## Inexpensive photonic crystal spectrometer for colorimetric sensing applications |

Optics Express, Vol. 21, Issue 4, pp. 4411-4423 (2013)

http://dx.doi.org/10.1364/OE.21.004411

Acrobat PDF (1653 KB)

### Abstract

Photonic crystal spectrometers possess significant size and cost advantages over traditional grating-based spectrometers. In a previous work [Pervez, *et al*, Opt. Express **18**, 8277 (2010)] we demonstrated a proof of this concept by implementing a 9-element array photonic crystal spectrometer with a resolution of 20nm. Here we demonstrate a photonic crystal spectrometer with improved performance. The dependence of the spectral recovery resolution on the number of photonic crystal arrays and the width of the response function from each photonic crystal is investigated. A mathematical treatment, regularization based on known information of the spectrum, is utilized in order to stabilize the spectral estimation inverse problem and achieve improved spectral recovery. Colorimetry applications, the measurement of CIE 1931 chromaticities and the color rendering index, are demonstrated with the improved spectrometer.

© 2013 OSA

## 1. Introduction

2. P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE **4300**, 25–32 (2000). [CrossRef]

4. W. M. Johnston, “Color measurement in dentistry,” J. Dent. **37**, e2–e6 (2009). [CrossRef] [PubMed]

6. J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. **41**, 2532–2548 (2002). [CrossRef]

9. O. Schmidt, P. Kiesel, and M. Bassler, “Performance of chip-size wavelength detectors,” Opt. Express **15**, 9701–9706 (2007). [CrossRef] [PubMed]

10. B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. **282**, 3168–3171 (2009). [CrossRef]

11. B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express **17**, 17060–17069 (2009). [CrossRef] [PubMed]

12. U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J. **11**, 1556–1563 (2011) [CrossRef]

13. S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater. **22**, 946–950 (2010). [CrossRef] [PubMed]

14. N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express **18**, 8277–8285 (2010). [CrossRef] [PubMed]

*μ*m by 30

*μ*m. This spatially resolved information can be directly monitored by a 2D imager such as a CCD or CMOS sensor. The resolution is no longer dependent on the length of the light path as in a grating spectrometer, and thus the photonic crystal spectrometer can be significantly smaller, as small as 1 cm

^{3}in an integrated format. The cost can be aggressively brought down from thousands of dollars to below ten dollars. Another advantage is the flexibility in customizing the photonic crystal spectrometers to address different applications with minimum cost. As mentioned previously, continuous spectrum recording is often overkill for applications where limited-channel representation is adequate, as in many colorimetry applications. The photonic crystal spectrometer can be customized to have varied resolution at different ranges of wavelengths.

## 2. Spectral estimation inverse problem

14. N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express **18**, 8277–8285 (2010). [CrossRef] [PubMed]

*m*output channels and let

*b*denote the output intensity of the

_{i}*i*th channel, 1 ≤

*i*≤

*m*. The spectrometer’s output can be modeled as where

*x*(

*λ*) is spectral intensity of the input light and

*A*(

_{i}*λ*) characterizes the response of the

*i*th channel to input wavelength

*λ*. The functions

*A*in this case are roughly Gaussian in shape, each with a different peak location. The goal is to recover

_{i}*x*(

*λ*) from the data

*b*

_{1},...,

*b*, an underdetermined problem.

_{m}**x**∈ ℝ

*encode the spectral intensity of the input light at*

^{n}*n*equispaced wavelengths

*λ*

_{1},...,

*λ*over some appropriate range. For actual computation we discretize Eq. (1) as where

_{n}**b**= (

*b*

_{1},...,

*b*)

_{m}*(superscript*

^{T}*T*denotes the transpose). Here

**A**is an

*m*×

*n*matrix whose (

*i*,

*k*) entry describes the output intensity of the

*i*th channel to input light near wavelength

*λ*; thus the

_{k}*i*th row of

**A**describes the response of the

*i*th channel over the wavelength range of interest. Recovery of

**x**from

**b**is still typically underdetermined, since

**x**is at least 100 dimensional (depending on wavelength spacing and range), while

**b**has at most 20 components.

## 3. Regularizing the inverse problem

**Ax**=

**b**is underdetermined, we can obtain a unique solution by imposing additional conditions on

**x**. These conditions should use a priori information about the true spectrum, for example, smoothness conditions, concentration in a certain range of wavelengths, nonnegativity, etc. Because the response functions

*A*peak at different locations, the matrix

_{i}**A**has full row rank, and so we might expect the Moore-Penrose pseudoinverse to do a reasonable job of estimating the spectrum

**x**from the data

**b**, as the minimum norm solution to

**Ax**=

**b**. However, the data here is sufficiently noisy that some additional regularization is necessary, especially as the number of channels increases.

**b**

*denote the measured (noisy) channel data. For the present work we regularize by seeking a vector*

^{meas}**x**

*that satisfies where ‖ · ‖*

^{rec}_{2}denotes the usual 2-norm; argmin

**(**

_{x}*f*) gives

**x**at which

*f*is minimized. Here Γ is an invertible

*n*×

*n*matrix chosen to penalize undesirable behavior in the minimizer and

*λ*> 0 is a regularization parameter. The minimizer of Eq. (3) is unique, for if we differentiate

**x**we see that any minimizer must satisfy the normal equation It’s easy to see that the symmetric matrix

**A**

^{T}**A**+

*λ*

^{2}Γ

*Γ is positive definite if Γ is invertible and*

^{T}*λ*> 0, so that Eq. (4) has a unique solution and Eq. (3) has a unique minimizer.

*λ*> 0 is chosen to strike a balance between fidelity to the original inverse problem (

*λ*∼ 0) and desirable qualities of the estimate

**x**

*, especially in the presence of noise. We use the simple “discrepancy principle” [16*

^{rec}16. P. C. Hansen, *Discrete Inverse Problems: Insight and Algorithms* (Society for Industrial and Applied Mathematics, 2010). [CrossRef]

17. J. C. Santamarina and D. Fratta, *Discrete Signals and Inverse Problems* (Wiley, 2005). [CrossRef]

*λ*, illustrated in Section 4.

**x**

*in Eq. (3) can cast as that of finding the least-squares solution to [16*

^{rec}16. P. C. Hansen, *Discrete Inverse Problems: Insight and Algorithms* (Society for Industrial and Applied Mathematics, 2010). [CrossRef]

**x**the incorporation of nonnegativity constraints on

*x*≥ 0 for the components of

_{i}**x**is also desirable. Specific examples illustrating the regularization process are shown in the next section.

## 4. Estimating the spectrum

**b**

*, even with a relatively small number of channels. In this section we illustrate this, and show how the regularization scheme outlined above can be used to provide stable spectral estimates.*

^{meas}**A**was characterized using a Xe arc light source, a monochromator, and a calibrated Si photodiode.

*λ*is determined using the discrepancy principle [17

17. J. C. Santamarina and D. Fratta, *Discrete Signals and Inverse Problems* (Wiley, 2005). [CrossRef]

16. P. C. Hansen, *Discrete Inverse Problems: Insight and Algorithms* (Society for Industrial and Applied Mathematics, 2010). [CrossRef]

**b**

*=*

^{meas}**b**

^{*}+

**e**, where

**b**

^{*}is the true (noiseless) channel intensities and

**e**is a noise vector with estimated magnitude ‖

**e**‖

_{2}=

*δ*. We select the regularization parameter

*λ*so that ‖

**Ax**

*−*

^{rec}**b**

*‖*

^{meas}_{2}=

*δ*, where

**x**

*is the solution to Eq. (3). Using the techniques in section 4.7 of [16*

^{rec}*Discrete Inverse Problems: Insight and Algorithms* (Society for Industrial and Applied Mathematics, 2010). [CrossRef]

*λ*is unique, provided

*δ*< ‖

**b**‖

_{2}.

### 4.1. Perfect data example

**x**

^{*}as our “ground-truth.” We then use

**x**

^{*}to compute the “true” channel intensities

**b**

^{*}=

**Ax**

^{*}. We then use the regularized reconstruction procedure from Section 3 to attempt to recover

**x**

^{*}from

**b**

^{*}.

**b**

^{*}is exact we use estimated noise level

*δ*= 1.0×10

^{−8}and choose

*λ*to obtain ‖

**Ax**

*−*

^{rec}**b**

*‖*

^{meas}_{2}=

*δ*(though any choice

*δ*< 1.0 × 10

^{−3}makes little difference). Reconstructions using each of the regularization matrices in Eq. (5) are shown in Fig. 2, as well as that obtained from Γ =

**I**. Second derivative regularization and first derivative regularization have almost identical recovery quality. Such quantification is done by calculating ‖

**x**

*−*

^{rec}**x**

^{*}‖

_{2}for first derivative regularization (0.125), second derivative regularization (0.121) and identity matrix regularization (0.205).

### 4.2. Real data examples

**I**. The noise level in the data is estimated as ‖

**b**

*−*

^{meas}**b**

^{*}‖

_{2}, and is equal to 12 percent of the magnitude of

**b**

^{*}; that is,

*δ*= 0.12‖

**b**

^{*}‖

_{2}in the discrepancy principle, and the regularization parameter

*λ*is chosen accordingly. Spectral recovery quality is quantified by calculating ‖

**x**

*−*

^{rec}**x**

^{*}‖

_{2}for first derivative regularization (0.211), second derivative regularization (0.208) and identity matrix regularization (0.290).

*δ*= 0.12‖

**b**

*‖*

^{*}_{2}and discrete first derivative regularization.

### 4.3. Spectrum estimation resolution

**A**is known and represented by multiple Gaussian peaks. We consider a test spectrum

*x*(

_{test}*λ*) consisting of two Dirac delta spikes at wavelengths

*λ*

^{*}and

*λ*

^{*}+ Δ

*λ*, so

*x*(

_{test}*λ*) =

*δ*(

*λ*−

*λ*

^{*}) +

*δ*(

*λ*−

*λ*

^{*}− Δ

*λ*). Similar to the approach in section IV (perfect data example), channel intensities can be calculated using this ground-truth

*x*and known

_{test}**A**, and these channel responses then used to recover an estimates of

*x*. The distance Δ

_{test}*λ*between the two delta functions is varied until the recovered spectrum does not resolve the two delta peaks. The two peaks are resolved when the valley between the two peaks is lower than half of the peak value. In Fig. 5 the resolution of spectrum recovery varies with the number of photonic crystal channels. With three different widths of response functions, the resolution all hit a plateau when there are sufficient number of channels (15 for width of 50nm, 30 for width of 20nm and 45 for width of 10nm).

## 5. Colorimetric applications

### 5.1. Estimation of tristimulus parameters

*X*,

*Y*, and

*Z*are computed from the spectral density function

*x*(

*λ*) as where

*W*denotes any of

*X*,

*Y*,

*Z*and

*w*(

*λ*) is an appropriate weighting function for the particular tristimulus parameter. One approach to computing these values is to estimate the spectrum

*x*(

*λ*) with one of the approaches of Section 4, then employ Eq. (7) (or a discrete version thereof) with the appropriate weighting function. However, we can estimate these values directly from the spectrometer output data vector

**b**without the intermediate step of directly estimating the spectrum itself, and regularize in a manner adapted to this setting.

**w**be a column-vector in ℝ

*that appropriately approximates the function*

^{n}*w*(

*λ*) on the interval [

*λ*,

_{min}*λ*], and let

_{max}**x**similarly be a vector approximating

*x*(

*λ*). The discrete version of Eq. (7) is In this case a smooth estimate

**x**is of less importance, and so we use Tikhonov regularization with Γ =

**I**in Eq. (3) or Eq. (4); let us also drop the non-negativity assumption on

**x**for now (since we aren’t estimating

**x**anyway). If

**A**has singular value decomposition

**A**=

**USV**

*with singular values*

^{T}*σ*

_{1}≥

*σ*

_{2}≥ ⋯ ≥

*σ*

_{1}≥ 0 then when

*λ*> 0 the solution to Eq. (3) can be expressed as where

*n*×

*m*matrix with diagonal entries

*i*≤

*m*and zero entries elsewhere; see Chapter 4 in [16

*Discrete Inverse Problems: Insight and Algorithms* (Society for Industrial and Applied Mathematics, 2010). [CrossRef]

*λ*= 0

*σ*> 0 and

_{i}**A**). From Eq. (8) with

**x**=

**x**

*we can estimate where*

^{rec}**b**is the spectrometer data vector and

**c**

*is the vector*

_{λ}*λ*= 0 the vector

**c**

_{0}may also be realized as the minimizer of

**A**

^{T}**c**−

**w**is given by precisely Eq. (11) with

*λ*= 0. That is,

**A**

^{T}**c**

_{0}is the best approximation to

**w**that can be constructed from the rows of

**A**. We might then consider

*W*in Eq. (8) that can be made with noiseless data

**b**

^{*}. The vector

**c**

*itself may be realized as the minimizer of*

_{λ}**c**

_{0}. In the presence of noise this regularization can be helpful.

*λ*, let

**b**

*=*

^{meas}**b**

^{*}+

**e**for some noise vector

**e**. We choose that value of

*λ*that minimizes the expected error

*E*((

*W*−

_{λ}*W*

^{*})

^{2}), under certain assumptions about

**e**. We have Let

**b**̃ =

**U**

^{T}**b**

^{*},

**ẽ**=

**U**

^{T}**e**and

**w**̃ =

**V**

^{T}**w**so that from Eq. (13)/Eq. (14) We assume that the components of

**e**have zero mean; let

**C**= Cov(

**e**). Squaring both sides of Eq. (15) and taking expected values yields In the case that the components of

**e**are independent with identical distribution (i.i.d.) and common variance

*σ*

^{2}we have

**U**

^{T}**CU**=

*σ*

^{2}

**I**and the right side of Eq. (16) can be written Of course we don’t have the true value of the

*b*̃

*, but rather the values*

_{k}*b*̃

*in the first term on the right in Eq. (17), with a slight correction. Under the i.i.d. assumption for the*

_{k}*e*one can compute that It is thus reasonable to replace

_{k}*b*̃

*in the first term on the right in Eq. (17) with the measured values*

_{k}*λ*that minimizes the computable quantity for

*λ*> 0 (or on some bounded interval 0 ≤

*λ*≤

*λ*).

_{max}*e*are independent with zero mean but with

_{k}*k*th channel has standard deviation proportional to the magnitude |

*b*|. In this case

_{k}**B**= diag(

*b*

_{1},...,

*b*).

_{m}### 5.2. Examples

*X*,

*Y*, and

*Z*using Eq. (10)/Eq. (11) with the regularization procedure of Section 5.1 and the same estimated noise level (‖

**e**‖

_{2}≈ 0.12‖

**b**

^{*}‖

_{2}) as in Subsection 4.2. We then use these values to compute the CIE 1931 chromaticity coordinates for each LED spectrum. We should note that for the case

*λ*= 0 the best approximations

**w**̃

*,*

_{X}**w**̃

*,*

_{Y}**w**̃

*for each of the tristimulus weighting vectors*

_{Z}**w**

*,*

_{X}**w**

*, and*

_{Y}**w**

*can be obtained using Eq. (12) and*

_{Z}**w**̃

*−*

_{X}**w**

*‖*

_{X}_{2}/‖

**w**

*‖*

_{X}_{2}= 0.0747, ‖

**w**̃

*−*

_{Y}**w**

*‖*

_{Y}_{2}/‖

**w**

*‖*

_{Y}_{2}= 0.0649, and ‖

**w**̃

*−*

_{Z}**w**

*‖*

_{Z}_{2}/‖

**w**

*‖*

_{Z}_{2}= 0.0750, thus we should expect at least comparable error in our tristimulus or chromaticity estimates, even with perfect noise-free data.

*x*=

*X*/(

*X*+

*Y*+

*Z*),

*y*=

*Y*/(

*X*+

*Y*+

*Z*) and are shown in Table 1. The “Actual channel Data” column contains the estimates using the measured 17-channel data; the “Noiseless channel Data” shows the estimates using perfect data computed as

**b**

^{*}=

**Ax**

^{*}, where

**x**

^{*}is measured using the commercial spectrometer and

**b**=

**b**

^{*}is used in Eq. (10). Finally, the column “True Values” contains the estimates obtained from Eq. (8) with

**x**=

**x**

^{*}and appropriate

**w**. Figure 6 plots these results on the CIE 1931 color space. The triangles are the true CIE 1931 chromaticities of the color LEDs (near the periphery of the space) and the superwhite LED (in the center of the space) as measured by Ocean Optics spectrometer (USB 4000, calibrated). The cross near the center of the space is the CIE (x,y) for the superwhite LED estimated based on measurement by photonic crystal spectrometer. The circles adjacent to the periphery of the color space are the CIE 1931 chromaticities of the response functions of the photonic crystal spectrometer. In other words, if these circles are connected by a line, the enclosed space represents the color space range that the photonic crystal spectrometer can measure.

*X*,

*Y*,

*Z*estimates due to noise, and the corresponding variation induced in the chromaticity coordinates. Under the i.i.d. assumption for the error

*e*with common variance

_{k}*σ*

^{2}we find For the tristimulus values with the 17 channel data with the superwhite LED we obtain approximate standard deviations of 0.038 for

*x*and 0.033 for

*y*.

### 5.3. Color rendering index

## 6. Conclusions

## Acknowledgments

## References and links

1. | H. Kipphan, |

2. | P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE |

3. | G. Celikiz and R. G. Kuehni, |

4. | W. M. Johnston, “Color measurement in dentistry,” J. Dent. |

5. | S. Ahuja and S. Scypinski, |

6. | J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng. |

7. | S. Gaurav, |

8. | S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys. |

9. | O. Schmidt, P. Kiesel, and M. Bassler, “Performance of chip-size wavelength detectors,” Opt. Express |

10. | B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun. |

11. | B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express |

12. | U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J. |

13. | S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater. |

14. | N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express |

15. | G. Wyszecki and W. S. Stiles, |

16. | P. C. Hansen, |

17. | J. C. Santamarina and D. Fratta, |

18. | G. H. Golub and C. F. Van Loan, |

**OCIS Codes**

(120.0280) Instrumentation, measurement, and metrology : Remote sensing and sensors

(130.6010) Integrated optics : Sensors

(330.1710) Vision, color, and visual optics : Color, measurement

(330.1730) Vision, color, and visual optics : Colorimetry

(130.5296) Integrated optics : Photonic crystal waveguides

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 1, 2012

Revised Manuscript: January 30, 2013

Manuscript Accepted: February 3, 2013

Published: February 13, 2013

**Virtual Issues**

Vol. 8, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Kurt M. Bryan, Zhang Jia, Nadia K. Pervez, Marshall P. Cox, Michael J. Gazes, and Ioannis Kymissis, "Inexpensive photonic crystal spectrometer for colorimetric sensing applications," Opt. Express **21**, 4411-4423 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4411

Sort: Year | Journal | Reset

### References

- H. Kipphan, Handbook of Print Media: Technologies and Production Methods (Springer, 2001).
- P. G. Herzog and F. Koenig, “Spectral scanner in the quality control of fabrics manufacturing,” Proc. SPIE4300, 25–32 (2000). [CrossRef]
- G. Celikiz and R. G. Kuehni, Color Technology in the Textile Industry (American Association of Textile Chemists and Colorists: 1983).
- W. M. Johnston, “Color measurement in dentistry,” J. Dent.37, e2–e6 (2009). [CrossRef] [PubMed]
- S. Ahuja and S. Scypinski, Handbook of Modern Pharmaceutical Analysis (Academic, 2010).
- J. Y. Hardeberg, F. Schmitt, and H. Brettel, “Multispectral color image capture using a liquid crystal tunable filter,” Opt. Eng.41, 2532–2548 (2002). [CrossRef]
- S. Gaurav, Digital Color Imaging Handbook (CRC, 2003).
- S. S. Murtaza and J. C. Campbell, “Effects of variations in layer thickness on the reflectivity spectra of semiconductor Bragg mirrors,” J. Appl. Phys.77, 3641–3644 (1995). [CrossRef]
- O. Schmidt, P. Kiesel, and M. Bassler, “Performance of chip-size wavelength detectors,” Opt. Express15, 9701–9706 (2007). [CrossRef] [PubMed]
- B. Momeni, E. S. Hosseini, M. Askari, M. Soltani, and A. Adibi, “Integrated photonic crystal spectrometers for sensing applications,” Opt. Commun.282, 3168–3171 (2009). [CrossRef]
- B. Momeni, E. S. Hosseini, and A. Adibi, “Planar photonic crystal microspectrometers in silicon-nitride for the visible range,” Opt. Express17, 17060–17069 (2009). [CrossRef] [PubMed]
- U. Kurokawa, B. I. Choi, and C-. C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sensors J.11, 1556–1563 (2011) [CrossRef]
- S. H. Kim, H. S. Park, J. H. Choi, J. W. Shim, and S. M. Yang, “Integration of colloidal photonic crystals toward miniaturized spectrometers,” Adv. Mater.22, 946–950 (2010). [CrossRef] [PubMed]
- N. K. Pervez, W. Cheng, Z. Jia, M. P. Cox, H. M. Edrees, and I. Kymissis, “Photonic crystal spectrometer,” Opt. Express18, 8277–8285 (2010). [CrossRef] [PubMed]
- G. Wyszecki and W. S. Stiles, Color Science, 2nd Ed. (Wiley, 1982).
- P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, 2010). [CrossRef]
- J. C. Santamarina and D. Fratta, Discrete Signals and Inverse Problems (Wiley, 2005). [CrossRef]
- G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Ed. (Johns Hopkins University,1996)

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.