Introduction

Liquid Crystal Tunable Filter (LCTF) is a kind of filter based on liquid crystal (LC) birefringence effect and electro-optical effect to realize the specific wavelength transmission, which takes LC as the electrically tunable optical material1,2,3. LCTF has been a research area of great interest due to its advantages of continuous electronically controlled tuning, wide tuning range, low power consumption, high tuning speed, compact structure and high reliability. Especially, the change of the optical properties of LC molecules under the electric field creates a new path for the industrial development in the fields of hyperspectral imaging4,5, computational imaging6 and polarimetric imaging7. The basic structures of LCTFs are mainly Lyot type8,9 and Solc type10,11, among others. The Solc-type LCTF is composed of a pair of parallel or orthogonal polarizers inserted into a liquid crystal filter with the same thickness and different azimuthal angles. It offers the advantages of high transmission; however, its precise design is fixed. In contrast, the Lyot-type LCTF offers the advantages of simple structure, wide tuning range, high spectral resolution, and small setting error. Currently, most of the researches for LCTF focus on improving spectral resolution12, enhancing switching speed13,14 and suppressing side-valve transmission9,15. However, these studies have been discussed for a single transmission wavelength and are not applicable to the detection of targets with dual-wavelength response characteristics. Dual-wavelength detection is a double-channel filter designed based on the response characteristics of the two characteristic peaks of the target16,17,18, which ultimately achieves two characteristic wavelength transmissions. The increase in the number of filter channels can effectively improve the energy transmission of the system, thus improving the signal-to-noise ratio (SNR) of the detected target19,20. The traditional dual-wavelength narrowband filters are primarily prepared based on optical film system theory16,21, Fabry-Perot cavity structure design22,23,24 and guided-mode resonance principle25,26, which are not able to achieve the real-time tuned filters for wide wavelength bands. Therefore, a corresponding dual-wavelength filter needs to be designed for each feature target. However, the reported tunable filter is a notch filter designed based on the hypersurface structure with different patterns from graphene material27,28, which can rapidly attenuate specific wavelength transmissions. Evidently, the notch filter prevents wavelength transmission in contrast to target detection applications. In summary, the.

available tunable filters are single-wavelength transmission, single-bandstop and dual-bandstop tunable filters, respectively, all of which cannot enable dual-wavelength tunable transmission. Therefore, this paper studies the dual-wavelength tunable filters based on Lyot-type structures for high SNR detection of the target.

Principle of filtering

The filtering principle of Lyot-type filter is shown in Fig. 1(a) is a structural form of single-cascade Lyot-type filter, where a pair of parallel polarisers P1 and P2 are inserted with a LC whose optical axis is at an angle of 45° to the transmission direction of the polarisers29. The linearly polarized light after the action of P1 propagates to LC, and then it is decomposed into two coherent linearly polarized light waves by the birefringence of LC, which ultimately interferes on P2 to form a series of interference fringes. If the thickness of the LC is d and ordinary and extraordinary refractive indices are no and ne, respectively, then the phase difference produced by the incident wave of wavelength λ passing through the LC is δ = 2π(ne-no)d/λ, and the transmission of the Lyot-type filter is:

$$T={\cos ^2}\frac{\delta }{2}$$
(1)

To obtain narrowband filtering characteristics with different half-height widths, it is necessary to cascade single-layer Lyot-type filters with different phase delay to obtain excellent transmission spectral characteristics30, as shown in Fig. 1(b). The phase delay is the primary factor in achieving the liquid crystal tuned filtering, and the phase differences of each layers of LC are progressive in an equal series δN = 2n−1δ1, so that the phase differences of the first to the Nth layer are δ1, 2δ1, 4δ1, ., 2N −1δ1, respectively31,then the spectral transmission of the multi-cascade LCTF is:

$$T=\prod_{n=1}^N\cos^2\frac{\delta_n}2,(n=1,2,3,\cdots,N)$$
(2)

where N denotes the number of layers in the cascade of the filter.

The phase delay of each cascaded LC in the multi-cascade Lyot-type structure satisfies the interferometric maxima condition at the transmission wavelength, then the phase delay of LC should satisfy:

$$\triangle_n=(n_e-n_o)d_n=2^{n-1}\lambda_c,\;\;\;\;(n=1,2,3,\cdots,N)$$
(3)

Then Eq. (2) can be rewritten as:

$$T={\cos ^2}\left( {\frac{\pi }{\lambda }{\lambda _c}} \right){\cos ^2}\left( {\frac{\pi }{\lambda }\cdot 2{\lambda _c}} \right){\cos ^2}\left( {\frac{\pi }{\lambda }\cdot 4{\lambda _c}} \right) \cdots {\cos ^2}\left( {\frac{\pi }{\lambda }\cdot {2^{N - 1}}{\lambda _c}} \right)$$
(4)

By varying the voltage applied to the LC layer, ne can be varied to achieve continuous tuning of the central wavelength λc, which is also the response wavelength of the detection target. Therefore, the phase delay Δn satisfy the interference maximum and interference minimum conditions at λc and the rest wavelengths, respectively, to achieve the effect of the desired wavelength transmission non-desired wavelength cut-off. Theoretical simulation analysis of Lyot-type filter can be carried out according to Eq. (3), and theoretically the LCTF is able to achieve 100% transmission at λc, as shown in Fig. 1(c).

Fig. 1
figure 1

Schematic diagram of Lyot-type filter. (a) Structure of single-cascade Lyot-type filters. (b) Structure of multi-cascade Lyot-type filter. (c) Transmittance diagram of the filter.

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Dual-wavelength LCTF

Implementation of dual-wavelength transmission

To achieve the dual-wavelength transmission, Eq. (3) is modified as follows:

$$\triangle_n=Q_p\bullet2^{n-1}\lambda_c,\;\;\;(n=1,2,3,\cdots,N)$$
(5)

here the coefficient Qp is introduced, which is called the transmission factor, and Qp is a positive integer greater than 1. Varying its value can change the delay difference of the o-light and e-light needed to meet the various layers of LC. According to the determined tuning central wavelength λc, a reasonable value of Qp can be designed to enable the co-transmission of the other transmission wavelength, and ultimately achieve the dual-wavelength transmission of LCTF in the free spectral range (FSR). This wavelength is referred to in this paper as the gain wavelength λg. λg is essentially the ± 1 order interference transmission wavelength accompanied by λc co-transmission. The FSR is the primary factor that limits the static spectral performance of the multi-cascade Lyot-type filter, and the number of transmission wavelengths within the FSR is determined by the Δ1 at the first-layer cascade LC of the system. Therefore, λg also satisfies the interference maxima condition Δ1 = (Qp±1) λg. Combined with Eq. (5), the gain wavelength of the dual-wavelength LCTF system can be solved as:

$${\lambda _g}=\frac{{{Q_p} \cdot {\lambda _c}}}{{{Q_p} \pm 1}}$$
(6)

In the above equation, ‘+’ and ‘-’ represent λg < λc and λg > λc, respectively. The different values of Qp lead to two cases of λg < λc and λg > λc for the position of λg in the range λmin-λmax of the tuning spectral. The dual-wavelength LCTF system is the presence of two transmission wavelengths within the FSR, so the value of Qp has a range of values. When λg > λc, the conditions shown in Eq. (7) need to be satisfied simultaneously in order to ensure that only one λg exists within the FSR:

$$\left\{\begin{array}{l}Q_p\lambda_c/(Q_p-1)<\lambda_{max}\\Q_p\lambda_c/(Q_p-2)>\lambda_{max}\\Q_p\lambda_c/(Q_p+1)<\lambda_{min}\end{array}\right.$$
(7)

Combining and simplifying the above limitations yields the condition for the value of Qp to be:

$$\left\{\begin{array}{l}\left\lfloor\frac{\lambda_{max}}{\lambda_{max}-\lambda_c}\right\rfloor<Q_p<\left\lceil\frac{2\lambda_{max}}{\lambda_{max}-\lambda_c}\right\rceil\\Q_p<\left\lceil\frac{\lambda_{min}}{\lambda_c-\lambda_{min}}\right\rceil\end{array}\right.$$
(8)

Similarly, the condition for the value of Qp when λg < λc is:

$$\left\{\begin{array}{l}\left\lfloor\frac{\lambda_{min}}{\lambda_c-\lambda_{min}}\right\rfloor<Q_p<\left\lceil\frac{2\lambda_{min}}{\lambda_c-\lambda_{min}}\right\rceil\\Q_p<\left\lceil\frac{\lambda_{max}}{\lambda_{max}-\lambda_c}\right\rceil\end{array}\right.$$
(9)

’ \(\left\lceil x \right\rceil\) ’ and ’ \(\left\lfloor x \right\rfloor\) ’ in Eqs. (8) and (9) denote upward and downward integers, respectively.

Based on the above theoretical equations, the design of a dual-wavelength LCTF system can be carried out. The FSR is defined as 450–750 nm, and for a particular tuning central wavelength λc, the range of Qp values can be calculated according to Eqs. (8) and (9) proposed in this paper. For multiple given wavelengths, the corresponding range of values of Qp is calculated as shown in Table 1.

Table 1 Range of values of Qp for various λc.
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According to the calculation results of Table 1, the central wavelength λc is 475 nm, and dual-wavelength transmission in the range of 450–750 nm can be achieved when Qp is taken as 3, 4 and 5 respectively, and λg are all in the position of λg > λc. To verify the effectiveness of this result, simulation analysis of the dual-wavelength transmission is carried out. The gain wavelength λg and the ∆n to be satisfied by each layer cascade LC are calculated according to Eqs. (6) and (5), respectively, and then, the FSR spectral transmission is simulated based on Eq. (4). The simulation results are shown in Fig. 2, when λg is 712.5 nm, 633.33 nm and 593.75 nm, which is consistent with the λg results analysed by Eq. (6). So the theoretical formula for LCTF dual-wavelength transmission proposed in this paper is effective. Therefore, the range of consistent values for Qp can be calculated when the central wavelength is determined, and further obtain the dual-wavelength transmission.

Fig. 2
figure 2

Dual-wavelength transmission with various Qp values for 475 nm.

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Dual-wavelength tuned spectral range analysis

For portable design and application of dual-wavelength tunable filters, the tuned spectral ranges (TSR) of λc and λg must be known. To quantitatively analyse the TSR of λg within the FSR, assuming that the interval of λg variation is λmin-λmax, then λg should satisfy λmin ≤ Qp·λc/(Qp±1) ≤λmax. Since there exists only one λg within the FSR, when λc < λg, the λg corresponding to the Qp-2 and Qp+1 orders should both lie outside the FSR, and combining this with Eq. (6), the TSR for λc can be obtained as:

$$\left\{\begin{array}{l}\lambda_{min}\leqslant\lambda_c\leqslant\lambda_{max}\\(1-1/Q_p)\lambda_{min}\leqslant\lambda_c\leqslant(1-1/Q_p)\lambda_{max}\\(1-2/Q_p)\lambda_{max}< \lambda_c <(1+1/Q_p)\lambda_{min}\end{array}\right.$$
(10)

Similarly it can be derived that the TSR of λc > λg for λc is:

$$\left\{\begin{array}{l}\lambda_{min}\leqslant\lambda_c\leqslant\lambda_{max}\\(1+1/Q_p)\lambda_{min}\leqslant\lambda_c\leqslant(1+1/Q_p)\lambda_{max}\\(1-1/Q_p)\lambda_{max}<\lambda_c<(1+2/Q_p)\lambda_{min}\end{array}\right.$$
(11)

From Eqs. (10) and (11), λmin and λmax can be determined when the FSR of the tunable filter is selected, so the TSR of λc can be solved for the given Qp. Based on the TSR of λc and Eq. (6), the TSR of λg can be solved. Using these formulas simplifies the dual-wavelength design of LCTFs with different tuning requirements.

To verify the effective of the above method, a simulation analysis is carried out. For this purpose, the theoretical TSRs of λc and λg are first calculated based on the above theoretical equations. This paper uses the data from Table 1 for theoretical calculations and the FSR is 450–750 nm. Figure 3 gives the TSR of λc and λg for Qp values varying from 2 to 5. The Fig. 3 shows that the red region indicates the FSR, the purple and green curves indicate the TSR of λc and λg, respectively. From Fig. 3, the TSRs of both λc and λg are located within the FSR. Specifically, (i) when Qp=2, the TSR of λc is 675–750 nm, and the corresponding TSR of λg is 450–500 nm. (ii) When Qp=3, the TSRs of λc are 450–500 nm and 600–750 nm, respectively, which corresponds to the TSRs of λg for 675–750 nm and 450–562.5 nm, respectively. (iii) When Qp=4, the TSR of λc is 450–675 nm, and that of λg is 600–750 nm and 450–540 nm, respectively, and the corresponding position relationships are λc < λg and λc > λg, respectively. (iv) When Qp=5, the TSRs of λc are 450–540 nm and 600–630 nm, and the corresponding TSRs of λg are 562.5–675 nm and 500–525 nm, respectively. Removing the λc repetitive coverage region, the Qp design values and TSR of the dual-wavelength LCTF system are shown in Table 2. Table 2 shows that the full band coverage of λc can be achieved by adjusting the value of Qp.

Fig. 3
figure 3

The TSR of λc and λg with different Qp in case of (a)λc < λg and (b)λc > λg, respectively.

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Table 2 Design value of Qp for dual-wavelength LCTF.
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The theoretical design results for Table 2 are verified by simulation using Eqs. (3) and (4), and finally the dual-wavelength transmission within the FSR of 450–750 nm is achieved. The value of λc is taken and simulated at intervals of 25 nm in the range of 450–750 nm as shown in Fig. 4. It can be seen that the gain wavelengths are all within the FSR and there is only one λg for each λc. In addition, the gain central wavelength obtained from the simulation and the gain wavelength value calculated by Eq. (6) are completely consistent, which verifies the effective of the above theoretical method. Therefore, the method in this paper can obtain the TSRs of λc and λg, thus simplifying the design of dual-wavelength LCTFs.

Fig. 4
figure 4

Dual-wavelength LCTF transmission curve.

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Dual-wavelength LCTF structural design

As can be seen from above, with the known ne and no of the LC material, the dual-wavelength transmission can be designed with the Eqs. (5)-(9). And then, the demand of LC thickness may be calculated with Eq. (3) and the concrete parameters of the LCTF will be obtained. Therefore, the proposed method is universal to design a dual-wavelength LCTF. To illustrate the effectiveness of the proposed method, with the typical nematic LC material E7 which has ne = 1.746 and no = 1.520, a dual-wavelength LCTF is designed with the working waveband of 450–675 nm and 3 layers. Firstly, according to Eq. (5) and the minimum and maximum λc of 450 nm and 675 nm respectively, the tunable range of ΔLC from the from the first-layer to the third-layer may be calculated as 1.8–2.7 μm, 3.6–5.4 μm and 7.2–10.8 μm, respectively, as shown in Table 3. Then LC thickness d may be computed with Eq. (3) and the ne and no of E7, and the ranges of d1-d3 are obtained as 7.96–11.95 μm, 15.93–23.89 μm, and 31.86–47.78 μm, respectively. However, actually, the thickness of LC thickness cell is limited and the typical cell thickness is not greater than 20 μm. Hence, in the LCTF device, the phase retardation is always realized with the combination of fixed phase retarder and a LC cell32.Thus, the LC tunable phase retarder only accomplishes the phase change section and then, the needed cell thickness is reduced greatly. As the first-layer may be realized with the LC cell with 12 μm, only the second-layer and third-layer need insert a suitable phase retarder. According to the ΔLC in Table 3, the phase changes of the 2nd and 3rd layers can be calculated as 1.8 μm and 3.6 μm, respectively. Thus, the corresponding LC thicknesses are 8 μm and 1.6 μm, respectively. All the theoretical and designed results are lists in Table 3, and the dual-wavelength LCTF may be fabricated with these parameters.

Table 3 Parameters of each structural device.
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To the actual devices, the energy loss should also be considered. The above-mentioned three-cascade dual-wavelength LCTF system consists of three LC elements, four polarizers and two phase retarders. The transmission of a LC cell is about 95% with the front and back surfaces coated with antireflection film. A typical visible polarizer WP25L-VIS produced by Thorlab is selected with the transmission of 90% or so. The transmission of the fixed phase retarder may be up to 99%. Then, the whole transmission of the three-cascade system may be evaluated as shown Fig. 5. It indicates that, the practical transmission of the three-cascade dual-wavelength LCTF system is higher than 50% at all wavelengths, and the highest transmission of the dual-wavelength LCTF system reaches 58.41% at 545 nm.

Fig. 5
figure 5

The practical transmission plot for various wavelengths of the three-cascade dual-wavelength LCTF.

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The above dual-wavelength LCTF structure is designed as an example of E7 and the results show that the LCTF may be fabricated with the designed parameters. For different LC materials, the birefringence Δn is different, and this only affect the thickness of LC. In other words, the dual-wavelength LCTF may be designed for different LC materials only with the changed LC thickness. Therefore, our proposed method is effective for individual LC materials. Furthermore, other characteristics of LC material will also influence the performance of the LCTF. For example, the rotational viscosity of the LC materials will affect the response time, different type LC materials possibly have different transmission ability at different waveband. Hence, the LC material characteristic should be considered to optimal design a suitable dual-wavelength LCTF.

Comparative analysis of LCTFs

The spectral bandwidth is characterized by the full width at half maximum (FWHM) of the transmission wavelength, which can reflect the spectral resolution of LCTF, and is another major factor limiting the static spectral performance of LCTF. The spectral bandwidth ∆λHW of the dual-wavelength LCTF system is primarily determined by the ∆N of the last layer LC:

$$\Delta {\lambda _{HW}}=\frac{{{2^N}{Q_p}{\lambda _c}\lambda _{{tran}}^{2}}}{{\left( {{2^N}{Q_p}{\lambda _c}+{\lambda _{tran}}} \right)\left( {{2^N}{Q_p}{\lambda _c} - {\lambda _{tran}}} \right)}}$$
(12)

where λtran refers to the spectral range transmission wavelength. The smaller the spectral bandwidth ∆λHW the higher the spectral resolution, but simultaneously the lower its transmission energy.

To compare and analyse the difference in transmission energy between dual-wavelength and single-wavelength LCTF systems, a calculation method for the transmission energy of dual-wavelength LCTF systems is firstly established, and the method is also applicable to single-wavelength LCTF systems. For the dual-wavelength LCTF system, the energy transmission at both wavelengths and the number of layers N needs to be considered simultaneously, and the final transmission energy can be calculated by the following formula:

$$E=\int_{{{\lambda _c} - \Delta {\lambda _{HW\_c}}}}^{{{\lambda _c}+\Delta {\lambda _{HW\_c}}}} {\left[ {\prod\limits_{{n=1}}^{N} {{{\cos }^2}\left( {\frac{{{2^n}\pi {Q_p}{\lambda _c}}}{{\lambda \left( s \right)}}} \right)} } \right]} ds+\int_{{{\lambda _g} - \Delta {\lambda _{HW\_g}}}}^{{{\lambda _g}+\Delta {\lambda _{HW\_g}}}} {\left[ {\prod\limits_{{n=1}}^{N} {{{\cos }^2}\left( {\frac{{{2^n}\pi {Q_p}{\lambda _g}}}{{\lambda \left( s \right)}}} \right)} } \right]} ds$$
(13)

where ∆λHW_c and ∆λHW_g are the spectral bandwidths at the central wavelength and gain wavelength, respectively.

To quantitatively compare and analyse the difference in spectral transmission energy between single-wavelength LCTF and dual-wavelength LCTF systems, it is assumed that they have the same spectral resolution, that is the same FWHM at the central wavelength λc for both the single-wavelength LCTF and the dual-wavelength LCTF. Also, the device losses of the system are ignored in the calculations. Firstly, the bandwidth of each transmission wavelength is calculated according to Eq. (12) and the results are shown in Fig. 6. It is evident that most of the central wavelength bandwidths of the five-cascade single-wavelength LCTF and three-cascade dual-wavelength LCTF systems are the same, and the Qp value of the dual-wavelength LCTF is 4 at this time. However, the condition of Eqs. (10) or (11) is not satisfied when Qp is taken to be 4 in the 675–750 nm band. Therefore, the dual-wavelength LCTF system is selected in this band with N = 4 and Qp = 3 for comparative analyses to make its spectral resolution approximately the same as that of the single-wavelength system.

Fig. 6
figure 6

Bandwidth of central wavelength for LCTFs.

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After determining the spectral resolution at each wavelength, the transmission energies E_sin and E_dua of the single- wavelength and dual-wavelength LCTF systems are calculated according to (13), and the results obtained by normalising based on the incident energy are shown in the bar chart in Fig. 7. Figure 7 represents the energy transmission capability of the two systems with the same central wavelength, furthermore, based on the feature that the dual-wavelength LCFT system has λg co-transmission with λc, so that E_dua is the sum of λc and λg transmission energy. Therefore, E_sin has a linear increasing relationship with λ, while E_dua shows a phase-increasing change with λ, and there is a case of 2 transmission energy jumps. The first jump is caused by the change in the position of λg the long-wavelength and short-wavelength energy transmission capacity causes the cliff-like decrease of E_dua. The second jump is caused by a sharp decrease in the system transmission energy due to the ∆λHW_c enhancement. This paper defines the system transmission energy improvement rate as η = (E_dua - E_sin)/E_sin to quantitatively analyse the superiority of dual-wavelength LCTF. The spectral resolution of the two systems in the 450–650 nm band can be visualised by the line graph in Fig. 7, which shows that the dual-wavelength LCTF can improve the transmission energy by about 1.8 times at short wavelengths and by about 70% at long wavelengths. The spectral resolution of the two systems in the 650–750 nm band is different, and the initial parameters of the dual-wavelength LCTF system are selected as N = 4 and Qp=3. At this time, the dual-wavelength system can improve its transmission energy by about 7% under the case of 33% improvement of the spectral resolution compared with that of the single-wavelength system. Evidently, the dual-wavelength LCTF has obvious advantages in both energy transmission and spectral resolution, and the transmission energy of the system can be significantly improved under the same resolution, thus improving the SNR of the system. Meanwhile, the problem of conflicting spectral resolution and system transmission energy in dual-wavelength target detection scenarios can also be effectively solved.

Fig. 7
figure 7

The energy transmission capacity for LCTFs.

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Conclusion

Based on the traditional Lyot-type LCTF structure, a dual-wavelength transmission system with high SNR is proposed in this paper. Dual-channel filtering can not only increase the effective information, but also improve the transmission energy of the system and the SNR of the system. The dual-channel filtering can increase both the effective information and the transmission energy, thus improving the SNR of the system. The proposed transmission factor Qp is the primary factor that affects the number of transmission wavelengths and spectral bandwidth within the FSR, and the corresponding theoretical equations are given and verified by simulation. Simulation results show that the method proposed in this paper can achieve dual-wavelength transmission of LCTF within the wide-band FSR. The results of the comparative analysis of single-wavelength and dual-wavelength LCTF systems show that, (1) when the central wavelength resolution is the same, the dual-wavelength LCTF can improve the transmission energy of the system by increasing filter channels. Moreover, the transmission energy at short-wavelength and long-wavelength can be increased by about 1.8 times and 70%, respectively. (2) With the spectral resolution at the central wavelength of the dual-wavelength LCTF at a significant advantage, that is, when the spectral resolution is increased by 33%, the system transmission energy is still able to be increased by about 7%. Obviously, the dual-wavelength LCTF can also effectively solve the problem of the contradiction between the spectral resolution and the transmission energy of the system, and has the universality and realizability. Therefore, the dual-wavelength transmission proposed in this paper can provide a reference for the LCTF system to improve the SNR, the problem of the contradiction between the spectral resolution and the transmission energy.