Three-stage Fabry–Perot liquid crystal tunable filter with extended spectral range

Zhenrong Zheng; Guowei Yang; Haifeng Li; Xu Liu

doi:10.1364/OE.19.002158

1. Introduction

Tunable optical filters, known for long time, have played an important role in many applications such as remote sensing [1], hyperspectral imaging [2], biomedical imaging [3,4], optical telecommunication [5,6], astronomical observation [7,8], and displays [9,10]. For tunable optical filters, electro-optic crystals (such as KDP and ADP) or liquid crystals (LC) are often used as variable birefringent retarders [11–13] to produce a narrow passband. LC tunable filters have been an active research area because of their advantages of continuous tuning, low power consumption, high tuned speed, compactness, and reliable performance without any movable components. LC tunable filters have been demonstrated in a variety of applications based on the Lyot/Solc filter [14,15] as well as the Fabry–Perot filter [16–18]. Compared with the Lyot/Solc filter, the Fabry–Perot filter usually has narrow FWHM, a higher transmission, but a small free-spectral range (FSR), which is typically a few tens of nm. The Fabry–Perot filter has been used widely in optical telecommunication because of the narrow FWHM, but it rarely has been used in many applications that need a wide FSR or a wide dynamic range, in particular for biomedical imaging. Much research has been reported to extend the FSR of an LC tunable optical filter by using a tandem technique. Hirabayashi presented a tunable wavelength selective filter at a wavelength of 1.47–1.6 μm that is stacked by two LC Fabry–Perot interferometers [19]. Aharon [20,21] used an additional LC variable retarder to extend the FSR of a Lyot/Solc tunable filter. The tandem technique is also reported for increasing the sharpness of the transmission profile [22]. Some reports have revealed that an improved channel shape [23,24] and flat-top LC tunable filter [25] can be obtained by using coupled Fabry–Perot cavities. The concept of a coupled-resonator filter is also used in silicon-on-insulator technology. Multistage racetrack resonator filters in silicon-on-insulator have been designed and manufactured [26–28]. A viable alternative to increase FSR using the theory of a series-coupled racetrack is reported to extend the FSR of a racetrack resonator, wherein 36 nm of FSR can be achieved [28]. For the tandem technique, the simplest case is the two tandem Fabry–Perot interferometers system. In order to minimize the interaction between the two interferometers, the second one is often used to make the separation distance between the two Fabry–Perot interferometers. However, a large FSR and a narrow FWHM are still in great demand to accomplish many applications such as remote sensing, astronomical observation, and biomedical imaging, which usually covers part of the visible range (450–700 nm) and part of the near infrared range (700–1000 nm).

In this paper, a novel concept (to our knowledge) for extending the spectral range of a tunable Fabry–Perot filter is proposed by cascading three-stage Fabry–Perot filters with an interleaving order. Two identical tunable Fabry–Perot filters are placed in tandem to extend the FSR by interleaving only one order of the transmission peak. The FSR can be extended to maximum because of using one order of difference between the first and second filters. Then an additional Fabry–Perot filter with a different FSR (with one order of difference, too) is cascaded to reduce the side lobes. A three-stage Fabry–Perot filter is achieved with 400 nm of FSR and 4 nm of FWHM. The results reveal that an extended tunable spectral range of a Fabry–Perot filter can be easily attained by this method.

2. Concept and design

A Fabry–Perot LC tunable filter can be achieved by enclosing a plane-parallel plate of the LC, shown in Fig. 1(a). The transmission of the LC Fabry–Perot filter has the form Fig. 1 (a) . The transmission of the LC Fabry–Perot filter has the form

T = \frac{I^{(t)}}{I^{(i)}} = \frac{1}{1 + F \sin^{2} \frac{δ}{2}},

where, I^(t)is the intensity of output light, and I⁽ⁱ⁾ is the intensity of input light. Finesse F has the form

F = \frac{4 \sqrt{R}}{{(1 - \sqrt{R})}^{2}}

; R is the reflectivity of two distributed Bragg reflectors (DBRs); δ is the phase in passing through the Fabry–Perot filter

δ = \frac{4 π}{λ} n d - 2 δ_{D B R}

, where n is the equivalent refractive index of liquid crystal; d is the thickness of LC layer; and δ_DBR is the reflective phase of two DBRs and can be considered as a constant value. The equivalent refractive index of the LC has the form

n = \frac{1}{d} {\int_{- \frac{d}{2}}^{\frac{d}{2}} n_{o} [1 - (1 - \frac{n_{o}^{2}}{n_{e}^{2}}) \sin^{2} θ]}^{- \frac{1}{2}} d z,

where n_e is the extraordinary index of refraction, which depends on the tilt angle of the LC molecules, θ, n_o is the ordinary index of refraction, and d is the LC layer thickness. The tilt angle profile θ(z) depends on the external voltage and on the surface anchoring conditions.

Fig. 1 (a) Fabry–Perot LC tunable filter; (b) layout of three-stage LC tunable Fabry–Perot filter.

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The FWHM and FSR of the Fabry–Perot filter are given by

λ_{p e a k} = \frac{2 n d}{m}; F S R = \frac{λ_{p e a k}^{2}}{2 n d} = \frac{2 n d}{m^{2}} = \frac{λ_{p e a k}}{m}; F W H M = \frac{2 λ}{m π} \cdot \frac{1}{\sqrt{F}},

where m = 1, 2, 3,…is the order of the transmission peak and λ_peak is the peak wavelength of the filter. From Eq. (2), the FSR of the filter is in inverse proportion to the order m. The FWHM is in inverse proportion to the order m². Different FSRs and FWHMs can be achieved at the fixed λ_peak when different orders of m are used. Then different peak wavelengths and FSRs can be achieved when the equivalent index of the LC is changed by adjusting the applied voltage of the LC layer.

The structure of a three-stage Fabry–Perot filter is shown in Fig. 1 (b). Two identical Fabry–Perot tunable filters and an additional tunable filter with a different FSR are cascaded to extend the tunable spectral range and reduce the side lobes. The first Fabry–Perot filter and the second filter have the same thickness of LC layer and are made of the same LC material (E44, purchased from Merck).

The transmission function of a three-stage Fabry–Perot filter has

\begin{matrix} T = \frac{1}{1 + F \sin^{2} \frac{δ_{1}}{2}} \cdot \frac{1}{1 + F \sin^{2} \frac{δ_{2}}{2}} \cdot \frac{1}{1 + F \sin^{2} \frac{δ_{3}}{2}} \\ = \frac{1}{1 + F \sin^{2} (\frac{2 π}{λ} n_{1} d)} \cdot \frac{1}{1 + F \sin^{2} (\frac{2 π}{λ} n_{2} d)} \cdot \frac{1}{1 + F \sin^{2} (\frac{2 π}{λ} n_{3} d_{3})}, \end{matrix}

where, n_1, n_2, and n₃ are the equivalent indices of the first, second, and third LC layers, respectively; d is the thickness of the first and second LC layers; and d₃ is the thickness of the third LC layer. Since a three-stage filter uses the same DBR layer, Finesse F has the same merit:

F = \frac{4 \sqrt{R}}{{(1 - \sqrt{R})}^{2}} .

When ${\begin{cases} \frac{2 π}{λ} n_{1} d = m π \\ \frac{2 π}{λ} n_{2} d = (m + 1) π, \\ \frac{2 π}{λ} n_{3} d_{3} = (m - 1) π \end{cases}$ the total transmission of the three-stage cascading filter is 1. Then, FSR of each filter is

F S R_{1} = \frac{λ_{p e a k}}{m}; F S R_{2} = \frac{λ_{p e a k}}{m + 1}; F S R_{3} = \frac{λ_{p e a k}}{m - 1} .

Three filters are working in different orders and have different FSRs after being cascaded, and the interval wavelength of next order is

λ_{int e r v a l} = \frac{λ_{p e a k}}{m} - \frac{λ_{p e a k}}{m + 1} = \frac{λ_{p e a k}}{m \cdot (m + 1)} .

So, the adjacent peak wavelength of the three-stage filter will be

λ_{p e a k}' = \frac{2 n_{1} d}{m + \frac{F S R_{1}}{λ_{int e r v a l}}} = \frac{2 n_{1} d}{2 m + 1} = λ_{p e a k} \cdot \frac{m}{2 m + 1},

where, λ_peak is the peak wavelength of the filter, λ_peak ’ is the adjacent peak wavelength, and m is the order of transmission peak.

So, the free spectral range of the whole three-stage filter is

F S R = λ_{p e a k} \cdot \frac{m + 1}{2m+1},

The first, second, and third Fabry–Perot filters are working at m, m + 1, and m-1 order, respectively; we simply called this the “jump order” filter. Equation (3) shows the transmission of the filter, and the FWHM can be determined exactly. Since the order difference of a three-stage filter is small, the phases of the filters have similar merit, then

T \approx {(\frac{1}{1 + F \sin^{2} \frac{δ_{1}}{2}})}^{3} .

It is easy to show that the finesse will be determined with a good approximation when the transmission function decreases to 50% of its maximum value when its phase is 2π/3.

Hence, the FWHM is approximately

F W H M \approx \frac{2}{3} \cdot \frac{2 λ_{p e a k}}{m π} \cdot \frac{1}{\sqrt{F}} .

From Eqs. (5) and (6), the FSR of the whole filter is extended to $\frac{m + 1}{2 m + 1} \cdot λ_{p e a k}$ and the FWHM improves almost 30% . Moreover, the side lobes of a three-stage Fabry–Perot LC filter are reduced because of the different FSRs by interleaving one order of the transmission peak.

The transmission spectra of a designed three-stage Fabry–Perot tunable filter are shown in Fig. 2 . Figure 2(a) shows the transmission of the first filter (pink) and second filter (blue). The thickness of the LC layer (E44, purchased from Merck) of two filters is exactly the same (3μm). The peak wavelength is tuned to 850 nm. The optical phase of the filter can be varied by adjusting the voltage of the LC layer. A different order of the first and second filters is used, and m = 11 for the first filter and m + 1 = 12 for the second filter. The total transmission of the two cascaded filters is shown in Fig. 2(b). We can see that the FSR is extended by interleaving only one order of two filters. To reduce the side lobes of the two cascaded filters, an additional filter is used. Figure 2(c) shows the transmission of the third filter in which the thickness of the LC is 2.7 μm, and m-1 = 10 is used by using different thicknesses and adjusting the voltage of the LC layer. The total output of the three-stage LC Fabry–Perot tunable filter is shown in Fig. 2(d). By using the method of cascading the three-stage filter, the FSR is extended to over 400 nm, and the FWHM is less than 4 nm and has lower-level side lobes.

Fig. 2 (a) Transmission spectra for the first (pink, d = 3000nm, m = 11) and second (blue, d = 3000 nm, m + 1 = 12) Fabry–Perot filter. (b) Transmission spectra of cascaded first and second filters. (c) Transmission spectra for the third (d = 2700nm, m-1 = 10) filter. (d) Total output of three-stage LC Fabry–Perot tunable filter.

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3. Experimental and results

A single LC tunable Fabry–Perot filter was built on glass substrates with an area of 20×20mm and a thickness of 2 mm. The two pieces of flat glass were coated with transparent electrically conducting electrodes made of indium tin oxide (ITO) and DBR. Then they were cleaned and spin coated with an alignment layer. The material of the alignment layer is polyimide (PI-5, purchased from Shanghai Jiaotong University). The spinner was set to 5000 rpm for 60 s to form 100 nm of uniform thickness. Then the polymer was baked in a vacuum oven at a temperature of 80°C for 30 min and then at a temperature of 180°C for 2 h. A rubbing method was used for the alignment of LC molecules. After rubbing, the two plates were celled together by a spacer with a fixed gap. The LC (E44, purchased from Merck) was filled in and then sealed with glue. The three stages of the LC Fabry–Perot filter were cascaded together, as shown in Fig. 3 . The measured transmission of the experimental three-stage filter is shown in Fig. 4 . The total optical loss was about 65%. The thicknesses we built were d1 = d2 = 3 μm for the first- and second-stage filters, and the thickness of the additional stage as the order eliminator was d3 = 2.7 μm. The transmission spectrum can be changed according to the design by adjusting the voltage on the LC layer. The loading voltages changed from 0v–20 V with a 2 K square wave signal.

Fig. 3 Experimental three-stage LC Fabry–Perot filter.

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Fig. 4 Five examples of measured spectra of three-stage LC Fabry–Perot tunable filter: the peak wavelength of the filter at 848 nm, 750 nm, 695nm, 625 nm and 490 nm when the loading voltages for the three stages were: {2.03V, 5.75V, 2.12V}, {8.49V, 12.58V, 7.86V}, {12.85V, 16.21V, 13.72V}, {8.23V, 15.45V, 9.16V}, {13.25V, 6.15V, 3.05V}, correspondingly.

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4. Conclusion

A method to extend the spectral range of tunable optical filters is demonstrated by cascading a three-stage Fabry–Perot filter. Two identical tunable Fabry–Perot filters and an additional tunable filter with a different FSR and order are cascaded. The design procedure and simulation are described. Over 400 nm of FSR and 4 nm of FWHM were achieved. An example tunable optical filter with visible and infrared spectrum from 450–850 nm is demonstrated. It is even possible to use our method with other types of filters and other spectra. Tunable Fabry–Perot filters with narrow FWHM and large FSR will play an important role in many applications such as remote sensing, biomedical imaging, and astronomical observation.

Acknowledgments

This work is partially supported by the National Basic Research Program of China (973 Program) (No. 2009CB320803) and the State Key Laboratory of Modern Optical Instrumentation Funding Program.

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Three-stage Fabry–Perot liquid crystal tunable filter with extended spectral range

Abstract

1. Introduction

2. Concept and design

3. Experimental and results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (11)

Optics Express