## Wire-grid diffraction gratings used as polarizing beam splitter for visible light and applied in liquid crystal on silicon

Optics Express, Vol. 13, Issue 7, pp. 2303-2320 (2005)

http://dx.doi.org/10.1364/OPEX.13.002303

Acrobat PDF (1024 KB)

### Abstract

The application of wire grid polarizers as efficient polarizing beam splitters for visible light is studied. The large differences between the transmissivity for different polarizations are explained qualitatively by using the theory of metallic wave guides. The results of rigorous calculations obtained by using the finite element method are compared with experiments for both classical and conical mount. Furthermore the application of wire-grid polarizers in liquid crystal on silicon display systems is considered.

© 2005 Optical Society of America

## 1. Introduction

*classical mount*. According to our coordinate system (see Fig. 2),

*ϕ*=90° for this case. When the incident light is linearly polarized such that the electric vector is parallel to the wires (or grooves) of the WGP, the light is almost totally reflected as if the WGP were a flat metallic mirror. This state of polarization is called s-polarization because the electric field is perpendicular (Senkrecht) to the plane of incidence through the incident wave vector and the

*z*-axis in Fig. 1. When the incident light is linearly polarized such that the electric field is perpendicular to the wires, most of the light is transmitted. This state of polarization is called p-polarization because the electric vector is parallel to the plane of incidence. It follows that when the incident light is unpolarized, the reflected light is predominantly s-polarized whereas the transmitted light is predominantly p-polarized. The WGP can thus be used as a polarizer. As such it is well known in applications in the infrared part of the spectrum. Using advances in nano fabrication technology, periodic metallic gratings with periods smaller than the wavelength are now finding applications in optics. A WGP has higher contrast and is less expensive than a conventional MacNeille polarizing beam splitter (PBS) based on thin layers on prismatic glass. Therefore, a WGP is a good alternative PBS in many applications such as projection systems in liquid crystal on silicon (LCoS) displays [1

1. Clark Pentico, Eric Gardner, Douglas Hansen, and Ray Perkins, “New, high performance, durable polarizers for projection displays,” SID 01 Digest1287–1289 (2001). [CrossRef]

*ϕ*is arbitrary. This case will be called the

*conical mount*as shown in Fig. 2(b). In this case, the s- and p-polarizations are coupled then the grating can not be used as a beam splitter.

3. X.J. Yu and H.S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. **93**, 4407–4412 (2003). [CrossRef]

4. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. **11**, 235–241 (1985). [CrossRef]

5. L. Li, “Multilayer diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A **11**, 2816–2828 (1994). [CrossRef]

6. M.G. Moharam and T.K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A **72**, 1385–1392 (1982). [CrossRef]

7. J.B. Judkins and R.W. Ziolkovski, “Finite-difference time-domain modeling of nonperfectly conduction metallic thinfilm gratings,” J. Opt. Soc. Am. A **12**, 1974–1983 (1995). [CrossRef]

8. H.P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM J. Numer. Anal. **28**, 697–710 (1991). [CrossRef]

7. J.B. Judkins and R.W. Ziolkovski, “Finite-difference time-domain modeling of nonperfectly conduction metallic thinfilm gratings,” J. Opt. Soc. Am. A **12**, 1974–1983 (1995). [CrossRef]

3. X.J. Yu and H.S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. **93**, 4407–4412 (2003). [CrossRef]

9. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. **40**, 553–573 (1993). [CrossRef]

3. X.J. Yu and H.S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. **93**, 4407–4412 (2003). [CrossRef]

*ϕ*<15°, the WGP is surprisingly efficient.

## 2. Qualitative analysis

### 2.1. Plasmons

*z*-axis is normal to the interface, the wave vector of the plasmon wave inside the dielectric is given by [10], [11

11. J.M. Brok and H.P. Urbach, “Rigorous model of the scattering of a focused spot by a grating and its application in optical recording,” J. Opt. Soc. Am. A **2**, 256–272 (2003). [CrossRef]

*n*is the refractive index of the dielectric and

*k*

_{0}is the wave number in a vacuum. In fact,

*k*

_{x}and

*k*

_{y}have small but nonzero imaginary part, hence the wave is slowly damped as in propagates along the surface. The magnetic field of the plasmon wave is perpendicular to the plane through the

*z*-axis and the real part of the vector (

**k**=(

*k*

_{x},

*k*

_{y},

*k*

_{z}) be the wave vector of a p-polarized wave that is incident on the WGP. The wave vector corresponding to the

*m*th reflected order is:

*λ*/2 with

*λ*the wavelength in the dielectric. Therefore we have that 2

*π*/

*p*is considerably larger than

*k*

_{0}

*n*. Hence (4) can never be satisfied for an integer

*m*. Hence plasmon resonances are not responsible for the large differences between the polarizations.

### 2.2. Metallic waveguide theory

*qualitatively*. The field in such a guide can be written as a linear superposition of so-called waveguide modes. In this section we shall only consider electromagnetic fields that are independent of

*y*.

*n*

_{1}consist of the same metal. The medium in region II is homogeneous dielectric with real refractive index

*n*

_{0}. The width of the middle region is

*w*. Then we can write the index of refraction as:

*z*by a factor exp(±

*iβ z*), where

*β*is the so-called propagation constant. We will consider only modes that are independent of

*y*. There are two types of modes, namely TE modes of which the electric field is parallel to the

*y*-direction and TM modes of which the magnetic field is parallel to the

*y*-direction.

*w*<

*λ*/(2

*n*

_{0}) all

*β*

_{m}are purely imaginary in the TE case. Then the amplitudes of all TE-polarized modes decrease exponentially with

*z*. In contrast, for TM there is always at least one real propagation constant, namely

*β*

_{1}=

*k*

_{0}

*n*

_{0}, no matter how small

*w*is.

*β*for aluminum are shown for a set of values of the width

*w*. The wavelength in the dielectric is 550 nm. It is seen that when w <

*λ*/2

*n*

_{0}the propagation constant

*β*

_{1}of Al for TE is always close to the imaginary axis whereas for TM the

*β*

_{1}is close to the real axis.

*w*<

*λ*/2

*n*

_{0}, hence the propagation constants of all TE-modes have large imaginary parts. The amplitude of the transmitted field is thus exponentially smaller than that of the incident field. Conversely, when the incident plane wave is p-polarized (again with wave vector in the plane perpendicular to the grooves), its magnetic field is parallel to the grooves and hence only the TM modes are excited in the guides. Since the fundamental mode has a propagation constant with only a small imaginary part, a large fraction of the incident energy is transmitted. This explains qualitatively the working of a WGP.

*p*=144 nm and the duty cycle

*D*=45%. The perpendicular incident plane wave is either s- or p-polarized. The field in one cell of the grating was calculated rigorously by using the rigorous finite element method with periodic boundary conditions imposed as described in Section 3.1. It is seen that the total field is strongly evanescent in the case of s-polarization, but propagates through the groove in the case of p-polarization.

## 3. Computational model and experimental procedure

### 3.1. Computational model of a wire-grid grating

8. H.P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM J. Numer. Anal. **28**, 697–710 (1991). [CrossRef]

11. J.M. Brok and H.P. Urbach, “Rigorous model of the scattering of a focused spot by a grating and its application in optical recording,” J. Opt. Soc. Am. A **2**, 256–272 (2003). [CrossRef]

*x*,

*y*,

*z*) the grating is periodic in the

*x*-direction and is translational invariant in the

*y*-direction. In such a configuration there are solutions of Maxwell’s equations of the form

*ω*and

*k*

_{y}and for some vector fields

**U**and

**V**that are periodic in

*x*apart from a common phase shift. It can be shown that for given

*ω*and

*k*

_{y}Maxwell’s equations are equivalent to a system of two coupled second order partial differential equations for only the components

*E*

_{y}and

*H*

_{y}. When we consider the scattering of an incident plane wave in classical mount, we have

*k*

_{y}=0. Then the two partial differential equations for

*E*

_{y}and

*H*

_{y}are uncoupled (one contains only

*E*

_{y}, the other only

*H*

_{y}) and when the incident plane wave is s- or p-polarized, the same is true for the total field. In conical mount however, there holds

*k*

_{y}≠0. Then the two partial differential equations are coupled and the polarizations mix. Hence, when the incident wave is s- or p-polarized, the total field is to some degree elliptically polarized. It is thus seen that the formulation in terms of the system of two, in general coupled, partial differential equations for

*E*

_{y}and

*H*

_{y}is valid for the general conical case and that the classical mount is only a special case of the general formulation.

*y*=0 which has width of one period in the

*x*-direction and is so large in the

*z*-direction that above and below Ω there are homogeneous half spaces. The boundary values for

*E*

_{y}and

*H*

_{y}on the boundaries

*x*=constant are derived from the periodicity of the tangential electric and magnetic field components. The boundary values on the boundaries

*z*=constant are derived by using plane wave expansions for the field in the half spaces above and below Ω. A non-local boundary condition is obtained which expresses the tangential magnetic field in terms of the tangential electric field components by means of a pseudo-differential operator. This boundary condition is rigorous in the sense that the boundary value problem on Ω is completely equivalent to the original scattering problem for Maxwell’s equations. It differs in this respect markedly from for instance approximative absorbing boundary conditions that are often used in other methods such as FDTD.

**93**, 4407–4412 (2003). [CrossRef]

5. L. Li, “Multilayer diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A **11**, 2816–2828 (1994). [CrossRef]

### 3.2. Experiment

13. Stephen Arnold, Eric Gardner, Douglas Hansen, and Ray Perkins, “An Improved Polarizing Beamsplitter LCOS Projection Display Based on Wire-Grid Polarizers,” SID 01 Digest, 1282–1285 (2001). [CrossRef]

*autronic-MELCHERS*detector [14] which can measure the intensity of the emitted light as function of wavelength was used in the experiments. The same detector is used to measure the incident light intensity and the reflected and transmitted light. A diffusive white light source is used for transmission measurements. For reflection measurements, collimated light is used.

*θ*changing from 25° to 65° in steps of 5° in order to measure the transmitted light at those angles. For reflection measurements, the light source and the detector are aligned in the plane perpendicular to the wire-grid. The angles

*θ*

_{i}and

*θ*

_{r}in Fig. 6(b) are the same and are varied between 25° to 65° in steps of 5°. In this way, the reflected intensity is obtained.

*θ*

_{i}=

*θ*

_{r}=45°, and the basis table rotates for varying conical angles from

*ϕ*=90° to

*ϕ*=0°. Because at the conical mount, the scattered field is not purely s- or p-polarized, but instead a mixture of the two. There is an other polarizer used in front of the detector in order to measure separately the s- or p-polarized components of the reflected light.

## 4. Results

### 4.1. General cases

#### 4.1.1. The Influence of the Depth of the Groove

*ϕ*=90°) is shown. The duty cycle has been chosen equal to 45%, hence, because the period was 144 nm, the width of the wires was 65 nm. The depth of the wires varies between 10 nm and 400 nm. The green curves in the figures are for normal incidence at

*θ*=0° and the blue curves correspond to oblique incidence at

*θ*=45°.

#### 4.1.2. Classical mount

#### 4.1.3. Metals

*θ*fixed at 45°, the simulated reflection and transmission as function of incident wavelength for the gratings of different metals is shown in Fig. 11. It is seen in Fig. 11 that the WGP of aluminum is better than that of gold and silver. The contrast ratios of transmission and reflection are much larger for Al than for the other two metals. In general, metals with refractive indices with larger imaginary and smaller real part give better results.

#### 4.1.4. Conical mount

*conical mount*(see Fig. 2). In this case, s- and p-polarizations are coupled, when the incident field is s- or p-polarized, the transmitted or reflected field is not purely s- or p-polarized. We consider the reflected and transmitted intensities for varying azimuthal angle

*ϕ*.

*θ*is kept at 45° while the azimuthal angle

*ϕ*is varied from 0° to 90° in steps of 15°. The incident rays are thus on a cone with its axis normal to the surface as depicted in Fig. 2. The simulated results for the transmission and reflection coefficients as function of

*ϕ*are plotted in Fig. 12. The transmission of p-polarized light drops as

*ϕ*is reduced from 90° to 0°, while the transmission of the s-polarization increases as expected. The reflection coefficients behave similarly. When

*ϕ*decreases, the reflection of s-polarized light decreases and that of p-polarized light increases.

*ϕ*=90°), the electric field of p-polarization is perpendicular to the wire grids. When the plane of incidence is parallel to the wire grids (

*ϕ*=0°), the electric field of p-polarization is not perpendicular to the wire grids. In this position the electric field of p-polarized light has a

*y*-component which is in the direction parallel to the wire grids and this results in a high reflection of p-polarization at this position.

*ϕ*for wavelength of 550 nm. The

*polar angle*

*θ*along the vertical axis is the propagation direction after scattering.

*ϕ*-axis. In the case of p-polarization, the electric field is along the

*θ*-direction. We can see in Fig. 13 that at the classical mount of

*ϕ*=90° and at the special conical mount of

*ϕ*=0°, the polarization states of the transmitted and reflected fields are the same as that of the incident field, i.e. they are purely s- or p-polarized. The polarization states for the cases 0<

*ϕ*<90° are elliptic with small ellipticity and a mix of s- and p-polarized signals. The ellipticity is important and hampers a good contrast. Furthermore, the numbers marked around the ellipses denote the total intensity of the transmitted or reflected light. At

*ϕ*=0° for s-polarization, although the transmitted and reflected field are also s-polarized, most of the light is transmitted while it is mostly reflected at

*ϕ*=90°.

*R*

_{ps}denotes the amplitude of the p-polarized component of the reflected field. Similar definitions holds for

*R*

_{ss},

*R*

_{sp}and

*R*

_{pp}. With the experimental set-up as shown in Fig. 6(b) these reflectivity components can be measured. The mounted polarizer on the detector in this conical experiment reduces the measured intensity significantly. The total reflected intensity at (

*ϕ*=90°) should equal the previous measurement taken at classical mount, i.e.

*R*

_{ss}+

*R*

_{ps}=

*R*

_{s}, hence the reduction ratio can be calculated and it is was found to be 64%. The comparison between measurements and simulations are shown in Fig. 14. The star-dotted computed lines are in good agreement with the measured solid lines.

*ϕ*at polar angle

*θ*=45°. Figure 15(a),(b) show the reflectivity and transmissivity components for incident s- and p-polarized light, respectively. Figure 16(a),(b) show the orientation of the polarization ellipse and the ellipticity of the reflected (R) and transmitted (T) light for the two polarizations of incident light. The solid line are calculated with the effective medium method and the dotted lines are calculated with the rigorous diffraction method. As can be seen, the agreement between the results obtained by the two methods is rather good and within the margin between experiments and calculations. Only for the case of the ellipticity of the reflected light the agreement is bad. Note, however, that in that case the intensity of the reflected light is low.

## 5. Application in LCoS optical assemblage

*θ*=45°,

*ϕ*=90° as shown in Fig. 17(b). For an F/2.0 pupil, the range of the angles of the cone would be from 30.5° to 59.5°. The cone of incident rays causes that the polarization of the reflected and the transmitted rays cannot be perfect.

*x*,

*z*)-plane are in the classical mount, and hence s- and p-polarized states are separated. For other rays, the reflected and/or transmitted field will be a mixture of two polarizations. We define a plane ∑ which is fixed by the wave vector

**k**of the chief ray (at

*θ*=45° and

*ϕ*=90°) and the

*y*-axis. The angle of

*α*is defined as the angle of an arbitrary ray deviating from the chief ray in the ∑-plane. With a ray moving away from the chief ray with angle

*α*, the corresponding

*ϕ*and

*θ*are given by:

*α*=5°,10°,15° and 20°, the corresponding values of

*ϕ*and

*θ*are listed in Table 1. For these cases, the states polarization of the reflected and transmitted field are shown in Fig. 18.

*ϕ*=90° are purely s- or p-polarization for the chief ray. When the deviation angle

*α*increases the ellipticity becomes larger. The result could be that the contrast at the system becomes too low.

## 6. Conclusions

## Acknowledgments

## References and links

1. | Clark Pentico, Eric Gardner, Douglas Hansen, and Ray Perkins, “New, high performance, durable polarizers for projection displays,” SID 01 Digest1287–1289 (2001). [CrossRef] |

2. | E. Hecht, |

3. | X.J. Yu and H.S. Kwok, “Optical wire-grid polarizers at oblique angles of incidence,” J. Appl. Phys. |

4. | J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. |

5. | L. Li, “Multilayer diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A |

6. | M.G. Moharam and T.K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A |

7. | J.B. Judkins and R.W. Ziolkovski, “Finite-difference time-domain modeling of nonperfectly conduction metallic thinfilm gratings,” J. Opt. Soc. Am. A |

8. | H.P. Urbach, “Convergence of the Galerkin method for two-dimensional electromagnetic problems,” SIAM J. Numer. Anal. |

9. | L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. |

10. | H. Raether, |

11. | J.M. Brok and H.P. Urbach, “Rigorous model of the scattering of a focused spot by a grating and its application in optical recording,” J. Opt. Soc. Am. A |

12. | J.D. Jackson, |

13. | Stephen Arnold, Eric Gardner, Douglas Hansen, and Ray Perkins, “An Improved Polarizing Beamsplitter LCOS Projection Display Based on Wire-Grid Polarizers,” SID 01 Digest, 1282–1285 (2001). [CrossRef] |

14. | |

15. | E. Palik and G. Ghosh, |

16. | Pochi Yeh and Claire Gu, |

17. | Pochi Yeh, “Generalized model for wire grid polarizers,” in |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(120.5700) Instrumentation, measurement, and metrology : Reflection

(120.7000) Instrumentation, measurement, and metrology : Transmission

(160.3900) Materials : Metals

(230.1360) Optical devices : Beam splitters

(230.3720) Optical devices : Liquid-crystal devices

(230.5440) Optical devices : Polarization-selective devices

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 1, 2005

Revised Manuscript: March 14, 2005

Published: April 4, 2005

**Citation**

M. Xu, H. Urbach, D. de Boer, and H. Cornelissen, "Wire-grid diffraction gratings used as polarizing beam splitter for visible light and applied in liquid crystal on silicon," Opt. Express **13**, 2303-2320 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2303

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### References

- Clark Pentico, Eric Gardner, Douglas Hansen, Ray Perkins, �??New, high performance, durable polarizers for projection displays,�?? SID 01 Digest 1287-1289 (2001). [CrossRef]
- E. Hecht, Optics, (Addison-Wesley Longman, New York, 1998).
- X.J. Yu and H.S. Kwok, �??Optical wire-grid polarizers at oblique angles of incidence,�?? J. Appl. Phys. 93, 4407-4412 (2003). [CrossRef]
- J. Chandezon, D. Maystre, and G. Raoult, �??A new theoretical method for diffraction gratings and its numerical application,�?? J. Opt. 11, 235-241 (1985). [CrossRef]
- L. Li, �??Multilayer diffraction gratings: differential method of Chandezon et al. revisited,�?? J. Opt. Soc. Am. A 11, 2816-2828 (1994). [CrossRef]
- M.G. Moharam and T.K. Gaylord, �??Diffraction analysis of dielectric surface-relief gratings,�?? J. Opt. Soc. Am. A 72, 1385-1392 (1982). [CrossRef]
- J.B. Judkins and R.W. Ziolkovski, �??Finite-difference time-domain modeling of nonperfectly conduction metallic thinfilm gratings,�?? J. Opt. Soc. Am. A 12, 1974-1983 (1995). [CrossRef]
- H.P. Urbach, �??Convergence of the Galerkin method for two-dimensional electromagnetic problems,�?? SIAM J. Numer. Anal. 28, 697-710 (1991). [CrossRef]
- L. Li, �??A modal analysis of lamellar diffraction gratings in conical mountings,�?? J. Mod. Opt. 40, 553-573 (1993). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1988).
- J.M. Brok and H.P. Urbach, �??Rigorous model of the scattering of a focused spot by a grating and its application in optical recording,�?? J. Opt. Soc. Am. A 2, 256-272 (2003). [CrossRef]
- J.D. Jackson, Classical Electrodynamics, (John Wiley & Sons, Inc., U.S. 1975).
- Stephen Arnold, Eric Gardner, Douglas Hansen, and Ray Perkins, �??An Improved Polarizing Beamsplitter LCOS Projection Display Based on Wire-Grid Polarizers,�?? SID 01 Digest, 1282-1285 (2001). [CrossRef]
- <a href= "http://www.autronic-melchers.com">http://www.autronic-melchers.com</a>
- E.Palik and G.Ghosh, Handbook of optical constants of solids, (Academic Press, New York, 1998).
- Pochi Yeh and Claire Gu, Optics of Liquide Crystal Displays, (John Wiley & Sons, Inc., New York, 1999).
- Pochi Yeh, �??Generalized model for wire grid polarizers,�?? in Polarizers and Applications, Giorgio B. Trapani. Bellingham, eds., Proc. SPIE 307, 13-21 (1981).

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