We propose an optical wave plate using a metal nano-grid. The wave plate operates in reflection mode. A single-mode truncated mode-matching theory is presented as a general method to design such nano-grid wave plates with the desired phase difference between the reflected TM and TE polarizations. This analytical theory allows angled incidence calculations as well, and numerical results agree-well with comprehensive finite-difference time-domain electromagnetic simulations. Due to the subwavelength path-length, the reflective wave plate is expected to have improved broad-band functionality over existing zero-order transmissive wave plates, for which an example is provided. The proposed wave plate is simple and compact, and it is amenable to existing nanofabrication techniques. The reflective geometry is especially promising for applications including liquid-crystal displays and laser feedback experiments.
©2009 Optical Society of America
Polarization sensitive optical devices are essential for many experiments and devices. A wave plate is an optical device through which orthogonal linear polarizations receive a relative phase shift. For example, a quarter-wave plate (QWP) introduces an extra π/2 radian phase difference between the two polarizations. Most wave plates are made of birefringent materials [1, 2, 3, 4, 5] or artificial birefringence materials consisting of dielectric grids [6, 7]. In those birefringent components, the perpendicular electric field polarizations obtain a relative phase-shift as light propagates through the material.
Unlike a wave-plate, a polarizer is an optical device that transmits only one polarization. Wire-grid polarizers have long been used for long-wavelengths [8, 9]. Recently, nanofabrication has allowed for wire-grid polarizers that operate in the UV to near-IR region [6, 7, 10, 11, 12, 13, 14, 15, 16, 17]. To the best of the authors’ knowledge, no wave plates have been implemented or proposed using comparable metal nano-grid structures.
In this paper, a metal nano-grid structure with reflecting substrate is proposed as a wave plate operating in reflection mode. We present an analytical method to design this wave plate, which agrees well with comprehensive numerical simulations. As light reflects from the proposed structure, the TM polarization receives an extra phase over the TE polarization. It is shown that this phase-difference can be tuned by changing the incident angle for a fixed structure, or the depth of the grid for different structures. The metal nano-grid structure is simple and compact. Since the wave plate operates in reflection geometry, instead of the transmission geometry, we believe that it will find new uses as an optical element, for example, as an ultra-compact component in reflection geometry liquid-crystal displays , or as a component to provide polarization rotated feedback in experiments on lasers . Furthermore, we believe that the proposed wave plate may be readily fabricated with existing nanofabrication techniques [20, 21, 22].
2. Analytical theory for design
Figure 1 shows the nano-grid structure analyzed in the paper. It is a metal grid structure surrounded by dielectric background. The widths of the dielectric and metal are dd and dm, and both are significantly smaller than a wavelength of the incident light. The working principle of this nano-grid wave plate is as follows: the TE polarization is mostly reflected at the top of the grid (at the interface between regions A and B) because it is cutoff from propagation within the grid region B. The TM polarization, however, can penetrate into the grid region B and it is reflected at the bottom metal interface to region C. In this way, the TM polarization receives an extra phase-shift due to the extra length of propagation into the grid region.
To design a wave plate with the desired phase shift between the TE and the TM polarizations, the propagation constants β in region B are required, which enables calculating the phases of reflection and of propagation. The TM polarization propagates in the grid in the form of a coupled surface plasmon polariton (SPP) mode; this is a periodic version of the gap mode, for which there have been several investigations since 1969 . To calculate the propagation constant β of the SPPs in a metallic grid array, we match the boundary conditions between the metal layers and the air layers in the grid. The following dispersion relation is found to be solved for βTM:
for the TM polarization and for βTE:
for the TE polarization, where εm and εd are the relative permittivities of metal and dielectric, and is the wave vector of free space. From these relations, the desired propagation constants in the grid can be calculated for the TM SPP-like mode and the TE cutoff mode. In addition, the following equation is set up by letting βTE = 0 in Eq. 2:
and the cutoff wavelength for the TE mode can be calculated by solving for k 0 using this equation.
With the propagation constant βTM, we can determine the phase of propagation inside the grid for the TM polarization. In addition, we need to calculate the reflection coefficients of both polarizations. This is done by the mode-matching technique; however, a single-mode truncation is used for every region due to the extreme subwavelength period. Furthermore, for the TM case, it is a good approximation to neglect multiple reflections within the grid region. The validity of these approximations will be discussed in the example.
To illustrate the mode-matching procedure, the reflection calculation for the TM polarization at the top of the grid is presented. The same calculation was used for both polarizations and at each interface. As a general case, linearly polarized light is incident in the plane of the grid’s grooves at an angle of θi. The sum of the incident and reflected fields in the free-space region (region A in Fig. 1(b)) above the grid are given by:
where rTMt is the reflection coefficient of the TM mode, with the superscript t denoting “top”, ω is the angular frequency and μ 0 is the free-space permeability. In keeping with the single-mode truncation, the reflection coefficients are for the lowest order mode. The magnetic fields within the grid (region B in Fig. 1(b)) are:
for (dielectric region) and
for (metal region), where tTMt is the transmission coefficient of the TM mode into the grid, θt is the transmission angle into the grid region, and βTM is the propagation constant of the TM polarization. The hyperbolic cosine dependence of the field along the x-axis in the grooves comes from the coupled surface plasmons on each metal-dielectric interface. Applying the boundary condition at the metal-dielectric interface of the grid, with Hm = 1, we obtain:
It is noted that the y component of the propagation constant ky remains unchanged in regions A and B since the structure is uniform in the y direction, therefore the transmission angle θt is given by:
The electric field EB within the grid (region B) is found by applying Ampere’s law.
Now we apply the mode matching criteria at the interface between uniform dielectric (region A) and the grid (region B) by equating the transverse electric and magnetic field components:
The subscript ⊥ denotes the transverse components, that is only the x and the y components of the fields. To use the orthogonality of the fields, we take the product of both sides of Eq. 10 with the complex conjugate of the magnetic field in region A, H A,⊥ * and integrate over the region (i.e. the region between the mid-points of a groove and an adjacent metal wire):
Similarly, we take the complex conjugate on both sides of Eq. 11, cross product both sides with the electric field in region B, E B,⊥, and integrate:
Eqs. 12 and 13 can now be solved using the expressions of the fields derived in 5, 4, 6 and 7 to get the complex reflection coefficient rTMt. The same approach can be used to find the reflection coefficients rTEt of the TE polarization at the top of the grid and rTMb of the TM polarization at the bottom of the grid (the superscript b stands for “bottom”). At the bottom of the grid, the TM mode in the grid is matched to the exponentially decaying plane wave in bulk metal instead of to the propagating plane wave in dielectric.
With βTM, rTEt and rTMb available, the phase-difference between the TE and TM modes reflected from the reflective grid wave plate can be determined as:
where Eq. 9 was used for the transmission angle θt.
3. Example: wave plate design and comprehensive numerical calculations
3.1. Normal incidence
As an example, we designed a QWP for a wavelength of λ 0 = 826.4 nm at perpendicular incidence. The widths of the nano-grid and the grooves were chosen to be dm = 50 nm and dd = 100 nm. The reason for this choice is that this groove width is below cutoff for the TE polarization at the chosen wavelength, and nano-metallic wire-grid at this scale can be readily fabricated for demonstration purposes . We chose to use gold (Au) as the material, which had a relative permittivity εm = -28.74+2.010i at λ 0 = 826.4nm. We chose gold as an exemplary material for the proposed structure because gold has a small material loss for the chosen near-IR wavelength and it does not oxidize readily in air; however, the approach may be extended to other wavelengths and materials. The surrounding material was assumed to be air with εd = 1. The cutoff wavelength of the TE mode in the grooves was found by using Eq. 3 to be λ c = 226.6 nm, This is well below the operating wavelength. Using the approach described above, the reflection coefficients were calculated to be rTEt = 0.984/_223.5° for the TE polarization at the top of the grid, rTMb = 0.985\_ 202.8° for the TM polarization at the bottom of the grid and rtTM = 0.0409\_182.5° for the TM polarization at the bottom of the grid. As expected, both rTEt and rTMt were close to unity in magnitude, and rtTM was close to zero in magnitude. Therefore, the approximation to neglect multiple reflections within the grid region is suitable; however, it may be incorporated in a straightforward way using the usual Fabry-Perot results. For the resulting QWP phase difference between the reflected TE and TM polarizations to be Δϕ = 90° at an incident angle of θi = 0° (perpendicular incidence), the required depth of the grid was calculated to be h = 107.4nm by using Eq. 14.
Figure 2(a) shows the comprehensive numerical calculations of the TE and TM normalized electric field perpendicularly reflected from the QWP designed above using a commercially available finite-difference time-domain (FDTD) electromagnetic solver. The reflected TM wave lags the TE wave by a quarter of the wavelength. (Media 1) for TM polarization and (Media 2) for TE polarization show animations of the FDTD calculated fields. It is clear from the animations the TM polarization travels into the grid and is reflected at the bottom interface, whereas the TE polarization is cutoff in the grid and is reflected at the top interface.
The design was modified to build a half-wave plate (HWP) by changing the height of the grid structure, for which Δϕ = 180° was used in Eq. 14. It was found that the corresponding height of the grid required is h = 174.7nm. Note that this is less than double the QWP because of the phases of reflection. The corresponding FDTD calculated result of this HWP is shown in Fig. 2(b), and the reflected TM wave lags the TE wave by half a wavelength, as required.
3.2. Angled incidence
Figure 3 showed the results for angled incidence. While keeping the height of the grid constant at h = 107.4 nm, the FDTD calculations were carried out for three additional angles: θi = 15°,30°,45°. Clearly, the analytical theory agrees-well with the FDTD calculations for angled incidence as well. This agreement supports the suitability of the single-mode truncation in the theory for comparable subwavelength structures.
The metal nano-grid structure used in the proposed wave plate plays a different role compared to previous work on gratings. One dimensional corrugations on metal surface have been used to couple light into SPP modes (for example, Refs. [24, 25]). In such cases, the incident light needs to be tilted along the corrugation direction (the x-z plane in Fig. 1) in order to match the propagation constant between the incident light and the SPP mode. In our structure, however, the gratings were not used for phase-matching. The gratings were used to tune the relative retardation between the TM and the TE mode. For the same groove depth, the relative retardation can be tuned by tilting the incident light along the translationally-invariant direction (the y-z plane in Fig. 1). Tilting in the x-z plane gives no change to the relative retardation due to the extremely subwavelength period of the grating. To confirm the irrelevance of the relative retardation to tilting in the x-z plane, we repeated the simulations with a 45° angled incident beam in the x-z plane by using the QWP designed above. The result still showed the reflected TM wave lagged the TE wave by a quarter wavelength (result not shown here). Therefore, angle changes in the x-z plane play little role in the proposed subwavelength structure.
The metal nano-grid reflective wave plate designed in this paper has several promising applications. The reflection geometry can be used to build more compact devices. For example, certain reflective LCDs use a QWP followed by a reflector . For such devices, the reflective nano-grid wave plate can be used instead of having discrete waveplate and reflector components; thereby forming a more compact final device. Similarly, a QWP and a reflector were used in an experiment on polarization self-modulation of vertical-cavity surface-emitting lasers, where the feedback length determined the modulation rate . The proposed reflective QWP is more compact and therefore it allows for a shorter feedback length so that higher-frequency modulation can be explored.
The reflective nano-grid wave plate is naturally zero-order. The response of this wave plate to wavelength variation is expected to be better than the existing zero-order wave plates made of birefringent materials, due to the shorter beam path within the wave plate. For example, a polyimide transmissive HWP was demonstrated at 15 wavelengths in thickness . For a 5% operating wavelength change, there will be a 0.75λ change to the beam path of a polyimide HWP. By comparison, a smaller 0.025λ change to the beam path is possible for the reflective wave plate. Therefore, the wave plate can have a wider operating band, although the further analysis is required.
The angle tuning of the reflective nano-grid wave plate offers potential for an even wider wavelength-range of operation. For example, if the different wavelengths are angularly dispersed by a prism or a grating, they can be incident at different angles. Therefore, it may be possible to compensate for the dispersion described in the previous paragraph. Alternatively, at a single wavelength, it is possible to tune between a QWP and HWP by changing the angle of incidence.
To investigate on the tolerance to the structure dimension fluctuations in fabrication, we considered a number of systematic fabrication errors. We performed three individual simulations with the depth, the periodicity and the filling factor of the grooves each changed by 10% in turn. In each of the three cases, the changes in the relative retardation were 14.3%, 1.4% and 2.9%. As expected, the results indicated that the proposed device was tolerant to fabrication errors in the periodicity and the filling factor, but is less tolerant to errors in the depth. Non-systematic fabrication errors could be studied by simulating structures whose dimensions were randomly varied over several periods; however, this is beyond the scope of this paper. Nevertheless, the changes in the relative retardation introduced by systematic dimension fluctuations can be “repaired” in-situ by the angle tuning described above.
In this paper, we proposed an optical wave plate that operates in reflection geometry by using metal nano-grid, or equivalently, metal nanometric grooves in a metal surface. We presented an analytical theory that can be used to design the wave plate structure. The theory agreed well with comprehensive FDTD numerical calculations for an example structure, both for normal and angled incidence. The proposed wave plate configuration is compact and it provides a useful reflective geometry, with potential for reflective LCD displays and laser-feedback experiments. The angle tuning of the phase presents potential for dispersion compensation or wave plate phase tuning. Furthermore, the proposed structure is amenable to different nanofabrication methods that have been used previously to create wire-grid polarizers in the visible and to create grooves in metals for channel SPP studies.
References and links
1. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Birefringence in two-dimensional bulk photonic crystals applied to the construction of quarter waveplates,” Opt. Express 11, 125–133 (2003). [CrossRef] [PubMed]
2. D. R. Solli, C. F. McCormick, R. Y. Chiao, and J. M. Hickmann, “Experimental demonstration of photonic crystal waveplates,” Appl. Phys. Lett. 82, 1036–1038 (2003). [CrossRef]
3. Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, and H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994). [CrossRef]
4. E. M. Korenic, S. D. Jacobs, J. K. Houghton, A. Schmid, and F. Kreuzer, “Nematic polymer liquid-crystal wave plate for high power lasers at 1054-nm,” Appl. Opt. 33, 1889–1899 (1994). [CrossRef] [PubMed]
5. A. M. Radojevic, R. M. Osgood, M. Levy, A. Kumar, and H. Bakhru, “Zeroth-order half-wave plates of LiNbO3 for integrated optics applications at 1.55 mu m,” IEEE Photon. Technol. Lett. 12, 1653–1655 (2000). [CrossRef]
6. D. Kim and E. Sim, “Segmented coupled-wave analysis of a curved wire-grid polarizer,” J. Opt. Soc. Am. A 25, 558–565 (2008). [CrossRef]
7. P. Deguzman and G. Nordin, “Stacked subwavelength gratings as circular polarization filters,” Appl. Opt. 40, 5731–5737 (2001). [CrossRef]
8. J. B. Young, H. A. Graham, and E. W. Peterson, “Wire grid infrared polarizer,” Appl. Opt. 4, 1023–1026 (1965). [CrossRef]
9. P. K. Cheo and C. D. Bass, “Efficient wire-grid duplexer polarizer for CO2 lasers,” Appl. Phys. Lett. 18, 565–567 (1971). [CrossRef]
11. J. Wang, J. Deng, X. Deng, F. Liu, P. Sciortino, A. N. L. Chen, and A. Graham, “Innovative high-performance nanowire-grid polarizers and integrated isolators,” IEEE J. Sel. Top. Quantum Electron. 11, 241–253 (2005). [CrossRef]
12. J. Wang, F. Liu, and X. Deng, “Monolithically integrated circular polarizers with two-layer nano-gratings fabricated by imprint lithography,” J. Vac. Sci. Technol. B 23, 3164–3167 (2005). [CrossRef]
15. A. Vengurlekar, “Polarization dependence of optical properties of metallodielectric gratings with subwavelength grooves in classical and conical mounts,” J. Appl. Phys. 104, 023,109–1–023,109–8 (2008). [CrossRef]
16. H. Tamada, T. Doumuki, T. Yamaguchi, and S. Matsumoto, “Al wire-grid polarizer using the s-polarization resonance effect at the 0.8-mu m-wavelength band,” Opt. Lett. 22, 419–421 (1997). [CrossRef] [PubMed]
18. H. S. Cole and R. A. Kashnow, “New reflective dichroic liquid-crystal display device,” Appl. Phys. Lett. 30, 619–621 (1977). [CrossRef]
19. S. J. Jiang, Z. Q. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 63, 3545–3547 (1993). [CrossRef]
20. X. Liu, X. Deng, J. P. Sciortino, M. Buonanno, F. Walters, R. Varghese, J. Bacon, L. Chen, N. O’Brien, and J. J. Wang, “Large area, 38 nm half-pitch grating fabrication by using atomic spacer lithography from aluminum wire grids,” Nano Lett. 6, 2723–2727 (2006). [CrossRef] [PubMed]
21. J. Wang, L. Chen, X. Liu, P. Sciortino, F. Liu, F. Walters, and X. Deng, “30-nm-wide aluminum nanowire grid for ultrahigh contrast and transmittance polarizers made by UV-nanoimprint lithography,” Appl. Phys. Lett. 89, 141,105–1–141,105–3 (2006).
22. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046,802–1–046,802–4 (2005). [CrossRef]
23. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969). [CrossRef]
24. Z. M. Zhu and T. G. Brown, “Nonperturbative analysis of cross coupling in corrugated metal films,” J. Opt. Soc. Am. A 17, 1798–1806 (2000). [CrossRef]