YANG T,L IC C,HO H P

(1.Jiangsu Key Laboratory forO rganic Electronics&Info rm ation D isplays,Nanjing University

of Posts and Telecomm unications,Nanjing(210000),China;

2.Department of Electronic Engineering,Centre for Advanced Research in Photonics,The Chinese University of Hong Kong,Shatin,N.T.,Hong Kong SAR,China)

1 Introduction

Absorption spectroscopy is an established method for the detection and analysis of chemical and biological samples extensively used in a wide range of industrial and research oriented applications[1-3]. Fourier transfor m spectroscopy is one of the numerous spectroscopy techniques,distinguished by its unprecedented spectral discr imination paired with the inherent sensitivity[4].However current Fourier transform spectrometers,particularly those using scanning mirror mechanis ms,do not fulfill the requirements of a s mall and easy-to-use sensor.Because such a compact and real-time operating analyzer could be used for monitoring the quality ofe.g.gasoline at gas stations,the quality and consistency of products(e.g.food and drug industry),the safety in fermentation environment(CO2),and many other out-of-the lab applications.Grating based spectrometers have potentially more commercial market applications due to their small size[5],but it also suffers disadvantages of low spectral resolution and expensive price.

In order to overcome the limitationsof traditional devices and explore a more economical,compact and high performance spectrometer,we propose a novel interferometer-based spectrometer.The design process of the new device requires only the solution of a linear system.The miniature spectrometer has the combined advantages of low cost,small size and high resolution.Itspolymer structure can be builton a chip using existing fabrication technologies such as moldingwith the help of Focused I on Beam(F IB)or Electron Beam Lithography(EBL),so the device is straight for ward to fabricate.

2 Operating principle of spectrometer

As shown in Fig.1,the design contains an interferometer array attached to a CCD chip.Each interferometer has two PMMA stages with a PMMA film as the substrate.The CCD pixelworks as a detector beneath the PMMA substrate for each interferometer.Considering the Signal-to-Noise Ratio(SNR)and the sensitivity,we only use part of a CCD pixel as the detector for an interferometer by shading the remaining part.

Fig.1 Schematic of the optical spectrometer.

We enlarge the cross-section of the interferometer in Fig.2 to demonstrate the principles of the device.When a plane wave illuminates the surfaces of the interferometer,the incoming beam is divided into two parts by the two stagesof the interferometer.Because the height of each stage is different,the phase of each beam portion is separately delayed.When the two beam portions with different phase changesmerge again,interference occurs and the interference intensity ismeasured by a CCD pixel underneath.

The detected intensity contains the spectral infor mation.Since each frequency component in the

Fig.2 Cross-section of the optical interferometer.

incoming beam corresponds to a unique phase difference of the two beam portions,the total intensity received by each CCD pixel,which results from the superposition of the interference signals from all the frequency components in the beam,should also be unique.

Fig.3 Source spectrum used in simulation.

If the incoming beam is uniformly divided intonfrequency componentsf1,f2,…,fnwith widthΔfas shown in Fig.3,the total power of the incoming beam can be calculated approximately by using integral calculus assumingnis large enough,i.e.

whereP(fx)is the spectrum amplitude offx.After passing through the interferometer,the measured power can be represented by whereC1,C2,…,Cnare trans mission coefficients for the frequency partsf1,f2,…,fn,respectively.

If the incoming beam illuminatesninterferometers,a seriesofpowers can bemeasured by the CCD array as follows

Therefore,given the trans mission coefficients and the powers from different CCD pixels,i.e.

we obtain a linear system

where

Consequently,the spectrum of the incoming beam can be obtained by fittingP(f1),P(f2),…,P(fn),which are the elements of the matrix

Because the trans mission coefficients can be determined by s imulation or measurements,the spectrum reconstruction is the solution to Eq.(6).However,any data errors in the matrixAdue to the limited signal-to-noise ratio(SNR)make this linear system poorly defined.It is then difficult to solve such a large system of linear equations by using atandard non-stationary iterative methodswithin the MATLAB environment.Here,we use the Tikhonov regularization method[6]to solve Eq.(8).

The reconstruction nor mally takes about 1 s whenn=2 000,so it enables real-timemeasurement for many applications.Although the fast Fourier transform(FFT)would require less time,its requirements are difficult to satisfy because beam portions coming from the two stages of an interferometer are not unifor m in intensity due to the confinement and absorption of the waveguide.At the same time,partial interference of the two beam portions due to the structure layout also makes the FFT unfeasible.

3 Simulation results and discussion

We have performed a series of 2D s imulation experiments by varying the height of one stage from 0 to 10 microns as demonstrated in Fig.2.FDTD solutions(Lumerical Solutions,Inc)with a minimum 8 nm mesh size are used to study the interferometer structure.In order to s mooth out edge effects[7],we use a plane wave source that acts as a total-field scattered-field(TFSF) source with perfectly matched layer(PML)boundaries.The values inCandAcan be obtained from a frequency domain power monitor by a series s imulation experiments and the initial data can be analyzed byMATLAB.

Fig.4 shows the final resultson the comparisons between the reconstructed and original spectra with different values ofαwhich is the data error ofA.In the figures,the solid lines are the original source spectrum;the dash lines are the reconstruction spectrum whenα is equal to 1×10-8;and the dotted lines are the reconstruction spectrum whenαis equal to 3.5×10-7. In Fig.4(a)and Fig.4(b),the dash lines almost coincide with the solid lines so that they are successful reconstructions.But the dotted line does not fully cover the solid line in Fig.4(b),which meansα=3.5×10-7is not a suitable parameter for this ill-posed problem.In Fig.4(c),neither the dotted line nor the dash line exactly covers the solid line.We can not reconstruct a perfect spectrum no matter how the value ofαis chosen.

Fig.4 Reconstruction with different values ofα.

The s imulation results also indicate that our device has the potential to achieve a very high resolution.Fig.4(a)represents 2 000 frequency components that are reconstructed in the range from 250 THz to 300 THz,so the resolution is 25 GHz in frequency or 0.083 3 nm in wavelength;Fig.4(b)represents 1 000 frequency components are reconstructed in the range from 750 THz to 760 THz,so the resolution is 10 GHz in frequency or 5.19 pm in wavelength;Fig.4(c)represents 1 000 frequency components are reconstructed in the range from 750 THz to 755 THz.The frequency interval is even narrower,but distortion occurs. It is not a problem to solve for 2 000 or even more linear equations by using the Tikhonov regularization method.The limit of the resolution mainly depends on the sensitivity and the SNR of the CCD pixels in the exper imentsor the significant digits kept for values in simulation and calculation.

Fig.5 Reconstruction of the original spectrum.

However,we also can investigate whether the shape of the original spectrum itself can affect the good reconstructed spectrum which we can get. If there are a lot of zeroes in the original spectrum and the resolution is close to the limit,distortion may occur.The spectra in Fig.4(a)and Fig.5(a)have the same resolution and frequency range,so both of them represent successful reconstructions.However,the spectra in Fig.4(b)and Fig.5(b)also have the same resolution and frequency range while there are more zeroes in the original spectrum,as shown in Fig.5(b),where an obvious distortion occurs because the resolution is close to the limit.

4 Other design considerations

One may also think about the crosstalk between the two interferometers. In the above s imulation,the interferometers are studied individually assuming that they are optically isolated.The illumination width of the incoming beam is only 3.5μm in the above simulation that is sufficient enough to cover the major structure ofthe interferometer. However,Fig.1 indicates that the interferometers should be assembled in an array configuration. In order to decrease the crosstalk between them,we need to separate the interferometers sufficiently.The distance between the two adjacent interferometers is 10μm,so that a 1 000×1 000 array only occupies 1 cm2.Therefore,the crosstalk problem can be solved by using a large enough spacing.In the following simulations,we setα=1 ×10-8and keep the illumination width at 10μm for one interferometer as shown in Fig.6.In order to explore the effects ofBeamCand BeamDin Fig.6,we reconstruct the same spectrum as shown in Fig.4(a)and Fig.4(b)with 10μm interval and show the results in Fig.7.By comparing Fig.7(a)with Fig.4(a)and Fig.7(b)with Fig.4(b),we find all the corresponding s imulations are a lmost identical.The original and reconstructed plots almost overlap with each other,so it can be concluded that the side illumination has little influence on the accuracy of reconstructions.

Fig.6 Optical interference effect with side illumination.

Fig.7 Reconstruction with a largespaceinterval(10μm).

We also explore other structures for the interferometer.Considering the difficulty in the fabricating of the extruded arrays,it would be s impler to adopt inverted pit structures.Both the extruded and thepits can generate the required phase delay,so they can be used to realize the same function.The main advantage of the pit structure is that it ismuch easy to fabricate with current optical storage technology,i.e.we can fabricate the pit arraywith a model similar to the fabrication of CDs and DVDs.In order to further s implify the structure,we can just use one pit for an interferometer as shown in Fig.8,because a single pit is sufficient to achieve the phase difference.Fig.9 shows the reconstructed results of the simulation.However,in contrast to Fig.9(a),there is a large distortion in Fig.9(b) and Fig.9(c). This means the resolution of the pit structure is inferior to that of the extruded structure.One explanation is that the positions of the monitors in the pit structure are further away from the polymer surface,so the detectors are more susceptible to distortion due to side illumination. Furthermore,the pits in the structure do not behave as waveguides,unlike the extruded structure.

Fig.9 Reconstruction for pit structure.

5 Conclusions

We have proposed an ultra-compact and high-resolution free space optical spectrometer and quantified its operation using FDTD simulations.The Tikhonov regularization method is used to deter mine the resolution down to the picometer level.For a relatively simple structure,the proposed device is low-cost and easy to fabricate.

6 Acknowledgements

The authors would like to extend their appreciation to Mr.Zewen Wang of the East China Institute of Technology for his fruitful discussions and help with the Tikhonov regularization method and also to gratefully acknowledge the research studentship support for T.Yang from Chinese Univesity of Hong Kong.

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