − θ] kiixiyiz kkkk icos ii sin j sin ii sin k cos i (6)

Appl. Sci. 2021, 11, 4789 T 3 of 11 ri=[]x jkyz (7) θ φ Here, αi and i are the azimuthal angle and the incident angle, respectively; i is T the polarization angle; k = 2πλ is the wavenumber;r = ix andjy λ k isz the wavelength of the la- (7) ser. As shown in Figure 1, the azimuth angles of the two non-coplanar coherent beams are Here, αi and θi are the azimuthal angle and the incident angle, respectively; φi is the α = 0 αα=<<()0 απ θ α 1 and polarization2 angle;, andk = they2π/ λincidentis thewavenumber; at the identical and angleλ is the. When wavelength is of the laser. equal to π ,As it turns shown into in Figurecoplanar1, theinterference. azimuth anglesUsing ofEquations the two (5)–(7), non-coplanar the contrast coherent for beams are two-beam interferenceα1 = 0 and canα2 = beα (further0 < α < reducedπ), and to they incident at the identical angle θ. When α is equal to π, it turns into coplanar interference. Using Equations (5)–(7), the contrast for two-beam sinαθ cos sin()φφ−+ cosα cos ()φφ − interference can be further reduced12 to 12 K ∝ (8) +−()αθφφ2 1 cos sin cos12 cos sin α cos θ sin(φ1 − φ2) + cos α cos(φ1 − φ2) K ∝ 2 (8) Equation (8) shows that for different azimut+(1 −h cosandα incident) sin θ cos angles,φ1 cos weφ2 need to choose different beam polarization modes to maximize the contrast of the interference field.

Figure 1. Wave vectors and the coordinate system for double beam interference. Both incident directions of the two beams Figure 1. Wave vectors and the coordinate system for double beam interference. Both incident are on the conedirections line of the of conical the two surface. beams are on the cone line of the conical surface. Equation (8) shows that for different azimuth and incident angles, we need to choose Owing to the incident surfaces of the two light waves not being coplanar, there is no different beam polarization modes to maximize the contrast of the interference ﬁeld. case where the horizontal polarization component is totally in the same direction. We can Owing to the incident surfaces of the two light waves not being coplanar, there is no only reduce the contrast loss by ensuring that their co-directional TE vibration compo- case where the horizontal polarization component is totally in the same direction. We can nents achieve the maximum value. Under this condition, the azimuth and the polarization only reduce the contrast loss by ensuring that their co-directional TE vibration components angles of the two beams satisfy the following equation: achieve the maximum value. Under this condition, the azimuth and the polarization angles φ =−α +φ of the two beams satisfy the following21 equation: (9)

In this case, the contrast can simplify to φ2 = −α + φ1 (9) ∝+22αθ() − αφ ()α −−φ θα 2 InK thiscos case, the sin contrast 1 cos can simplify cos11 cos to cos sin (10)

From Equation (10), each azimuth2 angle2 has a unique polarization co-direction angle2 K ∝ cos α + sin θ(1 − cos α) cos φ1 cos(α − φ1) − cos θ sin α (10) corresponding to it, which causes the contrast to obtain a maximum value. We can calculate that the conditionFrom for Equation maximum (10), contrast each azimuth is angle has a unique polarization co-direction angle corresponding to it, which causes theα contrast to obtain a maximum value. We can calculate that the condition for maximumφ contrast= is (11) 1 2 α φ = (11) This condition is verified in Figure 2. We simulated1 the2 contrast of all linear polarization orientations when the azimuth angles were 30°, 60°, 90°, 120°, 150°and 180°, and This condition is veriﬁed in Figure2. We simulated the contrast of all linear polarization orientations when the azimuth angles were 30◦, 60◦, 90◦, 120◦, 150◦and 180◦, and found that the corresponding polarization angle when the maximum contrast is achieved was 15◦, 30◦, 45◦, 60◦, 75◦, and 90◦. This shows that when the polarization angle is on the intersection of the vibration planes of two waves, the contrast is maximized. By comparing Figure2a,b, it can be seen that the polarization orientation has a great inﬂuence on the contrast, especially under the condition of a large angle of incidence. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 11

Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 11

found that the corresponding polarization angle when the maximum contrast is achieved was 15°, 30°, 45°, 60°, 75°, and 90°. This shows that when the polarization angle is on the intersectionfound that the of thecorresponding vibration planes polarization of two wa angleves, thewhen contrast the maximum is maximized. contrast By iscomparing achieved Figurewas 15°, 2a,b, 30°, it 45°, can 60°, be seen75°, and that 90°. the Thispolariza showstion that orientation when the has polarization a great influence angle is onon the contrast,intersection especially of the vibration under the planes condition of two of wa a largeves, the angle contrast of incidence. is maximized. By comparing Appl. Sci. 2021, 11, 4789 Figure 2a,b, it can be seen that the polarization orientation has a great influence4 of 11 on the contrast, especially under the condition of a large angle of incidence.

Figure 2. The relationship between the contrast and the polarization angle combinations with 6 azimuth angles. The interference contrast is most affected by polarization when the azimuth angles are orthogonal. The greater the incident angle, the greater the contrast loss caused by the po- FigureFigure 2. TheThe relationship relationship betweenbetween thethe contrastcontrast and and the the polarization polarization angle angle combinations combinations with with 6 larization. For (a), the incident angle is 15°, and for (b), the incident angle is 30°. azimuth6 azimuth angles. angles. The The interference contrast contrast is mostis most affected affected by polarization by polarization when the when azimuth the azimuth angles an- glesare orthogonal. are orthogonal. The greater The greater the incident the angle,incident the angle, greater the the contrastgreater lossthe causedcontrast by loss the polarization.caused by the po- Substituting Equation◦ (11) into Equation (10), the◦ maximum contrast that can be larization.For (a), the incidentFor (a), the angle incident is 15 , and angle for is (b ),15°, the and incident for (b angle), the is incident 30 . angle is 30°. achieved by double-beam interference is Substituting Equation (11) into Equation (10), the maximum contrast that can be Substituting Equation (11) into Equation (10), the maximum contrast that can be achieved by double-beam interference∝− is1 22θθθα + + 2 achieved by double-beam Kinterferencemax 1sincoscossin is (12) 2 1 2 2 2 Kmax ∝ 1 − sin1θ + cos θ + cos θ sin α (12) ∝−2 22θθθα + + 2()απ= As shown in Figure 3,K onlymax in1sincoscossin the case of coplanar interference and where(12) 2 two beamsAs shown are inincident Figure3 in, only TE- inTE the mode case will of coplanar the contrast interference not decrease(α = π )dueand to where the increase intwo the beamsAs incident shown are incident angle.in Figure inAsTE 3,the- TEonly azimuthmode in th wille anglecase the contrast ofincreases, coplanar not decreasethe interference contrast due toloss () theαπ =caused increase and by where large incidentin the incident angle angle.interference As the is azimuth considerable angle increases, until it increases the contrast to 90°, loss where caused the by largecontrast loss two beams are incident in TE-TE mode will the contrast not◦ decrease due to the increase causedinincident the incident by angle large interference angle.incident As angle is the considerable azimuth interference untilangle reaches it increases, increases the to maximum.the 90 ,contrast where theWhen loss contrast causedthe azimuth loss by large an- glecaused continues by large to incident increase angle beyond interference 90°, the reaches contrast the maximum.loss gradually When decreases. the azimuth Until angle coplanar incident angle interference is considerable◦ until it increases to 90°, where the contrast loss interferencecontinues to increaseis reached, beyond the 90contrast, the contrast loss caused loss gradually by the large decreases. incident Until angle coplanar completely causedinterference by large is reached, incident the angle contrast interference loss caused reaches by the the large maximum. incident angle When completely the azimuth an- disappears. As long as the incident angle is not greater than 35°, the contrast loss of any gledisappears. continues As to long increase as the beyond incident 90°, angle the is notcont greaterrast loss than gradually 35◦, the contrastdecreases. loss Until of any coplanar azimuthinterferenceazimuth non-coplanarnon-coplanar is reached, interference interference the contrast does does loss not not exceedcaused exceed 0.02. by 0.02.the However, large However, ifincident the incidentif the angle incident angle completely angle continuesdisappears.continues toto increase, Asincrease, long theas the the contrast contrast incident sharply sharply angle decreases. isdecreases. not greater than 35°, the contrast loss of any azimuth non-coplanar interference does not exceed 0.02. However, if the incident angle continues to increase, the contrast sharply decreases.

FigureFigure 3. The relationship between between the the contrast contrast of of the th two-beame two-beam interference interference light light ﬁeld field and theand the incidentincident angle under thethe optimaloptimal polarizationpolarization incident incident condition. condition. In In the the case case of of non-coplanar non-coplanar inter- ferenceinterference of two of two beams, beams, the the contrast contrast loss loss caused caused byby thethe increase increase in in the the incident incide anglent angle is inevitable. is inevitable. Figure 3. The relationship between the contrast of the two-beam interference light field and the incident angle under the optimal polarization incident condition. In the case of non-coplanar interference of two beams, the contrast loss caused by the increase in the incident angle is inevitable. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 11

Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 11

Appl. Sci. 2021, 11, 4789 5 of 11 In addition, the existence of polarization angle deviation will inevitably cause the loss In addition,of contrast. the existenceTo illustrate of polarization this impact, angle we assume deviation that will the inevitably polarization cause angle the deviations loss of δ δ of contrast.the To two illustrate beams are this impact,1 and we2 . In assume this case, that the the contrast polarization can be angle described deviations as of In addition,δ δ the existence of polarization angle deviation will inevitably cause the loss the two beams are 1 and 2 . In this case,αθ the contrast() δδ−+ can be α described () δδ − as of contrast. To illustratesin this impact,cos sin we12 assume cos that cos the 12polarization angle deviations of K ∝ the two beams areαθδ1 and δ12.() δδ In−+ this case,2 α the contrast () δδ − can be described as (13) sin cos+− sin()12αθδδαδδ cos cos ()() 12 − +− 1 cos sin cos12 cos 12 K ∝ 1 2 (13) +−() αθδδαδδ2 ()()( −− ) + +−( − ) 1 cossin sinα cos cosθ sin12δ1 δ cos2 cos 12α cos δ1 δ2 Under the sameK2 ∝ incident 1 angle, the contrast2 loss is largest when the azimuth angle (13)is + 2 (1 − cos α) sin θ cos(δ1 − δ2) cos(α + δ1 − δ2) Underequal the to same 90°. incident The polarization angle, the angle contrast deviation loss is largestat this pointwhen furtherthe azimuth increases angle the is contrast Under the same incident angle, the contrast loss is largest when the azimuth angle is equal to loss.90°. TheAs shownpolarization in Figure angle 4a,b, deviation when theat this po larizationpoint further angle increases deviation the of contrast the two beams equal to 90◦. The polarization angle deviation at this point further increases the contrast loss. As shownreaches in 5°, Figure the greater 4a,b, thewhen incident the po angle,larization the greaterangle deviation the loss of of contrast. the two beams loss. As shown in Figure4a,b, when the polarization angle deviation of the two beams reaches 5°, the greater the incident angle, the greater the loss of contrast. reaches 5◦, the greater the incident angle, the greater the loss of contrast.

Figure 4. Contrast loss caused by polarization angle deviation at different incident angles. The larger the incident angle, the more unstable the interference system. For (a), the incident angle is Figure 4. ContrastFigure loss 4. caused Contrast by polarization loss caused angleby polarization deviation atangle different deviation incident at different angles. The incident larger angles. the incident The angle, the 15°. For (b), the incident angle is 30°. ◦ ◦ more unstable thelarger interference the incident system. angle, For the (a more), the incidentunstable anglethe interference is 15 . For system. (b), the incidentFor (a), the angle incident is 30 .angle is 15°. For (b), the incident angle is 30°. TheThe incident incident angle, angle, azimuth, azimuth, and and polarization polarization for for non-coplanar non-coplanar interference interference contrast contrast The wereincidentwere analyzed analyzed angle, in azimuth, inturn. turn. As Asand shown shown polarization in inFigure Figure for5,5 considering ,non-coplanar considering all all interferencefactors factors at at a atime, contrast time, we we obtained obtained were analyzedthethe contrast contrastin turn. of ofAs two-beam two-beam shown in non-coplanar non-coplanarFigure 5, considering interference interference all factors at different at a time, incidentincident we obtained angles, angles, azimuth azimuth an- the contrastangles,gles, of two-beam andand polarizationpolarization non-coplanar angleangle deviations. deviations.interference TheThe at FiguredifferentFigure shows shows incident that that the angles,the polarization polarization azimuthorientation orienta- angles, andtionmust polarization must be be adjusted adjusted angle to deviations. reduceto redu thece theThe contrast contrast Figure loss shows loss caused caused that by the theby polarization increasethe increase in orienta- the in incidentthe incident angle. tion mustangle. beThe adjusted maximumThe maximum to redu co-directionalce co-directional the contrast polarization losspolarization caused component by component the modeincrease is mode not in suitablethe is incidentnot forsuitable large-angle for angle. Thelarge-angle incidentmaximum situations. incident co-directional situations. polarization component mode is not suitable for large-angle incident situations.

Figure 5. The contrast of two-beam non-coplanar interference at different incident angles, azimuth Figure 5. The contrast of two-beam non-coplanar interference at different incident angles, azimuth angles, and polarization angle deviations. The polarization mode of the incident light beam is the angles, and polarization angle deviations. The polarization mode of the incident light beam is the Figure 5. Themaximum contrast of co-directional two-beam non-coplanar polarization interf component.erence at different incident angles, azimuth maximum co-directional polarization component. angles, and polarization angle deviations. The polarization mode of the incident light beam is the maximum co-directional3. Contrast inpolarization Three-Beam component. Interference Patterns N-beam interference is the incoherent superposition of N(N − 1)/2 groups of two- beam non-coplanar interference. Therefore, the above analysis of non-coplanar interference Appl. Sci. 2021, 11, 4789 6 of 11

is applicable to N-beam interference. The intensity of N-beam interference can be calculated as N N 2 T IN−beam = ∑ Ai + 2∑ Ai Ajeiej cos ki − kj · r+ ϕi − ϕj (14) i=1 i

N T K ∝ ∑ eiej (15) i

It can be seen from Equation (15) that the polarization combination of the beams has a key effect on the contrast of the light ﬁeld. Three-beam interference is a typical case of N-beam interference, the related derivation process, and results are generally applicable in multi-beam interference. Therefore, this paper chooses three-beam interference as the contrast loss analysis model.

3.1. Co-Directional Component Polarization When the polarization common component is the largest, the relationship between the azimuth and the polarization angle of N-beam interference is as follows:

i − 1 α = 2π , φ = φ − α (16) i N i i where φ is the polarization angle of the beam whose azimuth is zero. The polarization vector can be written as follows:

T  i−1 i−1 i−1 i−1  i cos θ cos 2π N cos φ − 2π N − sin 2π N sin φ − 2π N   =  i−1 − i−1 + i−1 − i−1  ei  j cos θ sin 2π N cos φ 2π N cos 2π N sin φ 2π N  (17)  i−1  −k sin θ cos φ − 2π N

As shown in Figure6, the three incident beams follow a symmetrical conﬁguration ◦ ◦ ◦ with the azimuth of α1 = 0 , α2 = 120 , and α3 = 240 . The incident angles of the three beams are set as θ1 = θ2 = θ3 = θ. The polarization vectors of the three beams are e1 = i cos θ cos φ j sin φ −k sin θ cos φ  √ √ T i 1 cos θ cos φ − 3 cos θ sin φ + 3 sin φ + 3 cos φ 4√ 4 4 √4  3 3 1 3  e2 =  j − cos θ cos φ + cos θ sin φ + sin φ + cos φ   4 4 √ 4 4   1 3  k 2 sin θ cos φ − 2 sin θ sin φ (18)  √ √ T i 1 cos θ cos φ + 3 cos θ sin φ − 3 sin φ + 3 cos φ 4√ 4 4 √4  3 3 1 3  e3 =  j cos θ cos φ + cos θ sin φ + sin φ − cos φ   4 4 √ 4 4   1 3  k 2 sin θ cos φ + 2 sin θ sin φ

The three beams are symmetrically incident, and the vibration components of the x-y plane are along the same angle φ. Substituting Equation (18) into Equation (15), considering the polarization and incident angle one at a time, the contrast as a function of two factors is plot in Figure6. Similar to the two-beam non-coplanar interference, when interfering at a small angle, the polarization angle will not change the contrast. The contrast loss will only increase as the incident angle increases. As the incident angle continues to increase, the contrast is more sensitive to the changes in the polarization angle and is extremely susceptible to polarization angle deviations. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 11

=−[ θφ φ θφ] ei1 cos cos j sin k sin cos T  1333θφ−++ θφ φ φ i cos cos cos sin sin cos 44 44  3313 ej=−cosθφ cos + cos θφ sin + sin φ + cos φ 2 4444   13 k sinθφ cos− sin θφ sin  22 (18) T  133θφ+− θφ φφ+ 3 i cos cos cos sin sin cos 44 44  3313 e = j cosθφ cos++− cos θφ sin sin φ cos φ 3 4444   13 k sinθφ cos+ sin θφ sin  22

The three beams are symmetrically incident, and the vibration components of the x- y plane are along the same angleφ . Substituting Equation (18) into Equation (15), considering the polarization and incident angle one at a time, the contrast as a function of two factors is plot in Figure 6. Similar to the two-beam non-coplanar interference, when interfering at a small angle, the polarization angle will not change the contrast. The contrast loss will only increase as the incident angle increases. As the incident angle continues to increase, the contrast is more sensitive to the changes in the polarization angle and is ex- Appl. Sci. 2021, 11, 4789 7 of 11 tremely susceptible to polarization angle deviations.

FigureFigure 6. 6. OnOn the the left left is isa top a top view view of three of three beams beams incident, incident, and andthe arrows the arrows represent represent the polariza- the polar- tionization direction direction of the of thebeams beams on the on x-y the plane. x-y plane. The figure The ﬁgure on the on right the rightshows shows that the that maximum the maximum co- directionalco-directional polarization polarization component component is still is applicable still applicable to the to small-angle the small-angle incident incident situation situation of multi- of beammulti-beam interference. interference. However, However, as the incident as the incident angle increases, angle increases, the stability the stability of the multi-beam of the multi-beam inter- ferenceinterference system system will be will very be verypoor. poor.

InIn this this case, case, taking the the TE wave of one of the three beams as the common vibration directiondirection can ensureensure thatthat the the contrast contrast is is sufﬁciently sufficiently high high even even at aat large a large incident incident angle. angle. The polarization angle of beam 1 is φ =φφφ === 90◦. From Equation (16), the polarization vectors The polarization angle of beam 11 is 1 90 . From Equation (16), the polarization vec- torscan becan expressed be expressed as as = √ e√1 0 j 0 √ h 3 3 3 1 3 i e2 = i − cos θ + j cos θ + k − sin θ √4 √4 4 4 √ 2 (19) h 3 3 3 1 3 i e3 = i 4 cos θ − 4 j 4 cos θ + 4 k 2 sin θ

From Equation (14), the intensity distribution of three-beam interference can be calcu-

lated by √ 3 3 I(x, y, z) = 3 + (3 cos θ + 1) cos 2 k sin θ · x cos 2 k sin θ · y 2 √ (20) 9 1 + 4 cos θ + 3 − 2 cos 3k sin θ · y

According to the above equation, the maximum intensity of the interference field is related to the value of the incident angle. As shown in Figure7, after the incident angle is increased to 32◦, the contrast no longer decreases as the incident angle increases. At the same time, with the increase in incident angle, the pattern transitions from the hexagonal center maximum lattice to the hexagonal minimum lattice. The maximum value of the interference field intensity gradually decreases to half of the theoretical maximum value. This change in the maximum intensity value undoubtedly increases the difficulty of determining the exposure time in the photolithography process.

3.2. TE-TE-TE Polarization Both co-directional component TE polarization and co-directional component TM polarization were analyzed above. In both cases, neither the high contrast of the light ﬁeld nor the stability of the interference system can be guaranteed under the conditions of large-angle interference. As shown in Figure8d, another polarization mode is proposed for interference lithography under large-angle incidence. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 11

= [ ] ej1 00 33313  ei=−cosθθ + j cos + k − sin θ 2   44442 (19) 33313  ei=−cosθθθ j cos + k sin 3   44442  From Equation (14), the intensity distribution of three-beam interference can be calculated by  ()(=+θθ + )33 ⋅ θ ⋅ I xyz, , 3 3cos 1 cos k sin x cos k sin y 22 2 (20) 91 ++−cosθθ 2 cos() 3ky sin ⋅ 43 According to the above equation, the maximum intensity of the interference field is related to the value of the incident angle. As shown in Figure 7, after the incident angle is increased to 32°, the contrast no longer decreases as the incident angle increases. At the same time, with the increase in incident angle, the pattern transitions from the hexagonal center maximum lattice to the hexagonal minimum lattice. The maximum value of the interference field intensity gradually decreases to half of the theoretical maximum value. Appl. Sci. 2021, 11, 4789 8 of 11 This change in the maximum intensity value undoubtedly increases the difficulty of determining the exposure time in the photolithography process.

FigureFigure 7. 7. TheThe interference interference of of the the three three beams beams incident incident with with the the maximum maximum co-directional co-directional TE TE polari- polar- zation.ization. The The light light field field intensity intensity distribution distribution at at di differentfferent incident incident angles: angles: the the maximum maximum value value of of the the light field intensity changes with the incident angle and the relationship between the contrast and light field intensity changes with the incident angle and the relationship between the contrast and the incident angle. (a) The intensity distribution of the light field with an incident angle equal to the incident angle. (a) The intensity distribution of the light field with an incident angle equal to 15°. (b) The intensity distribution of the light field with an incident angle equal to 60°. (c,d) The Appl. Sci. 2021, 11, x FOR PEER REVIEW ◦ 9 of 11 ◦ maximum15 .(b) The value intensity of the distributionlight field intensity of the lightand the field contrast with an change incident due angle to the equal incident to 60 angle..(c,d ) The maximum value of the light field intensity and the contrast change due to the incident angle. 3.2. TE-TE-TE Polarization Both co-directional component TE polarization and co-directional component TM polarization were analyzed above. In both cases, neither the high contrast of the light field nor the stability of the interference system can be guaranteed under the conditions of large-angle interference. As shown in Figure 8d, another polarization mode is proposed for interference lithography under large-angle incidence.

Figure 8. SchematicFigure diagram 8. Schematic of polarization diagram of mode polarization of three-beam mode interference.of three-beam (a interference.) The incident (a azimuth) The incident of the interference of the three beams.azimuth (b– dof) Threethe interference different typesof the of three polarization beams. (b mode–d) Three combinations, different types the arrows of polarization represent mode the polarization direction of thecombinations, beams on the the x-y arrows plane. represent the polarization direction of the beams on the x-y plane.

When the polarizationWhen the polarizationof the three ofbeams the three are in beams TE-TE-TE are in mode,TE-TE-TE the polarizationmode, the polarization ◦ angle of threeφφφ=== beams is φ1 = φ2 = φ3 = 90 . The polarization vectors can be expressed as angle of three beams is 12390 . The polarization vectors can be expressed as e = [00j ]e1 = 0 j 0 1 h √ i e = −i 3 −j 1 312 2 2 0 (21) e =−ij −h √0 i 2 e = 3 − 1 223 i 2 j 2 0 (21)  From Equation (14), the=− intensity31 distribution of three-beam interference can be calcu- e3 ij0 lated by 22 √ ! From Equation (14), the intensity distributi3 on of three-beam3 interference can√ be cal- I(x, y, z) = 3 − 2 cos k sin θ · x cos k sin θ · y − cos 3k sin θ · y (22) culated by 2 2 33 ()It can=− be observed fromθθ Equation⋅ (22) that ⋅ the − maximum() intensity θ ⋅ remains unchanged Ixyz, , 3 2cos k sin x cos k sin y cos 3 k sin y (22) at 4.5 when the incident22 is in TE-TE-TE mode, which is only half of the theoretical value. The increase in the incident angle will not cause additional contrast loss and it makes it It can be observed from Equation (22) that the maximum intensity remains unchanged at 4.5 when the incident is in TE-TE-TE mode, which is only half of the theoretical value. The increase in the incident angle will not cause additional contrast loss and it makes it easier for us to control the exposure dose at different incident angles. Although the increase in exposure time may introduce additional environmental errors to the interference light field, it is clearly easier to obtain a pattern with a smaller feature size when the light field has a high contrast. It can be seen from Figure 6 that the contrast loss caused by the polarization angle deviation must be considered, especially under the condition of large-angle interference. δδδ Under TE-TE-TE incident condition, the polarization angle deviations are 123,,, and the φ =+δ φ =+δ φ =+δ polarization angle of three beams are 11223390 , 90 , 90 . The polarization vector can be expressed as =−[ θδ δ θδ] ei1111cos sin j cos k sin sin 13 31 ei=−−−cosθδ sin cos δ j cos θδ sin cos δ k sin θδ sin 222222 22 22 (23) 1331 ei=+cosθδ sin cos δ j cos θδ sin − cos δ k sin θδ sin 333333 2222 It can be concluded from Equation (23) that when there is a deviation in the polarization angle, the contrast loss is related to the incident angle. Under the condition that the δδ δ polarization angle deviations 12, and 3 are not more than±5°, we calculated the minimum value of contrast when the incident angle is increased from 5° to75°. As shown in Appl. Sci. 2021, 11, 4789 9 of 11

easier for us to control the exposure dose at different incident angles. Although the increase in exposure time may introduce additional environmental errors to the interference light field, it is clearly easier to obtain a pattern with a smaller feature size when the light field has a high contrast. It can be seen from Figure6 that the contrast loss caused by the polarization angle deviation must be considered, especially under the condition of large-angle interference. Under TE-TE-TE incident condition, the polarization angle deviations are δ1, δ2, δ3, and ◦ ◦ ◦ the polarization angle of three beams are φ1 = 90 + δ1, φ2 = 90 + δ2, φ3 = 90 + δ3. The polarization vector can be expressed as = − e1 √ i cos θ sin δ1 j√cos δ1 k sin θ sin δ1 h 1 3 3 1 i e2 = i 2 cos θ sin δ2 − 2 cos δ2 j − 2 cos θ sin δ2 − 2 cos δ2 k sin θ sin δ2 (23) h √ √ i Appl. Sci. 2021, 11, x FOR PEERe = REVIEWi 1 cos θ sin δ + 3 cos δ j 3 cos θ sin δ − 1 cos δ k sin θ sin δ 10 of 11 3 2 3 2 3 2 3 2 3 3 It can be concluded from Equation (23) that when there is a deviation in the polarization angle, the contrast loss is related to the incident angle. Under the condition that ◦ Figurethe polarization 9, when angle the incident deviations angleδ1, δ2 and is notδ3 are more not morethan40 than°, ±the5 , contrast we calculated loss the does not exceed minimum value of contrast when the incident angle is increased from 5◦ to 75◦. As shown 0.015. Even if the interference angle reaches60°, the contrast still has 0.975. At this time, in Figure9, when the incident angle is not more than 40 ◦, the contrast loss does not exceed we0.015. have Even a ifsufficient the interference contrast angle loss reaches margin 60◦, theto consider contrast still the has standing 0.975. At wave this time, effect caused by largewe have angle a sufficient incidence. contrast loss margin to consider the standing wave effect caused by large angle incidence.

Figure 9. The TE-TE-TE polarization mode is used for three-beam interference, and the contrast Figure 9. The TE-TE-TE polarization mode is used for three-beam interference, and the contrast at at different incident angles is obtained when the polarization direction of the three beams has a differentdeviation ofincident±5◦. angles is obtained when the polarization direction of the three beams has a deviation of ±5°. 4. Discussion 4. DiscussionThere is a contradiction between narrower line width and higher contrast in LIL. The smaller feature size of the pattern means a larger spatial incident angle, which also brings aboutThere the enhancement is a contradiction of the standing between wave effectnarrower and the lin sharpe width decrease and in higher the contrast contrast of in LIL. The smallerthe light field.feature When size the of coherence the pattern of the means light source a larger is not spatial considered, incident the contrast angle, of MILwhich also brings aboutdepends the on enhancement the polarization of state the of standing the beam, wave the angle effect of incidence, and the the sharp deviation decrease of the in the contrast ofpolarization the light angle,field. and When the stability the coherence of the beam of pointingthe light at thesource same is time. not In considered, this paper, by the contrast of constructing a MIL model, we obtained the optimal solution for the high-contrast light field MILof three-beam depends interference on the polarization lithography under state different of the beam, pattern linethe widths.angle of In theincidence, case of a the deviation ofsmall the incidentpolarization angle interference, angle, and the the co-directional stability of TE the polarization beam pointing incident combinationat the same time. In this paper,is adopted, by whichconstructing can reduce a theMIL exposure model, time we while obta obtainingined the a optimal high-contrast solution light field. for the high-con- trastWhen light the incident field of is three-beam at a large angle, interference the azimuthal lithography TE polarization under incident different combination pattern line widths. In the case of a small incident angle interference, the co-directional TE polarization incident combination is adopted, which can reduce the exposure time while obtaining a high- contrast light field. When the incident is at a large angle, the azimuthal TE polarization incident combination is used to obtain a high-contrast light field while allowing higher beam pointing stability deviations. This conclusion is also applicable to MIL, which provides a theoretical reference for the generation of narrow linewidth patterns and equipment development in interference lithography.

5. Conclusions In this work, the contrast loss model under non-coplanar interference conditions is proposed. It is demonstrated that the contrast under non-coplanar interference conditions will decrease with the increase in the incident angle. By changing the polarization combination mode of the beams, we avoided a large-scale contrast loss when fabricating narrow linewidth structures by LIL. According to the simulation result data, the deviation of the polarization angle and the polarization mode are important factors that cause a loss of contrast. The TE-TE-TE polarization mode is the best polarization combination for three- beam interference lithography to prepare sub-wavelength structures.

is used to obtain a high-contrast light ﬁeld while allowing higher beam pointing stability deviations. This conclusion is also applicable to MIL, which provides a theoretical reference for the generation of narrow linewidth patterns and equipment development in interference lithography.

5. Conclusions In this work, the contrast loss model under non-coplanar interference conditions is proposed. It is demonstrated that the contrast under non-coplanar interference conditions will decrease with the increase in the incident angle. By changing the polarization combination mode of the beams, we avoided a large-scale contrast loss when fabricating narrow linewidth structures by LIL. According to the simulation result data, the deviation of the polarization angle and the polarization mode are important factors that cause a loss of contrast. The TE-TE-TE polarization mode is the best polarization combination for three-beam interference lithography to prepare sub-wavelength structures.

Author Contributions: Conceptualization, F.P., J.D. (Jing Du)., and W.Y.; methodology, F.P. and W.Y.; software, F.P. and S.W.; validation, F.P. and S.W.; formal analysis, F.P. and J.D. (Jialin Du).; writing—original draft preparation, F.P., J.D. (Jing Du). and S.W.; writing—review and editing, F.P. and W.Y.; project administration, W.Y. and J.D. (Jialin Du).; funding acquisition, W.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Instrument Development of Chinese Academy of Sciences (Nos. YJKYYQ20180008, YJKYYQ20180006), Sichuan Science and Technology Program (Nos. 2020JDJQ0007, 2021JDRC0089, 2021JDRC0084). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Some or all data, models, or code used during the study can be requests directly from the author. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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