## Design and implementation of a sub-nm resolution microspectrometer based on a Linear-Variable Optical Filter |

Optics Express, Vol. 20, Issue 1, pp. 489-507 (2012)

http://dx.doi.org/10.1364/OE.20.000489

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### Abstract

In this paper the concept of a microspectrometer based on a Linear Variable Optical Filter (LVOF) for operation in the visible spectrum is presented and used in two different designs: the first is for the narrow spectral band between 610 nm and 680 nm, whereas the other is for the wider spectral band between 570 nm and 740 nm. Design considerations, fabrication and measurement results of the LVOF are presented. An iterative signal processing algorithm based on an initial calibration has been implemented to enhance the spectral resolution. Experimental validation is based on the spectrum of a Neon lamp. The results of measurements have been used to analyze the operating limits of the concept and to explain the sources of error in the algorithm. It is shown that the main benefits of a LVOF-based microspectrometer are in case of implementation in a narrowband application. The realized LVOF microspectrometers show a spectral resolution of 2.2 nm in the wideband design and 0.7 nm in the narrowband design.

© 2011 OSA

## 1. Introduction

1. R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. **15**(7), S145–S152 (2005). [CrossRef]

2. G. Minas, R. F. Wolffenbuttel, and J. H. Correia, “A lab-on-a-chip for spectrophotometric analysis of biological fluids,” Lab Chip **5**(11), 1303–1309 (2005). [CrossRef] [PubMed]

3. J. H. Correia, G. de Graaf, S. H. Kong, M. Bartek, and R. F. Wolffenbuttel, “Single-chip CMOS optical microspectrometer,” Sens. Actuators A Phys. **82**(1-3), 191–197 (2000). [CrossRef]

3. J. H. Correia, G. de Graaf, S. H. Kong, M. Bartek, and R. F. Wolffenbuttel, “Single-chip CMOS optical microspectrometer,” Sens. Actuators A Phys. **82**(1-3), 191–197 (2000). [CrossRef]

4. S. W. Wang, M. Li, C. S. Xia, H. Q. Wang, X. S. Chen, and W. Lu, “128 channels of integrated filter array rapidly fabricated by using the combinatorial deposition technique,” Appl. Phys. B **88**(2), 281–284 (2007). [CrossRef]

*N*optical resonators of different thickness (which is required for

*N*different spectral channels) is

^{2}logN. This approach becomes impractical for a large value of

*N*, especially when considering the thickness tolerances in the deposited or back-etched layers. A solution is the application of a Linear-Variable Optical Filter (LVOF). Figure 1 shows the schematic of a LVOF.

5. R. R. McLeod and T. Honda, “Improving the spectral resolution of wedged etalons and linear variable filters with incidence angle,” Opt. Lett. **30**(19), 2647–2649 (2005). [CrossRef] [PubMed]

6. A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “CMOS-compatible LVOF-based visible microspectrometer,” Proc. SPIE **7680**, 76800W (2010). [CrossRef]

7. A. Emadi, S. Grabarnik, H. Wu, G. de Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Spectral measurement using IC-compatible linear variable optical filter,” Proc. SPIE **7716**, 77162G (2010). [CrossRef]

8. A. Emadi, H. Wu, S. Grabarnik, G. De Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Fabrication and characterization of IC-compatible linear variable optical filters with application in a micro-spectrometer,” Sens. Actuators A Phys. **162**(2), 400–405 (2010). [CrossRef]

10. A. Emadi, H. Wu, S. Grabarnik, G. de Graaf, and R. F. Wolffenbuttel, “Vertically tapered layers for optical applications fabricated using resist reflow,” J. Micromech. Microeng. **19**(7), 074014 (2009). [CrossRef]

## 2. Design and fabrication of an LVOF-based microspectrometer for the visible spectrum

*N*, of a Fabry-Perot device can be expressed as:

*N*= 2n

*d*/λ

_{o}or

*N*=

*d*/(2×QWOT), where λ

_{o}denotes the reference wavelength in free space, n is the refractive index of the cavity,

*d*is the thickness of the cavity and QWOT (Quarter Wavelength Optical Thickness) is a quarter of the reference wavelength of the cavity. Higher resonance orders result in a smaller FWHM (Full Width Half Maximum), meaning sharper transmission peaks and thus a high resolving power. However, a high value for

*N*also results in a reduced operating bandwidth of the LVOF: the Free Spectral Range (FSR).

*N*. This is illustrated in Fig. 2 , which is the simulated transmission through a FP filter with

*d*=900 nm, n=1.5 and mirror reflectivity

*R*=0.9.

*N*, the transmission peak of the order

*N*+1 is always closer than the peak of the order

*N*-1. Therefore, we can define the FSR for order

*N*as the distance between the peaks of order

*N*and

*N*+1. This FSR

_{N}can be expressed as [9]:

*f*, results [11]:

*F*is the coefficient of finesse and depends on the reflectivity of the FP mirrors as:

*F*=4

*R*/(1-

*R*)

^{2}.

12. V. Krajicek and M. Vrbova, “Laser-induced fluorescence spectra of plants,” Remote Sens. Environ. **47**(1), 51–54 (1994). [CrossRef]

13. K. Burns, K. B. Adams, and J. Longwell, “Interference measurements in the spectra of neon and natural mercury,” J. Opt. Soc. Am. **40**(6), 339–344 (1950). [CrossRef]

14. NIST atomic spectra database, Online: http://www.nist.gov/pml/data/asd.cfm

^{th}order to meet the resolving power specification, which is incompatible with the coverage of the entire visible spectrum. Therefore, two special cases are pursued. The first one is a relatively wide band microspectrometer intended for 580 nm – 720 nm spectral range, whereas the second is for a smaller bandwidth of 615 nm – 680 nm. In both cases signal processing is used to achieve the highest possible spectral resolution, as is explained in Section 4.

_{0}is the reference wavelength and n

_{2}and n

_{1}are refractive indexes of the dielectric materials. Maximum value for Free Spectral Range (FSR) in turn is Δλ/2. TiO

_{2}and SiO

_{2}are selected as the high-n and low-n dielectric materials respectively. These two materials were the available dielectric materials with the highest refractive index difference. The refractive index of TiO

_{2}is about 2.25 and the refractive index of SiO

_{2}is about 1.45 in the visible spectral range. Applying this equation, results in Δλ = 720 nm – 470 nm = 250 nm. This is the reflection bandwidth of the Fabry-Perot mirrors, which is inherent to the use of dielectric materials. The operating wavelength range is in between 470 nm and 720 nm. The maximum possible FSR is Δλ/2 = 125 nm. However, the Fabry-Perot structure of the LVOF is intended for use at a higher-order mode.

## 3. LVOF fabrication

8. A. Emadi, H. Wu, S. Grabarnik, G. De Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Fabrication and characterization of IC-compatible linear variable optical filters with application in a micro-spectrometer,” Sens. Actuators A Phys. **162**(2), 400–405 (2010). [CrossRef]

10. A. Emadi, H. Wu, S. Grabarnik, G. de Graaf, and R. F. Wolffenbuttel, “Vertically tapered layers for optical applications fabricated using resist reflow,” J. Micromech. Microeng. **19**(7), 074014 (2009). [CrossRef]

## 4. Signal processing algorithm

*N*= FSR/FWHM. For the LVOF with layers defined in Table 1 and transmission spectra shown in Fig. 3, this equals to 40 channels, each 2.5 nm wide. This is a significant advantage compared to Fabry-Perot filters with a discrete number of elements [3

3. J. H. Correia, G. de Graaf, S. H. Kong, M. Bartek, and R. F. Wolffenbuttel, “Single-chip CMOS optical microspectrometer,” Sens. Actuators A Phys. **82**(1-3), 191–197 (2000). [CrossRef]

*C*in matrix C is defined as the response of channel

_{ij}*i*of the detector to component

*j*in the spectrum (

*i, j*= 1...N). The matrix C can be directly constructed from the data of a calibration measurement process. The maximum value of N is the number of the pixels on the camera, but can be limited by the spectral capability of the calibrating instrument (a monochromator). Hence, the measured intensity on the detector channels can be described as:

*d*denotes the measured intensity in channel

_{i}*i*and

*I*denotes the input light spectrum intensity in channel

_{i}*i*that has to be calculated. In other words, Matrix

*D*is the raw data recorded on the camera pixels, matrix

_{1N}*I*is the spectrum of incident light that has to be calculated and matrix

_{1N}*C*is the calibration matrix which is determined during the calibration process. Let us consider once again calibration procedure shown schematically in Fig. 5. The light from a broadband source (Xenon lamp) is filtered by a monochromator and the selected wavelength is varied in the spectral range of interest for all the N spectral channels. For each spectral channel from the above equations, the calibration step is equivalent to deliberately having:

_{NN}*m*is selected from the monochromator. In this case the recorded intensities on the pixels give the values of column

*m*of the C matrix:

*C*matrix. Plotting the rows and columns of the C matrix, Fig. 6 , gives good understanding about the spectral response of the LVOF. The

_{NN}*i*

^{th}row of the C matrix indicates the spectral response of the

*i*

^{th}channel in the detector array and the

*j*

^{th}column of the C matrix shows the recorded response of the detector array to the

*j*

^{th}monochromatic wavelength component.

*C*has no singularity and it is in principle possible to take the inverse transform of the matrix (the matrix

*I*can be calculated as:

*I*). However, since matrix

_{1N}= C_{NN}^{−1}.D_{1N}*D*is the result of actual measurement it contains uncertainties, which are primarily due to insufficient collimation and out of band signal. In case the uncertainties are significant as compared to the signal (low SNR), the direct calculation of matrix

*I*would result in negative values in some of the spectral channels, which is nonphysical. In this paper an iterative procedure was implemented to calculate the best approximation of matrix

*I*at the presence of uncertainties by minimizing matrix

*E*:

*Î*is an estimate of

*I*. The Least Mean Square (LMS) algorithm is implemented based on the following equations:

*μ*results in faster convergence of the algorithm. However, it can also result in completely instability of the algorithm with diverging results. The goal of the algorithm is to decrease the mean square of

*2nd/N*, whereas the dashed curve shows the illumination in case of a monochromatic component at

*2nd/(N-1).*Similar to the FWHM we can define the width of such illuminated regions as HPLW (Half-Power Line Width). The HPLW is the width of the region where the intensity of light is half of its maximum. We can note the relation between HPLW and FWHM is expressed as: HPLW≈FWHM/θ, where θ is the angle of the slope of the LVOF. The width of the regions that are illuminated on the detector array by the different monochromatic light sources is different. This is due to the fact that they have different FWHM, as expressed by Eq. (2). This difference allows the algorithm to differentiate between these two peaks, even when a linear combination of these two monochromatic light spots is used for illuminating the detector.

## 5. Spectral measurement for narrowband application

8. A. Emadi, H. Wu, S. Grabarnik, G. De Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Fabrication and characterization of IC-compatible linear variable optical filters with application in a micro-spectrometer,” Sens. Actuators A Phys. **162**(2), 400–405 (2010). [CrossRef]

*d*is the diameter of the aperture and

*φ*is maximum acceptable angle of incidence on the LVOF. Since these equations depend on

*φ*, transmission through the multilayered Fabry–Perot filter (which can be at any position along the length of the LVOF) is simulated at different angles. Figure 9b shows the result.

*φ*can be selected. For example, for spectral accuracy better than 0.5 nm,

*φ*= 3° is an acceptable choice. Hence, for our LVOF the following values are obtained: NA=0.4,

*f*=12.5 mm, D=10 mm and

*d*=1.3 mm. Consequently, the NA allows direct connection to a glass fiber (NA in the range 0.22 to 0.4, depending on type), which is not the as easily to achieve in grating-based systems. Based on these considerations, suitable collimating optics has been implemented in a C-mount holder to be put on the top of the CMOS camera, shown in Fig. 10 .

*N*channels. Since we are limited by the spectral bandwidth of the light component provided by the monochromator used, the spectral width of each channel (the wavelength increment) is 0.5 nm. Covering 615 to 665 with 0.5 increments implies:

*N*= 101. Consequently, N equally spaced pixels are selected along the length of the LVOF to result in the detector channels expressed by vector

*D*in Eq. (4).

_{1N}*I*), means finding weighing coefficients for measured intensity profiles in calibration in a way that the resulted superposition will converge to the recorded intensity in Fig. 13. This is illustrated in Fig. 14 by the top curve, with the dotted markers.

_{1N}*D*is the error in the measurement, which is expressed by vector

_{1N}*E*in Eq. (6). It is obvious that the level of noise and other unwanted effects in the measurement, determines the minimum achievable error. Figure 15a compares the blue dotted curve in Fig. 14 and

*D*. The numerical difference between these two is plotted in Fig. 15b.

_{1N}18. S. Grabarnik, A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “High-resolution microspectrometer with an aberration-correcting planar grating,” Appl. Opt. **47**(34), 6442–6447 (2008). [CrossRef] [PubMed]

18. S. Grabarnik, A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “High-resolution microspectrometer with an aberration-correcting planar grating,” Appl. Opt. **47**(34), 6442–6447 (2008). [CrossRef] [PubMed]

13. K. Burns, K. B. Adams, and J. Longwell, “Interference measurements in the spectra of neon and natural mercury,” J. Opt. Soc. Am. **40**(6), 339–344 (1950). [CrossRef]

14. NIST atomic spectra database, Online: http://www.nist.gov/pml/data/asd.cfm

## 6. Spectral measurement for wideband application

18. S. Grabarnik, A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “High-resolution microspectrometer with an aberration-correcting planar grating,” Appl. Opt. **47**(34), 6442–6447 (2008). [CrossRef] [PubMed]

## 6. Conclusions

## Acknowledgments:

## References and links

1. | R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. |

2. | G. Minas, R. F. Wolffenbuttel, and J. H. Correia, “A lab-on-a-chip for spectrophotometric analysis of biological fluids,” Lab Chip |

3. | J. H. Correia, G. de Graaf, S. H. Kong, M. Bartek, and R. F. Wolffenbuttel, “Single-chip CMOS optical microspectrometer,” Sens. Actuators A Phys. |

4. | S. W. Wang, M. Li, C. S. Xia, H. Q. Wang, X. S. Chen, and W. Lu, “128 channels of integrated filter array rapidly fabricated by using the combinatorial deposition technique,” Appl. Phys. B |

5. | R. R. McLeod and T. Honda, “Improving the spectral resolution of wedged etalons and linear variable filters with incidence angle,” Opt. Lett. |

6. | A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “CMOS-compatible LVOF-based visible microspectrometer,” Proc. SPIE |

7. | A. Emadi, S. Grabarnik, H. Wu, G. de Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Spectral measurement using IC-compatible linear variable optical filter,” Proc. SPIE |

8. | A. Emadi, H. Wu, S. Grabarnik, G. De Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Fabrication and characterization of IC-compatible linear variable optical filters with application in a micro-spectrometer,” Sens. Actuators A Phys. |

9. | M. Born and E. Wolf, |

10. | A. Emadi, H. Wu, S. Grabarnik, G. de Graaf, and R. F. Wolffenbuttel, “Vertically tapered layers for optical applications fabricated using resist reflow,” J. Micromech. Microeng. |

11. | A. Emadi, “Linear-variable optical filters for microspectrometer application,” PhD Thesis, Technical University of Delft (2010). |

12. | V. Krajicek and M. Vrbova, “Laser-induced fluorescence spectra of plants,” Remote Sens. Environ. |

13. | K. Burns, K. B. Adams, and J. Longwell, “Interference measurements in the spectra of neon and natural mercury,” J. Opt. Soc. Am. |

14. | NIST atomic spectra database, Online: http://www.nist.gov/pml/data/asd.cfm |

15. | M. P. Wisniewski, R. Z. Morawski, and A. Barwicz, “Algorithms for interpretation of spectrometric data- A comparative study,” Instrumentation and Measurement Technology Conference, 2000. IMTC 2000. Proceedings of the 17th IEEE, |

16. | D. Massicotte, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrometric data correction-Part I: an iterative algorithm of deconvolution,” IEEE Trans. Instrum. Meas. |

17. | M. H. Hayes and H. Monson, “Recursive least squares,” in |

18. | S. Grabarnik, A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “High-resolution microspectrometer with an aberration-correcting planar grating,” Appl. Opt. |

**OCIS Codes**

(220.0220) Optical design and fabrication : Optical design and fabrication

(300.6190) Spectroscopy : Spectrometers

(310.4165) Thin films : Multilayer design

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: September 28, 2011

Revised Manuscript: November 14, 2011

Manuscript Accepted: November 29, 2011

Published: December 21, 2011

**Citation**

Arvin Emadi, Huaiwen Wu, Ger de Graaf, and Reinoud Wolffenbuttel, "Design and implementation of a sub-nm resolution microspectrometer based on a Linear-Variable Optical Filter," Opt. Express **20**, 489-507 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-489

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### References

- R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng.15(7), S145–S152 (2005). [CrossRef]
- G. Minas, R. F. Wolffenbuttel, and J. H. Correia, “A lab-on-a-chip for spectrophotometric analysis of biological fluids,” Lab Chip5(11), 1303–1309 (2005). [CrossRef] [PubMed]
- J. H. Correia, G. de Graaf, S. H. Kong, M. Bartek, and R. F. Wolffenbuttel, “Single-chip CMOS optical microspectrometer,” Sens. Actuators A Phys.82(1-3), 191–197 (2000). [CrossRef]
- S. W. Wang, M. Li, C. S. Xia, H. Q. Wang, X. S. Chen, and W. Lu, “128 channels of integrated filter array rapidly fabricated by using the combinatorial deposition technique,” Appl. Phys. B88(2), 281–284 (2007). [CrossRef]
- R. R. McLeod and T. Honda, “Improving the spectral resolution of wedged etalons and linear variable filters with incidence angle,” Opt. Lett.30(19), 2647–2649 (2005). [CrossRef] [PubMed]
- A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “CMOS-compatible LVOF-based visible microspectrometer,” Proc. SPIE7680, 76800W (2010). [CrossRef]
- A. Emadi, S. Grabarnik, H. Wu, G. de Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Spectral measurement using IC-compatible linear variable optical filter,” Proc. SPIE7716, 77162G (2010). [CrossRef]
- A. Emadi, H. Wu, S. Grabarnik, G. De Graaf, K. Hedsten, P. Enoksson, J. H. Correia, and R. F. Wolffenbuttel, “Fabrication and characterization of IC-compatible linear variable optical filters with application in a micro-spectrometer,” Sens. Actuators A Phys.162(2), 400–405 (2010). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed., 360–376 (Cambridge University Press, 1999).
- A. Emadi, H. Wu, S. Grabarnik, G. de Graaf, and R. F. Wolffenbuttel, “Vertically tapered layers for optical applications fabricated using resist reflow,” J. Micromech. Microeng.19(7), 074014 (2009). [CrossRef]
- A. Emadi, “Linear-variable optical filters for microspectrometer application,” PhD Thesis, Technical University of Delft (2010).
- V. Krajicek and M. Vrbova, “Laser-induced fluorescence spectra of plants,” Remote Sens. Environ.47(1), 51–54 (1994). [CrossRef]
- K. Burns, K. B. Adams, and J. Longwell, “Interference measurements in the spectra of neon and natural mercury,” J. Opt. Soc. Am.40(6), 339–344 (1950). [CrossRef]
- NIST atomic spectra database, Online: http://www.nist.gov/pml/data/asd.cfm
- M. P. Wisniewski, R. Z. Morawski, and A. Barwicz, “Algorithms for interpretation of spectrometric data- A comparative study,” Instrumentation and Measurement Technology Conference, 2000. IMTC 2000. Proceedings of the 17th IEEE, 2, 703–706 (2000).
- D. Massicotte, R. Z. Morawski, and A. Barwicz, “Kalman-filter-based algorithms of spectrometric data correction-Part I: an iterative algorithm of deconvolution,” IEEE Trans. Instrum. Meas.46(3), 678–684 (1997). [CrossRef]
- M. H. Hayes and H. Monson, “Recursive least squares,” in Statistical Digital Signal Processing and Modeling (Wiley, 1996), ch. 9.4.
- S. Grabarnik, A. Emadi, H. Wu, G. de Graaf, and R. F. Wolffenbuttel, “High-resolution microspectrometer with an aberration-correcting planar grating,” Appl. Opt.47(34), 6442–6447 (2008). [CrossRef] [PubMed]

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