- Email: [email protected]
Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation
Optics Communications 244 (2005) 269–277 www.elsevier.com/locate/optcom
Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation Yi-Jun Jen *, Cheng-Hung Hsieh, Tsai-Sheng Lo Department of Electro-optical Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Rd., Taipei 106, Taiwan Received 3 February 2004; received in revised form 10 September 2004; accepted 21 September 2004
Abstract Non-symmetric reflection in anisotropic thin films is considered to correct the phase term of Airy formula. To measure the anisotropic optical constants of thin films accurately, the sensitivity of attenuated total reflection curve is calculated and analyzed. The analysis develops two curve fitting procedures to determine the optical constants of an anisotropic thin film. 2004 Elsevier B.V. All rights reserved. PACS: 78.20.e; 78.20.Ci; 78.20.Fm Keywords: Thin films; Optical constants; Birefringence
1. Introduction Anisotropic thin films recently have been applied in numerous optical devices, including antireflection coatings, high-reflection coatings and polarizers [1,2]. The columnar microstructure of a thin film exhibits birefringence and one of the principal axes follows the direction of columnar growth [3]. The index of refraction distribution is characterized by the orientation of the principal index and principal axes. The small difference *
Corresponding author. Tel./fax: +88634803113. E-mail address: [email protected] (Y.-J. Jen).
between the principal indices of refraction makes it necessary to accurately resolve and detect these indexes. Previous techniques [4,5] did not consider the non-symmetric internal reflection phenomenon [6], and thus the Airy formula for the reflection coefficient calculation of the anisotropic film must be corrected in this study. The correction indicated that the previous direct reflection and transmission measurements of air/anisotropic film/substrate system require extremely high sensitivity to distinct isotropic and weak anisotropic thin films. To measure anisotropy precisely, this work applied the Kretschmann configuration [7] to excite surface plasmons. Surface plasmon resonance
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.09.054
270
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
(SPR) is a sensitive method for measuring the optical constants of isotropic thin films. Optical constants can be determined easily by measuring the attenuated total reflection (ATR) curve [7–9]. This work considers the anisotropic properties resulting from the microstructures of thin films. This work examines the sensitivity of principal indices of refraction, tilted angle of columnar growth direction and anisotropic film thickness. Accordingly, sensitivity is calculated theoretically, by applying a range of anisotropic optical constants and calculating their sensitivity to the characteristics of ATR curve, including resonant angle, reflection minimum and half width of ATR curve.
2. An anisotropic thin film system This work considers a single layer of tilted-columnar birefrigent film, as shown in Fig. 1. One of the principal axes (x 0 , y 0 , z 0 ) is perpendicular to the plane of incidence, while the other two principal axes rotate at an angle of u to the film surface coordinates (x, y, z). For a p-polarized plane wave with a time-dependency given by exp(ixt), the incident plane is yz plane and the wave in the thin film is extraordinary. The principal refractive indices ny 0 , nz 0 and the angle u can be transformed to the equivalent refractive indices gy, gz and gyz in the surface coordinates [4]:
Following [6] in Eqs. (1)–(13), the wave vector distribution on the yz plane is elliptic for the extraordinary wave, as shown in Fig. 2. The component of the wave vector parallel to the interface b must be the same for incident, reflected and refracted ~ waves, so when the incident wave vector k0 ¼ þ b^ey þ k z0^ez is known, the wave vectors ~ k 1 and ~ k1 in the anisotropic medium can be obtained by visualizing the wave vector diagram, which is shown in Fig. 2: þ ~ k 1 ¼ b^ey þ k þ ez ; z ^
~ k 1 ¼ b^ey þ k ez : z ^
ð2Þ
Ifþ medium 1 is a thin film, the wave vectors ~ k 1 correspond to the internal multik 1 and ~ reflected phase vectors in the thin film. Where kþ z and k z are represented as: 0 kþ z ¼ þk z K DK;
0 k z ¼ k z K DK;
ð3Þ
g2y ¼ n2y 0 cos2 u þ n2z0 sin2 u; g2z ¼ n2y 0 sin2 u þ n2z0 cos2 u; gyz ¼
ðn2y 0
ð1Þ
n2z0 Þ sin u cos u: z’
Fig. 2. Elliptical surface of wave vector distribution in an anisotropic medium.
z
y’
ϕ
y
x x’
Fig. 1. The principal axes (x 0 , y 0 , z 0 ) and the film surface coordinates (x, y, z).
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
k 0z ¼ ðgy =gz Þðk 20 g2z b2 Þ
1=2
2 1=2
K ¼ ð1 ½gyz =ðgy gz Þ Þ
;
ð4Þ
;
ð5Þ
DK ¼ ðbgyz Þ=g2z :
r02 ¼
r01 þ r12 expðiDÞ : 1 þ r01 r12 expðiDÞ
ð7Þ
Fig. 3 shows the extraordinary wave propagating in the anisotropic thin film. The phase term D in Airy formula can be derived by subtracting the optical path length of the propagation of the plane wave AE from the optical path length of the propagation of the plane wave ABC D ¼ dABC dAE :
dAE ¼ k 0 AE ¼ k 0 sin h0 ðd tan h þ d tan hþ Þ;
sec h þ
kþ 1
sec h Þ;
ð10Þ
hþ ¼ tan1
b ; kþ z
þ 2 2 dABC ¼ dðk z sec h þ k z sec hþ Þ;
ð11Þ
ð12Þ
the factor D representing the phase change of the plane wave travelling the film with thickness d should be modified as [6] 0 D ¼ kþ z d k z d ¼ 2k z Kd ¼ 2K
gy 2 2 1=2 ðk g b2 Þ : gz 0 z ð13Þ
This correction makes the form of the reflection coefficient resemble that of an isotropic thin film. The coefficients r01 and r12 for the p-polarized light are represented as: rp01
¼
n20 2 ½n0 ðb=k 0 Þ2 1=2 n20 ½n20 ðb=k 0 Þ2 1=2 gy gz K
rp12 ¼ þ
b ; k z
thus
ð9Þ
þ dABC ¼ ðk 1 AB þ k 1 BCÞ
¼
h ¼ tan1
ð8Þ
The phase changes of optical path length AE and ABC are represented as:
dðk 1
where h1 and h2 denote the incident and reflected angles for internal reflected wave vectors:
ð6Þ
Previous papers [4,5] have calculated the terms k 0z ; K and DK but ignored the other wave vector solutions ~ k 1 in the thin film. The reflection coefficient of the single thin film system can be represented by Airy formula
271
½g2z ðb=k 0 Þ2 1=2 gy gz K ½g2z ðb=k 0 Þ2 1=2
g gK
y z ½g2 ðb=k z
0Þ
2 1=2
gy gz K
þ ½g2 ðb=k z
0Þ
2 1=2
0Þ
2 1=2
0Þ
2 1=2
;
ð14Þ
:
ð15Þ
n2
½n2 ðb=k2 2
n22
þ ½n2 ðb=k 2
For a weak anisotropic thin film, the following approximate relations exist: gy g2z ffi gy gz K and K ffi 1: ð16Þ gz
Fig. 3. The plane wave travelling the anisotropic thin film with thickness d.
For a film with a thickness of less than a wavelength, detecting the variation of weak anisotropy is difficult (the difference between the refractive indices is about 0.02). WangÕs system (air/anisotropic SiO2 film/BK7 substrate) has optical constants ny 0 = 1.395, nz 0 = 1.368 and thickness d = 80 nm. According to the reflectance calculation, this system requires reflectance resolution of 1.0 · 105 to resolve the tilted angle u variation 1. To increase the sensitivity of the anisotropy measurement, this work examines a sensitive method in optical constants determination of thin films: surface plasmon resonance (SPR).
272
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
3. Sensitivity calculation and analysis Fig. 4 shows the optical setup for SPR, Kretschmann configuration [7] is applied as a two layer system (0/1/2/3: 0, prism; 1, metal film; 2, anisotropic film; 3, air). The reflection coefficient of the system is represented as r0123 r01 þ r123 exp½iH ; 1 þ r01 r123 exp½iH r12 þ r23 exp½iD ; ¼ 1 þ r12 r23 exp½iD
r0123 ¼ r123
ð17Þ
where H denotes the phase change for the plane wave reflected from the metal film, and the coefficient r123 represents the reflection coefficient of the sub-system (1/2/3). The incident vapor direction is assumed to be located on the y–z plane. When p-polarized light is applied to derive the reflection spectrum, the wave in the anisotropic thin film is extraordinary and the index distribution is characterized by n0y ; n0z and /. Surface plasmon can be excited provided that the equivalent admittance (index of refraction) of anisotropic thin film/air system is smaller than the admittance (index of refraction) of the prism. The surface plasmon resonance is measured by its reflectance angular spectrum: ATR curve. In the region of resonance, reflectance can be represented as a function of b and the ATR curve can be approximately a Lorentzian curve [7] R¼1
4Ci Crad 2
ðb bsp Þ þ ðCi þ Crad Þ
2
;
ð18Þ
where bsp is the resonant parallel interface wave vector, and Ci and Crad are the internal damping and radiation damping, respectively. The ATR curve can be characterized by its resonant wave vector bsp, minimum reflectance Rmin and half width (HW). The ‘‘depth’’ of the ATR curve is determined by the difference between Ci and Crad. The half width of the ATR curve is represented as 2(Ci + Crad). Therefore, it is reasonable to analyze the resolution of SPR method by calculating sensitivity of anisotropic optical constants to the characteristics of ATR curve, including resonant angle, reflection minimum and half width of ATR curve. The ATR curve of the sample with SiO2 film is used to determine the optical constants n0x ; n0y ; n0z ; d 2 , and the tilt angle /. When s-polarized light are applied to measure the reflectance spectrum, the SiO2 film is taken as an isotropic film. Since the SiO2 film is assumed to be a birefringent thin film, the two principal indices of refraction ðn0x ¼ n0y and n0z Þ are derived from ATR curve fitting procedures. From s-polarized measurement, we can derive the initial values of thickness d2 and index of refraction ni for the first fitting ATR curve procedure. Meanwhile, the other three optical constants (n0y ; n0z and /) are determined from the p-polarized light spectrum (ATR curve). The sensitivity is calculated and analyzed by simulating ATR curves for wavelength k = 632.8 nm and various SiO2 columns tilt angles / (10– 80) and thicknesses (50–100 nm). The simulated optical constants of the metal film are thickness: 40 nm, index of refraction 0.088 and index of
Fig. 4. The Kretschmann configuration system (0: prism/1: metal film/2: anisotropic film/3: isotropic medium).
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
273
extinction 4.030. The refractive index of the prism is 1.515. Fig. 5(a) shows the calculated resonant angle as a function of the tilted angle u and the thickness of SiO2 d2 under the condition that the principal indices of refraction, n0y and n0z equal 1.380 and 1.360, respectively, and Fig. 5(b) and (c) shows that hmin requires a resolution of 1.6 · 102 deg to resolve the thickness variation 0.1 nm. Moreover, for thickness d2 = 100 nm, hmin requires resolution of 6.1 · 103 deg to resolve the
tilted angle variation of 1. The sensitivity of hmin to tilted angle u increases with the thickness d2. Fig. 6(a) shows calculated resonant reflection minimum as a function of tilted angle u and the thickness of SiO2 d2. Fig. 6(b) and (c) shows that Rmin requires resolution 6.0 · 105 to resolve the thickness variation 0.1 nm. For thickness d2 = 100 nm, Rmin requires a resolution of 2.9 · 105 to resolve the tilted angle variation of 1. The sensitivity of Rmin to tilted angle u also increases with thickness d2. Fig. 7(a) shows the half width of the ATR curve as a function of the tilted angle u and the thickness of SiO2 d2. Fig. 7(b) and (c) shows that HW requires a resolution of 3 · 103 deg to
Fig. 5. Relation among the resonant angle hmin and the anisotropic tilt angle u and the thickness of SiO2 film.
Fig. 6. Relation among the resonant angle Rmin and the anisotropic tilt angle u and the thickness of SiO2 film.
274
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
to use reflectance minimum to resolve a weak anisotropic thin film from an isotropic thin film. For the first order approximation, the ATR curve is sensitive only within a small range of incident angles and the anisotropic refractive index in that range tends to be uniform. The columnar orientation angle variation means the isotropic index of refraction variation at the resonant angle. We calculated the sensitivity of ATR curve to the columnar orientation angle to have that it is difficult to use reflectance minimum to resolve a weak anisotropic thin film from an isotropic thin film. For an instrument with angular resolution of 0.001, the anisotropic thin film initially can be treated as an isotropic film with index of refraction ni. By fitting the measured ATR curve with Airy formula Eq. (17) at the reflectance minimum and resonant angle, the simulated curve will not completely match the half width of the ATR curve. To fit the measured curve completely, anisotropy must be considered. The two principal indexes n0y and n0z of the index ellipsoid are represented as ni + Dn1 and ni Dn2. The initial value of / can be predicted by the tangent rule [10]: relation between the directions of incident vapor and columnar growth in the coating process. Three variables, Dn1, Dn2 and /, are determined by fitting the measured curve with Airy formula Eq. (17) once again. Following the second fitting procedure, the simulated curve approaches the measured curve more closely than the isotropic film simulation. Fig. 8 shows the ATR curve of the two layer system (0/1/2/3: 0, BK7 prism; 1, Ag film; 2, SiO2 Fig. 7. Relation among the half width HW and the anisotropic tilt angle u and the thickness of SiO2 film.
Reflectance
0.8
resolve the thickness variation 0.1 nm. Moreover, for thickness d2 = 100 nm, HW requires resolution of 1.0 · 103 deg to resolve the tilted angle variation of 1. The sensitivity of Rmin clearly is much lower than that of hmin and HW. The sensitivity of HW to tilted angle u is proportional to the thickness d2 and also to HW itself. From the above analysis, the half width has higher measurement sensitivity than reflectance minimum. The sensitivity of reflectance minimum to the tilted angle u is undetectable. It is difficult
0.6
0.4
0.2 50
55
60
65 angle
70
75
Fig. 8. The ATR curve of the two layers system (0/1/2/3: 0, BK7 prism; 1, Ag film; 2, SiO2; 3, air) at wavelength 632.8 nm.
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
film; 3, air) at wavelength 632.8nm. In our experiment, the light source is intensity stabilized laser beamand the intensity variation is within ±0.1%. For a bare prism without coating, the detected total reflection intensity variation is less than ±0.02% at angles larger than the critical angle. From spolarized measurement, we can derive the initial values of thickness d2 = 100 nm and index of
275
refraction ni = 1.376 for the first fitting ATR curve procedure. The SiO2 film is prepared to be an anisotropic thin film with principal axes orientation of u = 30. The optical constants of the Ag film are determined by the ATR curve without coating the SiO2 film. Moreover, the indices of refraction and extinction of the Ag film with thickness 40.2 nm are 0.088 and 4.030. For s-polarized
0.27
Reflectance
0.26
0.25
0.24
0.23 0.22
0.21
58
58.2
58.4
(a)
58.6
58.8
59
angle 0.64
0.62
Reflectance
0.6 0.58
0.56
0.54 0.52
0.5
60
(b)
60.2
60.4
60.6
60.8
61
angle
Fig. 9. The measured ATR curve (dash line) and the simulated ATR curve (solid line) in the first fitting procedure.
276
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
light, the reflectance difference between thickness 99 and 100 nm is 0.14% at incident angle 38.00. From the reflectance angular spectrum of s polarization, it is adequate to have SiO2 film thickness 100 nm. During the first fitting procedure, the initial value of isotropic refractive index ni and thickness d2 are modulated to fit the minimum point of ATR curve (including minimum reflectance and
resonant angle). For p polarized light, the reflectance difference between thickness 99.7 and 100 nm is 0.10% at incident angle 59.29. It is adequate to have the SiO2 film thickness 99.7 nm and ni = 1.368 from the first fitting procedure. The first fitting procedure lets the simulated curve closely fit the reflectance minimum. As illustrated in Fig. 9, the close fitting of the reflectance minimum causes
0.27
Reflectance
0.26 0.25
0.24
0.23
0.22
0.21 58
58.2
58.4
(a)
58.6
58.8
59
angle 0.64
0.62
Reflectance
0. 6 0.58 0.56 0.54
0.52
0. 5 60
(b)
60.2
60.6
60.4
60.8
61
angle
Fig. 10. The measured ATR curve (dash line) and the simulated ATR curve (solid line) in the second first fitting procedure.
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
277
poor fitting of the half width (the deviation between two curves is 0.0015). The second fitting procedure is applied to fit the curve completely by simulating the two principal indices ni + Dn1 and ni Dn2 of SiO2. Fig. 10 shows that the second simulated curve completely fits the ATR curve (the deviation between two curves is less than 105 deg), and the principal indices are determined to be n0y ¼ 1:381 and n0z ¼ 1:360.
Acknowledgements
4. Conclusion
[1] I.J. Hodgkinson, Q.H. Wu, Opt. Lett. 23 (1998) 1553. [2] I.J. Hodgkinson, Q.H. Wu, Opt. Eng. 37 (9) (1998) 2630. [3] F. Horowitz, Structure-Induced Optical Anisotropy in thin film, Ph.D. Dissertation, University of Arizona, Optical Science Center, 1983. [4] H. Wang, J. Mod. Opt. 42 (3) (1995) 497. [5] H. Wang, J. Mod. Opt. 42 (11) (1995) 2173. [6] Y.J. Jen, C.C. Lee, Opt. Lett. 26 (4) (2001) 190. [7] Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer-Verleg, Berlin, Heidelberg, Germany, 1988, pp. 10–16. [8] W.P. Chen, J.M. Chen, J. Opt. Soc. Am. 71 (2) (1981) 189. [9] L. Levesque, B.E. Paton, S.H. Payne, Appl. Opt. 33 (34) (1994) 8036. [10] I.J. Hodgkinson, Q.H. Wu, J. Hazel, Appl. Opt. 37 (13) (1998) 2653.
This study corrects the Airy formula by considering the non-symmetric reflection phenomenon in anisotropic thin films. Under this correction, previous direct reflection/transmission measurements require highly resolution to detect anisotropic optical constants. This study analyzes the surface plasmon resonance method by sensitivity calculation. The analysis demonstrates that the SPR method is a more sensitive and feasible method. Two fitting procedures can determine the weak anisotropic optical properties of a thin film. This study concludes that the SPR technique can be applied to study the optical properties of the microstructures of thin films with small thickness.
The authors thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 92-2215-E-027-005.
References
Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation Yi-Jun Jen *, Cheng-Hung Hsieh, Tsai-Sheng Lo Department of Electro-optical Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Rd., Taipei 106, Taiwan Received 3 February 2004; received in revised form 10 September 2004; accepted 21 September 2004
Abstract Non-symmetric reflection in anisotropic thin films is considered to correct the phase term of Airy formula. To measure the anisotropic optical constants of thin films accurately, the sensitivity of attenuated total reflection curve is calculated and analyzed. The analysis develops two curve fitting procedures to determine the optical constants of an anisotropic thin film. 2004 Elsevier B.V. All rights reserved. PACS: 78.20.e; 78.20.Ci; 78.20.Fm Keywords: Thin films; Optical constants; Birefringence
1. Introduction Anisotropic thin films recently have been applied in numerous optical devices, including antireflection coatings, high-reflection coatings and polarizers [1,2]. The columnar microstructure of a thin film exhibits birefringence and one of the principal axes follows the direction of columnar growth [3]. The index of refraction distribution is characterized by the orientation of the principal index and principal axes. The small difference *
Corresponding author. Tel./fax: +88634803113. E-mail address: [email protected] (Y.-J. Jen).
between the principal indices of refraction makes it necessary to accurately resolve and detect these indexes. Previous techniques [4,5] did not consider the non-symmetric internal reflection phenomenon [6], and thus the Airy formula for the reflection coefficient calculation of the anisotropic film must be corrected in this study. The correction indicated that the previous direct reflection and transmission measurements of air/anisotropic film/substrate system require extremely high sensitivity to distinct isotropic and weak anisotropic thin films. To measure anisotropy precisely, this work applied the Kretschmann configuration [7] to excite surface plasmons. Surface plasmon resonance
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.09.054
270
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
(SPR) is a sensitive method for measuring the optical constants of isotropic thin films. Optical constants can be determined easily by measuring the attenuated total reflection (ATR) curve [7–9]. This work considers the anisotropic properties resulting from the microstructures of thin films. This work examines the sensitivity of principal indices of refraction, tilted angle of columnar growth direction and anisotropic film thickness. Accordingly, sensitivity is calculated theoretically, by applying a range of anisotropic optical constants and calculating their sensitivity to the characteristics of ATR curve, including resonant angle, reflection minimum and half width of ATR curve.
2. An anisotropic thin film system This work considers a single layer of tilted-columnar birefrigent film, as shown in Fig. 1. One of the principal axes (x 0 , y 0 , z 0 ) is perpendicular to the plane of incidence, while the other two principal axes rotate at an angle of u to the film surface coordinates (x, y, z). For a p-polarized plane wave with a time-dependency given by exp(ixt), the incident plane is yz plane and the wave in the thin film is extraordinary. The principal refractive indices ny 0 , nz 0 and the angle u can be transformed to the equivalent refractive indices gy, gz and gyz in the surface coordinates [4]:
Following [6] in Eqs. (1)–(13), the wave vector distribution on the yz plane is elliptic for the extraordinary wave, as shown in Fig. 2. The component of the wave vector parallel to the interface b must be the same for incident, reflected and refracted ~ waves, so when the incident wave vector k0 ¼ þ b^ey þ k z0^ez is known, the wave vectors ~ k 1 and ~ k1 in the anisotropic medium can be obtained by visualizing the wave vector diagram, which is shown in Fig. 2: þ ~ k 1 ¼ b^ey þ k þ ez ; z ^
~ k 1 ¼ b^ey þ k ez : z ^
ð2Þ
Ifþ medium 1 is a thin film, the wave vectors ~ k 1 correspond to the internal multik 1 and ~ reflected phase vectors in the thin film. Where kþ z and k z are represented as: 0 kþ z ¼ þk z K DK;
0 k z ¼ k z K DK;
ð3Þ
g2y ¼ n2y 0 cos2 u þ n2z0 sin2 u; g2z ¼ n2y 0 sin2 u þ n2z0 cos2 u; gyz ¼
ðn2y 0
ð1Þ
n2z0 Þ sin u cos u: z’
Fig. 2. Elliptical surface of wave vector distribution in an anisotropic medium.
z
y’
ϕ
y
x x’
Fig. 1. The principal axes (x 0 , y 0 , z 0 ) and the film surface coordinates (x, y, z).
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
k 0z ¼ ðgy =gz Þðk 20 g2z b2 Þ
1=2
2 1=2
K ¼ ð1 ½gyz =ðgy gz Þ Þ
;
ð4Þ
;
ð5Þ
DK ¼ ðbgyz Þ=g2z :
r02 ¼
r01 þ r12 expðiDÞ : 1 þ r01 r12 expðiDÞ
ð7Þ
Fig. 3 shows the extraordinary wave propagating in the anisotropic thin film. The phase term D in Airy formula can be derived by subtracting the optical path length of the propagation of the plane wave AE from the optical path length of the propagation of the plane wave ABC D ¼ dABC dAE :
dAE ¼ k 0 AE ¼ k 0 sin h0 ðd tan h þ d tan hþ Þ;
sec h þ
kþ 1
sec h Þ;
ð10Þ
hþ ¼ tan1
b ; kþ z
þ 2 2 dABC ¼ dðk z sec h þ k z sec hþ Þ;
ð11Þ
ð12Þ
the factor D representing the phase change of the plane wave travelling the film with thickness d should be modified as [6] 0 D ¼ kþ z d k z d ¼ 2k z Kd ¼ 2K
gy 2 2 1=2 ðk g b2 Þ : gz 0 z ð13Þ
This correction makes the form of the reflection coefficient resemble that of an isotropic thin film. The coefficients r01 and r12 for the p-polarized light are represented as: rp01
¼
n20 2 ½n0 ðb=k 0 Þ2 1=2 n20 ½n20 ðb=k 0 Þ2 1=2 gy gz K
rp12 ¼ þ
b ; k z
thus
ð9Þ
þ dABC ¼ ðk 1 AB þ k 1 BCÞ
¼
h ¼ tan1
ð8Þ
The phase changes of optical path length AE and ABC are represented as:
dðk 1
where h1 and h2 denote the incident and reflected angles for internal reflected wave vectors:
ð6Þ
Previous papers [4,5] have calculated the terms k 0z ; K and DK but ignored the other wave vector solutions ~ k 1 in the thin film. The reflection coefficient of the single thin film system can be represented by Airy formula
271
½g2z ðb=k 0 Þ2 1=2 gy gz K ½g2z ðb=k 0 Þ2 1=2
g gK
y z ½g2 ðb=k z
0Þ
2 1=2
gy gz K
þ ½g2 ðb=k z
0Þ
2 1=2
0Þ
2 1=2
0Þ
2 1=2
;
ð14Þ
:
ð15Þ
n2
½n2 ðb=k2 2
n22
þ ½n2 ðb=k 2
For a weak anisotropic thin film, the following approximate relations exist: gy g2z ffi gy gz K and K ffi 1: ð16Þ gz
Fig. 3. The plane wave travelling the anisotropic thin film with thickness d.
For a film with a thickness of less than a wavelength, detecting the variation of weak anisotropy is difficult (the difference between the refractive indices is about 0.02). WangÕs system (air/anisotropic SiO2 film/BK7 substrate) has optical constants ny 0 = 1.395, nz 0 = 1.368 and thickness d = 80 nm. According to the reflectance calculation, this system requires reflectance resolution of 1.0 · 105 to resolve the tilted angle u variation 1. To increase the sensitivity of the anisotropy measurement, this work examines a sensitive method in optical constants determination of thin films: surface plasmon resonance (SPR).
272
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
3. Sensitivity calculation and analysis Fig. 4 shows the optical setup for SPR, Kretschmann configuration [7] is applied as a two layer system (0/1/2/3: 0, prism; 1, metal film; 2, anisotropic film; 3, air). The reflection coefficient of the system is represented as r0123 r01 þ r123 exp½iH ; 1 þ r01 r123 exp½iH r12 þ r23 exp½iD ; ¼ 1 þ r12 r23 exp½iD
r0123 ¼ r123
ð17Þ
where H denotes the phase change for the plane wave reflected from the metal film, and the coefficient r123 represents the reflection coefficient of the sub-system (1/2/3). The incident vapor direction is assumed to be located on the y–z plane. When p-polarized light is applied to derive the reflection spectrum, the wave in the anisotropic thin film is extraordinary and the index distribution is characterized by n0y ; n0z and /. Surface plasmon can be excited provided that the equivalent admittance (index of refraction) of anisotropic thin film/air system is smaller than the admittance (index of refraction) of the prism. The surface plasmon resonance is measured by its reflectance angular spectrum: ATR curve. In the region of resonance, reflectance can be represented as a function of b and the ATR curve can be approximately a Lorentzian curve [7] R¼1
4Ci Crad 2
ðb bsp Þ þ ðCi þ Crad Þ
2
;
ð18Þ
where bsp is the resonant parallel interface wave vector, and Ci and Crad are the internal damping and radiation damping, respectively. The ATR curve can be characterized by its resonant wave vector bsp, minimum reflectance Rmin and half width (HW). The ‘‘depth’’ of the ATR curve is determined by the difference between Ci and Crad. The half width of the ATR curve is represented as 2(Ci + Crad). Therefore, it is reasonable to analyze the resolution of SPR method by calculating sensitivity of anisotropic optical constants to the characteristics of ATR curve, including resonant angle, reflection minimum and half width of ATR curve. The ATR curve of the sample with SiO2 film is used to determine the optical constants n0x ; n0y ; n0z ; d 2 , and the tilt angle /. When s-polarized light are applied to measure the reflectance spectrum, the SiO2 film is taken as an isotropic film. Since the SiO2 film is assumed to be a birefringent thin film, the two principal indices of refraction ðn0x ¼ n0y and n0z Þ are derived from ATR curve fitting procedures. From s-polarized measurement, we can derive the initial values of thickness d2 and index of refraction ni for the first fitting ATR curve procedure. Meanwhile, the other three optical constants (n0y ; n0z and /) are determined from the p-polarized light spectrum (ATR curve). The sensitivity is calculated and analyzed by simulating ATR curves for wavelength k = 632.8 nm and various SiO2 columns tilt angles / (10– 80) and thicknesses (50–100 nm). The simulated optical constants of the metal film are thickness: 40 nm, index of refraction 0.088 and index of
Fig. 4. The Kretschmann configuration system (0: prism/1: metal film/2: anisotropic film/3: isotropic medium).
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
273
extinction 4.030. The refractive index of the prism is 1.515. Fig. 5(a) shows the calculated resonant angle as a function of the tilted angle u and the thickness of SiO2 d2 under the condition that the principal indices of refraction, n0y and n0z equal 1.380 and 1.360, respectively, and Fig. 5(b) and (c) shows that hmin requires a resolution of 1.6 · 102 deg to resolve the thickness variation 0.1 nm. Moreover, for thickness d2 = 100 nm, hmin requires resolution of 6.1 · 103 deg to resolve the
tilted angle variation of 1. The sensitivity of hmin to tilted angle u increases with the thickness d2. Fig. 6(a) shows calculated resonant reflection minimum as a function of tilted angle u and the thickness of SiO2 d2. Fig. 6(b) and (c) shows that Rmin requires resolution 6.0 · 105 to resolve the thickness variation 0.1 nm. For thickness d2 = 100 nm, Rmin requires a resolution of 2.9 · 105 to resolve the tilted angle variation of 1. The sensitivity of Rmin to tilted angle u also increases with thickness d2. Fig. 7(a) shows the half width of the ATR curve as a function of the tilted angle u and the thickness of SiO2 d2. Fig. 7(b) and (c) shows that HW requires a resolution of 3 · 103 deg to
Fig. 5. Relation among the resonant angle hmin and the anisotropic tilt angle u and the thickness of SiO2 film.
Fig. 6. Relation among the resonant angle Rmin and the anisotropic tilt angle u and the thickness of SiO2 film.
274
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
to use reflectance minimum to resolve a weak anisotropic thin film from an isotropic thin film. For the first order approximation, the ATR curve is sensitive only within a small range of incident angles and the anisotropic refractive index in that range tends to be uniform. The columnar orientation angle variation means the isotropic index of refraction variation at the resonant angle. We calculated the sensitivity of ATR curve to the columnar orientation angle to have that it is difficult to use reflectance minimum to resolve a weak anisotropic thin film from an isotropic thin film. For an instrument with angular resolution of 0.001, the anisotropic thin film initially can be treated as an isotropic film with index of refraction ni. By fitting the measured ATR curve with Airy formula Eq. (17) at the reflectance minimum and resonant angle, the simulated curve will not completely match the half width of the ATR curve. To fit the measured curve completely, anisotropy must be considered. The two principal indexes n0y and n0z of the index ellipsoid are represented as ni + Dn1 and ni Dn2. The initial value of / can be predicted by the tangent rule [10]: relation between the directions of incident vapor and columnar growth in the coating process. Three variables, Dn1, Dn2 and /, are determined by fitting the measured curve with Airy formula Eq. (17) once again. Following the second fitting procedure, the simulated curve approaches the measured curve more closely than the isotropic film simulation. Fig. 8 shows the ATR curve of the two layer system (0/1/2/3: 0, BK7 prism; 1, Ag film; 2, SiO2 Fig. 7. Relation among the half width HW and the anisotropic tilt angle u and the thickness of SiO2 film.
Reflectance
0.8
resolve the thickness variation 0.1 nm. Moreover, for thickness d2 = 100 nm, HW requires resolution of 1.0 · 103 deg to resolve the tilted angle variation of 1. The sensitivity of Rmin clearly is much lower than that of hmin and HW. The sensitivity of HW to tilted angle u is proportional to the thickness d2 and also to HW itself. From the above analysis, the half width has higher measurement sensitivity than reflectance minimum. The sensitivity of reflectance minimum to the tilted angle u is undetectable. It is difficult
0.6
0.4
0.2 50
55
60
65 angle
70
75
Fig. 8. The ATR curve of the two layers system (0/1/2/3: 0, BK7 prism; 1, Ag film; 2, SiO2; 3, air) at wavelength 632.8 nm.
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
film; 3, air) at wavelength 632.8nm. In our experiment, the light source is intensity stabilized laser beamand the intensity variation is within ±0.1%. For a bare prism without coating, the detected total reflection intensity variation is less than ±0.02% at angles larger than the critical angle. From spolarized measurement, we can derive the initial values of thickness d2 = 100 nm and index of
275
refraction ni = 1.376 for the first fitting ATR curve procedure. The SiO2 film is prepared to be an anisotropic thin film with principal axes orientation of u = 30. The optical constants of the Ag film are determined by the ATR curve without coating the SiO2 film. Moreover, the indices of refraction and extinction of the Ag film with thickness 40.2 nm are 0.088 and 4.030. For s-polarized
0.27
Reflectance
0.26
0.25
0.24
0.23 0.22
0.21
58
58.2
58.4
(a)
58.6
58.8
59
angle 0.64
0.62
Reflectance
0.6 0.58
0.56
0.54 0.52
0.5
60
(b)
60.2
60.4
60.6
60.8
61
angle
Fig. 9. The measured ATR curve (dash line) and the simulated ATR curve (solid line) in the first fitting procedure.
276
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
light, the reflectance difference between thickness 99 and 100 nm is 0.14% at incident angle 38.00. From the reflectance angular spectrum of s polarization, it is adequate to have SiO2 film thickness 100 nm. During the first fitting procedure, the initial value of isotropic refractive index ni and thickness d2 are modulated to fit the minimum point of ATR curve (including minimum reflectance and
resonant angle). For p polarized light, the reflectance difference between thickness 99.7 and 100 nm is 0.10% at incident angle 59.29. It is adequate to have the SiO2 film thickness 99.7 nm and ni = 1.368 from the first fitting procedure. The first fitting procedure lets the simulated curve closely fit the reflectance minimum. As illustrated in Fig. 9, the close fitting of the reflectance minimum causes
0.27
Reflectance
0.26 0.25
0.24
0.23
0.22
0.21 58
58.2
58.4
(a)
58.6
58.8
59
angle 0.64
0.62
Reflectance
0. 6 0.58 0.56 0.54
0.52
0. 5 60
(b)
60.2
60.6
60.4
60.8
61
angle
Fig. 10. The measured ATR curve (dash line) and the simulated ATR curve (solid line) in the second first fitting procedure.
Y.-J. Jen et al. / Optics Communications 244 (2005) 269–277
277
poor fitting of the half width (the deviation between two curves is 0.0015). The second fitting procedure is applied to fit the curve completely by simulating the two principal indices ni + Dn1 and ni Dn2 of SiO2. Fig. 10 shows that the second simulated curve completely fits the ATR curve (the deviation between two curves is less than 105 deg), and the principal indices are determined to be n0y ¼ 1:381 and n0z ¼ 1:360.
Acknowledgements
4. Conclusion
[1] I.J. Hodgkinson, Q.H. Wu, Opt. Lett. 23 (1998) 1553. [2] I.J. Hodgkinson, Q.H. Wu, Opt. Eng. 37 (9) (1998) 2630. [3] F. Horowitz, Structure-Induced Optical Anisotropy in thin film, Ph.D. Dissertation, University of Arizona, Optical Science Center, 1983. [4] H. Wang, J. Mod. Opt. 42 (3) (1995) 497. [5] H. Wang, J. Mod. Opt. 42 (11) (1995) 2173. [6] Y.J. Jen, C.C. Lee, Opt. Lett. 26 (4) (2001) 190. [7] Heinz Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer-Verleg, Berlin, Heidelberg, Germany, 1988, pp. 10–16. [8] W.P. Chen, J.M. Chen, J. Opt. Soc. Am. 71 (2) (1981) 189. [9] L. Levesque, B.E. Paton, S.H. Payne, Appl. Opt. 33 (34) (1994) 8036. [10] I.J. Hodgkinson, Q.H. Wu, J. Hazel, Appl. Opt. 37 (13) (1998) 2653.
This study corrects the Airy formula by considering the non-symmetric reflection phenomenon in anisotropic thin films. Under this correction, previous direct reflection/transmission measurements require highly resolution to detect anisotropic optical constants. This study analyzes the surface plasmon resonance method by sensitivity calculation. The analysis demonstrates that the SPR method is a more sensitive and feasible method. Two fitting procedures can determine the weak anisotropic optical properties of a thin film. This study concludes that the SPR technique can be applied to study the optical properties of the microstructures of thin films with small thickness.
The authors thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 92-2215-E-027-005.
References