- Email: [email protected]
Vanadium dioxide as a material to control light polarization in the visible and near infrared
Optics Communications 382 (2017) 80–85
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Vanadium dioxide as a material to control light polarization in the visible and near infrared Patrick Cormier a, Tran Vinh Son b, Jacques Thibodeau a, Alexandre Doucet a, Vo-Van Truong b, Alain Haché a,n a b
Département de physique et d’astronomie, Université de Moncton, E1A 8T1, Canada Department of Physics, Concordia University, Montreal H3G 1M8, Canada
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2016 Received in revised form 23 July 2016 Accepted 25 July 2016
We report on the possible use of vanadium dioxide to produce ultrathin ( o100 nm) adjustable phase retarders working over a wide spectral range. The refractive index of vanadium dioxide undergoes large changes when the material undergoes a phase transition from semiconductor to metal at a temperature of 68 °C. In a thin film, the resulting optical phase shift is different for s- and p-polarizations in both reflection and transmission, and under certain conditions the polarization state changes between linear or circular or between linear polarizations oriented differently when the material phase transitions. Specific ultrathin modulators are proposed based on the results. & 2016 Elsevier B.V. All rights reserved.
Keywords: Vanadium dioxide Transition metal oxide Phase control Polarization control Adjustable phase retarder Light modulator
1. Introduction Vanadium dioxide (VO2) is an insulator that acquires metallic properties when heated to temperatures above 68 °C, owing to a rearrangement of its crystalline structure [1]. This results in refractive index changes on the order of unity, which makes the material suitable for a variety of optical applications, including tunable metamaterials [2–4], spectrally selective filters [5,6], optical switches and limiters [7–10] and infrared imaging systems [11,12]. So far, VO2 has been considered of interest only in the context of filtering and amplitude modulation, in particular at wavelengths above 1000 nm where optical properties vary the most during phase transition. Recently, we explored a different avenue by considering effects on the optical phase of light during the material's transition. We demonstrated optical phase control [13] and ultrathin flat lenses [14], the latter also suggesting applications of VO2 may extend to the visible spectrum. In this paper, we report VO2 thin films as a way to control the polarization state of light, including switching from linear to circular and vice-versa, or to rotate linear polarization. We investigate the combination of material properties and optical conditions for which such control is possible. Using this effect in n
Corresponding author. E-mail address: [email protected] (A. Haché).
http://dx.doi.org/10.1016/j.optcom.2016.07.070 0030-4018/& 2016 Elsevier B.V. All rights reserved.
combination with polarizers could lead to highly efficient optical modulators working over a wide spectral range.
2. Modeling We calculate changes in the polarization state of a light beam interacting with a layer of VO2 as the material undergoes a phase transition. To this end, we assume a monochromatic plane wave incident on a uniform and homogeneous layer of VO2 deposited on a semi-infinite dielectric substrate. Reflection and transmission coefficients at air/VO2 and VO2/substrate interfaces are given by Fresnel's equations and assume the refractive indices to be uniform throughout the film. We will also assume that the polarization of incident light can be prepared in any arbitrary state. Fig. 1 defines the parameters of the problem. Here θi is the incident angle, Eo⃑ , Er⃑ and Et⃑ the complex electric field of the oncoming, reflected and transmitted waves, respectively. The VO2 layer is assumed to have a complex refractive index n + ik and be deposited on a dielectric substrate. The electric field component parallel to the plane of incidence is the p-polarization and the component perpendicular to it is the s-polarization. For convenience, we chose to express the electric field in the basis of sand p-polarizations. We first treat the case of reflection, then expand the results to
P. Cormier et al. / Optics Communications 382 (2017) 80–85
81
temperature TB . ⃑ and ErB ⃑ in Of particular interest is the difference between ErA terms of their polarization state. If the phase between the s- and p-components varies with the thin film's temperature, the polarization state will be altered. We can express the difference be⃑ and ErB ⃑ in terms of a “switching matrix” rAB defined as tween ErA
⃑ = rABErA ⃑ ErB
(5)
where
rAB
⎛ rBp ⎞ 0⎟ ⎜ r Ap ⎟. = rBrA−1 = ⎜⎜ rBs ⎟ ⎜ 0 ⎟ rAs ⎠ ⎝
(6)
We can write this matrix in the form Fig. 1. Definitions of polarizations states and incoming and outgoing light beams on a film.
⎛1 0⎞ rAB = a r ⎜ ⎟, ⎝ 0 zr ⎠
(7)
with
ar =
rBp rAp
(8)
rBs rAp . rAs rBp
(9)
and
zr =
⃑ equally, it Since ar affects both the s- and p-components of ErA does not change the relative phase between the two and the polarization state remains the same. On the other hand, the phase of zr (or arg( zr )) will have an effect on the polarization state. Note that parameter zr is related to the commonly used ellipsometry parameters Ψ and Δ defined as
rp rs
= tanΨeiΔ ,
(10)
and Fig. 2. Reflected electric fields at temperatures TA and TB in the p- and s-polarization basis.
⃑ the reflected electric field when the VO2 transmission. Let's call ErA ⃑ the corresponding field at film is at low temperature TA and ErB some higher temperature TB . In the Jones matrix formalism, the electric field in the basis of s- and p-polarizations is written
⎛ Ep ⎞ E ⃑ = ⎜⎜ ⎟⎟. ⎝ Es ⎠
(1)
The reflected field is related to the incident field via
⃑ = rAEo⃑ ErA
(2)
⃑ = rBEo⃑ ErB
(3)
where rA is the thin film's reflection matrix at temperature TA which takes the form
⎛ rAp 0 ⎞ rA = ⎜⎜ ⎟⎟, ⎝ 0 rAs ⎠
zr =
tanΨA i( ΔA −ΔB ) e . tanΨB
(11)
In the most general case, the effect of rAB would be to transform ⃑ into differently polarized but still an elliptically polarized field ErA ⃑ elliptical ErB . Indeed, since the phase difference between rs and rp in a VO2 layer is generally not a multiple of 2 π , the reflected field is generally not linearly polarized even if Eo⃑ is linearly polarized. But in order to attain the greatest contrasts of modulation using polarizers, it is best to operate with linear polarizations. To this end, a phase compensator (e.g. birefringent crystal) can be used to pre⃑ (or ErB ⃑ ) is linearly polarized, in which pare Eo⃑ in such a way that ErA case high-contrast modulation through polarizers would be possible. ⃑ is linearly polarized, two options are particularly proIf ErA mising: rotation of the linear polarization by some angle when arg( zr )=π (1 + 2m) with m = 0, ± 1, ± 2… and conversion to circular polarization when arg( zr ) = π (1/2+m) with =0, ± 1, ± 2… . We examine these two situations in more details in the following sections. 2.1. Rotation of linear polarization
(4)
where rAp and rAs are the (complex) coefficients of reflection of the VO2 layer for the p- and s-polarizations, respectively, at temperature TA . Matrix rB has the same structure but with coefficients for
⃑ in the s–p plane, thus defining γ the angle of Fig. 2 draws ErA A the field with respect to the s-axis. (Note that γA should not take values of 0 or π/2 as they represent polarization states purely in s or p, a condition that does not allow polarization control). A phase
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P. Cormier et al. / Optics Communications 382 (2017) 80–85
4
4 Cold [13] Cold [15] Cold [17] Cold [16] Hot [13] Hot [15] Hot [17] Hot [16]
3
3
Extinction coefficient
Refractive index
3.5
Cold [13] Cold [15] Cold [17] Cold [16] Hot [13] Hot [15] Hot [17] Hot [16]
3.5
2.5
2
2.5 2 1.5 1
1.5
1
0.5
0
500
1000
1500
2000
0
2500
0
500
Wavelength (nm)
1000
1500
2000
2500
Wavelength (nm)
Fig. 3. Reported complex refractive indices (real part left and imaginary part right) of VO2 thin films. Dashed lines correspond to the metallic state at temperatures above 68 °C (hot) and solid lines correspond to the insulating state (cold).
Fig. 4. Refractive index of VO2 at 1550 nm and 800 nm as a function of temperature. Arrows give indicate heating or cooling of the sample.
shift of arg( zr )=π results in zr =− zr and from Eqs. (5) and (7) we obtain
⎛ EAp ⎞ ⎛ EBp ⎞ ⎟ ⎜⎜ ⎟⎟ = a r ⎜ ⎜− z E ⎟ ⎝ EBs ⎠ r As ⎠ ⎝
tanγA =
zr .
(15)
(12) 2.2. Transformation between linear and circular polarization
so that
tanγB =
and with Eq. (13) we find the condition
EBp EBs
=
EAp − zr E
As
=−
1 tanγA zr
(13)
⃑ is to become circularly polarized from an initially linear If ErB ⃑ , it implies that the s- and p- components have the same amErA plitudes but phase-shifted by arg( zr ) = ±π /2:
from which the rotation angle ∆γ =γA − γB is obtained. The condition under which the rotation angle becomes ±90° is
EBp
tanγAtanγB= − 1
which leads to the following additional condition
(14)
EBs
=±i =
EAp z r EAs
=
1 tanγA zr
(16)
P. Cormier et al. / Optics Communications 382 (2017) 80–85
83
Fig. 5. Films thickness and angles of incidence producing polarization switching in VO2 using data of Ref. [17]: A and B for phase switch of arg (zr )=π /2 and C and D for arg (zr ) = π .
tanγA= z r .
(17)
Transformation between linear to circular polarizations requires arg( zr ) = π (1/2+m) with m = 0, ± 1, ± 2… with the condition of Eq. (17) becoming
2.3. Theory in transmission
tanγA= zt .
The previous results can be extended to light beams transmitted through the VO2 film. In transmission, the switching matrix equivalent to that of Eq. (7) becomes
3. Calculations with reported data on VO2
⎛1 0⎞ tAB = at ⎜ ⎟ ⎝ 0 zt ⎠
(18)
where
at =
tBp tAp
(19)
tBs tAp . tAs tBp
(20)
and
zt =
Rotation of linear polarization by 90° is obtained when arg( zt )=π (1 + 2m) with m = 0, ± 1, ± 2…with the condition analogous to Eq. (15) becoming
tanγA =
zt .
(21)
(22)
The refractive indices of VO2 films have been measured in our group and in previous studies for both the metallic (hot) and insulating (cold) states of the material. Some of these results are plotted in Fig. 3, showing the real and imaginary parts of the refractive index from Refs. [13,15–17], all superimposed on the same graphs. Each study measured refractive indices over a different wavelength range, which explains why some curves extend farther than others. The data are very similar in all cases, with typical variations on the order of 75%. Fig. 4 shows a typical dependence of the refractive index of VO2 on temperature at two different wavelengths. The curves exhibit the usual hysteresis observed in VO2: during the cooling portion of the cycle, the critical temperature is shifted down by about 5 °C. In this study, hysteresis is not a factor because we consider optical effects at temperatures well outside the hysteresis region, e.g. 20 °C for insulating VO2 and 100 °C for metallic VO2.
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P. Cormier et al. / Optics Communications 382 (2017) 80–85
Fig. 6. Phase and amplitude parameters for reflection (A) and transmission (B) are calculated from Ref. [17] with a film of 180 nm in thickness. A phase switch of arg (zr )=π /2 is found in reflection but not in transmission.
Table 1 Examples of optimized parameters for applications at commonly used laser wavelengths in reflection. Wavelength (nm)
Film thickness (nm)
Angle of incidence (deg)
Angle γA (deg)
Effect
532 532 1064 1064 1550 1550
125 125 50 50 50 50
72.5 69.9 71.6 65.6 71.8 64.7
24.8 60.5 31.3 36.9 36.2 26.5
90° linear rotation linear-to-circular 90° linear rotation linear-to-circular 90° linear rotation linear-to-circular
From these data sets, and using Fresnel equations for a thin VO2 film on a glass substrate, reflection and transmission coefficients were calculated as well as parameters ar , t and zr , t . From these results, we investigated the conditions of film thickness, incidence angle and wavelength which made polarization switching possible. Using refractive indices of Ref. [17], we determined the conditions for which polarization switching was possible. For these calculations and the following, we assumed a substrate similar to glass with n ¼1.5. Figs. 5(A) and (B) reveal the combination of film thickness and angle of incidence leading to a transformation from linear to circular polarization during the film's phase transition. At
each wavelength, we chose random pairs of film thickness and angle of incidence and calculated the resulting optical phase difference: each dot on the graph represent a successful result, namely a case where arg (zr )=π /2 to within a 72% error (used as a criterion of success). Interference effects are periodic functions of space, and as a result, we see in Fig. 5(A) four bands of possible solutions for the film thickness. Note that each one of these bands has a finite thickness since another parameter, namely the incidence angle, is simultaneously adjusted. Since it is typically easier to create high-quality VO2 in the form of thinner films, generally speaking it would be optimal to choose a solution in the bottom band. As Fig. 5(B) shows, successful trials occur at
P. Cormier et al. / Optics Communications 382 (2017) 80–85
incidence angles between 60 and 80°. These incidence angles are near Brewster's angle of the air/VO2 interface. However, the concept of Brewster's angle (the angle of null reflection in p-polarization) is valid for a single interface and does not apply to a thin film. We studied polarization switching by VO2 in the case where the temperature goes from well below Tc to well above Tc, and we did not consider intermediate situations where temperature is varied over a lesser range. Although it would be interesting, it is not possible here because of lack of available data on refractive indices of VO2 as a function of temperature. Figs. 5(C) and (D) give the same results for rotation of linear polarization (when arg (zr ) = π )). Interestingly, the effect appears at wavelengths as short as 500 nm, in contrast with most applications of VO2 films, which are typically found at 1300 nm and above. The film may also be as thin as 100 nm or less. We also carried out calculations using refractive indices from other studies [15,16] as well as our own [13] and found similar results. Fig. 6 further explores the range of optical phase shifts for a specific film thickness of 180 nm, the film studied in Ref. [17]. With the thickness fixed, parameters arg (zr ), zr and ar are calculated as a function of wavelength and angle of incidence. Fig. 6(A) plots the phase shift amplitude and a line curve where the phase shift is constant at π/2. Within that area, centered at around 700 nm wavelength and 68° incidence angle, the phase shifts is highest with arg ( z r ≃π . Also in that area zr ≃1 so that the reflected po-
)
larization is at a convenient angle γA ≃ 45o and ar ≃1, implying that reflectance remains relatively constant during phase transition. This particular film would therefore be suitable for applications in the 600–800 nm spectral range, with simple angle-tuning from 66° to 72° to adjust to a specific wavelength. As seen in Fig. 5(B), the phase shifts are much less pronounced in reflection. To give an example of specific device applications, we chose the commonly-used wavelengths of the Nd:YAG laser (1064 nm), its second harmonic (532 nm) and the 1550 nm telecommunication wavelength. Table 1 summarizes the results for linear-circular and linear-rotator devices. In addition to having easy to achieve specifications in terms of film thickness, incidence angle and polarization angles, the device would work in reflection, thus avoiding dispersion problems in the substrate. On the other hand, unlike dielectric mirrors, which typically operate at 0° or 45° incidence angle, the proposed polarization rotator would operate at high incidence angles, near Brewster's angle. We performed calculations in transmission with the same data set with various thicknesses but did not find relative phase shifts larger than 0.2π rad. Since this is below the π/2 minimum level to achieve circular polarization, so we did not pursue this route further.
4. Conclusions Vanadium dioxide shows promises for light polarization
85
control applications using thin films. Because of unequal optical phase shifts in s- and p-polarizations during phase transition, the polarization state of reflected light can change considerably. The effect originates from large differences in refractive indices between the material's insulating and metallic states, and manifests itself in both reflection and transmission through thin films. We derived a theory to calculate phase shifts and its effect to the polarization and used reported measurements of refractive indices to show that films as thin as 50 nm could transform polarization from linear to circular (and vice-versa) or rotate linear polarization by 90°. This renders possible optical switches and devices many times thinner than optical means such as liquid crystals. Switching is also possible at wavelengths of 500 nm and above, making it one of the few applications of VO2 in the visible spectrum. Work is currently under way to demonstrate the effect experimentally.
References [1] F.J. Morin, Oxides which show a metal-to-insulator transition at the Néel temperature, Phys. Rev. Lett. 3 (1959) 34–36. [2] G. Dayal, S.A. Ramakrishna, Metamaterial saturable absorber mirror”, Opt. Lett. 38 (2013) 272–274. [3] M.J. Dicken, K. Aydin, I.M. Pryce1, L.A. Sweatlock, E.M. Boyd, S. Walavalkar, J. Ma, H.A. Atwater, Frequency tunable near-infrared metamaterials based on VO2 phase transition, Opt. Exp. 17 (2009) 18330–18339. [4] W. Huang, X. Yin, C. Huang, Q. Wang, T. Miao, Y. Zhul, Optical switching of a metamaterial by temperature controlling, Appl. Phys. Lett. 96 (2010) 261908. [5] A. Hendaoui, N. Émond, M. Chaker, É. Haddad, Highly tunable-emittance radiator based on semiconductor-metal transition of VO2 thin films, Appl. Phys. Lett. 102 (2013) 061107. [6] A.I. Sidorov, Controllable VO2 mirrors based on a three-mirror interferometer for the spectral range 0.5–2.5 um, J. Opt. Technol. 67 (2000) 532–537. [7] G. Seo, B. Kim, Y.W. Lee, H. Kim, Photo-assisted bistable switching using Mott transition in two-terminal VO2 device, Appl. Phys. Lett. 100 (2012) 011908. [8] J.D. Ryckman, K.A. Hallman, R.E. Marvel, R.F. Haglund Jr., S.M. Weiss, Ultracompact silicon photonic devices reconfigured by an optically induced semiconductor-to-metal transition, Opt. Exp. 21 (2013) 10753–10763. [9] M.A. Kats, D. Sharma, J. Lin, P. Genevet, R. Blanchard, Z. Yang, M. Mumtaz Qazilbash, D.N. Basov, S. Ramanathan, F. Capasso, Ultra-thin perfect absorber employing a tunable phase change material, Appl. Phys. Lett. 101 (2012) 221101. [10] S. Chena, H. Maa, X. Yib, T. Xiong, H. Wang, C. Ke, Smart VO2 thin film for protection of sensitive infrared detectors from strong laser radiation, Sens. Actuators A: Phys. 115 (2004) 28–31. [11] S. Bonora, U. Bortolozzo, S. Residori, R. Balu, P.V. Ashrit, Mid-IR to near-IR image conversion by thermally induced optical switching in vanadium dioxide, Opt. Lett. 35 (2010) 103–105. [12] S. Bonora, G. Beydaghyan, A. Haché, P.V. Ashrit, Mid-IR laser beam quality measurement through vanadium dioxide optical switching, Opt. Lett. 38 (2013) 1554–1556. [13] T.V. Son, K. Zongo, C. Ba, G. Beydaghyan, A. Haché, Pure optical phase control in vanadium dioxide thin films, Opt. Comm. 320 (2014) 151–155. [14] T.V. Son, C.O.F. Ba, R. Vallée, A. Haché, Nanometer-thick flat lens with adjustable focus, Appl. Phys. Lett. 105 (2014) 231120. [15] H. Kakiuchida, P. Jin, S. Nakao, M. Tazawa, Optical properties of vanadium dioxide films during semiconductive-metallic phase transition, Jpn. J. Appl. Phys. 46 (2007) L113. [16] J.B. Kana, J.M. Ndjaka, G. Vignaud, A. Gibaud, M. Maaza, Thermally tunable optical constants of vanadium dioxide thin films measured by spectroscopic ellipsometry, Opt. Commun. 284 (2011) 807–812. [17] A. Joushaghani, Micro- and nano-scale optoelectronic devices using vanadium dioxide, Graduate Department of Electrical and Computer Engineering, University of Toronto, 2014.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Vanadium dioxide as a material to control light polarization in the visible and near infrared Patrick Cormier a, Tran Vinh Son b, Jacques Thibodeau a, Alexandre Doucet a, Vo-Van Truong b, Alain Haché a,n a b
Département de physique et d’astronomie, Université de Moncton, E1A 8T1, Canada Department of Physics, Concordia University, Montreal H3G 1M8, Canada
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2016 Received in revised form 23 July 2016 Accepted 25 July 2016
We report on the possible use of vanadium dioxide to produce ultrathin ( o100 nm) adjustable phase retarders working over a wide spectral range. The refractive index of vanadium dioxide undergoes large changes when the material undergoes a phase transition from semiconductor to metal at a temperature of 68 °C. In a thin film, the resulting optical phase shift is different for s- and p-polarizations in both reflection and transmission, and under certain conditions the polarization state changes between linear or circular or between linear polarizations oriented differently when the material phase transitions. Specific ultrathin modulators are proposed based on the results. & 2016 Elsevier B.V. All rights reserved.
Keywords: Vanadium dioxide Transition metal oxide Phase control Polarization control Adjustable phase retarder Light modulator
1. Introduction Vanadium dioxide (VO2) is an insulator that acquires metallic properties when heated to temperatures above 68 °C, owing to a rearrangement of its crystalline structure [1]. This results in refractive index changes on the order of unity, which makes the material suitable for a variety of optical applications, including tunable metamaterials [2–4], spectrally selective filters [5,6], optical switches and limiters [7–10] and infrared imaging systems [11,12]. So far, VO2 has been considered of interest only in the context of filtering and amplitude modulation, in particular at wavelengths above 1000 nm where optical properties vary the most during phase transition. Recently, we explored a different avenue by considering effects on the optical phase of light during the material's transition. We demonstrated optical phase control [13] and ultrathin flat lenses [14], the latter also suggesting applications of VO2 may extend to the visible spectrum. In this paper, we report VO2 thin films as a way to control the polarization state of light, including switching from linear to circular and vice-versa, or to rotate linear polarization. We investigate the combination of material properties and optical conditions for which such control is possible. Using this effect in n
Corresponding author. E-mail address: [email protected] (A. Haché).
http://dx.doi.org/10.1016/j.optcom.2016.07.070 0030-4018/& 2016 Elsevier B.V. All rights reserved.
combination with polarizers could lead to highly efficient optical modulators working over a wide spectral range.
2. Modeling We calculate changes in the polarization state of a light beam interacting with a layer of VO2 as the material undergoes a phase transition. To this end, we assume a monochromatic plane wave incident on a uniform and homogeneous layer of VO2 deposited on a semi-infinite dielectric substrate. Reflection and transmission coefficients at air/VO2 and VO2/substrate interfaces are given by Fresnel's equations and assume the refractive indices to be uniform throughout the film. We will also assume that the polarization of incident light can be prepared in any arbitrary state. Fig. 1 defines the parameters of the problem. Here θi is the incident angle, Eo⃑ , Er⃑ and Et⃑ the complex electric field of the oncoming, reflected and transmitted waves, respectively. The VO2 layer is assumed to have a complex refractive index n + ik and be deposited on a dielectric substrate. The electric field component parallel to the plane of incidence is the p-polarization and the component perpendicular to it is the s-polarization. For convenience, we chose to express the electric field in the basis of sand p-polarizations. We first treat the case of reflection, then expand the results to
P. Cormier et al. / Optics Communications 382 (2017) 80–85
81
temperature TB . ⃑ and ErB ⃑ in Of particular interest is the difference between ErA terms of their polarization state. If the phase between the s- and p-components varies with the thin film's temperature, the polarization state will be altered. We can express the difference be⃑ and ErB ⃑ in terms of a “switching matrix” rAB defined as tween ErA
⃑ = rABErA ⃑ ErB
(5)
where
rAB
⎛ rBp ⎞ 0⎟ ⎜ r Ap ⎟. = rBrA−1 = ⎜⎜ rBs ⎟ ⎜ 0 ⎟ rAs ⎠ ⎝
(6)
We can write this matrix in the form Fig. 1. Definitions of polarizations states and incoming and outgoing light beams on a film.
⎛1 0⎞ rAB = a r ⎜ ⎟, ⎝ 0 zr ⎠
(7)
with
ar =
rBp rAp
(8)
rBs rAp . rAs rBp
(9)
and
zr =
⃑ equally, it Since ar affects both the s- and p-components of ErA does not change the relative phase between the two and the polarization state remains the same. On the other hand, the phase of zr (or arg( zr )) will have an effect on the polarization state. Note that parameter zr is related to the commonly used ellipsometry parameters Ψ and Δ defined as
rp rs
= tanΨeiΔ ,
(10)
and Fig. 2. Reflected electric fields at temperatures TA and TB in the p- and s-polarization basis.
⃑ the reflected electric field when the VO2 transmission. Let's call ErA ⃑ the corresponding field at film is at low temperature TA and ErB some higher temperature TB . In the Jones matrix formalism, the electric field in the basis of s- and p-polarizations is written
⎛ Ep ⎞ E ⃑ = ⎜⎜ ⎟⎟. ⎝ Es ⎠
(1)
The reflected field is related to the incident field via
⃑ = rAEo⃑ ErA
(2)
⃑ = rBEo⃑ ErB
(3)
where rA is the thin film's reflection matrix at temperature TA which takes the form
⎛ rAp 0 ⎞ rA = ⎜⎜ ⎟⎟, ⎝ 0 rAs ⎠
zr =
tanΨA i( ΔA −ΔB ) e . tanΨB
(11)
In the most general case, the effect of rAB would be to transform ⃑ into differently polarized but still an elliptically polarized field ErA ⃑ elliptical ErB . Indeed, since the phase difference between rs and rp in a VO2 layer is generally not a multiple of 2 π , the reflected field is generally not linearly polarized even if Eo⃑ is linearly polarized. But in order to attain the greatest contrasts of modulation using polarizers, it is best to operate with linear polarizations. To this end, a phase compensator (e.g. birefringent crystal) can be used to pre⃑ (or ErB ⃑ ) is linearly polarized, in which pare Eo⃑ in such a way that ErA case high-contrast modulation through polarizers would be possible. ⃑ is linearly polarized, two options are particularly proIf ErA mising: rotation of the linear polarization by some angle when arg( zr )=π (1 + 2m) with m = 0, ± 1, ± 2… and conversion to circular polarization when arg( zr ) = π (1/2+m) with =0, ± 1, ± 2… . We examine these two situations in more details in the following sections. 2.1. Rotation of linear polarization
(4)
where rAp and rAs are the (complex) coefficients of reflection of the VO2 layer for the p- and s-polarizations, respectively, at temperature TA . Matrix rB has the same structure but with coefficients for
⃑ in the s–p plane, thus defining γ the angle of Fig. 2 draws ErA A the field with respect to the s-axis. (Note that γA should not take values of 0 or π/2 as they represent polarization states purely in s or p, a condition that does not allow polarization control). A phase
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P. Cormier et al. / Optics Communications 382 (2017) 80–85
4
4 Cold [13] Cold [15] Cold [17] Cold [16] Hot [13] Hot [15] Hot [17] Hot [16]
3
3
Extinction coefficient
Refractive index
3.5
Cold [13] Cold [15] Cold [17] Cold [16] Hot [13] Hot [15] Hot [17] Hot [16]
3.5
2.5
2
2.5 2 1.5 1
1.5
1
0.5
0
500
1000
1500
2000
0
2500
0
500
Wavelength (nm)
1000
1500
2000
2500
Wavelength (nm)
Fig. 3. Reported complex refractive indices (real part left and imaginary part right) of VO2 thin films. Dashed lines correspond to the metallic state at temperatures above 68 °C (hot) and solid lines correspond to the insulating state (cold).
Fig. 4. Refractive index of VO2 at 1550 nm and 800 nm as a function of temperature. Arrows give indicate heating or cooling of the sample.
shift of arg( zr )=π results in zr =− zr and from Eqs. (5) and (7) we obtain
⎛ EAp ⎞ ⎛ EBp ⎞ ⎟ ⎜⎜ ⎟⎟ = a r ⎜ ⎜− z E ⎟ ⎝ EBs ⎠ r As ⎠ ⎝
tanγA =
zr .
(15)
(12) 2.2. Transformation between linear and circular polarization
so that
tanγB =
and with Eq. (13) we find the condition
EBp EBs
=
EAp − zr E
As
=−
1 tanγA zr
(13)
⃑ is to become circularly polarized from an initially linear If ErB ⃑ , it implies that the s- and p- components have the same amErA plitudes but phase-shifted by arg( zr ) = ±π /2:
from which the rotation angle ∆γ =γA − γB is obtained. The condition under which the rotation angle becomes ±90° is
EBp
tanγAtanγB= − 1
which leads to the following additional condition
(14)
EBs
=±i =
EAp z r EAs
=
1 tanγA zr
(16)
P. Cormier et al. / Optics Communications 382 (2017) 80–85
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Fig. 5. Films thickness and angles of incidence producing polarization switching in VO2 using data of Ref. [17]: A and B for phase switch of arg (zr )=π /2 and C and D for arg (zr ) = π .
tanγA= z r .
(17)
Transformation between linear to circular polarizations requires arg( zr ) = π (1/2+m) with m = 0, ± 1, ± 2… with the condition of Eq. (17) becoming
2.3. Theory in transmission
tanγA= zt .
The previous results can be extended to light beams transmitted through the VO2 film. In transmission, the switching matrix equivalent to that of Eq. (7) becomes
3. Calculations with reported data on VO2
⎛1 0⎞ tAB = at ⎜ ⎟ ⎝ 0 zt ⎠
(18)
where
at =
tBp tAp
(19)
tBs tAp . tAs tBp
(20)
and
zt =
Rotation of linear polarization by 90° is obtained when arg( zt )=π (1 + 2m) with m = 0, ± 1, ± 2…with the condition analogous to Eq. (15) becoming
tanγA =
zt .
(21)
(22)
The refractive indices of VO2 films have been measured in our group and in previous studies for both the metallic (hot) and insulating (cold) states of the material. Some of these results are plotted in Fig. 3, showing the real and imaginary parts of the refractive index from Refs. [13,15–17], all superimposed on the same graphs. Each study measured refractive indices over a different wavelength range, which explains why some curves extend farther than others. The data are very similar in all cases, with typical variations on the order of 75%. Fig. 4 shows a typical dependence of the refractive index of VO2 on temperature at two different wavelengths. The curves exhibit the usual hysteresis observed in VO2: during the cooling portion of the cycle, the critical temperature is shifted down by about 5 °C. In this study, hysteresis is not a factor because we consider optical effects at temperatures well outside the hysteresis region, e.g. 20 °C for insulating VO2 and 100 °C for metallic VO2.
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P. Cormier et al. / Optics Communications 382 (2017) 80–85
Fig. 6. Phase and amplitude parameters for reflection (A) and transmission (B) are calculated from Ref. [17] with a film of 180 nm in thickness. A phase switch of arg (zr )=π /2 is found in reflection but not in transmission.
Table 1 Examples of optimized parameters for applications at commonly used laser wavelengths in reflection. Wavelength (nm)
Film thickness (nm)
Angle of incidence (deg)
Angle γA (deg)
Effect
532 532 1064 1064 1550 1550
125 125 50 50 50 50
72.5 69.9 71.6 65.6 71.8 64.7
24.8 60.5 31.3 36.9 36.2 26.5
90° linear rotation linear-to-circular 90° linear rotation linear-to-circular 90° linear rotation linear-to-circular
From these data sets, and using Fresnel equations for a thin VO2 film on a glass substrate, reflection and transmission coefficients were calculated as well as parameters ar , t and zr , t . From these results, we investigated the conditions of film thickness, incidence angle and wavelength which made polarization switching possible. Using refractive indices of Ref. [17], we determined the conditions for which polarization switching was possible. For these calculations and the following, we assumed a substrate similar to glass with n ¼1.5. Figs. 5(A) and (B) reveal the combination of film thickness and angle of incidence leading to a transformation from linear to circular polarization during the film's phase transition. At
each wavelength, we chose random pairs of film thickness and angle of incidence and calculated the resulting optical phase difference: each dot on the graph represent a successful result, namely a case where arg (zr )=π /2 to within a 72% error (used as a criterion of success). Interference effects are periodic functions of space, and as a result, we see in Fig. 5(A) four bands of possible solutions for the film thickness. Note that each one of these bands has a finite thickness since another parameter, namely the incidence angle, is simultaneously adjusted. Since it is typically easier to create high-quality VO2 in the form of thinner films, generally speaking it would be optimal to choose a solution in the bottom band. As Fig. 5(B) shows, successful trials occur at
P. Cormier et al. / Optics Communications 382 (2017) 80–85
incidence angles between 60 and 80°. These incidence angles are near Brewster's angle of the air/VO2 interface. However, the concept of Brewster's angle (the angle of null reflection in p-polarization) is valid for a single interface and does not apply to a thin film. We studied polarization switching by VO2 in the case where the temperature goes from well below Tc to well above Tc, and we did not consider intermediate situations where temperature is varied over a lesser range. Although it would be interesting, it is not possible here because of lack of available data on refractive indices of VO2 as a function of temperature. Figs. 5(C) and (D) give the same results for rotation of linear polarization (when arg (zr ) = π )). Interestingly, the effect appears at wavelengths as short as 500 nm, in contrast with most applications of VO2 films, which are typically found at 1300 nm and above. The film may also be as thin as 100 nm or less. We also carried out calculations using refractive indices from other studies [15,16] as well as our own [13] and found similar results. Fig. 6 further explores the range of optical phase shifts for a specific film thickness of 180 nm, the film studied in Ref. [17]. With the thickness fixed, parameters arg (zr ), zr and ar are calculated as a function of wavelength and angle of incidence. Fig. 6(A) plots the phase shift amplitude and a line curve where the phase shift is constant at π/2. Within that area, centered at around 700 nm wavelength and 68° incidence angle, the phase shifts is highest with arg ( z r ≃π . Also in that area zr ≃1 so that the reflected po-
)
larization is at a convenient angle γA ≃ 45o and ar ≃1, implying that reflectance remains relatively constant during phase transition. This particular film would therefore be suitable for applications in the 600–800 nm spectral range, with simple angle-tuning from 66° to 72° to adjust to a specific wavelength. As seen in Fig. 5(B), the phase shifts are much less pronounced in reflection. To give an example of specific device applications, we chose the commonly-used wavelengths of the Nd:YAG laser (1064 nm), its second harmonic (532 nm) and the 1550 nm telecommunication wavelength. Table 1 summarizes the results for linear-circular and linear-rotator devices. In addition to having easy to achieve specifications in terms of film thickness, incidence angle and polarization angles, the device would work in reflection, thus avoiding dispersion problems in the substrate. On the other hand, unlike dielectric mirrors, which typically operate at 0° or 45° incidence angle, the proposed polarization rotator would operate at high incidence angles, near Brewster's angle. We performed calculations in transmission with the same data set with various thicknesses but did not find relative phase shifts larger than 0.2π rad. Since this is below the π/2 minimum level to achieve circular polarization, so we did not pursue this route further.
4. Conclusions Vanadium dioxide shows promises for light polarization
85
control applications using thin films. Because of unequal optical phase shifts in s- and p-polarizations during phase transition, the polarization state of reflected light can change considerably. The effect originates from large differences in refractive indices between the material's insulating and metallic states, and manifests itself in both reflection and transmission through thin films. We derived a theory to calculate phase shifts and its effect to the polarization and used reported measurements of refractive indices to show that films as thin as 50 nm could transform polarization from linear to circular (and vice-versa) or rotate linear polarization by 90°. This renders possible optical switches and devices many times thinner than optical means such as liquid crystals. Switching is also possible at wavelengths of 500 nm and above, making it one of the few applications of VO2 in the visible spectrum. Work is currently under way to demonstrate the effect experimentally.
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