## Abstract

Extension of the dynamic range of liquid crystal tunable Lyot filter is demonstrated by incorporating with it a liquid crystal variable retarder as an eliminator for the third and fourth order peaks. The filter is continuously tunable in the range 500 nm to 900 nm with a nominal width in the range 50nm-100nm. Design procedure is described including the exact solution to the LC director profile and the suitability for biomedical optical imaging applications. Flexibility in the design is proposed by applying different voltages to the different liquid crystal retarders thus compensating for small thickness deviations from the nominal values and obtaining the high dynamic range.

© 2009 Optical Society of America

## 1. Introduction

Birefringent filters are known for long time [1], however lately they gained high practical importance due to the possibility of tuning them by a small voltage when made from liquid crystals (LCs). There is a wide variety of electro optical effects in LCs [2], based on which tunable filters have been built for different applications such as in displays [3,4,5], in biomedical imaging [6], in hyperspectral imaging [7,8, 9], in optical telecommunications [10], in Raman chemical imaging [11] and in fluorescence microscopy [12]. Among the advantages of liquid crystal tunable filters (LCTFs) are the ease of fabrication, low operation voltages, no currents involved, high switching speed and compactness [2]. LCTFs were demonstrated both using nematic LCs [13,14,15,16,17] and using ferroelectric LCs [18,19,20] in variety of configurations based on the classical filter concepts of Lyot-Öhman [21,22] or Šolc [1] as well as the Fabry-Perot etalon. LCTFs based on nematic LCs exhibit response time in the tens of msec range depending on the configuration used, while higher speeds in the tens of microseconds range can be obtained using ferroelectric liquid crystals. Although the speed is important for certain applications, for biomedical optical imaging few tens of msec is enough for the majority of applications. Similarly, the flat top requirement [23,24] for optical telecommunication applications is not required for biomedical imaging applications. The requirement on the full width at half maximum (FWHM) is moderate for biomedical optical applications as one can work with FWHM of tens of nm for example in color and multispectral imaging [25]. An important design parameter is the free spectral range (FSR) or the dynamic range which is required usually to cover part of the near infrared and part of the visible range (500–1100nm).

For biomedical imaging [26] the interest is usually in the therapeutic window (750–1300 nm) because the penetration depth in the tissue is high in this range; however for analytes functional imaging such as for oxygen blood saturation measurement through the eye [27] one or more measurements are required in the visible range as well. This and together with the fact that Si detectors and CCD or CMOS cameras are reasonably sensitive up to 900nm, a practically desirable range is 500–900nm. As it is well known, the spectrum of the Lyot filter has several sharp output wavelengths corresponding to different interference orders. By nature of this filter the FWHM varies with the wavelength, the longer the wavelength the wider the FWHM. The main problem we wish to address is the demand for wider FSR but at the same time keeping the major order of the Lyot filter prominent. In this paper we demonstrate an extension of the FSR of Lyot-based filter by integrating with it an additional variable LC retarder that eliminates actively the unwanted interference orders.

## 2. Concept and design

Lyot filter consists of N birefringent plates (retarders) each between parallel polarizers oriented such that the polarizer axis makes an angle of 45 degs to the optic axis (Fig. 1(a)). The thickness d of the thinnest plate is such that it satisfies the full wave condition: *d*=*mλ*/*Δn* where *Δn*=(*n _{e}-n_{o}*) is the birefringence and m is a positive integer. The thickness of the j

^{th}plate is 2

^{j-1}

*d*. The transfer function of the j

^{th}birefringent plate between parallel polarizers has the form:

*T*=

_{j}*cos*(

^{2}*2*) where

^{j-2}Γ*Γ*=

*2πdΔn/λ*is the phase retardation accumulated in passing through the thinnest retarder. The transfer function for the N plates is:

This transfer function gives peak wavelengths, FWHM and FSR given by:

where m=1,2,3,…, and *λ _{peak}* is the peak wavelength. The angular field of view is given by: n

_{o}(FWHM/

*λ*)

_{peak}^{0.5}defined as the angle of incidence when the peak shifts by half the width.

The concept of a two stage Lyot filter is shown in Figs. 1(a) and 1(c), where *d* is the first retarder thickness and 2*d* is for the 2^{nd}, both between parallel polarizers having extinction ratio of 1000. Transmission spectra for the 1^{st} (dotted, *d*=8000 nm) and 2^{nd} (dashed) retarders within the Lyot filter as well as their multiplication output (bold) are shown in Fig. 1(c). The width and dynamic range of the Lyot filter are determined mainly by the first retarder following Eq. (2). The spectral bandwidth can be decreased by increasing the number of retarders N, however the dynamic range does not change. In working with the first order peak one can get wide dynamic range, however the FWHM is larger by factor of 4 than the 2^{nd} order. For example taking *λ _{peak}*=900nm, the FSR for m=1 using Eq. (2) is 450nm however the FWHM will be 225nm if two stages (N=2) are used. Working with the 2

^{nd}order peak m=2 yields FSR=300nm and FWHM=125nm. Adding more Lyot stages means adding retarders with thicknesses of 4

*d*, 8

*d*or more which makes the tuning speed very slow and increases the attenuation due to the additional interfaces and polarizers. In our design we overcome this problem by working with higher order peaks to get smaller spectral width and eliminating the nearby orders to get wider dynamic range. The additional retarder between crossed polarizers shown in Fig. 1(b) is designed so that it eliminates the 3

^{rd}order peak as shown in Fig. 1(d).

For designing the filter we relied on the properties of Nematic LC materials of the E-series of Merck. Specifically we used the *4-pentyl-4′-cyanobiphenyl* (E44) purchased from Merck. With strong anchoring where the LC molecules orientations on the surfaces are fixed the LC director profile is nonuniform along the normal to the substrates z when a voltage V is applied. The total retardation [2] should then be calculated by the integral over the LC layer:

where λ is the wavelength in vacuum, *z* is the coordinate normal to the substrates, *V* is the applied voltage, *n _{e}* is the extraordinary index of refraction which depends on the LC molecules tilt angle

*θ, n*is the ordinary index of refraction, and

_{o}*d*is the LC layer thickness. The tilt angle profile

*θ*(

*z*) depends on the external voltage and on the surface anchoring conditions. In reality the angle of the LC molecules is governed by a nonlinear differential equation [2] which results in a larger angle at the middle of the LC layer compared to the facets at

*z*=0 and

*z*=

*d*.

The design algorithm composed of determining the set of retardations {*δ _{j}*} with

*j*=1,2,3,…,N where

*j*is the retarder number in the filter structure and N is the total number of retarders. Once this set of retardations is found for each wavelength, the LC dynamic equation is solved and the retardation integral in Eq. (3) is calculated as a function of the applied voltage in order to determine the corresponding set of voltages {

*V*}. The dispersion of the LC refractive indices calculated based on the Sellmeier type relations [28],

_{j}*n*

_{⊥,‖}=((

*A*

_{⊥,‖}

*λ*

^{2}-1)/(

*B*

_{⊥,‖}

*λ*

^{2}-1))0.5, which for E44 the constants are given as follows when the wavelength is in nm:

*A*

_{⊥}=9.8468×10

^{-5}nm

^{-2};

*B*

_{⊥}=4.3937×10

^{-5}

*nm*

^{-2};

*A*

_{‖}=6.7553×10

^{-5}

*nm*

^{-2};

*B*

_{‖}=2.3057×10

^{-5}

*nm*

^{-2}.

## 3. Filter transfer function

The transfer function for an LCR between two parallel polarizers with the optic axis oriented at 45° with respect to the polarizer axis is cos^{2}(*δ*/2). Similarly, when it is between crossed polarizers the transfer function is sin^{2}(*δ*/2). The filter transfer function for the general case is then:

where *δ*
_{e} is the phase retardation from the eliminator retarder. For the system of two LCRs between parallel polarizers and the additional LCR between crossed polarizers (third LCR) we get the transmission function:

Where δ_{j} with j=1,2,3, for LCR j corresponds to the retardation calculated using the integral in Eq. (3) with the corresponding thickness for each LCR. The expression in square brackets describes the transfer function for a two stage Lyot filter (the first two LCRs) which gives a peak at: *λ _{peak}*=

*dΔn/m*with spectral bandwidth FWHM=

*d/4m*and

^{2}*FSR*=

*λ*/(

_{peak}*m*+

*1*), where

*m*is a positive integer. The question now is how the multiplication by the transfer function of the third LCR improves the FSR?. The basic idea is to design this third retarder (the eliminator) so that its minima coincide with the 3

^{rd}order peak of the Lyot filter. It is also possible to cancel to some degree (10%-20%) two high orders simultaneously, thus increasing the dynamic range even more. We are looking for a mathematical condition when sin

^{2}(δ

_{3}/2)=0 for the eliminator retarder (Fig. 1b) and cos

^{2}(δ

_{1}/2)=1 for the first Lyot’s retarder. The latest condition gives: δ

_{1}=2πm, which imposes the condition on the eliminator LCR: δ

_{3}=2

*πl*where

*l*is another positive integer. Assuming we want to eliminate the m

^{th}peak of the Lyot filter then using:

*λ*=

_{m}*dΔnl/m*we get the following condition:

where here *d _{e}*=

*d*is the thickness of the eliminator retarder and Δ

_{3}*n*=Δ

_{e}*n*

_{3}is its total birefringence. As discussed before, the set of voltages that we find for each wavelength should assure that this equation holds, meaning that the total effective birefringence values Δ

*n*,

_{j}*j*=1,3 should be found numerically corresponding to the voltage values

*V*,

_{j}*j*=1,3 where Δ

*n*is calculated from the integral over the whole LC retarder as in Eq. (3) without the 2

_{j}*π/λ*term.

First option of eliminating an order and broadening the FSR is given in Fig. 2, which shows the Lyot filter (Fig. 1(a)) with the additional retarder (Fig. 1(b)), of thickness *d _{e}*=2666 nm. The first minimum of its transfer function coincides with the third order peak of the Lyot filter output, thus improving the FSR without damaging the second order severely (around ~970 nm in Fig. 1(d)). The first order peak tuning can also be considered if one is interested in the IR range. The order eliminator minima can then be situated at the second order peak and the FSR will increase to 1000 nm. In this case the order eliminator will have a thickness of 4µm and can be tuned to get the elimination of the second, third, and the forth orders as necessary depending on the spectral range of interest.

Second option, may be adding another retarder between crossed polarizers to eliminate the fourth Lyot’s order. Figure 2(a) shows the output of two eliminating retarders, one to eliminate the third order and the other eliminates the fourth order of Lyot filter output. The total output of the Lyot filter with the two eliminators is shown in Fig. 2(b). This configuration has flexibilities of choosing different orders and different retarders thicknesses or voltages for different regions of the spectrum. Its disadvantage is that it reduces the total output of the resulting filtered peak.

## 4. Experimental and results

The LC retarders were built using UV photoalignment on glass substrates of area 15mmx17mm and thickness of 2mm. The two pieces of flat glass coated with transparent electrically conducting electrodes made of indium tin oxide (ITO) were cleaned and spin coated with Rolic photoalignment polymer LPP. The spinner was set to 4000rpm for 60 sec to form 50 nm of uniform polymer thickness. Then the polymer was baked in a vacuum oven at temperature of 90°C for 30 minutes and then at temperature of 250°C for 1 hour. Then the two polymer coated substrates were irradiated by a collimated UV polarized light at normal incidence. The LC we used is *4-pentyl-4′-cyanobiphenyl* (E44) purchased from Merck. After UV irradiation, glass spacers mixed with UV glue Norland 68 were applied near the edges of the glass substrates. Assembling the two glass plates was then achieved with a specially designed mechanical jig to obtain uniform gap by observing the interference colors reflected from the empty cell with green light. The cell was then filled in vacuum at temperature above the clearing temperature (101°C in our case) and then cooled slowly to room temperature. The cell was then sealed with the UV glue Norland 68 and metal wires connected to the electrodes using sliver paste and epoxy glue. Characterization of the LC thickness was done by spectral measurement of the retarder between crossed polarizers and using fitting of the measured output spectrum with the theoretical spectrum. Our polarizers are limited to the wavelength regime between 450 nm to 850 nm. The retarders were aligned in a setup which enabled us to rotate them freely and modify their voltage. The output transmission spectrum was read by a spectrometer manufactured by StellarNet Inc. Each air-glass interface attenuates 4% (transmittance 0.962 for each cell) and the polarizer attenuates 30% (transmits 0.7) thus the total transmission is 17%. One of the drawbacks of the Lyot filter is indeed the high attenuation mainly due to the polarizers, however for bioimaging applications especially when nowadays the CCD cameras are very sensitive, it is not a problem to compensate with stronger light source. It is also possible to use higher throughput polarizers such as those made of metal wire grids with transmission above 90% bringing the filter output to nearly 50%. The filter control is established using LabView program code with National Instrument PCICIA Card NI6715 with eight output programmable voltage channels at its outputs.

Figure 3 shows measured spectra of the filter at different voltages together with the calculated ones using Eq. (6) normalized by the maximum transmission factor of 0.17. In the case of Fig. 3, the peaks are at 838 nm, 740 nm, 634 nm and 560 nm. The sets of the voltages in the captions are pointing to those voltages applied to the first, the second Lyot LCR and the additional eliminator LCR. By varying the voltage sets the major peak (Lyot’s second order) moves to shorter wavelengths. The tuning of the 2^{nd} order peak of the Lyot filter starts with voltage pre-calibration for each retarder. At the same time we changed the voltage of each LCRs until we reached the desired spectrum. The process can be optimized and completed within up to approximately 20 steps over the spectral range (450nm–850nm). For better resolution an interpolation can be executed and stored as the driver of the filter for future use. The degradation of the major peak when moving to the shorter wavelength occurs due to absorption in the LC and in the polarizers. Scattering in the LC is negligible in the homogeneous geometry in particular when the alignment is of high quality.

The experimental filter outputs shown in Fig. 3 closely resemble the designed filter especially in the visible regime since our polarizers have good extinction ratio in the range from 450nm to 850nm. The retarders thicknesses we built were *d _{1}*=8899 nm,

*d*=16043 nm for the first and the second retarders of the Lyot filter, and the thickness of the additional retarder as the order eliminator was

_{2}*d*=2600 nm. The thicknesses

_{e}*d*and

_{1}*d*are not at the nominal values of 8000 nm and 16000 nm, yet we were able to construct the transmission spectrum according to the design by having the flexibility in driving each LCR at different voltage. The second LCR with

_{2}*d*=16043 nm was actually made of two retarders each of thickness nearly 8000 nm. Although this has the disadvantage of increasing the number of glass interfaces, its advantage is in keeping fast switching speed.

_{2}The slowest switching time depends on the thickness of 8000 nm following the approximate equation [2]:

here *τ _{ν}* is the visco-elastic time constant which equals 197 msec for an LCR with thickness of 8000nm filled with LC material E44 at room temperature. The main limitation in nematic LC switching speeds [2] is this time constant which becomes dominant when the voltage is turned off as the fall time will simply become equals to τ

_{ν}. However there are ways for driving schemes that shorten the rise and fall times of nematic LCs as proposed by Wu [2,16]. In Fig. 4 experimental measurements for the rise time (10%–90%) of an 8000 nm retarder are presented. The applied voltage at 1 kHz was pulsed with low frequency pulses (~5 Hz) between 2 V in the low voltage cycle and voltages up to 7 V in the higher voltage cycle. Based on these results we expect the filter switching time that corresponds to the operation voltages used in the tuning presented in Fig. 3 to be in the range of 70 ms–200 ms. Further detailed study of this point will be carried out and published in a future longer article.

## 5. Conclusions

An LC tunable filter with an improved FSR by about factor of 2 was demonstrated based on Lyot filter integrated with an additional LC tunable retarder between crossed polarizers. An additional retarder widens the FSR by eliminating higher order peaks of the Lyot filter. A design procedure was described by which the set of retardations of the individual retarders is calculated and the corresponding set of voltages found by simulating the LC director profile. Reasonable agreement is found between the simulated and measured outputs of the filter. To compensate for small variations in the retarders thicknesses, multichannel voltage supply was provided so that each retarder is under different voltage. Besides providing a filter with larger dynamic range the approach presented here allows flexibility in the design of filters. The same concept can be applied to obtain narrow band high dynamic range tunable filtering by simply increasing the number of Lyot stages. It is even possible to use our approach with other types of filters such as Fabry-Perot filters which suffer even more from the small dynamic range (FSR of few tens of nm). Adding a tunable retarder with the suitable design to eliminate actively the higher order cavity modes will increase its dynamic range. This topic is under investigation and will be published separately. Presently the integration of the filter demonstrated in this article is going on within a skin spectral imaging system in our lab. For this particular application the spectral width of few tens of nm is adequate to enable under skin imaging of abnormalities [26].

## Acknowledgements

This work is supported by the Israeli Ministry of Science under “Tashtiot” funding program.

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