## Abstract

A standing wave Fourier transform spectrometer is realized. The spectrometer consists of an ultra thin and partially transparent photodetector and a tunable mirror. The incident light forms a standing wave in front of the mirror, which is sampled by the ultra thin optical detector. The thickness of the photodetector is significantly smaller than the wavelength of the incident light. The spectral information of the incident light is determined by the Fourier transform of the detector signal. The linear arrangement of the optical detector and the mirror enables the realization of spectrometer arrays and optical cameras with high spectral resolution. For the first time a complete optical model of the standing wave spectrometer is presented and compared with experimental results. The influence of the design of the optical detector on the performance of the spectrometer is discussed.

©2010 Optical Society of America

## 1. Introduction

Advancement in optical micro-electro-mechanical systems (MEMS) technology has lead to the miniaturization of interferometers and spectrometers [1]. The development of micro-spectrometers has enabled practical applications of spectroscopy in chemical and medical analysis, food and color inspection, wavelength detection and multispectral imaging. These sensing applications require compact and portable devices, low manufacturing costs and good performance. Several types of spectrometers have been transferred from macro to micro scale such as Fabry-Perot [2], grating [3] and Fourier transform spectrometer [4].

In this paper we will present a Fourier transform spectrometer operating in the visible part of the optical spectrum. The spectral information is determined by measuring the intensity profile of a standing wave created in front of a mirror. The standing wave is detected by an ultra thin and partially transparent optical detector. The classical setup of a Fourier transform spectrometer is based on a Michelson interferometer (Fig. 1a ), which does not allow for the dense two-dimensional integration of micro spectrometers on a single sensor chip. The operation principle of the novel spectrometer allows for the integration of two-dimensional (2D) spectrometer arrays on a sensor chip and the realization of very compact spectrometers and spectrometer arrays.

The concept of wavelength sensitive devices based on standing wave detection is known for many years [5], but the realization of such devices was limited by technological reasons. Only in recent years the first standing wave interferometers and spectrometers have been realized [6–9]. Sampling of a standing wave by a semiconductor device requires (i) a partially transparent and (ii) an ultra thin detector with (iii) an adequate photosensitivity which is realized on a transparent substrate. The working principle of a standing wave transform spectrometer is based on the detecting of a standing wave profile [7,8]. The standing wave is created in front of a planar tunable mirror and its intensity profile is detected with an ultra thin and partially transparent detector.

A schematic sketch of the standing wave spectrometer setup is given in Fig. 1b. The linear arrangement of the sensor and the mirror enables the dense integration of sensor elements on a single sensor chip, so that compact 1D and 2D spectrometer arrays can be realized. This novel spectrometer fills the gap between standard cameras with only three-color channels, but with high spatial resolution, and precision spectrometers with high spectral resolution, but with no spatial resolution.

The key component of the novel spectrometer is an ultra thin and partially transparent optical detector. The high transmission of the detector is required for the formation of a standing wave in front of the tunable mirror. On the other hand, a certain fraction of the incident light has to be absorbed, so that a photocurrent can be generated. However, the thickness of the sensor absorbing region has to be smaller than the wavelength of the incoming light, so that the photocurrent is modulated by the displacement of the mirror. The spectral information of the incoming light is determined by the Fourier transform of the sensor signal.

The ultra thin photodetector consists of a p-i-n photodiode, sandwiched between two transparent conductive oxide (TCO) layers. The photodiodes were prepared at low temperatures on a glass substrate. The TCO layers act as electrical contacts of the device and antireflection coatings. The photodiode was realized by amorphous silicon, which exhibits excellent optical properties in the visible part of the optical spectrum [10].

The performance of the standing wave spectrometer is determined by the optoelectronic properties of the photodetector and the interaction of the photodetector with the standing wave. In this manuscript we describe for the first time a complete model of a standing wave spectrometer.

The basic theory of standing wave spectrometry is described in section 2 of the manuscript. An optical model of the standing wave spectrometer is presented in section 3, which is divided in two parts. Section 3.1 describes the model of an ideal standing wave spectrometer. The real standing wave spectrometer is described in section 3.2 taking the optics of the ultra thin photodetector into account. The fabrication of the optical detector is described in section 4. The optical models are compared to experimental results in section 5. The results are discussed in section 6 followed by a summary of the work in section 7.

## 2. Standing wave spectroscopy

The interference of two monochromatic optical waves can be described by the superposition principle:

where ${\overrightarrow{E}}_{1}$ and ${\overrightarrow{E}}_{2}$ are the complex amplitudes of the electric field of the two monochromatic waves. The parameter ${\overrightarrow{E}}_{s}$ represents the complex amplitude of the resulting wave. The intensity of the monochromatic waves can be calculated by:where c is the speed of the light in vacuum, ε_{0}is the dielectric permeability of the vacuum and η is the real part of the refractive index. The intensity of the resulting wave is given by:

_{1}and I

_{2}are the intensities of the two interfering waves, while ϕ represents the phase difference between them.

In the case of a Fourier spectrometer based on a Michelson interferometer setup the two superimposed waves are propagating in the same direction, while in the case of a standing wave spectrometer they are propagating in opposite directions. The incoming monochromatic wave is reflected by the mirror. Due to the superposition of the forward and backward propagating wave a standing wave is created in front of the mirror. Assuming that all the light is reflected by the mirror, the intensity of the forward and backward waves are equal, so that
I_{1} = I_{2} = I_{0}. The phase difference between the forward and backward propagating wave can be expressed as the path difference between the waves plus the phase change due to the reflection by the mirror:

_{0}is the wave number of the incoming monochromatic light (k

_{0}= 2π/λ) and 2x

_{m}is the optical path difference between the forward and backward wave. The parameter x

_{m}represents the distance between the mirror and the point at which the intensity is measured. Assuming that the mirror can be described by a perfect conductor (φ

_{m}= π), the intensity of the standing wave is given by:

From the Fourier transform spectroscopy it is known that the AC component of the recorded intensity profile and the spectrum of the light source form a Fourier transform pair [11]. The spectrum of the monochromatic light source can be determined by calculating the Fourier transform of the AC component of the standing wave intensity. According to Eq. (6) the AC component of the standing wave is given by:

## 3. Optical model of the standing wave spectrometer

#### 3.1 The ideal standing wave spectrometer

The basic operational principle of a standing wave spectrometer is schematically depicted in Fig. 2 . Incoming light is reflected by the mirror, which leads to the creation of a standing wave. The ultra thin and partially transparent photodetector is introduced in the standing wave pattern. The change of the distance between the detector and the mirror leads to a change of the power loss profile within the detector and thus the photocurrent.

Under monochromatic illumination the output of the spectrometer in the wave number domain represents the instrument profile of the spectrometer. The instrument profile describes the influence of the spectrometer over the spectrum and determines the spectrometer ability to resolve nearby wavelengths. For a broad spectrum light source, the output of the spectrometer is a convolution of the input spectrum and the instrument profile. To illustrate the operational principles of the standing wave spectrometer, we will analyze the case of an ideal standing wave spectrometer, which is depicted in Fig. 2.

In this case the optical detector is described only by the absorbing region of the p-i-n photodiode. We will denote the thickness of the absorbing layer with d_{j} and its complex refractive index with n_{j} = η_{j} - iκ_{j}. The electrical field within the detector is a sum of the forward and the backward (reflected) propagating monochromatic waves. Assuming that there is no interaction between the detector and the standing wave, the total electrical field inside the active layer is given by:

_{0}represents the electric field of the incoming light, x represents the position within the detector (0 ≤ x ≤ d

_{j}) and ϕ represents the phase difference between the forward and backward propagating wave (see Eq. (5)). The first term in Eq. (9) represent the forward propagating wave, while the second term describes the backward propagating wave.

The intensity of the standing wave inside the detector is calculated according to Eq. (2) and, assuming that we have a perfect mirror again, it is given by:

_{j}represents time averaged power loss and α

_{j}is the absorption coefficient of the detector (α

_{j}= 4πκ

_{j}/λ). The absorption coefficient of amorphous silicon, which was used to realize the optical sensor, is plotted together with the absorption coefficient of crystalline silicon in Fig. 3 . Amorphous silicon shows a high absorption in the visible part of the optical spectra and, therefore, it is an excellent material for realizing optical detectors. Assuming that all photogenerated electron-hole pairs can be collected, the photocurrent is determined by:

Since we assumed a monochromatic light source, the Fourier transform of the photocurrent AC component gives us the instrument profile of the spectrometer. However, when determining the instrumental profile two aspects have to be taken into account. The standing wave pattern can be recorded only for finite mirror movements and in discrete spatial points. Consequently the AC component of the photocurrent can be represented by:

_{s}represents the scan length of the mirror and Δx represents the distance between sampling points. The Fourier transform of Eq. (14) gives us the instrumental profile of the ideal standing wave spectrometer:

_{s}/Δx).

The instrument profile of the ideal standing wave spectrometer is a sinc-function shifted to the wave number of the monochromatic light source. The resolution of the standing wave spectrometer is determined by the instrument profile and for the ideal case it is identical with the resolution of a classical Fourier transform spectrometer based on a Michelson interferometer. The resolution of the sinc-function is given by the Rayleigh criterion:

The resolution of the standing wave spectrometer in the wave number domain depends only on the scan length, whereas in the wavelength domain the resolution depends on the scan length and the wavelength of the monochromatic wave:From Eq. (16) and Eq. (17) it is clear that the spectrometer ability to distinguish nearby wavelengths increases for longer scan lengths. The instrument profile of an ideal standing wave spectrometer is depicted in Fig. 4 . The instrument profile is plotted for a scan length of 10 μm for two different wavelengths of 650 nm and 525 nm to illustrate the difference in the resolution for the wave number and the wavelength domain. To show the influence of the scan length on the resolution, the instrument profile is plotted for the wavelength of 525 nm and a scan length of 20 μm.

#### 3.2 The real standing wave spectrometer

The analytical expressions in the previous section were derived under the assumptions, that the optical detector can be described by a single absorbing layer. Also, the interaction between the detector and the standing wave was neglected. In order to develop a realistic model of the standing wave spectrometer, we must describe the optical detector more accurately. For this purpose, we will take into account the complete layer stack that composes the detector, which includes a glass substrate, TCO layers and a p-i-n photodiode sandwiched between them. The real standing wave spectrometer is schematically presented in Fig. 5 .

In order to determine the propagation of the optical wave through the standing wave spectrometer, the transfer matrix method was used. Figure 6a
exhibits the normalized electric field intensity of a monochromatic optical wave in the standing wave spectrometer for different distances between the photodetector and the mirror obtained with the transfer matrix method. |E_{s}|^{2} represents the intensity of the standing wave and |E_{0}|^{2} represents the intensity of the incident wave. A certain fraction of the standing wave is absorbed in the active region of the photodetector. Figure 6b shows the power loss profile within the individual layers of the optical detector for different distances between the mirror and the photodetector. Since the optical constants for the different regions of the p-i-n photodetector are different, discontinuities in the power loss profile can be observed. It can be clearly seen that the standing wave pattern moves through the photodetector when changing the distance between the mirror and the photodetector.

The transfer matrix method was used to derive the expression for the electric field within the absorbing region of the detector, while taking the influence of the additional layers of the photodetector into account. The electric field within absorbing layer is given by:

The derived expression in Eq. (18) is similar to Eq. (9) developed for an ideal standing wave spectrometer. Due to the interaction between the detector and the standing wave, only a certain fraction of the incoming light will reach the absorbing region of the detector. Therefore, the incident electric field, E_{0}, from Eq. (9) is replaced by E_{0}∙t_{c}, where t_{c} represents a transmission coefficient of the layer system placed in front of the active layer. The transmission coefficient is a complex function consisting of a real and an imaginary part. The complex transmission coefficient describes the wave propagation through the glass substrate, the front TCO-layer and the p-layer of the photodiode. The phase of the complex transmission coefficient defines the phase change introduced by the front layer system, while its amplitude determines the amount of light that is transmitted through the layer system placed in front of the active layer.

The phase difference factor exp(-iϕ) in the backward term in Eq. (9) is replaced by r_{c} in Eq. (18), where r_{c} is the reflection coefficient of the layer system placed behind the active region. The layer system placed behind the active region consists of the n-layer, the back TCO-layer, an air gap and the mirror. The reflection coefficient is also a complex function. The phase of the complex reflection coefficient defines the phase difference between the forward and backward wave and its amplitude determines the amount of light that is reflected back into the active region by the back layer system.

The transmission and the reflection coefficient are calculated with the transfer matrix method and their analytical expressions are provided in literature [12]. In the case of an ideal standing wave spectrometer, a change in the distance between the mirror and the detector results only in a change of the phase difference between the forward and the backward wave. The transfer matrix calculations show that by changing the distance between the mirror and the detector, we are changing not only the phase difference between the forward and the backward wave, but also the amplitudes of the complex reflection and transmission coefficient. The amplitudes of the complex reflection and transmission coefficient are periodic functions of the distance between the detector and the mirror. This effect is caused by Fabry-Perot oscillations, which are formed between the mirror and the detector. This means that the real standing wave spectrometer can be described as a combination of an ideal standing wave spectrometer and a Fabry-Perot spectrometer.

Knowing the electric field within the detector, the intensity of the standing wave is easily obtained according to Eq. (2) and it is given by:

_{m}) represents the amplitude of the electric field transmission coefficient, r(x

_{m}) represents the amplitude of the electric field reflection coefficient and θ(x

_{m}) represents the phase of the reflection coefficient. If we compare Eq. (19) with Eq. (10), we can see that the expressions are slightly different due to the transmission and reflection coefficients.

In order to obtain an expression for the photocurrent, the power loss is determined by using Eq. (11). Since the p-i-n photodiode is very thin, we can assume that only photogenerated charges from the active i-layer contribute to the overall photocurrent and that all generated charges are collected. Consequently, the photocurrent is calculated from Eq. (12) and it is given by:

## 4. Experimental

The standing wave spectrometer is composed of two main components, a tunable mirror and an ultra thin and partially transparent detector. The partially transparent detector is realized by an amorphous silicon p-i-n photodiode sandwiched between two transparent conductive oxide (TCO) layers, which act as front and back contacts. The overall thickness of the complete optical detector is less than 600 nm, while the thickness of the p-i-n photodiode is less than 100 nm. The thickness of the active i-layer is less than 50 nm. The partially transparent optical detector is realized on glass substrate and optimized for a wavelength of 633 nm. Further details on the used materials and the device fabrication are given in reference [13-17]. The tunable mirror is realized by mounting of a silver mirror on a piezo microactuator. In order to experimentally determine the instrumental profile of the standing wave spectrometer, a helium neon laser emitting at a wavelength of 633 nm was used. The photocurrent was amplified using a low noise current amplifier and sampled by a digital oscilloscope. The “photocurrent versus time” curves obtained with the oscilloscope were mapped to “photocurrent versus mirror displacement” curve, accounting for nonlinear mirror motion.

## 5. Results

The measured photocurrent of the partially transparent optical detector as a function of the mirror displacement is shown in Fig. 7a and Fig. 7b. For comparison, we plotted the measured photocurrent together with the calculated photocurrent for the model of the ideal standing wave spectrometer (Fig. 7a) and with the calculated photocurrent for the model of the real spectrometer (Fig. 7b). The calculated photocurrent of the ideal detector is a cosine function, whereas the calculated photocurrent of the real spectrometer and the measured photocurrent slightly deviate from the ideal cosine function. Figure 7b shows a better agreement between the experimentally obtained and the calculated photocurrent for the model of the real device. The photocurrent obtained from simulations is also a periodical function slightly different from the cosine function. From the model of the real standing wave spectrometer, we can conclude that these deviations are due to the Fabry-Perot oscillations formed between the detector and the mirror and, consequently, the periodic nature of the transmission and the reflection coefficients.

To determine the instrument profile of the standing wave spectrometer, the discrete Fourier transform of the photocurrents was obtained. In the ideal case, the instrument profile is given by a sinc-function as it is shown in Fig. 8a . Figure 8b exhibits the instrument profile based on the real model of the standing wave spectrometer, while the experimentally obtained instrument profile is shown in Fig. 8c.

## 6. Discussion

A novel Fourier transform spectrometer, consisting of an ultra thin and partially transparent photodetector and a tunable mirror, was realized. The main advantages of the Fourier transform spectrometers are multiplexing (simultaneous detection of multiple wavelengths) and high throughput (spectrometer optical system efficiency to gather light), which make them useful for measuring both weak and broad spectra and, therefore, for inspection and multispectral imaging. For applications like multispectral imaging, it is desired to acquire not only the spectral information, but also the spatial information of the object, which requires an integration of two-dimensional spectrometer arrays on a sensor chip.

The instrument profile of the spectrometer can be described in the Fourier domain as a convolution of a rect-function, which describes the finite mirror movement, and a sampled signal that represents the measured photocurrent. The Fourier transform of the finite mirror movement is a sinc-function, while the Fourier transform of the photocurrent of the ideal standing wave spectrometer is a Dirac function shifted to the wave number of a monochromatic light. Consequently, the instrument profile of the ideal standing wave spectrometer is a sinc-function shifted to the wave number of the incoming light as it is shown in Fig. 8(a) and described by Eq. (15). Figure 8(b) shows the instrument profile calculated for the optical model of a real standing wave spectrometer, while the experimentally obtained instrument profile is depicted in Fig. 8(c). In addition to the peak at 633 nm, additional peaks appear at integer multiples of the wave number of the incoming light. The calculated photocurrent for the model of the real spectrometer is a periodic function, which slightly deviates from the cosine function. Therefore, the photocurrent can be represented by a Fourier series. The same applies for the experimentally obtained photocurrent. The Fourier transform of the photocurrent is a sum of Dirac functions shifted to the original wave number and its harmonics. The instrument profile is, consequently, a sum of sinc-functions shifted to the original wave number and higher order harmonics. The amplitude of the higher order harmonics depends on the deviation of the photocurrent from the ideal cosine function. The difference between the optically modeled and the experimentally determined instrument profile might be caused by a slight misalignment of the experimental setup.

The calculated and measured instrument profile shows that the operation of the standing wave spectrometer is limited by a fundamental operating range. In the case of an ideal standing wave spectrometer, higher harmonics do not appear in the Fourier spectra and, therefore, the operating range is determined only by the optical properties of the ultra thin and partially transparent photodetector. In the case of an optical detector realized by amorphous silicon, the operating range is from 350 nm to 725 nm. For short wavelengths, a very small fraction of the light is transmitted through the photodetector and, subsequently, the detected AC component of the photocurrent is very small. For longer wavelengths, the photo energy of the standing wave is smaller than the bandgap of amorphous silicon (E_{G}=1.72 eV [10]) and only few photons are absorbed. By changing the optical bandgap of the material, the operating range can be changed to other regions of the optical spectra. For example, by using a crystalline silicon detector (E_{G}=1.12 eV), the operating range can be extended to the near infrared part of the optical spectra.

The operating range of a real standing wave spectrometer is limited by the optical properties of the ultra-thin and partially transparent photodetector and the Free Spectral Range of the spectrometer. The Free Spectral Range of the standing wave spectrometer is given by λ/m, where m is the order of the higher harmonics. For short wavelengths, the operating range is limited by the optical properties of the detector, which define the lower operating limit to 350 nm. For longer wavelengths, the real spectrometer is limited by the Free Spectral Range, which results in an upper operating limit of 700 nm. In the case of an ultra thin and partially transparent photodetector based on amorphous silicon, the difference between an ideal and a real standing wave, in terms of the operating range, is very small. For other materials, like crystalline silicon, the difference is significantly larger.

The standing wave spectrometer combines the properties of a Michelson spectrometer and a Fabry-Perot spectrometer. The resolution of the standing wave spectrometer is determined by the scan length of the mirror (Eq. (16), Eq. (17)) just like in the case of the Michelson spectrometer. A resolution of 6 nm has been demonstrated experimentally for a scan length of 33 μm. The experimentally determined resolution is in very good agreement with the expected resolution according to Eq. (17) [8]. In terms of the operating range, the standing wave spectrometer has similar behavior like a Fabry-Perot spectrometer.

## 7. Summary

A novel type of Fourier transform spectrometer was realized. The spectrometer consists of an ultra thin and partially transparent photodetector and a tunable mirror. The operating principle of the novel Fourier transform spectrometer is based on the sampling of a standing wave created in front of a tunable mirror with an ultra thin and partially transparent photodetector. The spectral information of the incident light is determined by the Fourier transform of the detector signal. The linear arrangement of the ultra thin detector and the mirror enables the realization of spectrometer arrays and optical cameras with high spectral resolution. A partially transparent detector, which operates in the visible part of the optical spectrum, was realized. A complete optical model of the standing wave spectrometer was presented along with a model of the ideal standing wave spectrometer. Furthermore, the instrument profile of the spectrometer was derived. The optical model was compared with experimental results and a good agreement was observed. The resolution of the standing wave spectrometer is determined by the scan length of the tunable mirror, similar to classical Fourier transform spectrometers based on Michelson interferometers. The operating range of the standing wave spectrometer is affected by Fabry-Perot oscillations between the partially transparent photodetector and the tunable mirror. Therefore, the operation range is comparable to classical Fabry-Perot spectrometers. The experimentally realized standing wave spectrometer exhibits an operation range from 350 nm to 700 nm, covering exactly the visible spectrum. Therefore, standing wave spectrometers are optimal candidates for spectral imaging applications.

## Acknowledgements

The authors would like to acknowledge the Institute of Energy Research (IEF-5) Photovoltaics, Research Center Jülich. Specifically, the authors thank H. Stiebig and E. Bunte for providing optical detectors and helpful discussions.

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