1. Introduction

Biomedical applications of optical spectroscopy and spectral imaging deal with light covering a broad optical spectrum. Optical filters, such as various longpass, shortpass and bandpass interference filters, or color absorptive filters [1], are widely used to transmit light at the wavelengths of interest and block unwanted light at other wavelengths. These thin-film interference filters are advantageous in their extremely high transmission in the passbands and steep spectral edges at the cutoff wavelengths. Unfortunately, they will not work when dynamic filtering or complex transmittance spectra are required. Switching between different filters using a filter wheel is an option for dynamic filtering but it has limitations in speed and flexibility.

Tunable optical filters, originally invented for wavelength division multiplexing in fiber based communication, have now been exploited in other applications including spectral imaging [2–5], optical computing [6] and compressive sensing [7,8]. Researchers have reported on the implementation of tunable optical filters using various techniques including liquid crystals [9–11], acoustic optics [12–14] and Fabry-pérot interferometers [15,16]. These tunable filters were mainly used as bandpass filters with Gaussian-like transmittance spectra, while more complex transmittance spectra such as those mimicking the Raman spectra of biological samples were beyond their reach.

Recent years have seen a growing popularity in the use of digital micromirror devices (DMDs) as a powerful tool for light manipulation in both commercial and research applications [17,18]. Manufactured with a huge number of individually addressable micromirrors, a DMD has a high spatial resolution, a fast pattern alternating speed and is easily programmable. When coupled with a light disperser (a prism or a diffraction grating), the DMD can function as an optical filter capable of real-time spectral modulation. In early studies, DMD based optical filters were also employed as bandpass filters, focusing on the selection of one or a few separate spectral bands. For example, Wilcox and his colleagues [7] used a DMD coupled with a volume holographic grating (VHG) to generate binary filter patterns for compressive chemical quantification based on Raman spectroscopy. Later, the functionality of DMD based filters was extended to manipulation of the full spectral range for implementing an optical attenuator [19] and optical equalizer [20]. Recently, Luo et al developed a programmable light source using a supercontinuum laser as the light source and a DMD as the spectral modulator [21]. The combination of a prism and an echelle grating was employed to realize a two-dimensional light dispersion on the DMD with a high spectral resolution of 1 nm. The DMD was divided into numerous independent channels and the intensity spectrum of every individual channel was characterized in sequence for subsequent spectral synthesis. Due to the poor signal to noise ratio, each channel actually consisted of a block of 5 × 5 pixels. Even so, only a small portion of the channels had enough signals to be used as the effective basis for spectral synthesis.

The sequential channel scanning procedure mentioned above is a straightforward method for filter calibration. In principle, to get better accuracy and resolution in spectral tuning, more independent channels in the DMD will be necessary, with the maximum channel number achieved when each channel consists of only one micromirror. However, the decrease in signal intensity with fewer pixels in one channel can lead to larger calibration errors due to the limited signal-to-noise ratio of the detector for filter calibration. To address this problem, we propose a Hadamard transform based calibration method for DMD based programmable optical filters. The advantage of Hadamard transform in signal multiplexing has been demonstrated for signal to noise ratio enhancement in other applications where the signal level is low [22,23]. In this paper, a DMD based programmable optical filter system is established and calibrated with both the sequential scanning method and the proposed Hadamard transform based method for comparison. After calibration, three representative types of filters, including the bandpass filter, Gaussian filter and principal component based filter, are synthesized to evaluate the spectral tuning accuracy.

2. Methods

2.1 Experimental setup

The schematic layout of the experimental setup is shown in Fig. 1. The programmable optical filter consists mainly of three parts: a VHG (WP-1200/840-50.8, Wasatch Photonics, USA) acting as a dispersive element to disperse incoming light of each wavelength to a different angle; an achromatic lens (AC508-075-B, Thorlabs, USA) focusing the dispersed light onto a DMD (DLP Light Crafter 4500, Texas Instruments, USA); the DMD responsible for spectral selection and filter transmittance synthesis. The row spacing of the DMD is smaller than its column spacing, so the DMD is placed in the portrait orientation to achieve a higher spectral tuning resolution. The light selected by the DMD will come back in the same optical path as the incoming light. A 50:50 beam splitter is used to separate the incoming and outgoing beams.

 

Fig. 1 Schematic layout of the experimental setup for implementing and evaluating the programmable optical filter. Note that the module inside the dashed box is the programmable optical filter and the rest is for evaluating it.

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For the calibration of the optical filter, a light-emitting diode (LED; OSL1-EC, Thorlabs, USA) is used as the light source and a spectrometer (SR303i, Andor Technology, UK) is used as the detection device. The spectral region of interest is set to be 823 to 914 nm, corresponding to a Raman shift range of 600 – 1800 cm^–1 when the excitation wavelength is 785 nm. The spectral resolution of the programmable filter, which depends on the focal spot size of monochromatic light of a single wavelength on the DMD and the dispersive power of the VHG, is measured to be about 2 nm.

2.2 System calibration

The above optical filter is first calibrated in the sequential scanning scheme. Assume that the wavelength range of interest is sampled at m discrete points in the detection device. For a spectrometer, m is the number of wavelengths in a recorded spectrum. A transmittance spectrum is an $m\times 1$ column vector specifying the transmittance at each wavelength.

The DMD is divided into n independently programmable channels, each corresponding to one row of DMD pixels, along the wavelength direction. For the $i^{th}$channel, $i\in \left\{1,2,\mathrm{...}n\right\}$, its transmission spectrum $T_i$can be measured by assigning 1 to all rows of DMD pixels for this channel and 0 to the rest of DMD pixels. Scanning across all channels, we can get an ensemble of the transmittance spectra, which composes the calibration matrix of the programmable filter, i.e.:

For Hadamard transform based calibration with an n channel division on the DMD, a Sylvester matrix of order n, denoted by S, is used for generating the Hadamard base patterns. The first row of S is produced by quadratic residue construction and S is a left circulant matrix in which each row is obtained by cyclically shifting the previous row to the left by one position [24]. As an example, Fig. 2 gives the 11th-order S matrix and the corresponding DMD patterns in a checkerboard-like graph.

 

Fig. 2 (a) The 11th order S matrix and (b) a checkerboard-like illustration of the corresponding DMD pattern sequence with 11 channels. Each row of the checkerboard represents a DMD pattern and each column represents a DMD channel. White and black squares stand for the ‘on’, i.e. value 1 in the S matrix, and ‘off’, i.e. value 0 in the S matrix, status of the channels, respectively.

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The inverse of S is easily obtained by:

For comparison, the programmable filter is calibrated using both the sequential scanning method and the Hadamard transform based method. In each calibration method, a series of channel numbers are used to examine how the resulting spectral accuracy evolves with the channel number.

2.3 Filter function synthesis

Normally, a DMD is used as a digital spatial filter. At any moment, the incident light onto a DMD pixel, which is essentially a micromirror, is directed toward (when the mirror is assigned a value of 1) or away from (when the mirror is assigned a value of 0) the detection device, making a binary transmittance of either 1 or 0. However, if we consider the average transmittance over the working period of the DMD, any transmittance value between 0 and 1 can be achieved by programming the duty cycle of the micromirror, defined as the proportion of one period in which the mirror is assigned as 1.

If the duty cycles of the DMD channels are denoted as$D={[d_1,d_2,\mathrm{...},d_n]}^T$, the overall transmission spectrum, or filter function, F, can be expressed as the linear combination of the transmission spectrum of each individual DMD channel in the form of:

It should be noted here that a real programmable optical filter system suffers from energy loss intrinsic of the constituent elements such as the DMD, dispersive gratings and lenses. There is a maximum transmittance at each wavelength that the programmable filter can possibly reach. This transmittance ceiling spectrum, as shown by the green solid curve in Fig. 3, can be obtained by recording the transmittance spectrum when all channels are assigned as 1 at all times within the specified working period. A target transmittance spectrum as shown by the blue dotted curve needs to be scaled down by a proper factor to generate a scaled version as shown by the red dashed curve before Eq. (7) is applied. Otherwise some calculated duty cycles might exceed 1. The scaling factor is the minimum value of the wavelength-wise ratio of the target transmittance spectrum to the transmittance ceiling spectrum.

 

Fig. 3 Target transmittance spectrum after scaling down.

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Ultimately, the duty cycles of all channels can be converted into a DMD pattern sequence for display by time multiplexing. The integration time of each pattern was 100 ms. The synthesized transmittance spectrum was measured and compared to the target transmittance spectrum by calculating the relative root mean square error (RMSE) which was defined as

3. Results

3.1 Performance of the programmable filter in terms of relative RMSE

After the calibration procedure, the programmable filter is configured to synthesize three common types of optical filters including the bandpass filter, Gaussian filter and principal-component based filter. Note that the transmittance spectrum of the last filter was derived out of the principal component analysis of cell spectra, which can optimize the accuracy in the spectral reconstruction of Raman spectra from narrow-band measurements [25,26]. The performance of the programmable filter is evaluated in terms of the relative RMSE of the synthesized transmittance spectrum relative to the target transmittance spectrum. Figure 4 plots the relative RMSE for the synthesis of different types of filters as a function of channel number with either the sequential scanning method or the proposed Hadamard transform based method used in calibration. For the sequential scanning method, the five channel numbers used are 30, 60, 114, 380 and 1140, corresponding to 38, 19, 10, 3, and 1 row(s) of DMD pixels per channel, respectively. When the channel number is not greater than 114, there is a clear downward trend of relative RMSE with an increasing channel number for all three filter types. However, when the channel number goes beyond 114, the relative RMSE increases dramatically, suggesting that the signal-to-noise ratio is too low to achieve decent calibration. The above results indicate that the best spectral accuracy for the sequential scanning calibration method is limited by the signal reaching the spectrometer relative to the noise. For the Hadamard transform based calibration method, since the S matrix is produced with quadratic residue construction, its order has to be a prime number in the form of 4 × m + 3 where m is a non-negative integer. Therefore, the channel numbers used for the Hadamard transform-based calibration method are 31, 59, 107, 379 and 1123, corresponding to 37, 19, 10, 3, and 1 row(s) of DMD pixels per channel, respectively. For all three filter types, the relative RMSE monotonically decreases with an increasing channel number, with the smallest relative RMSE values achieved at the maximum channel number. Moreover, these values are smaller than the smallest achieved relative RMSE with the sequential scanning calibration method, demonstrating that the proposed Hadamard transform calibration method is effective in enhancing the signal-to-noise ratio during calibration and improving the spectral accuracy of the programmable filter.

 

Fig. 4 Performance of the programmable filter in terms of relative RMSE for the synthesis of three different types of filters as a function of channel number used for both the sequential scanning calibration method and the Hadamard transform based method. Note that both the x and y axes are in the log-10 scale.

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3.2 Gaussian filter

The transmittance spectrum of a typical Gaussian filter contains one peak and the transition from the peak to the tail regions is smooth. The synthesized spectrum of a Gaussian filter using the Hadamard transform-based calibration method with 1123 channels is shown in Fig. 5. It can be seen that the synthesized filter spectrum matches almost perfectly with the target spectrum with a tiny relative RMSE of 0.11%.

 

Fig. 5 Transmittance spectrum of a Gaussian filter synthesized using the Hadamard transform-based calibration method with 1123 channels and the target spectrum. The red curve and the blue curve represent the target transmittance spectrum and the synthesized transmittance spectrum, respectively.

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3.3 Bandpass filter

Figure 6 depicts the transmittance spectrum of a bandpass filter synthesized using the Hadamard transform based calibration method with 1123 channels. Although well calibrated, the synthesized transmittance spectrum still has a considerable deviation from the target spectrum. The relative RMSE is 5.53% in this case and comes mainly from the ringing artifacts near the two cutoff wavelengths. This means that the limiting factor for the spectral accuracy is the spectral resolution of the DMD which depends mainly on the dispersive power of the VHG.

 

Fig. 6 Transmittance spectrum of a bandpass filter synthesized using the Hadamard transform-based calibration method with 1123 channels and the target spectrum. The red curve and the blue curve stand for the target transmittance spectrum and the synthesized transmittance spectrum, respectively.

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3.4 Principal component based filter

Figure 7 depicts the transmittance spectrum of a principal component based filter synthesized using the Hadamard transform based calibration method with 1123 channels. The transmittance spectrum is complex since it mimics the Raman spectrum of cells with multiple sharp peaks [24]. The relative RMSE is estimated to be 1.94%, which manifests excellent agreement between the synthesized spectrum and the target spectrum.

 

Fig. 7 Transmittance spectrum of a principal component based filter synthesized using the Hadamard transform-based calibration method with 1123 channels and the target spectrum. The red curve and the blue curve stand for the target transmittance spectrum and the synthesized transmittance spectrum, respectively.

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4. Discussion

In this paper we experimentally demonstrated the effectiveness of a Hadamard transform based calibration method for enhancing the spectral tuning accuracy of a DMD based programmable optical filter. Since this work intended to focus on the calibration methodology, the optical setup was not optimized for light transmission efficiency. A 50:50 beam splitter was used in the optical path to simplify system alignment. However, the double pass of light through the beam splitter reduced the transmitted light power by four folds, which yielded a theoretical maximum transmittance of 25%. Together with the efficiency loss caused by the gratings and the DMD, the overall efficiency was slightly below 20% in the current setup. In the future work, we could use a total internal reflection (TIR) prism to direct the outgoing beam from the DMD onto another set of the achromatic lens and VHG. In this way, the beam splitter can be removed and the overall efficiency could reach 70 ~80%.

Another aspect worth noting is the time multiplexing fashion in the process of filter synthesis. In our work, after the duty cycles are determined, they are quantized into 5-bit digits, which correspond to 32 patterns in one working period of the DMD. The display duration of each pattern equals the inverse of the binary pattern alternating frequency of the micromirrors. Increasing the pattern number will reduce error in the quantization process, but then the working period will be longer and the corresponding refreshing frequency will be lower, which will be disadvantageous for capturing dynamic events. For this reason, a tradeoff must be made between the accuracy of quantization and the refreshing frequency according to the specific requirements of different applications.

Besides the quantization error, the spectral resolution of a programmable filter is also a key factor influencing the spectral accuracy. The spectral resolution depends on the focal spot size for monochromatic light at a single wavelength on the DMD. It is about 2 nm in our setup. Figure 6 indicates that although well calibrated with our Hadamard transform method, there is still considerable relative RMSE in the synthesized transmittance spectrum relative to the target spectrum. This is attributed mainly to the regions near the cutoff wavelengths of the bandpass filters. By further increasing the spectral resolution, i.e. the sharpness of the spectral edges, the relative RMSE can be further reduced.

Nowadays, a number of applications benefit from tunable optical filters, such as biomedical imaging, chemical analysis, optical communication, optical sensing, tunable laser systems and optical computing. There is a rising demand for tunable filters with low energy loss, large tuning range, fast tuning speed and high spectral tuning accuracy. In this context, DMD based programmable optical filters, although limited in transmission efficiency when compared to conventional thin-film filters, are excellent in other properties. The proposed Hadamard transform calibration method is expected to play an important role in achieving better spectral accuracy in those applications demanding high spectral accuracy such as optical computing and optical communication. Moreover, the proposed could be also useful in programmable optical filters based on other dynamic optical elements such as spatial light modulators.

5. Conclusion

We presented a Hadamard transform based calibration method for DMD based programmable optical filters and achieved enhanced spectral accuracy in filter synthesis in case of high spectral resolution compared to the traditional sequential scanning method. The advantage of the proposed method is demonstrated for the synthesis of three typical filters including the bandpass filter, Gaussian filter and principal component based filter. The results show that when calibrated with the proposed method, the programmable filter exhibits a consistent decrease in the relative RMSE with an increasing channel number for all three filters. The smallest relative RMSE values are therefore achieved when each channel contains only one DMD pixel. In contrast, for the sequential scanning method, the relative RMSE increases dramatically when each channel contains three or fewer DMD pixels.