## Abstract

We present a detailed examination of the design and expected operation of an artificially birefringent material based around the nanostructured stack-and-draw fabrication technique developed recently. The expected degree of birefringence is estimated using a Finite Difference Time Domain simulation of the physical system and is shown to be in agreement with that predicted by a second order effective medium theory treatment of the nanostructured material. The effects of finite device dimensions are studied and an estimate of the required device thickness for a half-wave retardation is made.

© 2011 OSA

## 1. Introduction

The recent implementation of the nanostructured micro-optical element fabrication technology [1], based around the photonic crystal fibre stack-and-draw technique, has opened up hitherto unexplored areas of micro-optical design. In particular, areas which have required the use of costly, low volume and time-consuming fabrication methods (e.g. direct-write electron beam lithography), such as high numerical aperture microlenses and form birefringent devices, can now be explored using a conceptually simple and low cost fabrication methodology. In this paper, we investigate the suitability of the nanostructured micro-optical fabrication process for the design and fabrication of birefringent materials in order to develop optical devices with a customised level of phase retardation between orthogonal linear polarisations. The level of birefringence customisation that can be produced by means of nanostructured distributions of glasses of different refractive indices gives the designer almost arbitrary control over the performance of the optical components in terms of overall transmission and phase retardation. Although the use of high frequency micro-structured surfaces as form birefringent materials is well known [2], these components have generally required the use of direct electron beam lithography fabrication techniques, increasing the cost per unit and limiting the deployment of these components to highly specialised applications. The nanostructuring fabrication technology, which has been sucessfully used in the fabrication of high numerical aperture micro-lenses, allows the accurate mass replication of the basic design for a wide range of wavelengths and applications from a single preform assembly.

## 2. Nanostructured micro-optic element fabrication

The nanostructuring process starts with the design of a structure consisting of individual glass rods with a diameter in the range 50–500nm drawn from a basis set of two or more soft glasses with different refractive indices and similar thermal and mechanical properties. The precise arrangement of the glasses composing this nanostructure is calculated by means of an optimisation of the desired optical properties produced by a full electromagnetic simulation of the nanostructure. The basis set of soft glasses most commonly used in the nanostructuring process can be divided into two separate families, one (LIC) with a low refractive index contrast (defined in this paper as the difference between the refractive indices of the individual glasses in each family) and the other (HIC) with a greater refractive index contrast. It should be noted that these two families of glasses have incompatible thermal and mechanical properties and individual glasses from the different families cannot be combined in a single element. Examplars of these two glass families are shown in Table 1 with their chemical composition, refractive index (*n _{d}*) and glass transition temperatures (

*T*).

_{g}The precise choice of electromagnetic simulation method used in the optimisation procedure is dependent upon the total number of rods used in the initial design, the mode of operation of the target optical system (i.e. continuous wave or pulsed) and the overall incident optical power levels. In general, a Fourier decomposition method such as the Fourier Modal Method (FMM) [3, 4] is more useful for cw low power level designs with a large number of rods in the initial preform, whereas the Finite Difference Time Domain (FDTD) method [5,6] is necessary when pulsed operation and non-linear effects have to be considered. For the purposes of the designs presented in this paper, which lie somewhere between these two extreme cases (in that a relatively low number of rods will be used in the initial preform and there are no non-linear effects or pulsed operation considered), we have used the FDTD method with cross-verification of the results being achieved with the FMM. This cross-verification, using two independently implemented code bases, gives us a high degree of confidence in the simulated results, which is critical for the successful design of the nanostructured devices. It should be noted that within this simulation regime, the results from both the numerical methods employed are identical as both produce a rigourous solution to Maxwell’s curl equations - provided the necessary conditions for numerical and algorithmic stability are satisfied [3, 5].

After the design has been finalised, it is used to create an initial preform, made up of the requisite glass rods with a diameter of 1–2mm, which is then loaded into a fibre drawing tower and drawn down, by a factor of 2–5 times, to create an intermediate preform. All of the individual glass rods within the intermediate preform are fused to create a composite whole with individual features on the order of a few hundred micrometres. The intermediate preform is then cut to short (100–1000mm) lengths and these lengths are restacked with appropriate non-structured buffering rods to create a composite intermediate preform. This composite preform, which can contain either single or multiple instances of the design, is then drawn down several more times until the final refractive index variation scale matches that of the final nanostructured design. Figure 1 shows a nanostructured microlens at different stages during the fabrication process - by using a relatively gentle drawing procedure (with a small draw-down factor at each stage) the initial structure of the preform is maintained throughout the whole fabrication run. The nanostructuring fabrication technology has been successfully used in the fabrication of a range of short (10–100*μ*m) focal length microlenses. These optical components, which have been shown to exhibit diffraction limited achromatic performance [7], are ideally suited to optical fibre launch and micro-laser collimation [8].

## 3. Birefringence in one-dimensional nanometre-scale microstructures

By varying the relative proportion of one glass (refractive index n_{1}) to the other (refractive index n_{2}) within a slab of one dimensional nanostructured material, the refractive index of the composite slab can be varied across the full range of refractive indices from n_{1} to n_{2}. First order effective medium theory [9] predicts the average relative permittivity of the composite slab by the Maxwell-Garnett formula

*δ*is the relative proportion of the high index material

*ɛ*. In order to test the validity of this theory, a series of 2D FDTD simulations (with the non-interacting groups of EM field components (

_{i}*H*,

_{x}*H*,

_{y}*E*) and (

_{z}*E*,

_{x}*E*,

_{y}*H*)) of the set of one dimensional arrangements of the two LIC glasses - shown in Fig. 2 - were performed for both transverse electric illumination (TE) - where the electric field component is perpendicular to the index variation in the 1D structure - and transverse magnetic illumination (TM) - where the electric field component is parallel to the 1D structure index variation. The total number of separate inclusions in these simulations was set to be ten - this number adequately demonstrates the desired effect (which requires an inclusion size of <

_{z}*λ*/5 [7]) while keeping the total simulation time on an acceptable timescale.

The variation of the optical phase through the slab as a function of thickness for TM polarised light is shown in Fig. 3a. This demonstrates the change in the effective refractive index of the nanostructure with increasing number of high index inclusions. The corresponding effective refractive index for an increasing number of high index inclusions for both TE and TM polarisations is shown in Fig. 3b. It can be seen from these figures that the effective index for the TM polarisation differs from the first order result (which is plotted in Fig. 3b and is identical to the curve for the TE polarisation) as the overall fill factor (*δ*) moves towards 0.5 (corresponding, for the simulations presented here, to 5 high index inclusions). At this point, the first order Maxwell-Garnett formula is no longer valid and instead a 2* ^{nd}* order theory [10] must be used to accurately model the different polarisations.

*λ*is the wavelength of the incident illumination. It should be noted that the second order effective medium theory is only valid where Λ <

*λ*-when this condition is not satisfied a fully vectorial solution to Maxwell’s curl equations must be used to calculate the effective refractive index of the material.

The point of maximal effective refractive index difference in Fig. 3b corresponds to a 50/50 high/low index structure and is ideally suited to the nanostructured fibre drawing process. It consists of alternating slabs of high and low refractive index glass with the direction of variation lying perpendicular to the direction of propagation as shown in Fig. 4. The initial preform of this structure can be assembled from 1mm thick slabs of the constituent glasses giving a considerably simplified preform assembly stage when compared to that required by the nanostructured lens designs. Furthermore, the final index variation scale is on the order of one-half to one-quarter of the incident illumination resulting in a much reduced draw-down requirement for this material compared to that required for the nanostructured microlens where the final index variation scale is on the order of one-tenth to one-twentieth of the incident illumination.

The 1D structure of Fig. 4 was simulated using both the FMM and FDTD methods. The refractive index dispersion of the low index contrast glasses in the modelled nanostructures are given by the Sellmeier coefficients shown in Table 2. Figure 5 shows the change in the predicted level of birefringence as a function of wavelength for an infinite slab of birefringent material composed of alternating bands of F2 and NC21A. The general trend of the curve is primarily determined by the relative differences between the glass dispersion curves (shown by the red line in Fig. 5) and, by careful selection of the operating wavelength range of the material, an effectively constant level of birefringence (defined as the optical phase difference (2*πdnt*)/*λ* between the TE and TM polarisations) over many tens of nanometres can be achieved. The green curve in Fig. 5 is a 1/*λ* fit to the *dn* curve for 300nm wavelength bands (i.e. 500–800nm, 800–1100nm, 1100–1400nm, 1400–1700nm and 1700–2000nm). The inverse wavelength fit is excellent in the upper wavelength region (> 800nm) demonstrating that the material is capable of giving constant birefringent operation over several hundred nanometres in the near infra-red.

In order to determine the final index variation scale of the artificially birefringent material, the variation in the birefringence as a function of the overall period of the nanostructure was studied by means of a FDTD simulation for a number of different wavelengths of incident light (500nm, 1000nm, 1500nm and 2000nm). The results of this study are shown in Fig. 6 - each curve in the figure has been scaled by the peak value of dn calculated by the second order effective medium theory for a nanostructure period of *λ*/20. It can be seen that for each wavelength of light, there is a definite threshold beyond which the expected level of refractive index difference is significantly reduced. This reduction in dn is due to the structure ceasing to operate as a true effective medium and instead beginning to function as a scalar domain diffraction grating - as shown by the inset figures for an incident wavelength of light of 1000nm. The value of this threshold was determined numerically by fitting a sigmoid-like curve to the simulated dn curves. The fitting curves, shown by the solid lines in Fig. 6, are

_{Λ}is the large period normalised steady-state dn,

*T*

_{Λ}is the threshold level, Λ is the nanostructure period and

*λ*is the illumination wavelength of light. The variation of the large period normalised steady state dn and the threshold level from the fitting curves (solid lines in Fig. 6) obey a linear trend and are given by, It can be readily seen from Fig. 6 that the large period steady state dn is not in actual fact a constant value, but the fitting curves give an over-estimate of the large period birefringence and therefore this value should be regarded as an upper bound to the expected large period birefringence.

Figure 7 shows the phase difference between the TE and TM
polarisations for a finite (in the propagation direction) slab of the birefringent
material for both high and low index contrast family nanostructures at a wavelength
of 1*μm*. It can be seen that, as expected, the phase
difference increases linearly with the device thickness and the higher index
contrast device has a greater rate of increase. There appears to be an oscillation
on top of the general linear trend and, in order to reduce the probability of this
being due to a numerical artifact in the FMM simulation, the low index contrast
simulation was repeated using the FDTD method. For these simulations the
birefringent structure was placed within a free space domain bounded by perfectly
matched layers [11] to
eliminate spurious back reflections from the computational boundaries. Due to the
numerical stability contraints of the FDTD method the simulated structures were
limited to a device thickness of a few waves but it can be seen from Fig. 8 that even over this limited range the
same oscillatory behaviour is present. In order to determine the source of this
oscillation, an identical structure was simulated in an infinite mode with the
structure being allowed to run into the perfectly matched layers placed at the
boundaries of the computational domain. As shown by the blue curve in Fig. 8, the linear trend is identical to that
for the finite case without the oscillation implying that the source of the
oscillation is related to the finite nature of the initial birefringent structure. A
plot of the amplitude transmission of the finite slab for both TE (red dots) and TM
(blue dots) polarisations (Fig. 9) shows that
the finite slab is acting as a low finesse cavity with the reflectivities of the
facets determined by the effective refractive index of the composite material. The
transmission of the cavity, shown by the solid lines in Fig. 9, is determined by the relevant effective refractive
index and is given by

*n̄*) is derived from the effective relative permittivities of Eqs. (2) and (3) and is equal to 1.568498 and 1.565276 for TE and TM polarisations respectively. The total phase delay across the nanostructured material is calculated from the amplitude transmission function of the low finesse cavity and is equal to Figure 10a shows the fit of Eq. (8) to the FDTD simulation results - to show the accuracy of fit more clearly the positive linear trend observed in Figs. 7 and 8 has been removed from both of the graphs in Fig. 10. In order to further test the veracity of the model [Eqs. (2), (3), (7) and (8)] the magnitude of the phase delay oscillation was estimated up to a thickness of 200 waves. As was observed in the FMM simulations, the oscillation builds up and then decays until the magnitude of the oscillation is equal to zero. At this point the polarisation dependent phase difference for the finite device is equal to that for the infinite device and the phase difference between the two polarisations is equal to

*π*. The thickness of the birefringent nanostructured device to achieve a phase delay of

*π*is given by,

*π*thickness of 25 micrometres and polarisation dependent cavity finesses of 0.303 (TE)/0.289 (TM) compared to 0.217 (TE)/0.215 (TM) for the low index contrast structure.

## 4. Conclusions

In this paper, we have presented a design for the fabrication of a birefringent nanostructured optical device compatible with the nanostructured micro-optical fabrication technology. This conceptually simple design, which is based around a one dimensional variation in the refractive index, has been shown to produce a controllable level of birefringence which depends upon the thickness of the sample, the relative frequency of the refractive index variation and the effective refractive index produced by the composite structure. Applying second order effective medium theory to the birefringent structure has been shown to give good agreement to both the phase difference between polarisations and the low-finesse cavity effects observed in the more rigorous (FMM/FDTD) electromagnetic simulations. The accuracy of this simplified model allows the rapid design of customised birefringent devices without the need for time consuming rigorous electromagnetic simulations. Furthermore, due to the flexible nature of the draw-down process, where different final feature sizes can be achieved from a single preform assembly by simply changing the draw parameters, the assembly of a single generic preform will allow us to produce birefringent components covering the entire transparent region of the appropriate glass family. Currently, the necessary samples of the low-index contrast glass family needed to fabricate the nanostructured design have been produced and the assembly of the initial preform is underway.

## Acknowledgments

The authors acknowledge the financial contribution of Scottish Enterprise through the Proof of Concept Fund to the development of this research.

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