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Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings
Microelectronic Engineering 88 (2011) 170–174
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Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings E.A. Bezus ⇑, L.L. Doskolovich, N.L. Kazanskiy Image Processing Systems Institute of the Russian Academy of Sciences, Molodogvardeyskaya 151, 443001 Samara, Russia Technical Cybernetics Department, Samara State Aerospace University, Molodogvardeyskaya 151, 443001 Samara, Russia
a r t i c l e
i n f o
Article history: Received 18 June 2010 Received in revised form 9 September 2010 Accepted 3 October 2010 Available online 8 October 2010 Keywords: Nanolithography Guided-mode resonant grating Evanescent wave Interference pattern
a b s t r a c t Generation of interference patterns of the evanescent electromagnetic waves with significantly subwavelength period using guided-mode resonant gratings is studied. The potential application field is the nearfield interference lithography aimed at fabrication of periodic structures with nanoscale features. Calculations, based on the rigorous coupled-wave analysis of Maxwell’s equations, demonstrate the possibility of obtaining high-quality interference patterns by means of evanescent diffraction orders enhancement at the resonance conditions. The interference pattern feature size is up to 8 times smaller than the incident wavelength and the intensity at the interference maxima exceeds the incident wave intensity by an order-of-magnitude. The possibility of generating high-frequency interference patterns using low-frequency diffraction gratings is demonstrated. The 1D and 2D interference patterns with the periods 6 times smaller than those of the used diffraction gratings are generated. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The rapid growth of nanoscience and technology in the recent years requires the development of the methods for fabricating nanostructures. Nowadays, projection photolithography is one of the main techniques of creating micro- and nanostructures. In this case the minimal achievable detail size is diffraction-limited and is about one quarter of the wavelength. One of the ways to increase the resolution in photolithography systems is the reduction of the used wavelength, i.e. shifting to the far UV or X-ray range [1–3]. The main disadvantage of this approach is the high complexity and cost of the used systems. As an alternative, various nearfield photolithography techniques, based on the recording of the interference patterns of the evanescent electromagnetic waves (EEW) [4–7] or surface plasmon polaritons (SPP) [8–15] have been proposed. Using EEW and SPP allows one to overcome the diffraction limit and to generate periodic structures with detail size by an order-of-magnitude smaller than the wavelength of the used light. In [4] the possibility to obtain one-dimensional EEW interference patterns corresponding to the ± 1-st orders of subwavelength diffraction grating was shown for the first time. According to the results of the rigorous electromagnetic simulation, the intensity at the interference pattern maxima was found to be 3–4 times higher than the incident wave intensity whereas its period was ⇑ Corresponding author at: Image Processing Systems Institute of the Russian Academy of Sciences, Molodogvardeyskaya 151, 443001 Samara, Russia. E-mail address: [email protected] (E.A. Bezus). 0167-9317/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2010.10.006
2 times smaller than the diffraction grating period and was close to a half of the wavelength [4]. In [5–7] the authors suggested the methods for generating 1D and 2D EEW interference patterns generated at the total internal reflection of two and four incident waves, respectively. The period of the generated 2D EEW patterns in [5–7] was 2–4 times smaller than the incident wavelength. The main disadvantage of the method [6,7] is the utilization of a complex optical system producing four different beams with preset polarization and phase. In [8–14] the photolithography techniques based on generating SPP interference patterns were considered. In [8,9], the SPP interference pattern was generated on the surface of the perforated metal film. In [10–12] the diffraction structures consisting of a diffraction grating and a metal layer applied in the substrate region were used for generating SPP interference patterns. The diffraction grating was designed to excite several SPPs, which generate the interference pattern at the bottom interface of the metal film. In [13,14] the authors considered similar structures, but the system of metal-dielectric layers was used instead of a single metal layer. The periods of the interference patterns in [8–14] are 2–5 times smaller than the incident wave length and are thus comparable with the EEW interference patterns in [5–7]. The SPP interference patterns are characterized by the significantly increased field enhancement as opposed to the EEW interference patterns. Particularly, in [15] the comparison of the SPP interference patterns generated in Kretschmann geometry and EEW interference patterns generated at the total internal reflection was given. In [15] the intensity of SPP interference patterns was
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shown to be 6–8 times higher and the contrast by 20–30% higher than those of EEW interference patterns. Moreover, in [11–13] the higher diffraction orders of the grating (the orders with numbers n, n > 1) were used for SPP excitation. Using the higher orders with numbers n allows one to generate high-frequency interference patterns of SPP utilizing low-frequency diffraction gratings with period 2n times larger than that of the interference pattern. This significantly reduces the requirements for a technological implementation of the structure. Using the higher diffraction orders to generate the EEW interference patterns in the dielectric diffraction grating seemed to be improbable. In fact, in [11–14] the metal layer or the system of metal-dielectric layers underneath the grating functions as a filter, which transmits the diffraction orders that excite SPP and ‘‘blocks” all the other orders. It is the filtering property of the metal layer or the multilayer system that provides the generating of high-contrast SPP interference patterns [10–14]. In the present paper we study theoretically the possibility to generate the high-frequency EEW interference patterns using the guided-mode resonant gratings. These structures have been intensively investigated as the optical filters during the last decade by many research groups [16–18], however, relatively little attention has been paid to the near-field distribution issues under resonance conditions [19,20]. Authors believe this paper to be the first one to show the high efficiency of these structures in solving the problem of generating high-frequency EEW interference patterns. It is shown on the basis of the numerical simulation that guided-mode resonant gratings allow generating of high-contrast and highintensity EEW interference patterns. The possibility to generate high-quality interference patterns using the higher diffraction orders is also demonstrated. 2. The geometry of the structure and generation of the interference patterns of the evanescent electromagnetic waves The considered structure consists of a binary dielectric grating with a waveguide layer situated on a dielectric substrate (Fig. 1). Such diffraction structures are traditionally used as narrow-band spectral filters having high reflection (close to 100%) in the vicinity of a certain wavelength of incident light [17]. The mentioned high reflection effect is caused by resonant excitation of the quasiwaveguide modes in the structure [18]. The reason for using the structure shown in Fig. 1 for generating the EEW interference patterns is the following. At the certain conditions provided by choosing the geometrical and physical parameters of the structure, the structure modes are localized in the waveguide layer and their propagation constants match the
propagation constants of the grating diffraction orders. At the normal incidence, two quasiwaveguide modes having the opposite directions and corresponding to the symmetric diffraction orders with numbers ±n can be excited. In the substrate region the quasiwaveguide modes have an exponentially evanescent form and thus the EEW interference pattern will be generated directly under the waveguide layer. Let us notice that in this case the interfering EEW will correspond to the evanescent diffraction orders with numbers ±n in the substrate area. At the resonance conditions, the electric field within the structure can be enhanced significantly [19,20]. The electric field amplitude at the maxima of the interference patterns of the excited modes can be by an order-of-magnitude larger than the incident wave amplitude [19,20]. Thus when modes are excited, one should also expect a significant increase of the amplitude of the evanescent waves (diffraction orders) in the near field underneath the waveguide layer/substrate interface. Generally, the field in the substrate region is represented as a superposition of the infinite number of transmitted orders (propagating and evanescent). However, due to the field enhancement caused by the resonance [19,20], generating of the distribution corresponding to the superposition of two evanescent diffraction orders is possible in the near field. Let us consider the expressions for the field intensity corresponding to the superposition of the ±n-th transmitted orders. In the case of TE-polarization the field intensity in the substrate region is TE 2 2 2 ITE n ðx; zÞ ¼ jAn j cos ðkx;n xÞj expðikz;n zÞj ;
where kx; n ¼ 2pn=d is the propagation constant of the n-th diffracqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 tion order, d is the grating period, kz;n ¼ i kx;n k0 esub , ATE n are the complex amplitudes of ±n-th orders. Let us note that the chosen form of kz;n corresponds to evanescent waves in the substrate region. In the case of TM-polarization of the incident wave the field intensity is
ITM n ðx; zÞ ¼
d
k
w
ε super
εl 0
2
k0 esub
2
2
½2kx;n k0 esub 2
ð2Þ
According to (1) and (2), the period of the interference pattern of the ±n-th diffraction orders is defined by the expression
dip;n ¼ p=kx; n ¼ d=2n:
ð3Þ
It is worth mentioning that in the case of TE-polarization the theoretical estimate of the contrast of the interference pattern (1) is 1. For the case of TM-polarization the value of the contrast can be easily obtained from (2) as
esub 2ðkx;n =k0 Þ2 esub
:
ð4Þ
The expression (4) shows that while kx;n is increasing (and, consequently, while the interference pattern period is decreasing) the contrast of the generated pattern is also decreasing. Let us note that complex configuration modes corresponding to the superposition of the large number of diffraction orders can also exist in the structure in Fig. 1. For such modes EEW interference pattern will have a complex form. The complex interference patterns are of less interest for practical applications, thus our consideration is restricted to generating the patterns of the form (1), (2) only.
ε gr
h gr
2 jATM n j
cosð2kx;n xÞk0 esub j expðikz;n zÞj2 :
K TM;n ¼
z
ð1Þ
hl
x
ε sub Fig. 1. Geometry of the structure.
3. Simulation results and discussion The study of the possibility of generating the interference patterns (1), (2) was carried out for the diffraction structure with
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the period d ¼ 240 nm and the coherent illumination with the wavelength k ¼ 441:6 nm. The following values (see Fig. 1): esuper ¼ 1 (air), egr ¼ el ¼ 4:41 (corresponds to the dielectric permittivity of the zinc oxide ZnO), esub ¼ 2:56 (corresponding to the photoresist) were used as the dielectric permittivities of the materials. In this case at the normal incidence all diffraction orders except the zeroth reflected and zeroth transmitted ones are evanescent. The values of the geometrical parameters of the structure w; hgr ; hl (Fig. 1) were determined as a result of numerical optimization of the following merit function
maxfIn ðx; pÞg x
d
0
0 12 Iðx; pÞ I ðx; pÞ n @ A dx ! min maxfIn ðt; pÞg
ð5Þ
t
where p ¼ ðw; hgr ; hl Þ, Iðx; pÞ is the calculated intensity, generated by the structure at the interface between the substrate and the waveguide layer (at z = 0), In ðx; pÞ is the intensity of the interference pattern of the ±n-th diffraction orders at z = 0, determined from the expressions (1), (2) (i.e. the ‘‘ideal” interference pattern). The first multiplier in (5) is responsible for the maximization of the field intensity at the interference pattern maxima. The second multiplier in (5) is the proximity measure of the calculated interference pattern intensity to the ‘‘ideal” interference pattern of ± n-th diffraction orders. To calculate the interference pattern intensity Iðx; pÞ we used the rigorous coupled-wave analysis as described in [22–24]. First, the diffraction structures for generating the EEW interference patterns corresponding to ±1-th diffraction orders were calculated. In this case the interference pattern period dip;1 ¼ d=2 ¼ 120 nm. Fig. 2a shows the distribution of the electric field in the structure optimized for the TE-polarization of the incident wave. The values of the parameters obtained while minimizing (5) are given in the figure caption. The interference pattern at the interface between the waveguide layer and substrate (photoresist) is shown in Fig. 2b. The form of the interference pattern is fully consistent with Eq. (1). The values of the intensity in Fig. 2a and b are normalized by the incident wave intensity. The maximum intensity at the interface exceeds the incident wave intensity by 30 times and thus is comparable to the intensity of the interference patterns of the SPP [11,12]. Fig. 2c shows the contrast and intensity at the interference pattern maxima versus the distance from the interface between the waveguide layer and substrate. According to Fig. 2c, the contrast of the generated pattern is close to 1 as it follows from Eq. (1).
0.3 0.2 0.1 0
−0.1 −0.2 (a )
0
0.2
0.4 x, µm
1
30
0.9999
20
0.9998
10
0.9997
2
E0 )
2
20
2
0.4
25
max( E
z, µm
0.5
40
30
E0
0.6
2
0.7
35
E
55 50 45 40 35 30 25 20 15 10 5
0.8
15 10
K
gðpÞ ¼
Z
1
Fig. 3a and b show the electric field distribution and the interference pattern generated by the structure optimized for the case of the TM-polarization of the incident wave. The interference pattern corresponds to the expression (2) and the intensity of the interference maxima is 27 times higher than that of the incident wave. The value of the interference pattern contrast coincides with the theoretical estimate (4) and is 0.608 (Fig. 3c). It should be noted that the intensity of the 0-th transmitted order is less than 0.0001 for both structures and thus the contrast of the generated interference patterns does not change significantly at the distance from the interface. Now let us consider the possibility of generating the EEW interference patterns in the case when the modes of the structure are excited by the diffraction orders with numbers n, n > 1. In this case the period of the EEW interference pattern (3) will be 2n times smaller than the grating period. Thus using higher orders reduces the requirements for the technological implementation of the structure and allows generating high-frequency EEW interference patterns by using a low-frequency diffractive relief. Since the contrast of the generated interference patterns is higher in the case of TE-polarization, further we will consider structures optimized for the TE-polarized incident wave only. The calculation of the EEW interference patterns was carried out at the grating period d ¼ 720 nm for n = ±3. In this case the period of the generated interference pattern dip;3 ¼ d=6 ¼ 120 nm coincides with the previous example and the grating period is 3 times larger. As previously, the geometrical parameters of the structure were obtained as a result of optimizing the merit function (5). Fig. 4a shows the distribution of the electric field in the calculated structure (the parameters of the structure are given in the figure caption). The interference pattern at the interface between the waveguide layer and the substrate (photoresist) is shown in Fig. 4b. The maximum intensity at the interface exceeds the incident wave intensity by more than 25 times. The dependence of the contrast and electric field intensity at the interference pattern maxima is given in Fig. 4c. Let us note that for this structure the total intensity of the propagating transmitted orders is 0.23 as opposed to the structures, in which the modes are excited by the ± 1-st diffraction orders. As a result of this the contrast of the generated interference pattern is decreasing away from the waveguide layer/photoresist interface. Near the interface the contrast is close to unity and at the distance of 120 nm decreases to 0.4. This value of the contrast is the minimal required value to record the interference pattern using the standard positive photoresists [14,25]. It should be noted that the photoresist thickness of
5 0 0
0.6 (b)
0.05
0.1
x, µm
0.15
0 0
0.2 (c)
0.05 0.1 0.15 Distance from the interface, µm
0.9996 0.2
Fig. 2. (a) The distribution of the electric field in the structure at the normal incidence of the TE-polarized wave (three periods of the grating are shown, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the period is 240 nm, the grating height is 123 nm, the layer thickness is 545 nm, the width of the ridge is 98 nm, the mode is excited by ±1 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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30
0.6085
45
20
0.608
10
0.6075
25
10
15 10
0
2
2
20
0.2
15
2
25
20
max( E
0.4
E0
30
2
35
E
0.6
E0 )
40
K
0.8
z, µm
30
50
5
5 −0.2 0
0.2
(a)
0.4 x, µm
0 0
0.6
0 0.05
0.1
(b)
x, µm
0.15
0.2
0
0.05 0.1 0.15 Distance from the interface, µm
(c)
0.607 0.2
Fig. 3. (a) The distribution of the electric field in the structure at the normal incidence of TM-polarized wave (three periods of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 290 nm, the layer thickness is 456 nm, the width of the ridge is 166 nm, the mode is excited by ±1 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
30
0.75
20
0.5
2
E
15
10
0.25
0
0.2
0.4 0.6 x, µm
K
10
2
0
20
10
(a)
1
E0 )
20
2
0.2
25
max( E
30
2
0.4
30 E0
0.6
40
−0.2
40
35 50
0.8
z, µm
40
60
1
5 0 0 (b)
0.1
0.2
0.3
0.4 x, µm
0.5
0.6
0 0
0.7 (c)
0.02
0.04 0.06 0.08 0.1 0.12 Distance from the interface, µm
0 0.14
Fig. 4. (a) The distribution of the electric field in the structure at the normal incidence of TE-polarized wave (one period of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 155 nm, the layer thickness is 763 nm, the width of the ridge is 349 nm, the mode is excited by ±3 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
approximately 100 nm is sufficient for patterning nanoscale features as it was experimentally demonstrated in [26]. Thus Fig. 4 shows the possibility to generate the interference patterns of the higher evanescent diffraction orders with high intensity and contrast. In the above considered examples the superstrate material is air (esuper ¼ 1). In this case the structures are supposed to be deposited directly on the resist. The case when the superstrate material corresponds to dielectric with esuper > 1 is more interesting in terms of technology. In this case, the diffraction structure can be fabricated on the top of the given material and then brought into contact with the resist surface. It should be noted that intimate optical contact between the structure and the resist can be achieved using index-matching liquid [26]. Fig. 5 presents the structure performance computed for esuper ¼ 1:69 (Teflon), grating period d ¼ 660 nm and n = ±3. The other geometrical parameters of the structure were obtained as a result of optimizing the merit function (5) and are given in the figure caption. For the specified parameters the period of the generated interference pattern is dip;3 ¼ d=6 ¼ 110 nm which corresponds to the feature size of about k=8. The intensity of interference maxima at the interface exceeds the incident wave intensity by more than 100 times (Fig. 5b). The contrast is close to unity at the interface and at the distance of 120 nm decreases to 0.4 (Fig. 5c).
The suggested approach can be generalized onto the case of generating the 2D EEW interference patterns. In this case a 3D diffraction grating is used (Fig. 6a), and the modes of the structure are excited by four diffraction orders with the numbers ðn; 0Þ, ð0; nÞ. The diffraction structure with period dx ¼ dy ¼ 720 nm for generating 2D interference pattern at n = 3 was calculated as an example. The values of the dielectric permittivities esuper , egr , el , esub coincide with those used in examples in Figs. 2–4. The geometrical parameters of the structure were obtained as the result of minimization of the merit function similar to (5). It should be noted that the configuration of the generated EEW interference patterns depends on the incident wave polarization. Fig. 5b shows the interference pattern at the interface between the waveguide layer and photoresist generated for the case of the circular polarization of the incident wave. In this case the period of the generated interference pattern is dip;3 ¼ d=6 ¼ 120 nm. Let us notice that the interference patterns presented in [12] have similar structure. The intensity of the interference peaks in Fig. 6b exceeds the incident wave intensity by more than 20 times, and the contrast of the generated pattern exceeds 0.45. 4. Conclusion The technique for generating EEW interference patterns using guided-mode resonant gratings is proposed. For the considered
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200
0.2
150
0 0.2 0.4 0.6 x, µm
100 0.5
K
2
E0 )
40
50 0.25
100 –0.2 0 ( a)
60
2
250
0.4
max( E
300
0.6
0.75 80 2
350 E0
0.8
1
100
2
400
E
z, µm
450 1
150
120
500
1.2
20
50 0
0 0
0.1
0.2
(b)
0.3 0.4 x, µm
0.5
0 0
0.6
0.02
(c)
0.04 0.06 0.08 0.1 0.12 Distance from the interface, µm
0 0.14
Fig. 5. (a) The distribution of the electric field in the structure with esuper ¼ 1:69 at the normal incidence of TE-polarized wave (one period of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 140 nm, the layer thickness is 961 nm, the width of the ridge is 355 nm, the mode is excited by ±3 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
z
εsuper
k
wx
ments for the grating fabrication technology and allows generating high-frequency interference patterns of evanescent waves using a low-frequency grating. The suggested approach can be used for creating the periodic structures with nanoscale details by the contact photolithography techniques.
dy wy
y
dx
εgr
h gr
εl
hl
Acknowledgements
x
This work was financially supported by RF Presidential grant (NSh-7414.2010.9), RFBR grants 09-07-92421, 09-07-12147, 1002-01391, 10-07-00553 and Basic Research and Higher Education grant (CRDF PG08-014-1).
εsub
(a)
References
E
2
E0
2
30 20 10 0 0.8
0.8 0.6
0.6 0.4
y, µm
(b)
0.4 0.2
0.2 0
x, µm
0
Fig. 6. (a) The diffraction structure for generating 2D EEW interference patterns. (b) 2D interference pattern at the interface of the waveguide layer/substrate interface (one period of the grating). The parameters of the structure: the grating height is 285 nm, the layer thickness is 1469 nm, the size of the grating hole is 349 nm, the mode is excited by (±3, 0), (0, ±3) orders of the grating.
wavelength of the incident light (441.6 nm), the achieved period of the EEW interference patterns is 110 nm, which corresponds to significantly subwavelength detail size of about 55 nm (k/8). According to the results of numerical simulation, the contrast of the interference patterns generated in the case of TE-polarization of the incident wave is close to unity, and the intensity at the interference peaks is more than 30 times higher than that of the incident wave. The possibility of generating 1D and 2D interference patterns corresponding to the higher evanescent diffraction orders is shown. Using the higher diffraction orders reduces the require-
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Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings E.A. Bezus ⇑, L.L. Doskolovich, N.L. Kazanskiy Image Processing Systems Institute of the Russian Academy of Sciences, Molodogvardeyskaya 151, 443001 Samara, Russia Technical Cybernetics Department, Samara State Aerospace University, Molodogvardeyskaya 151, 443001 Samara, Russia
a r t i c l e
i n f o
Article history: Received 18 June 2010 Received in revised form 9 September 2010 Accepted 3 October 2010 Available online 8 October 2010 Keywords: Nanolithography Guided-mode resonant grating Evanescent wave Interference pattern
a b s t r a c t Generation of interference patterns of the evanescent electromagnetic waves with significantly subwavelength period using guided-mode resonant gratings is studied. The potential application field is the nearfield interference lithography aimed at fabrication of periodic structures with nanoscale features. Calculations, based on the rigorous coupled-wave analysis of Maxwell’s equations, demonstrate the possibility of obtaining high-quality interference patterns by means of evanescent diffraction orders enhancement at the resonance conditions. The interference pattern feature size is up to 8 times smaller than the incident wavelength and the intensity at the interference maxima exceeds the incident wave intensity by an order-of-magnitude. The possibility of generating high-frequency interference patterns using low-frequency diffraction gratings is demonstrated. The 1D and 2D interference patterns with the periods 6 times smaller than those of the used diffraction gratings are generated. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The rapid growth of nanoscience and technology in the recent years requires the development of the methods for fabricating nanostructures. Nowadays, projection photolithography is one of the main techniques of creating micro- and nanostructures. In this case the minimal achievable detail size is diffraction-limited and is about one quarter of the wavelength. One of the ways to increase the resolution in photolithography systems is the reduction of the used wavelength, i.e. shifting to the far UV or X-ray range [1–3]. The main disadvantage of this approach is the high complexity and cost of the used systems. As an alternative, various nearfield photolithography techniques, based on the recording of the interference patterns of the evanescent electromagnetic waves (EEW) [4–7] or surface plasmon polaritons (SPP) [8–15] have been proposed. Using EEW and SPP allows one to overcome the diffraction limit and to generate periodic structures with detail size by an order-of-magnitude smaller than the wavelength of the used light. In [4] the possibility to obtain one-dimensional EEW interference patterns corresponding to the ± 1-st orders of subwavelength diffraction grating was shown for the first time. According to the results of the rigorous electromagnetic simulation, the intensity at the interference pattern maxima was found to be 3–4 times higher than the incident wave intensity whereas its period was ⇑ Corresponding author at: Image Processing Systems Institute of the Russian Academy of Sciences, Molodogvardeyskaya 151, 443001 Samara, Russia. E-mail address: [email protected] (E.A. Bezus). 0167-9317/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2010.10.006
2 times smaller than the diffraction grating period and was close to a half of the wavelength [4]. In [5–7] the authors suggested the methods for generating 1D and 2D EEW interference patterns generated at the total internal reflection of two and four incident waves, respectively. The period of the generated 2D EEW patterns in [5–7] was 2–4 times smaller than the incident wavelength. The main disadvantage of the method [6,7] is the utilization of a complex optical system producing four different beams with preset polarization and phase. In [8–14] the photolithography techniques based on generating SPP interference patterns were considered. In [8,9], the SPP interference pattern was generated on the surface of the perforated metal film. In [10–12] the diffraction structures consisting of a diffraction grating and a metal layer applied in the substrate region were used for generating SPP interference patterns. The diffraction grating was designed to excite several SPPs, which generate the interference pattern at the bottom interface of the metal film. In [13,14] the authors considered similar structures, but the system of metal-dielectric layers was used instead of a single metal layer. The periods of the interference patterns in [8–14] are 2–5 times smaller than the incident wave length and are thus comparable with the EEW interference patterns in [5–7]. The SPP interference patterns are characterized by the significantly increased field enhancement as opposed to the EEW interference patterns. Particularly, in [15] the comparison of the SPP interference patterns generated in Kretschmann geometry and EEW interference patterns generated at the total internal reflection was given. In [15] the intensity of SPP interference patterns was
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shown to be 6–8 times higher and the contrast by 20–30% higher than those of EEW interference patterns. Moreover, in [11–13] the higher diffraction orders of the grating (the orders with numbers n, n > 1) were used for SPP excitation. Using the higher orders with numbers n allows one to generate high-frequency interference patterns of SPP utilizing low-frequency diffraction gratings with period 2n times larger than that of the interference pattern. This significantly reduces the requirements for a technological implementation of the structure. Using the higher diffraction orders to generate the EEW interference patterns in the dielectric diffraction grating seemed to be improbable. In fact, in [11–14] the metal layer or the system of metal-dielectric layers underneath the grating functions as a filter, which transmits the diffraction orders that excite SPP and ‘‘blocks” all the other orders. It is the filtering property of the metal layer or the multilayer system that provides the generating of high-contrast SPP interference patterns [10–14]. In the present paper we study theoretically the possibility to generate the high-frequency EEW interference patterns using the guided-mode resonant gratings. These structures have been intensively investigated as the optical filters during the last decade by many research groups [16–18], however, relatively little attention has been paid to the near-field distribution issues under resonance conditions [19,20]. Authors believe this paper to be the first one to show the high efficiency of these structures in solving the problem of generating high-frequency EEW interference patterns. It is shown on the basis of the numerical simulation that guided-mode resonant gratings allow generating of high-contrast and highintensity EEW interference patterns. The possibility to generate high-quality interference patterns using the higher diffraction orders is also demonstrated. 2. The geometry of the structure and generation of the interference patterns of the evanescent electromagnetic waves The considered structure consists of a binary dielectric grating with a waveguide layer situated on a dielectric substrate (Fig. 1). Such diffraction structures are traditionally used as narrow-band spectral filters having high reflection (close to 100%) in the vicinity of a certain wavelength of incident light [17]. The mentioned high reflection effect is caused by resonant excitation of the quasiwaveguide modes in the structure [18]. The reason for using the structure shown in Fig. 1 for generating the EEW interference patterns is the following. At the certain conditions provided by choosing the geometrical and physical parameters of the structure, the structure modes are localized in the waveguide layer and their propagation constants match the
propagation constants of the grating diffraction orders. At the normal incidence, two quasiwaveguide modes having the opposite directions and corresponding to the symmetric diffraction orders with numbers ±n can be excited. In the substrate region the quasiwaveguide modes have an exponentially evanescent form and thus the EEW interference pattern will be generated directly under the waveguide layer. Let us notice that in this case the interfering EEW will correspond to the evanescent diffraction orders with numbers ±n in the substrate area. At the resonance conditions, the electric field within the structure can be enhanced significantly [19,20]. The electric field amplitude at the maxima of the interference patterns of the excited modes can be by an order-of-magnitude larger than the incident wave amplitude [19,20]. Thus when modes are excited, one should also expect a significant increase of the amplitude of the evanescent waves (diffraction orders) in the near field underneath the waveguide layer/substrate interface. Generally, the field in the substrate region is represented as a superposition of the infinite number of transmitted orders (propagating and evanescent). However, due to the field enhancement caused by the resonance [19,20], generating of the distribution corresponding to the superposition of two evanescent diffraction orders is possible in the near field. Let us consider the expressions for the field intensity corresponding to the superposition of the ±n-th transmitted orders. In the case of TE-polarization the field intensity in the substrate region is TE 2 2 2 ITE n ðx; zÞ ¼ jAn j cos ðkx;n xÞj expðikz;n zÞj ;
where kx; n ¼ 2pn=d is the propagation constant of the n-th diffracqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 tion order, d is the grating period, kz;n ¼ i kx;n k0 esub , ATE n are the complex amplitudes of ±n-th orders. Let us note that the chosen form of kz;n corresponds to evanescent waves in the substrate region. In the case of TM-polarization of the incident wave the field intensity is
ITM n ðx; zÞ ¼
d
k
w
ε super
εl 0
2
k0 esub
2
2
½2kx;n k0 esub 2
ð2Þ
According to (1) and (2), the period of the interference pattern of the ±n-th diffraction orders is defined by the expression
dip;n ¼ p=kx; n ¼ d=2n:
ð3Þ
It is worth mentioning that in the case of TE-polarization the theoretical estimate of the contrast of the interference pattern (1) is 1. For the case of TM-polarization the value of the contrast can be easily obtained from (2) as
esub 2ðkx;n =k0 Þ2 esub
:
ð4Þ
The expression (4) shows that while kx;n is increasing (and, consequently, while the interference pattern period is decreasing) the contrast of the generated pattern is also decreasing. Let us note that complex configuration modes corresponding to the superposition of the large number of diffraction orders can also exist in the structure in Fig. 1. For such modes EEW interference pattern will have a complex form. The complex interference patterns are of less interest for practical applications, thus our consideration is restricted to generating the patterns of the form (1), (2) only.
ε gr
h gr
2 jATM n j
cosð2kx;n xÞk0 esub j expðikz;n zÞj2 :
K TM;n ¼
z
ð1Þ
hl
x
ε sub Fig. 1. Geometry of the structure.
3. Simulation results and discussion The study of the possibility of generating the interference patterns (1), (2) was carried out for the diffraction structure with
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the period d ¼ 240 nm and the coherent illumination with the wavelength k ¼ 441:6 nm. The following values (see Fig. 1): esuper ¼ 1 (air), egr ¼ el ¼ 4:41 (corresponds to the dielectric permittivity of the zinc oxide ZnO), esub ¼ 2:56 (corresponding to the photoresist) were used as the dielectric permittivities of the materials. In this case at the normal incidence all diffraction orders except the zeroth reflected and zeroth transmitted ones are evanescent. The values of the geometrical parameters of the structure w; hgr ; hl (Fig. 1) were determined as a result of numerical optimization of the following merit function
maxfIn ðx; pÞg x
d
0
0 12 Iðx; pÞ I ðx; pÞ n @ A dx ! min maxfIn ðt; pÞg
ð5Þ
t
where p ¼ ðw; hgr ; hl Þ, Iðx; pÞ is the calculated intensity, generated by the structure at the interface between the substrate and the waveguide layer (at z = 0), In ðx; pÞ is the intensity of the interference pattern of the ±n-th diffraction orders at z = 0, determined from the expressions (1), (2) (i.e. the ‘‘ideal” interference pattern). The first multiplier in (5) is responsible for the maximization of the field intensity at the interference pattern maxima. The second multiplier in (5) is the proximity measure of the calculated interference pattern intensity to the ‘‘ideal” interference pattern of ± n-th diffraction orders. To calculate the interference pattern intensity Iðx; pÞ we used the rigorous coupled-wave analysis as described in [22–24]. First, the diffraction structures for generating the EEW interference patterns corresponding to ±1-th diffraction orders were calculated. In this case the interference pattern period dip;1 ¼ d=2 ¼ 120 nm. Fig. 2a shows the distribution of the electric field in the structure optimized for the TE-polarization of the incident wave. The values of the parameters obtained while minimizing (5) are given in the figure caption. The interference pattern at the interface between the waveguide layer and substrate (photoresist) is shown in Fig. 2b. The form of the interference pattern is fully consistent with Eq. (1). The values of the intensity in Fig. 2a and b are normalized by the incident wave intensity. The maximum intensity at the interface exceeds the incident wave intensity by 30 times and thus is comparable to the intensity of the interference patterns of the SPP [11,12]. Fig. 2c shows the contrast and intensity at the interference pattern maxima versus the distance from the interface between the waveguide layer and substrate. According to Fig. 2c, the contrast of the generated pattern is close to 1 as it follows from Eq. (1).
0.3 0.2 0.1 0
−0.1 −0.2 (a )
0
0.2
0.4 x, µm
1
30
0.9999
20
0.9998
10
0.9997
2
E0 )
2
20
2
0.4
25
max( E
z, µm
0.5
40
30
E0
0.6
2
0.7
35
E
55 50 45 40 35 30 25 20 15 10 5
0.8
15 10
K
gðpÞ ¼
Z
1
Fig. 3a and b show the electric field distribution and the interference pattern generated by the structure optimized for the case of the TM-polarization of the incident wave. The interference pattern corresponds to the expression (2) and the intensity of the interference maxima is 27 times higher than that of the incident wave. The value of the interference pattern contrast coincides with the theoretical estimate (4) and is 0.608 (Fig. 3c). It should be noted that the intensity of the 0-th transmitted order is less than 0.0001 for both structures and thus the contrast of the generated interference patterns does not change significantly at the distance from the interface. Now let us consider the possibility of generating the EEW interference patterns in the case when the modes of the structure are excited by the diffraction orders with numbers n, n > 1. In this case the period of the EEW interference pattern (3) will be 2n times smaller than the grating period. Thus using higher orders reduces the requirements for the technological implementation of the structure and allows generating high-frequency EEW interference patterns by using a low-frequency diffractive relief. Since the contrast of the generated interference patterns is higher in the case of TE-polarization, further we will consider structures optimized for the TE-polarized incident wave only. The calculation of the EEW interference patterns was carried out at the grating period d ¼ 720 nm for n = ±3. In this case the period of the generated interference pattern dip;3 ¼ d=6 ¼ 120 nm coincides with the previous example and the grating period is 3 times larger. As previously, the geometrical parameters of the structure were obtained as a result of optimizing the merit function (5). Fig. 4a shows the distribution of the electric field in the calculated structure (the parameters of the structure are given in the figure caption). The interference pattern at the interface between the waveguide layer and the substrate (photoresist) is shown in Fig. 4b. The maximum intensity at the interface exceeds the incident wave intensity by more than 25 times. The dependence of the contrast and electric field intensity at the interference pattern maxima is given in Fig. 4c. Let us note that for this structure the total intensity of the propagating transmitted orders is 0.23 as opposed to the structures, in which the modes are excited by the ± 1-st diffraction orders. As a result of this the contrast of the generated interference pattern is decreasing away from the waveguide layer/photoresist interface. Near the interface the contrast is close to unity and at the distance of 120 nm decreases to 0.4. This value of the contrast is the minimal required value to record the interference pattern using the standard positive photoresists [14,25]. It should be noted that the photoresist thickness of
5 0 0
0.6 (b)
0.05
0.1
x, µm
0.15
0 0
0.2 (c)
0.05 0.1 0.15 Distance from the interface, µm
0.9996 0.2
Fig. 2. (a) The distribution of the electric field in the structure at the normal incidence of the TE-polarized wave (three periods of the grating are shown, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the period is 240 nm, the grating height is 123 nm, the layer thickness is 545 nm, the width of the ridge is 98 nm, the mode is excited by ±1 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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30
0.6085
45
20
0.608
10
0.6075
25
10
15 10
0
2
2
20
0.2
15
2
25
20
max( E
0.4
E0
30
2
35
E
0.6
E0 )
40
K
0.8
z, µm
30
50
5
5 −0.2 0
0.2
(a)
0.4 x, µm
0 0
0.6
0 0.05
0.1
(b)
x, µm
0.15
0.2
0
0.05 0.1 0.15 Distance from the interface, µm
(c)
0.607 0.2
Fig. 3. (a) The distribution of the electric field in the structure at the normal incidence of TM-polarized wave (three periods of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 290 nm, the layer thickness is 456 nm, the width of the ridge is 166 nm, the mode is excited by ±1 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
30
0.75
20
0.5
2
E
15
10
0.25
0
0.2
0.4 0.6 x, µm
K
10
2
0
20
10
(a)
1
E0 )
20
2
0.2
25
max( E
30
2
0.4
30 E0
0.6
40
−0.2
40
35 50
0.8
z, µm
40
60
1
5 0 0 (b)
0.1
0.2
0.3
0.4 x, µm
0.5
0.6
0 0
0.7 (c)
0.02
0.04 0.06 0.08 0.1 0.12 Distance from the interface, µm
0 0.14
Fig. 4. (a) The distribution of the electric field in the structure at the normal incidence of TE-polarized wave (one period of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 155 nm, the layer thickness is 763 nm, the width of the ridge is 349 nm, the mode is excited by ±3 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
approximately 100 nm is sufficient for patterning nanoscale features as it was experimentally demonstrated in [26]. Thus Fig. 4 shows the possibility to generate the interference patterns of the higher evanescent diffraction orders with high intensity and contrast. In the above considered examples the superstrate material is air (esuper ¼ 1). In this case the structures are supposed to be deposited directly on the resist. The case when the superstrate material corresponds to dielectric with esuper > 1 is more interesting in terms of technology. In this case, the diffraction structure can be fabricated on the top of the given material and then brought into contact with the resist surface. It should be noted that intimate optical contact between the structure and the resist can be achieved using index-matching liquid [26]. Fig. 5 presents the structure performance computed for esuper ¼ 1:69 (Teflon), grating period d ¼ 660 nm and n = ±3. The other geometrical parameters of the structure were obtained as a result of optimizing the merit function (5) and are given in the figure caption. For the specified parameters the period of the generated interference pattern is dip;3 ¼ d=6 ¼ 110 nm which corresponds to the feature size of about k=8. The intensity of interference maxima at the interface exceeds the incident wave intensity by more than 100 times (Fig. 5b). The contrast is close to unity at the interface and at the distance of 120 nm decreases to 0.4 (Fig. 5c).
The suggested approach can be generalized onto the case of generating the 2D EEW interference patterns. In this case a 3D diffraction grating is used (Fig. 6a), and the modes of the structure are excited by four diffraction orders with the numbers ðn; 0Þ, ð0; nÞ. The diffraction structure with period dx ¼ dy ¼ 720 nm for generating 2D interference pattern at n = 3 was calculated as an example. The values of the dielectric permittivities esuper , egr , el , esub coincide with those used in examples in Figs. 2–4. The geometrical parameters of the structure were obtained as the result of minimization of the merit function similar to (5). It should be noted that the configuration of the generated EEW interference patterns depends on the incident wave polarization. Fig. 5b shows the interference pattern at the interface between the waveguide layer and photoresist generated for the case of the circular polarization of the incident wave. In this case the period of the generated interference pattern is dip;3 ¼ d=6 ¼ 120 nm. Let us notice that the interference patterns presented in [12] have similar structure. The intensity of the interference peaks in Fig. 6b exceeds the incident wave intensity by more than 20 times, and the contrast of the generated pattern exceeds 0.45. 4. Conclusion The technique for generating EEW interference patterns using guided-mode resonant gratings is proposed. For the considered
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200
0.2
150
0 0.2 0.4 0.6 x, µm
100 0.5
K
2
E0 )
40
50 0.25
100 –0.2 0 ( a)
60
2
250
0.4
max( E
300
0.6
0.75 80 2
350 E0
0.8
1
100
2
400
E
z, µm
450 1
150
120
500
1.2
20
50 0
0 0
0.1
0.2
(b)
0.3 0.4 x, µm
0.5
0 0
0.6
0.02
(c)
0.04 0.06 0.08 0.1 0.12 Distance from the interface, µm
0 0.14
Fig. 5. (a) The distribution of the electric field in the structure with esuper ¼ 1:69 at the normal incidence of TE-polarized wave (one period of the grating, the grating is displayed by the white dashed line). (b) The interference pattern at the waveguide layer/substrate interface (one period of the grating). (c) The contrast (green dashed line) and the electric field intensity (blue solid line) vs. the distance from the interface. The parameters of the structure: the grating height is 140 nm, the layer thickness is 961 nm, the width of the ridge is 355 nm, the mode is excited by ±3 orders of the grating. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
z
εsuper
k
wx
ments for the grating fabrication technology and allows generating high-frequency interference patterns of evanescent waves using a low-frequency grating. The suggested approach can be used for creating the periodic structures with nanoscale details by the contact photolithography techniques.
dy wy
y
dx
εgr
h gr
εl
hl
Acknowledgements
x
This work was financially supported by RF Presidential grant (NSh-7414.2010.9), RFBR grants 09-07-92421, 09-07-12147, 1002-01391, 10-07-00553 and Basic Research and Higher Education grant (CRDF PG08-014-1).
εsub
(a)
References
E
2
E0
2
30 20 10 0 0.8
0.8 0.6
0.6 0.4
y, µm
(b)
0.4 0.2
0.2 0
x, µm
0
Fig. 6. (a) The diffraction structure for generating 2D EEW interference patterns. (b) 2D interference pattern at the interface of the waveguide layer/substrate interface (one period of the grating). The parameters of the structure: the grating height is 285 nm, the layer thickness is 1469 nm, the size of the grating hole is 349 nm, the mode is excited by (±3, 0), (0, ±3) orders of the grating.
wavelength of the incident light (441.6 nm), the achieved period of the EEW interference patterns is 110 nm, which corresponds to significantly subwavelength detail size of about 55 nm (k/8). According to the results of numerical simulation, the contrast of the interference patterns generated in the case of TE-polarization of the incident wave is close to unity, and the intensity at the interference peaks is more than 30 times higher than that of the incident wave. The possibility of generating 1D and 2D interference patterns corresponding to the higher evanescent diffraction orders is shown. Using the higher diffraction orders reduces the require-
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