1. Introduction

A wave-plate is an essential part of modern optics and photonics applications. In the long wavelength infrared (LWIR) regime, a quarter-wave plate (QWP) and a half-wave plate (HWP) can find applications in industrial CO₂-laser machining to control the polarization state for offering more uniform and efficient cutting [1]. A HWP can also be found in beam splitters for polarization-sensitive imaging applications [2, 3]. A conventional LWIR wave-plate is usually made by polished anisotropic dielectric materials such as Cadmium Thiogallate with thicknesses in the millimeter range including the mount [4]. The miniaturization of photonic devices and sensors has generated an interest for wave-plate integration which requires the development of thin wave-plates. In this regard, recent research in nano-metallic structures has provided a low cost method to fabricate thin wave-plates. Reflective QWPs and HWPs have been introduced in [5–8]; however, a QWP or a HWP working in the transmission mode can be more desirable. A classical transmitting single-layer meanderline QWP operating in the range 8μm –12μm has been reported in [9], but the transmitted power is rather low at 23%.

In this paper, we propose thin multilayer designs of a QWP and a HWP that are compatible with existing fabrication technology. The wave plates are designed based on microwave phase-shifter theory [10]. They operate in the long wavelength infrared (LWIR) region in the transmission mode with a design frequency of 30THz. High transmission and low reflection can be achieved over a reasonable bandwidth. One advantage of the proposed designs is that they are readily scalable to any frequency as long as the metal remains reasonably opaque. Moreover, the materials used in the QWP and the HWP designs are common dielectrics of ZnSe and YbF₃, and metallic layers made of gold. The QWP and the HWP designs with such materials have high laser-induced damage threshold (LIDT) [11] and may be used with high power CO₂ lasers at 10.6μm.

2. Theory and design

In [12], the 90° phase difference required for a QWP is achieved by using two perpendicular detuned dipoles in a periodic fashion. The longer dipole acts as an inductor and introduces a phase delay in the transmitted field, whereas the shorter dipole acts as a capacitor and introduces a phase advance. By adjusting the lengths of the two perpendicular dipoles, a 90° phase difference is achieved between two orthogonal polarizations. Some other wave-plate designs working in the optical regime, based on single layer nano-metallic structures utilizing strips, slits and corrugated gratings are demonstrated in [12–17]. To scale those single layer metallic periodic structures into the LWIR regime, we can treat the layers as frequency selective surfaces (FSS). A FSS is equivalent to a filter and the transmission through it can be described by a corresponding transfer function H(s) [18]. Such an AAS of detuned dipoles has a transfer function equivalent to a FSS modeled as a series resistor-inductor-capacitor (RLC) band-reject filter. An AAS with two perpendicular detuned dipoles arranged in a cross can be modeled with two separate transfer functions, one for each polarization. To introduce a 90° phase difference between two polarizations, transfer function H_x(s) for the X-polarization has to introduce a +45° degree phase shift and transfer function H_y(s) for the Y-polarization has to introduce a −45° phase shift, as shown in Fig. 1(a). H_x(s) and H_y(s) are band-reject filters with different rejection frequencies. The magnitudes of H_x(s) and H_y(s) are exactly $1/\sqrt{2}$ when the corresponding phases are ±45° [19], indicating a total 50% of transmitted power which constitutes the upper bound for this type of single-layer AAS based QWP. Similarly, for a HWP, a ±90° phase shift is required for the two transfer functions, as shown in Fig. 1(b). However, as the phase difference approaches 180°, the magnitudes of both transfer functions approach zero. In fact, it is not possible to achieve exactly 180° phase difference with a thin single layer AAS made of cross dipoles. A similar statement also applies to single layer cross-slot designs which can be modeled as band-pass filters. The transmission limitation of a single layer QWP or HWP can be overcame by using multilayer designs [18]. This multilayer concept has recently been employed in [20] to overcome this transmission limitation in the optical regime.

 

Fig. 1 Magnitude and phase of the transmitted fields of a QWP and a HWP made from single layer AAS of cross dipoles. The intersection of the magnitude curves determines the ideal operating frequency since the same power is transmitted for both polarizations. (a) For a QWP, to achieve a 90° phase difference at the operating frequency, at most 50% power is transmitted for both polarizations. (b) For a HWP, as the phase difference approaches 180°, the transmission magnitudes approach zero.

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2.1. Design and results of a quarter-wave plate

The transmission through a two-layer AAS from the input to the output plane can be modeled as a transmission line loaded with lumped reactances as shown in Fig. 2, where the dielectric is modeled as a transmission line having an impedance of Z₀ and the AAS are modeled as shunt reactances. A single AAS layer provides a reactance jB_x for the X-polarized field and a reactance jB_y for the Y-polarized field. It should be pointed out that in the model of Fig. 2, the corresponding transmission-line impedances should be adjusted according to the angle of incidence and polarization state. For simplicity, we only consider here the case of normal incidence. For unity transmission, the phase difference Δϕ between the transmitted E_x and E_y fields can be computed by Eq. (1) [10], with B_x and B_y normalized with respect to Z₀.

By assuming that reactance B_x gives a +45° phase shift and reactance B_y gives a −45° phase shift, the normalized values of B_x and B_y are calculated to be +4 and −4. In free space, B_x is 1500Ω and B_y is −1500Ω. We can relate B_x and B_y to the self-reactance of a dipole and approximate the corresponding dipole lengths from [21]. The dipole lengths needed are approximated to be 0.75λ and 0.25λ respectively. Therefore, by using an array of cross dipoles of lengths 0.75λ and 0.25λ for each layer, a QWP with transmission close to unity can be implemented.

 

Fig. 2 The circuit model for a normally incident field E₀ containing both X and Y polarizations with equal magnitude passing through a 2-layer AAS. The dielectric is modeled as a transmission line with impedance Z₀ and the AAS layers are modeled as shunt reactances. The AAS layer has an admittance jB_x for the X-polarized field and jB_y for the Y-polarized field. Depending on the values of jB_x and jB_y, a 90° differential phase shift can be achieved with a combined transmission magnitude of unity.

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Figure 3 shows a double layer AAS design with elliptical patches. The working principle of the patch is the same as for the cross dipoles: Current flow along the two axes L_x and L_y resembles current flowing along a long dipole and a short dipole. The elliptical patch design is inspired from the hemispherical dipole as in [21] for increasing the bandwidth. The patch is made of gold of 50nm thickness. The behavior of the gold is characterized as a Drude-Lorentz model with parameters described in [22]. The major axis L_x is 3.3μm and the minor axis L_y is 1.3μm. The bottom gold layer sits on a common dielectric substrate intended for LWIR applications such as ZnSe with ε₁ = 5.8 [23]. Between the bottom and top layer gold, a 1.1μm ZnSe thin film which acts as a quarter-wavelength transmission line at the design frequency of 30THz can be deposited with the CVD method [24]. The lengths of L_x and L_y correspond to 0.77λ and 0.3λ in the dielectric respectively, which match well to the approximated cross dipole lengths of 0.75λ and 0.25λ computed in the previous section. A 1.6μm quarter-wave transformer made of YbF₃ with ε₂ = 2.3 [25] is deposited on top of another 0.2μm ZnSe to match to free space. The total thickness of the design is 2.9μm.

 

Fig. 3 A unit cell design of the proposed QWP using double layer elliptical patches made of gold with dielectrics of ZnSe with ε₁ = 5.8 and YbF₃ with ε₂ = 2.3.

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The simulated result of the transmitted field for each polarization is shown in Fig. 4. The transmission magnitudes of both the X and Y components are about 0.95, leading to 90% transmitted power. The axial ratio (AR) and phase difference between the two polarizations are shown in Fig. 5. The bandwidth of the QWP can be defined by satisfying both the AR limit and the phase difference limit which are 0.8 – 1.2 and 80° – 100° respectively. This bandwidth of the QWP is 8.6μm – 10.4μm, or 19% of the design frequency.

 

Fig. 4 Magnitude and phase of the transmitted fields in the X and Y polarizations.

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Fig. 5 The axial ratio and phase difference between the X and Y polarizations. The bandwidth of the QWP is 8.6μm–10.4μm, which is the frequency range where both AR and phase difference curves stay within the gray region.

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2.2. Design and results of a half-wave plate

For a HWP design, the required phase difference Δϕ is π in Eq. (1). However, the arctan function has a periodicity of π, which leads to the same value for jB_x and jB_y. As a result, it is not possible to design a HWP of unity transmission using 2 layers. This problem can be solved by cascading one more layer [20,26]. By adding one more quarter-wavelength dielectric layer and gold elliptical patches, the HWP can be designed with good transmission.

Figure 6 shows a triple layer AAS design with elliptical patches. The major axis L_x of the elliptical patch remains 3.3μm but the minor axis L_y is changed slightly to 1.45μm for the required 180° phase difference. The total thickness of the HWP is 3.8μm. The transmission of each polarization is shown in Fig. 7. The transmission magnitudes at the operating frequency are about 0.9 leading to 80% of transmitted power. The phase difference is 180° as desired. We can define the bandwidth of this HWP by satisfying both the AR and the phase difference limits which are 0.8 – 1.2 and 160° – 200° respectively. As shown in Fig. 8, this bandwidth of the HWP is from 9.2μm – 11μm.

 

Fig. 6 A unit cell design of a HWP using three layers of gold elliptical patches.

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Fig. 7 Magnitude and phase of the transmitted fields in the X and Y polarizations.

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Fig. 8 The axial ratio and phase difference between the X and Y polarizations. The bandwidth of this HWP is 9.2μm – 11μm, or 17% of the operating frequency.

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Both the QWP and the HWP designs presented utilize gold layers with the same patterning (homogenous loading) and fixed interlayer spacing of λ/4. The specific choice of λ/4 is to achieve the widest bandwidth possible, since in this case the incident and reflected waves are out of phase and cancel each together. This is consistent with [10] which shows that indeed the bandwidth is widest when the interspacing is λ/4 between the loading elements. As the QWP or the HWP is sandwiched between a glass substrate and air with different impedances, a quarter-wave transformer is required to achieve good matching. It may be possible that with inhomogeneous loading (different metallization patterning) and different dielectric interspace materials the quarter-wave matching could be eliminated. This is even more likely for more broadband designs that contain more layers, as it is well known from microwave filter theory. However, in this example of only having 2 or 3 layers, we used a homogenous loading for simplicity and ease of fabrication.

3. Conclusion

In this paper, we have presented a thin quarter-wave plate and a half-wave plate design based on microwave phase-shifter theory. Both wave-plates operate in the long wavelength infrared region at 30THz with 90% and 80% transmitted power respectively. Good axial ratios and phase differences can be maintained over bandwidths of 19% and 17% for the QWP and HWP respectively. The advantages of the proposed plates are that (a) they are compatible with current planar fabrication technology and potentially are scalable up to the near IR regime where metal remains reasonably opaque (b) they are typically less than one wavelength thick (c) they offer high transmission since they are well matched due to their multilayers structure (d) they exhibit a good bandwidth and (e) gold layers with ZnSe and YbF₃ dielectrics have low absorption in the far infrared region which leads to high laser-induced damage threshold. Hence QWP and HWP designs with such materials can be potentially used with high power CO₂ lasers. In addition, it should be pointed out that larger bandwidths can be achieved by cascading more layers. Finally, one could envision 2D inhomogeneous metallization of elliptical patches in the transverse plane for general wavefront manipulation [27].