Polarity-variable birefringence on hyperlens structure

Y.J. Hung; M. H. Shih; J.H. Liou; J.Y. Tai

doi:10.1364/OE.18.027606

1. Introduction

The optical superlensing effect has been caused attentions since J. Pendry reports the negative lens concept [1]. Various kinds of superlens have been discussed [2–5]. It is interesting to note that the two dimensional hyperlens structure proposed by Narimanov et.al. in 2006 has hyperbolic dispersion characteristic [2]. This hyperlens can carry very large k-vector and achieve super-resolution effect by tolerating the infinite propagation wave-vectors. Except the imaging function, the proposed hyperlens structure acts like a uniaxial plasmonic crystal. A more detailed explanation can be referred to [6–10]. The Dyakonov states on uniaxial crystal film [6–8] were adopted to explain the super-resolution effect demonstrated by Smolyaninov [4] more specifically. The 1D grating structures inherit birefringence property and surface plasmon polariton (SPP) experiences different in-plane dielectric constants. Here we measured the birefringence Δn^eff of the 1D straitened hyperlens structure as shown in Fig. 1 .(a). A similar analysis on anomalous phase from periodic birefringent material without SPP has been discussed in [11,12].

Fig. 1 (a) Structure of the sample with Au thickness 50nm and PMMA thickness 150nm. (b) Grating pitch is 500nm. Polarization of the incident light is rotated along x-y plane where z is the direction of light incidence.

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The interesting point of our birefringence observation is that Δn^eff is usually either positive or negative once crystal orientation is fixed. Here we report Δn^eff can change from positive to negative simply by changing the operation wavelength without changing other parameters like periodicity and grating orientation. If this crystal is made from Poly(methyl methacrylate) (PMMA) material, there should be no any chromatic dispersion which causes Δn^eff flip sign unless SPP waves at the interface between the Au film and the dielectric gratings play a role inside. An elliptical index sphere is used to describe uniaxial crystal material and it is naturally to believe that it should have fixed long and short axes once the grating orientation is determined. The expression of the dielectric constants on a finely dielectric layered structure can be found in [13]. It shows that the roles of long axis and short axis never swap when the dielectric constants of both layers are positive.

2. Experiment

PMMA stripe gratings with 150nm thickness have been patterned on a 50nm thick gold film on a cover glass substrate. The Au film is intact under the PMMA gratings which periodicity is 500nm as shown in Fig. 1(a). 515nm Ar laser light is normally incident from the substrate side with polarization angle θ varied from 0° to 180°. TM and TE modes are defined as E-field along x and y axes respectively as shown in Fig. 1(b).

Some different PMMA stripe widths with same periodicity have been fabricated and shown in Fig. 2 . The optical measurement setup is shown in Fig. 3 . The open aperture (A) is around 2mm*2mm placed behind the objective lens. The magnified image has been projected onto the photo-detector and no other scattering light from non-grating area was collected. In order to get the four Stokes parameters, we use a quarter wave plate, half wave plate and polarizers to get the following operations:

S_{0} = I_{0} + I_{90}

S_{1} = I_{0} - I_{90}

S_{2} = I_{45} - I_{135}

S_{3} = I_{R H C} - I_{L H C}

(S₁, S₂, S₃) is the (x,y,z) position of the transmitted signal on the Poincare sphere. RHC and LHC present right-hand circular and left-hand circular polarizations, respectively. I_x is the optical intensity of polarization at x-degree. The polarization state of a light can be determined after the three orthogonal indexes (S₁, S₂, S₃) being determined.

Fig. 2 PMMA width: Air width (a) 1:1 (b) 1:2 (c) 1:3.

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Fig. 3 Optical setup for Poincare sphere measurement. λ/2: half wave-plate, P₁: polarizer, Obj: 20X objective, λ/4: quarter wave-plate, P₂: analyzer, D: detector.

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Referring to Fig. 1(b), suppose the intrinsic polarization of the laser output is horizontal (θ = 0°), which is expressed as θ(Ε_laser) = 0°. If we want to check what the response of the sample is when the polarization of the incident light E_in at the sample surface is rotated to θ = 15° (which is expressed as θ(Ε_in) = 15°), the optical elements before the sample should be arranged as θ(Ε_laser) = 0°, θ(λ/2−fast axis) = 7.5°, and θ(P₁ transmission axis) = 15°. This gives us θ(Ε_in) = 15°. The λ/2 wave plate is like a mirror which flips the polarization of the laser output from 0° to 15° without losing optical power. P₁ is used to guarantee that the polarization of the light is the same as what we expected. The rotational angles of the analysis elements after the sample are arranged as shown in Table 1 . I_RHC is measured when the analyzer P₂ is at θ(P₂) = θ(E_in) + 135° while I_LHC is acquired when P₂ is at θ(P₂) = θ(E_in) + 45°. When the relative angle between the fast axis of λ/4 wave plate and P₂ is fixed at 135°, this optical set λ/4 -P₂ should be viewed as a part which can completely block out LHC polarization light. In other words, when λ/4 -P₂ has 45° difference, the optical set is orthogonal to RHC polarization. The detailed polarization measurement can be found in [14] or any textbooks related to Jones vectors.

Table 1. The orientations of the analysis components λ/4 and P₂ when θ(Ε_in) = 15°.

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In the measurement setup and above descriptions, we actually fix the sample and rotate the coordinates of the input and output optical components simultaneously in counterclockwise direction. It is same as the sample being rotated 15° clockwise. In fact, the data points are taken with every 15° rotation of the optical elements from 0° to 180°.

3. Results and discussions

The degree of polarization (DOP) is (S₁+S₂+S₃)/S₀. In our measurement, DOP ranges from 0.98 to 1.0 which shows that the sample is a linear crystal. If a crystal is isotropic, its Poincare path will just stick to the same point and does not move due to no birefringence. In our case, if the sample is air, the Poincare path will just stay at (1,0,0) after we took the data from 0° to 180°. To understand further what the Poincare sphere represents, we first look at the Stokes parameters of an elliptical polarized light. The elliptical light has E field components on x and y axis:

E_{x}^{} = E_{x}^{0} \cos (ω t - k z - δ_{x})

E_{y}^{} = E_{y}^{0} \cos (ω t - k z - δ_{y})

The definitions of the four Stokes parameters are:

S_{0} = {(E_{x}^{0})}^{2} + {(E_{y}^{0})}^{2}

S_{1} = {(E_{x}^{0})}^{2} - {(E_{y}^{0})}^{2}

S_{2} = 2 (E_{x}^{0}) (E_{y}^{0}) \cos (δ_{y} - δ_{x})

S_{3} = 2 (E_{x}^{0}) (E_{y}^{0}) \sin (δ_{y} - δ_{x})

The handness of the polarization is determined by the sign of phase difference δ_y-δ_x between E_y and E_x. The ellipticity is determined by the ratio of E_x to E_y. By calculating the four Stokes parameters, the polarization state of the light can be determined and marked on the Poincare sphere surface as shown in Fig. 4(a) . Each position represents different handness, phase difference, and ellipticity. To illustrate further, Fig. 4(b) shows the Poincare paths for λ/8, λ/4, and λ/2 wave plates by rotating them every 15° from 0° to 90°. If the wave plate provides enough phase difference, the locus of the light can circle around the equator; otherwise, it will just go around a loop. If it is λ/4 wave plate, the light can reach RHC polarization point at θ =45°.

Fig. 4 (a) The corresponding polarization states on the Poincare sphere (b) The Poincare paths for λ/8, λ/4, and λ/2 wave plates by rotating them every 15° from 0° to 90°.

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The measurement on polarization state of the transmission light through the PMMA surface gratings has been depicted in Fig. 5 . It is interesting to note that δ_y-δ_x(=Δn^eff*k_o*d) flips signs when λ changes from 515nm to 633nm on Au sample. In our case, δ_y-δ_x > 0 means right-hand elliptical polarization. Four conditions in Fig. 5 all show birefringence responses. It is interesting to note that the birefringence of the PMMA gratings on Au substrate is more eminent than that on ITO substrates. There is still some birefringence effect (with same polarity) on ITO substrate as shown in Fig. 5 (c) and (d). Light at shorter wavelength shows stronger Δn^eff which can be explained by the band behavior of 1D periodic layered structure [15].

Fig. 5 Poincare sphere plot of PMMA surface gratings on different substrates with different width ratio (a) Au film substrate, 515nm, (b) Au substrate, 633nm(c) ITO substrate, 515nm, (d) ITO substrate, 633nm.

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To analyze the system in an easier way, we fix the coordinates of the optical elements but rotate the sample gratings in clockwise direction. Assume the matrix of our PMMA surface gratings is ABCD matrix. The matrix expressions for our system setup are as follows:

Input P 1 = [\begin{matrix} 1 \\ 0 \end{matrix}]

Sample matrix W(θ) = [\begin{matrix} \cos (θ) & \sin (θ) \\ - \sin (θ) & \cos (θ) \end{matrix}] [\begin{matrix} A & B \\ C & D \end{matrix}] [\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}]

Analyzer P 2 (ψ) = [\begin{matrix} \cos (ψ) & \sin (ψ) \\ - \sin (ψ) & \cos (ψ) \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} \cos (ψ) & - \sin (ψ) \\ \sin (ψ) & \cos (ψ) \end{matrix}] = [\begin{matrix} \cos^{2} (ψ) & - \sin (ψ) \cos (ψ) \\ - \sin (ψ) \cos (ψ) & \sin^{2} (ψ) \end{matrix}]

QWP = [\begin{matrix} e^{jπ/4} & o \\ 0 & e^{-jπ/4} \end{matrix}] fast axis on \hat{x} axis with e^{j (ω t - k z)} convention

θ and ψ are the rotational angles of the sample and the analyzer P₂ with respect to the polarizer P₁. For example, if the sample is rotated θ degrees clockwise, the intensity of the right-hand circular polarization light can be measured as:

Analyzer P₂(135°)*QWP*Sample (-θ) *Input P₁(0°), i.e.

I_{R H C} = [\begin{matrix} \cos^{2} (135) & - \sin (135) \cos (135) \\ - \sin (135) \cos (135) & \sin^{2} (135) \end{matrix}] [\begin{matrix} e^{j π / 4} & o \\ 0 & e^{- j π / 4} \end{matrix}] * W (- θ) * [\begin{matrix} 1 \\ 0 \end{matrix}]

For simplicity, Fig. 5 just shows the data with the sample rotated from 0° to 90°. Table 2 shows the fitting parameters of the ABCD sample matrix. The mean deviation of the 7 data points is expressed as: $\sum_{i = 1}^{i = 7} {[{(S_{1} - S_{1}^{'})}^{2} + {(S_{2} - S_{2}^{'})}^{2} + {(S_{3} - S_{3}^{'})}^{2}]}^{1 / 2} / 7$ . For better understanding, the off-diagonal coefficients are set zero. Except the 1:1 case on Au substrate, others show little birefringence. At 515nm, the amplitudes of A and D coefficients on 1:1 Au substrate are different. It means that except the phase difference along x and y axis, the scattering strength is also different on both axes.

Table 2. Fitting data of ABCD matrix of the sample element.

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From [6], when the operation wavelength is at 633nm, the dispersion curve of the Dyakonov SPP waves is a closed ellipse, just like a normal dispersion curve on a layer of uniaxial crystal film. But the mode fields are Dyakonov SPP waves with wave vectors defined as $q_{| |}$ and $q_{⊥}$ . In Fig. 1 (b), $q_{| |}$ means the wave vector is along x-axis. The ellipse has specific long and short axes. The polarization states of the transmission light are like the ensemble of the Dyakonov SPP waves excited on the grating structure. These SPP waves get scattered out in z-direction and the output phase we measured in z-direction is the combination of in-plane waves E_x and E_y with SPP wave vectors $(q_{| |}, q_{⊥})$ . On average, $(q_{| |} - q_{⊥})$ is almost zero if the dispersion curve is a closed circle. Figure 6(a) shows that if the phase difference is small, the closed dispersion curve in center is almost a circle. We assume the incident light excites $(q_{| |}, q_{⊥})$ in the first quadrant in Fig. 6 due to no perfect illumination on the sample. With the aid of Au film, the anisotropy has been enhanced as shown in Fig. 5 (a) and (b). According to [6], the anisotropy at 633nm shown in Fig. 5(b) should come from the existence of both hyperbolic dispersion and circular dispersion curves as is shown in Fig. 6(a). In Fig. 5(a) the anisotropy comes from the hyperbolic dispersion curve only and it is shown in Fig. 6(b). Our measurement in Fig. 5(a) and (b) show that the roles of the long and short axes of the ellipse swap when wavelength is tuned from 515nm to 633nm. That means Δn^eff*k_o*d changed sign or the handness of the sample changed due to wavelength change. It also means that the sign of the total integral of $(q_{| |} - q_{⊥})$ at 515nm is opposite to that at 633nm.

Fig. 6 The dispersion curves of the 1D PMMA gratings on Au film according to our experimental data. (a) Δn^eff*k_o*d ~-2π/16 at 633nm. (b) Δn^eff*k_o*d ~2π/7 at 515nm.

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It can be deducted that the anisotropy of the hyperlens system demonstrated in [4] is much more effective at 515nm as well as the cone of the energy flow is much well-defined compared to the device at 633nm. In [4], the superresolution is achieved when the energy propagates along x-axis. With different width ratio between air gap and PMMA stripe, Δn^eff changes and large Δn^eff can be achieved at 1:1 in our demonstration. In fact, maximum Δn^eff can be determined by judging from the band diagrams of this system. If the operation wavelength is tuned, slight change in Δn^eff will result in different ellipse in Fig. 6(b). To get maximum ellipticity, the optimization on the width ratio might be needed. According to our measurement, 1:1 width ratio is good for imaging function in [4]. With larger index contrast between stripes and air, the practical superresolution limit can be pushed further.

4. Conclusion

A variable optical birefringence effect on surface gratings has been observed. The refractive index difference Δn^eff of the uniaxial crystal-like grating covers positive and negative regions by changing the operation wavelength. The superresolution effect in [4] and the Dyakonov states in [6] have been discussed. The measured birefringence suggests that when PMMA width: Air width =1:1, we can get large birefringence from our sample. The resolution capability can be improved.

Acknowledgments

We thank Professor Yi-Jen Chiu for the assistance on Au film deposition, Professor Tsung-Hsien Lin for polarimetry measurement discussion. Both are from Dept. of Photonics, National Sun Yat-sen University. We thank Professor Christopher C. Davis from University of Maryland College Park for suggestions on polarimetry measurement. This work is supported by National Science Council of the Republic of China (Taiwan), (Contract No. NSC 99-2112-M-100-001-MY3)

References and links

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4. I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]

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7. R. Li, C. Cheng, F. F. Ren, J. Chen, Y. X. Fan, J. Ding, and H. T. Wang, “Hybridized surface plasmon polaritons at an interface between a metal and a uniaxial crystal,” Appl. Phys. Lett. 92(14), 141115 (2008). [CrossRef]

8. M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714 (1988).

9. A. Drezet, C. Genet, and T. W. Ebbesen, “Miniature plasmonic wave plates,” Phys. Rev. Lett. 101(4), 043902 (2008). [CrossRef] [PubMed]

10. L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96(4), 041112 (2010). [CrossRef]

11. A. Mandatori, C. Sibilia, M. Bertolotti, S. Zhukovsky, J. W. Haus, and M. Scalora, “Anomalous phase in one-dimensional, multilayer, periodic structures with birefringent materials,” Phys. Rev. B 70(16), 165107 (2004). [CrossRef]

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14. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999), pp. 22–41.

15. P. Yeh, Optical waves in layered media (Wiley, 1991), pp. 118–138.

Orientation angle θ	I₀	I₉₀	I₄₅	I₁₃₅	I_RHC	I_LHC
λ/4 fast axis	absent	absent	absent	absent	$15^{°}$	$15^{°}$
P₂ transmission axis	$15^{°}$	$105^{°}$	$60^{°}$	$150^{°}$	$150^{°}$	$60^{°}$

Λ	515nm			633nm			515nm
Substrate width ratio	Au 2:1	Au 1:1	Au 1:2	Au 2:1	Au 1:1	Au 1:2	ITO 1:1
A	1.7 $e^{ j \frac{π}{30}}$	2.5 $e^{ j \frac{π}{7}}$	3.5 $e^{ j \frac{π}{32}}$	2.0 $e^{ - j \frac{π}{33}}$	2.0 $e^{ - j \frac{π}{16}}$	1.8 $e^{ - j \frac{π}{70}}$	2.0 $e^{ - j \frac{π}{82}}$
B	0	0	0	0	0	0	0
C	0	0	0	0	0	0	0
D	2.0 $e^{ - j \frac{π}{30}}$	1.1 $e^{ - j \frac{π}{7}}$	1.1 $e^{ - j \frac{π}{32}}$	1.9 $e^{ j \frac{π}{33}}$	1.8 $e^{ j \frac{π}{16}}$	2.0 $e^{ j \frac{π}{70}}$	1.4 $e^{ j \frac{π}{82}}$
$\sum_{j = 1}^{j = 7} \sqrt{Δ S_{j}^{2}} / N$	0.0111	0.0522	0.0864	0.0314	0.0419	0.0304	0.0279

Orientation angle θ	I₀	I₉₀	I₄₅	I₁₃₅	I_RHC	I_LHC
λ/4 fast axis	absent	absent	absent	absent	$15^{°}$	$15^{°}$
P₂ transmission axis	$15^{°}$	$105^{°}$	$60^{°}$	$150^{°}$	$150^{°}$	$60^{°}$

Λ	515nm			633nm			515nm
Substrate width ratio	Au 2:1	Au 1:1	Au 1:2	Au 2:1	Au 1:1	Au 1:2	ITO 1:1
A	1.7 $e^{ j \frac{π}{30}}$	2.5 $e^{ j \frac{π}{7}}$	3.5 $e^{ j \frac{π}{32}}$	2.0 $e^{ - j \frac{π}{33}}$	2.0 $e^{ - j \frac{π}{16}}$	1.8 $e^{ - j \frac{π}{70}}$	2.0 $e^{ - j \frac{π}{82}}$
B	0	0	0	0	0	0	0
C	0	0	0	0	0	0	0
D	2.0 $e^{ - j \frac{π}{30}}$	1.1 $e^{ - j \frac{π}{7}}$	1.1 $e^{ - j \frac{π}{32}}$	1.9 $e^{ j \frac{π}{33}}$	1.8 $e^{ j \frac{π}{16}}$	2.0 $e^{ j \frac{π}{70}}$	1.4 $e^{ j \frac{π}{82}}$
$\sum_{j = 1}^{j = 7} \sqrt{Δ S_{j}^{2}} / N$	0.0111	0.0522	0.0864	0.0314	0.0419	0.0304	0.0279

Polarity-variable birefringence on hyperlens structure

Abstract

1. Introduction

2. Experiment

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Tables (2)

Equations (15)

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