Analysis of System



Chapter 3.

IHOS MODEL.

3.1. Introduction

The goal of this chapter is to present a comprehensive model for the IHOS (Integrated Heterodyne Optical System). The chapter provides an introduction to the MOEMS (Micro-Opto-Electro-Mechanical-System) modulator, the photodetector, and the amplifier.

First, we discuss the system electrical parameters. We model the signal and noise for the various components of the system. Firstly, wWe introduce the background that definesfor the noise mechanisms of the silicon photodetectors and the transresistance amplifier. In addition, we simulate the photodetector and amplifier stages in order to predict the signal and noise voltage spectral densities at the output of the system. InNext addition, we provide the model for the MOEMS, which is used as the basis for the design chapter. Figure 3.3.11 illustrates the four stages that will be described in detail.

Lastly, we simulate the photodetector and amplifier stages in order to predict the signal and noise voltage spectral densities at the output of the system. The novelty in modeling the entire system is based on the unique representation of different physical variables that describe the combined behavior of MOEMS, and semiconductor parameters into a single model that can be utilized as a design tool.

Thermal noise and shot noise are present mainly in the photo-detecting stages generally showing a uniform spectral distribution. Thermal noise is also present at the input of the amplifier and in the feedback network. The electronic system can be cooled down in order to reduce the thermal noise. Modulation prior to photodetection provides the most effective method to overcome the low-frequency noise (flicker noise) manifested in the subsequent stages. The low-frequency noise of the photodiode and amplifier stages can be overcome by modulating the input signal in the low kHz. A bandpass filter further improves the SNR as the system bandwidth is reduced to the desired frequencies of interest. Such filter can be implemented with a lock-in amplifier operating synchronously at the same frequency as the MOEMS modulator.

– Noel why f0= 1kHz ???

[pic]

Figure 3.1: Integrate heterodyne optical system adapted for detection of light emitting whole-cell biosensors.

[pic]

Figure 3.1: Block diagram of the systeIHOS including different noise sourcem. Modutlation is achieved prior to photodetection. Noise is added in the photodiode and amplifier stages. The bandpass filter can be implemented with a lock-in amplifier operating at the same frequency as the modulator., including noise

33.2. Noise in the Photodetection Stage

Noise in Photodiodes: The Motivation for Modulation

The detection limit of optical signal processing depends strongly on the noise features of photodiodes. The total noise of a photodiode includes thermal noise (Johnson noise), shot noise, flicker noise (1/f noise), and generation-recombination (G-R) noise [182]. Figure 3.2 shows the equivalent circuit for a photodiode. The photodiode can be modeled as a resistor, Rd in parallel with a capacitor, Cd, and both in series with a resistor Rs. Shot noise is and thermal noise are a ffundamental noise mechanisms, as is thermal noise. It is caused by the random arrival of carriers crossing a diode junction. The shot noise is the result of the fluctuations in the flux of the electron and hole currents that carry the electrical current. The shot noise is proportional to both the forward, backward and the photo currents. For a photodiode that is reverse bias, the Shshot noise current is can be described by from the following expression [83]:

[pic] (13.1)

Wwhere q is the electron charge, [pic] is the photodiode current, and [pic] is the snoise equivalent bandwidth (NEB)ystem bandwidth.

In a photodiode tThermal noise is mainly caused by the random motion in a conductor, or semiconductor due to the contact and parasitic series and parallel resistorsresistance. The thermal voltage can be calculated by from the following expression [84]:

is the total noise in a system where the noise equivalent bandwidth (NEB) is Δf. When the spectrum is flat the NEB is equal to Δf othersie it is more complicated. You also missedt he resistance !!!!

[pic]

= 4kTRDf you missed the R !!!!!!

(3.2)

Wwhere [pic] is the Boltzmann constant, [pic] is the temperature, Rs is the total series resistance, i.e. contact pads, and [pic] is the system bandwidthNEB.

The shot noise is the result of the fluctuations in the flux of the electron and hole currents that carry the electrical current. The shot noise is proportional to both the forward, backward and the photo currents and is give by:

=2q(ID+2 IR + Iph)Δf (*)

Where ID is the total diode current , IR is the reverse current at equilibrium and Iph is the photo current.

T The thermal current noise can be calculated from the following expressions [85]:

[pic] (3.3)

The gGeneration-recombination (G-R) noise is another noise mechanism affecting photodiodes [286]. The source of this noise is based on fluctuations in the carrier flux due to generation, recombination, and trapping of carriers in semiconductors. These fluctuations cause the number of free carrier to vary, thereby leading to random variations in conductivity that manifest as noise. Generation-recombination noise appears similar to thermal noise, although the physical mechanism is different. While in G-R noise the number of carriers varies, in thermal noise the number of carriers is constant, whereas the instantaneous velocity of the carriers fluctuate. The spectral response distribution of of G-R noise is almost uniform up to a frequency determined by the lifetime of the carriers in the photodetectors. In most cases, this noise mechanism is negligible.

Flicker noise dominates at low frequencies. Surface defects and metal contacts are among the main contributors to the low-frequency noise [387,88], which is the limiting factor in detection sensitivity. The current spectral density of the Fflicker noise is can be determined by the following semi-empirical expression:

[pic] (3.34)

where Idark is the diode dark current, C1 K is a coefficient of the photodiode fabrication process (quality and technology), β value is an empirical constant related to doping profile (usually equals to 2), ( γ is the flicker noise empirical constant (usually equals to 1), f is the frequency of operation. Photodiode outputs can be measured as a current, or converted into a voltage. The difference in the noise spectral density amplitude between low-frequency detection (~1 Hz) and frequencies higher than 1 kHz is at least three orders of magnitude (30 dB).

Figure 3.2: Equivalent circuit for the photodiode.

After photo-detection, the output signal can be fed into a pre-amplifier for subsequent signal conditioning.

Thermal noise is the only type of noise that is independent of current bias. All other types of noise sources are dependent on current. Shot noise is proportional to the square root of the current, whole G-R noise and 1/f noises are proportional to the photodiode current.

You are mixing current noise, voltage noise, current noise power density and voltage noise power density.

Thermal and shot noise sources have a uniform spectral distribution, while G-R and 1/f noise have a characteristic frequency response. For low-frequency signal detection, Flicker noise is the most dominant noise mechanism. In addition, measurement systems composed of connectors, coaxial cables, amplifiers all seem to demonstrate a high noise content at low frequencies. measurement systems composed of connectors, coaxial cables, amplifiers all seem to demonstrate high noise content at low frequencies, such as the power line coupling (50 or 60 Hz). Thus, it is preferable to overcome the low-frequency noise source entirely when measuring low intensity signals in general, such as bioluminescence.

We have modeled the noise following the work developed by Hamstra and Wendland [89], which neglects the series resistor of the photodiode. The photodiode noise can be approximated by the following expression:

[pic] (3.5)

Figure 3.2 shows a typical spectral response of the output noise of a transresistance amplifier connected to a photodiode. The signal was measured with a spectrum analyzer with a 16 Hz bandwidth (for more information on experimental results, the reader is referred to the Chapter 5).

[pic]

Figure 3.2: Spectral noise in of PN photodiode (Hammatsu 1226, Hammatsu Corportation, Japan) connected to a transresistnace amplifier (Hammatsu C9051), measured with a system bandwidth ofNEB of 16 Hz. (Which photodiode ? under what conditions ? What system? What is the meaning of measurement at 50Hz with a 16 Hz bandwidth ? The measurements below 100 Hz are questionable.

The photodetector stage is implemented by a photodiode in reverse bias connected to a transimpedance amplifier with a resistor and capacitor in the feedback loop. The feedback network is based on the parallel connection of the capacitor with the resistor, which provides frequency compensation. Figure 3.2 shows a schematic diagram of an amplifier with the integrated noise sources. The transresistance amplifier, however, also suffers from various types of noise mechanisms. We have used the noise model developed by Hamstra and Wendland [8967]. Herein, we refer the reader to Appendix I for comprehensive discussion of the noise in photodiodes and amplifiers, and all corresponding definitions. In this chapter, we simply summarize the results.Figure 3.3 shows a schematic diagram of an amplifier with the integrated noise sources.

The noise present at the amplifier circuit output must be taken into account as integral part of the analysis. Fluctuations in voltage from the power supply are not considered as part of this analysis. Assuming a dominant pole, an amplifier forward transfer function (output voltage versus input voltage) can be modeled as follows:

[pic] (3.6) Where ao is the low frequency gain (V/V), (c is the cut-off frequency. The next step is to formulate the expression for the output voltage due to photocurrent is defined as follows:

[pic] (3.7)

Where a1 is defined as the transfer function of the amplifier to an input current, which is defined as follows:

[pic] (3.8)

Where Rf is defined as the feedback resistor, Ci is the input capacitor, Cf is defined as the feedback network capacitor, and (H is defined as the bandwidth-gain product when the gain, ao (V/V), equals unity.

The amplifier suffers from three main noise sources: [pic] is the amplifier input current noise, [pic] is the amplifier input voltage noise, which are give by the manufacturer, see Table 3.1. In addition, we need to take into account the current noise in the amplifier feedback network, [pic], defined by the following expression:

[pic] (3.9)

The output voltage noise due to these noise sources is defined as follows:

[pic] (3.10)

Where a1 has already been defined as the transfer function for an input current (Equation 1), a2 is the defined as the transfer function due to an input voltage, a3 is the transfer function due to feedback network noise. The corresponding transfer functions defined as follows:

[pic], (3.11)

[pic] (3.12)

The actual magnitude of the noise at the output of the amplifer can be calculated using the following expression: [pic] (3.131) Where ao is the internal gain of the amplifier, a1 is the transfer function of the amplifier for the input current noise due to the photodiode, inpd, and the input current noise of the amplifier, in; a2 is transfer function of the amplifier for an input voltage noise, en; and a3 is the transfer function due to the feedback network noise, in_f. The total theoutput voltage noise is obtained by integratingted over the entire system bandwidth ([pic]).

[pic][pic]

Figure 3.32: Photodiode connected to a transresistance amplifier.

In order to calculate the SNR ratio, we formulate the ratio of the output voltage due to the photocurrent, ipd, that enjoys a specific operating frequency determined by the transmissive MOEMS modulator, and output noise voltage integrated over the desired frequency range. Such expression is given by:

[pic] (3.142)

The bandpass filter can be implemented electronically using a standard filter that provides a selective bandwidth. Among the several options exist to implement a lock-in amplifier provides a very narrow bandwidth and a simple method to demodulate the signal. The selectivity of the bandpass filter allows to further reduce the noise. Furthermore, a MEMS filter can also be implemented in order to obtain a high-Q, several references can be found in [9068,69,70,71,72]. The SNR is the figure that ultimately determines the performance of the system.

Next, we have simulated the noise contribution of each stage in order to predict the minimum detectable signal (MDS). In order to achieve this task, we simulated the noise of the photodetector and transresistance, using Matlab (Matlab Corporation, USA). Table 3.1 summarizes all the physical values used for the system. Figure 3.3 illustrates the noise sources for each stage. In this first model, we have neglected the noise contribution of the MOEMS modulator.

[pic]

Figure 3.3: Block diagram of the IHOS including different noise source. Modulation is performed prior to photodetection. Noise is added in the photodiode and amplifier stages.

In order to estimate the noise magnitude it is necessary to integrate the total of the square voltage spectral density over the desired frequency range. A simple method to estimate the magnitude of the noise at the output of the amplifier is based on multiplying the noise equivalent bandwidth (NEB) by the magnitude of the voltage density at each frequency. In this way, it is possible to estimate the minimum detectable signal by dividing the output noise by the gain of the amplifier. Figure 3.4 shows the noise signal at the output of the amplifier. Each curve corresponds to a different NEB. The minimum detectable signal (MDS) of the photocurrent as a function of the bandpass filter bandwidth, which was simulated as 25 mHz, 100 mHz, 1 Hz, 5 Hz, and 10 Hz. A lock-in amplifier can implement such narrow bandwidth. per system bandwidth can be estimated from these curves.

[pic] [pic]

Figure 3.4: Calculated noise signal as a function of bandpass filter quality factor.minimum detectable signal as a function of bandwidth. Green is 1025 mHz, cyan Light Blue is 20100 mHz , Red is 251 Hz, Blue is 505 Hz, and Yellow is 10 Hz0.

Table 3.1: Values of the photodiode and amplifier used for the simulations.

|Stages |Physical Variables |Values |

| |Active Area (A) |8 mm2 |

|Photodiode | | |

| | | |

| |Operating Voltage (Vd) |-1 V |

| |Saturation Current (Isat) |10-11 A |

| |Quantum Efficiency (() |0.95 |

| |Parallel Resistance (rRd) |10 GΩ |

|Amplifier |Capacitance (Cd) |10-9 F |

| |Input Impedance (Rin) |100 MΩ |

| |Input Capacitance (Cd) |10-12 F |

| |Feedback Resistor (Rf) |15 kΩ |

| |Feedback Capacitance (Cd) |10-9 F |

| |Open loop Gain (Rf) |15 kV/A |

| |Gain Product Bandwidth (ωH) |10 MHz |

| | Amplifier Input Noise (en) |10 [pic]pV |

| |Amplifier Input Noise (in) |0.5 [pic] |

3.43. MOEMS Modulator

The MOEMS modulator is based on the design presented in the Chapter 4, which its schematics are shown here. The device is comprised of a comb-drive actuators connected to set of shutters that resonate in plane to allow transmissive modulation. The modulator is excited using a time dependent voltage signal [pic], which is composed of a dc-offset signal and an ac-sinusoidal signal in such a way that [pic]. Here [pic]denotes the excitation "electrical" frequency of the electrical signal applied to the transducer as the excitation. Since the mechanical force produced by the comb-drive-based transducer is proportional to the square of the voltage, the transducer is characterized by the following force-voltage relation:

(3.153)[pic]

Where [pic] represents the permittivity of the free space, [pic] represents the height of the actuator, d is the gap between the comb fingers, and [pic] is number of combs. One observes that as the electro-static transducer is driven by a single frequency electrical harmonic signal, the mechanical force, which is related to the voltage squared, is produced at two frequency scales: a. the excitation frequency and b. twice this frequency.

The mechanical dynamic behavior of the modulator under harmonic excitation is described using a simplest one degree of freedom model:

(2) [pic] (3.164)

Here Where [pic] (the typical value is 1.86.10-5 [N.S/m2]) are the equivalent effective mass of the modulator, the total suspension stiffness in the stroke direction (axial displacement), and the damping coefficient (the typical value is 1.86.10-5 [N.S/m2]), respectively., [pic]is the amplitude of the excitation force; an example for the calculation of the equivalent effective mass of a comb-drive actuator and stiffness of the folded suspension can be found in [491]. The steady state solution of Equation (2) can be written in the form [73]:

(3) , [pic] (3.175)

where

(4) ,[pic], (3.186)

is the amplitude, and

(5 ) ,[pic] (3.197)

is the phase angle. Here, [pic]is the "mechanical" natural frequency of the modulator and [pic] is the Q-factor. Substituting Equation (1) for the mechanical force into Equation (2) and taking into account Equations (3)-(5), we obtain the steady-state response of the modulator:

(36203.8) ,[pic]

where

(7) ,[pic] (3.219)

is the static displacement,

(3.223.108) ,[pic]

(93.2311) , [pic]

and

[pic] (103.2412)

The first mechanical scale occurs at the electrical signal frequency, [pic], while the second one occurs at twice that frequency, [pic]. As a result, the resonant excitation can be achieved as the device is driven by an electrical harmonic signal at half damped resonant frequency of the modulator, i.e. [pic], as well as at the damped resonant frequency itself, i.e. [pic]. In the former case, the mechanical response frequency differs significantly from the frequency of the electrical signal. The effective mass of such comb-drive actuator [592] can be calculated according to the following expression [74]:

[pic] (113.2513)

Wwhere mshuttle is the mass of the actuators, shutters and moving portion of the actuators; mbeam is the mass of a single beam of the folded flexure; mtruss is the mass of the truss of the folded suspension (typical design values are described in the subsequent in Chapter 4).

The suspension stiffness can be calculated from the expression of a folded flexure [75]:

[pic] [pic] (123.2614)

where i is the total number of spring structures (generally taken in pairs), and lb are and l are the height and length of the folded-flexure beam, E is the Young’s modulus of silicon (the value is taken as 169 GPa). If we take into account that the MOEMS design is comprised of either 8 or 10 springs (the reader is referred to Chapter 5), then the The natural frequency of the system can then be calculated as:

[pic] [pic] [pic] (3.132715)

3.54. Photodetection of Modulated Signal

The characteristic equation of silicon photodiodes is defined as:

[pic] (14)

where Isat is defined as the saturation current, q is the electron charge, V is the bias voltage

kB is the Boltzmann constant, T is the temperature [K], Ip is the photocurrent. As As described earlier, the mechanical response of the device enjoys two main frequency scales. The first one occurs at the frequency of electrical excitation,[pic], whereas the second one occurs at twice that frequency, [pic]. Figure 3.45 shows the cross-section of shuttersMOEMS modulator, where the shutterswhich are driven by the actuators over the etched cavities that act as optical vias. Given this configuration, the frequency of the photodetected signal doubles the mechanical frequency. Therefore, the photodetected signal enjoys two main spectral components that occur at twice the electrical excitation frequency,[pic], and at four times the electrical excitation frequency, [pic]. Figure 3.56 shows a schematic representation of the frequency response of the modulated signal. Note that the arrows represent the Fourier components for the frequency spectrum of the electrical, mechanical, and optical responses.

As the active area of a photodiode is exposed to a photon-flux, the current due to photon-generation is defined as:

[pic] (3.152816)

where Φ is the incident photon-flux [photons/sec-cm2], η is the quantum efficiency (i.e. 0.9), q is the electron charge, A is the active area of the photodiode. Since the active area of the photodiode is shuttered periodically as described by Equations 6-7, the photodetected signal varies as a function of time.

As discussed later in Chapter 4, Tthe MOEMS design of the MOEMS modualtator is comprised of either 20 or 40 shutters. Hence, the photodetected signal can be calculated from the following expression:

[pic] (3.2917)

[pic].

Where Ls is the length of the shutters, u is the number of shutters. We have assumed that the displacement magnitude of the shutters is 50 µm, which is exactly the width of the optical vias. At an excitation frequency of half resonance, Ithe photodetected signal can be expressed as followsf we take into account the optical modulator stage, the actual photo-current is defined as:

[pic] [pic] ((3.163018) [pic]

In this expression we disregarded the phase since it is not relevant for measuring bioluminescence from whole-cell biosensors.

[pic]

[pic]

Figure 3.45: Cross section of shutter configuration showing the input and output fluxes.

[pic]

Figure 3.56: Frequency Spectrum of the Eelectrical excitation (red), mechanical (blue) and optical (green) responses.

Note that the second harmonic of the optical response is at 4(e, which is much higher than (e. Hence, the unwanted electrical coupling from the modulator at (e can be avoided.

For noise calculations the photodiode can be modeled as a resistor, rp, in parallel with a capacitor, cp, and both in series with a resistor, rs; rp is defined as:

[pic], (17) and rs is defined as:

[pic] (18)

where id is the diode current, ed is the diode voltage measured at large current to have its value deviate from the ideal forward voltage, ei, by at least 100 percent. For frequencies below 100 kHz, rs can be considered negligible [6]. The photodiode power spectral density noise can be mathematically represented as the summation of all relevant noise mechanisms previously discussed, and summarized by the following expression:

[pic] (19)

Where the first term is the low-frequency noise (flicker), the second term is thermal noise, and the third term represents the shot noise. Figure 3.3 shows the equivalent diagram of the photodiode, including all the noise sources. In order to properly calculate the thermal and shot noise, it is important to note that the term,[pic], is related to the system bandwidth. It is note-worthy that this bandwidth factor seems to be easily confused and misused in numerous literature sources when calculating the various types of sources. For example, in order to calculate the spectral power of photodiode noise sources, it is crucial to note that equation (19) is the actual power spectral density. Hence, it is necessary to integrate the spectral density for the frequency range of interest. A graphic method to calculate noise is presented in the Simulation Section.

Figure 3.3: Equivalent circuit for the photodiode. rRs together with en can be neglected altogether for frequencies below 100 kHz. It is a little tricky, the thermal noise is due to parasiticresistance and you should be careful not to confuse it with the shot noise.

3.6. Amplifier Noise

The photodetector stage is implemented by a photodiode in reverse bias connected to a transimpedance amplifier with a resistor and capacitor in the feedback loop, as illustrated in Figure 3.4. The feedback network is based on the parallel connection of the capacitor with the resistor, which provides frequency compensation. If chosen carefully, the transfer function of the amplifier will provide a very fast roll-off at the frequency of interest. The transresistance amplifier, however, also suffers from various types of noise mechanisms. The noise present at the amplifier circuit must be taken into account as integral part of the analysis. Fluctuations in voltage from the power supply are not taken into account as part of this analysis. In general form, an amplifier open-loop transfer function (output voltage over input current signal) can be represented by the following expression:

[pic] (20)

where G(ω) represents the output voltage over the input current signals, A is the open-loop gain, [pic] H ???Rf is the feedback resistor, and Cf is the feedback capacitor. The product RfCf is the low-pass cut-off frequency of the amplifier. Figure 3.4 shows a schematic diagram of an amplifier with the integrated noise sources. Equating the amplifier currents at point ‘P’ in Figure 3.4, we obtain the following expression:and

[pic] (21)

Ssolving for eout/ iin in order to obtain the total transfer function of the amplifier to calculate the frequency response to photocurrent, following expression [7] is obtained: You miss a J here !!!

[pic] (212)

where ωH represents the gain-bandwidth product, Rf is the feedback resistor, Cf is the feedback resistor, Cin is the input capacitance. The feedback resistor also introduces thermal noise represented in the form:

[pic] (23)

The noise response across the feedback resistor [8], efeedback/ ifeedback, can be calculated by:

[pic] (24)

[pic]

Figure 3.4: Photodiode connected to a transresistance amplifier.

In this analysis, we assume that the input current noise of the amplifier is neglibnegligiblele, as instead we consider the input voltage noise., If we take into account the amplifier noise at the output, excited by the input voltage noise given by en, we obtain the current expressions at point 'P':

[pic] (25)

Where en is defined as the input thermal noise defined as:

[pic] (265)

Where en is defined as the input thermal noise defined. Although the spectral distribution of en is uniquely modeled as uniform in this case, in several occasions the spectral distribution rolls off very slowly (for more details, see Simulations Section). We can take into account the amplifier noise at the output. TThe noise response, eout/ en, can then be calculated as follows:

[pic] (276)

The total noise of the system at the output is therefore defined as the summation of the noise sources of the photodetector and amplifier. Therefore, total noise signal is given by:

[pic] (287) where the noise is integrated over the entire system bandwidth [pic]. The signal of the system can be obtained by following expression:

[pic] (298)

The bandpass filter can be implemented electronically using a standard filter that provides a selective bandwidth. Among the several options exist to implement a very narrow bandpass filter, a lock-in amplifier provides a mHz bandpass windows. Furthermore, a MEMS filter can also be implemented in order to obtain a high-Q, several references can be found in [9]. The SNR is the figure that ultimately determines the efficiency of the system. The ratio between the signal power and the noise power figure can computed, given in the form:

[pic] (3029)

33.86. Noise Simulations

The goal of this section is to simulate the noise contribution of each stage of the photodetection in order to predict the minimum detectable signal (MDS). In order to achieve this task, we simulated the noise of the photodetector and transresistance, using Matlab (Matlab Corporation, USA). Table 3.1 1 summarizes all the physical expressions values used for the system. Table 2 summarizes all the design parameters involved in the simulation of each stage. Figure 3.6 illustrates the noise sources for each stage.

[pic]

Figure 3.6: Block diagram of the IHOS including different noise source. Modulation is performed prior to photodetection. Noise is added in the photodiode and amplifier stages.

The first step of the simulation was to introduce noise sources of the photodiodes, and noise sources of the amplifier, as shown in Figure 3.3.57. In this case, we plotted the current and voltage noise densities in the same graph in order to show the spectral distribution of each type of noise. The flicker noise current is depicted in red, Equation (3); the shot noise dand thermal noise are added and depicted in is depicted in green, Equations (1) and (2). The thermal noise due to the feedback resistor, Equation (3), is depicted in yellow, while input noise voltage of the amplifier, Equation (6) is depicted in cyanblue. The flicker noise dominates the lower part of the spectrum, decaying at 10 dB per decade (power density units), while the shot and thermal noise are characterized by their approximate uniform distribution. In addition, the amplifier input voltage is plotted as a slow decaying function, as it can be empirically gathered that its distribution is not completely uniform, but it rather decays obtained at 1 dB per decade. Moreover, a simple method to estimate this figure in the lower part of the spectrum can performed by measuring the output voltage at the output of the transresistance amplifier, without any input and in dark conditions, and subsequently dividing this figure by the gain of the amplifier (for further information about the noise distribution, the reader is referred to Chapter 3).

The next step involved simulating the transfer functions of the amplifier: Aa1(j[pic]),(f), Ba2(f),(j[pic]), and a3C(f), (j[pic]), corresponding to Equations (22), (24), (27), respectively, illustrated in Figure 3.3.68. It is possible to observe that the contribution of the feedback transfer function, Ba2(fj[pic]), and the input current output voltage transfer function, Aa1(j[pic]),f) dominate the spectrum. Furthermore, the frequency response of the amplifier is dictated by the cut-off frequency of the feedback network. The transfer function of the amplifier is shown in Figure 3.9.

In order to graphically compute the noise, first we plot in a log-log graph the voltage-square and current-square densities of the noise mechanism. Thereafter, it is possible towe plot the product of each transfer function squared by its input-square noise contribution. The resulting graphs are illustrated in Figure 3.310. In order to calculate the total noise contribution, it is necessary add the spectral densities at the output of the transresistance amplifier, shown as a blue trace.

Finally, in order to estimate the noise magnitude it is necessary to integrate the total of the square voltage spectral density over the desired frequency range. A simple method to estimate the magnitude of the noise at the output of the amplifier is based on multypling athe noise equivalent bandwidth (NEB) narrow bandwidth by the magnitude of the voltage density at each frequency. The width of the window is equivalent to the system bandwidth. Figure 3.4 11 shows the noise signal at the output of the amplifier. Each curve corresponds to a different bandwidthNEB. The range of values was chosen according to the standard bandwidth values defined by our lock-in amplifier (the reader is referred to experimental results for more information). The minimum detectable signal (MDS) per system bandwidth can be estimated from these curves.

[pic]

Figure 3. 3.76: Spectral Rresponse of the photodiode current noise (current noiseflicker noise and shot noise are depicted in red and green, correspondingly), and and voltage noise generators of photodiode and transresistance amplifier input noise (blue).

.

[pic]

Figure 3. 3.87: Transfer function for each noise generator. Green, violet, and red correspond to the transfer functions of A(j[pic])a1(f), B(j[pic])a2(f), and C (j[pic])a3(f), correspondinglyrespectively.

[pic]

Figure 3. 3.98: Frequency response of transresistance amplifier.

[pic]

Figure 3.3.109: Transfer functions, Spectral Density of Signal Response, and Noise.

[pic]

Figure 3. 3.1011: Noise signal as a function of bandpass filter quality factor. Green is 10, Light Blue is 20, Red is 25, Blue is 50, and Yellow is 100.

|Stage/ Variables |Input Function |Output Function |

|Amplifier Response to |[pic][pic] |[pic] |

|Photodiode Noise |[pic] | |

| |[pic] | |

|In = photodiode noise | | |

|Amplifier Response to Feedback |[pic] [pic] |[pic] |

|Noise | | |

| | | |

|Iff = feedback noise | | |

|Amplifier response to |[pic] [pic] |[pic] |

|Input Voltage Noise | | |

| | | |

| | | |

|Vn = amplifier input voltage | | |

|Total Noise |[pic], [pic], [pic] |[pic] |

T

Table 1: Equations used for noise simulation

able 3.1: Values of the photodiode, amplifier, and bandpass filter used for simulation.

|Stages |Physical Variables |Values |

| |Active Area (A) |8 mm2 |

|Photodiode | | |

| | | |

|Amplifier | | |

| |Operating Voltage (V) |-1 V |

| |Saturation Current (Isat) |10-11 A |

| |Quantum Efficiency (() |0.95 |

| |Parallel Resistance (rd) |10 GΩ |

|Amplifier |Capacitance (Cd)Parallel Resistance (Rd) |10-9 F10 GΩ |

| |Input Impedance (Rin) |100 MΩ |

| |Input Capacitance (Cd)Capacitance (Cd) |10-12 F10-9 F |

| |Feedback Resistor (Rf)Input Impedance (Rin) |15 kΩ100 MΩ |

| |Feedback Capacitance (Cd) Input Capacitance |10-9 F10-12 F |

| |(Cd) | |

| |Open loop Gain (Rf) Feedback Resistor (Rf) |15 kV/A15 kΩ |

| |Gain Product Bandwidth (ωH)Feedback Capacitance |10 MHz10-9 F |

| |(Cd) | |

| | Amplifier Input Noise (en)Open loop Gain (Rf)|10 [pic]pV15 kV/A |

| |Amplifier Input Noise (in)Gain Product Bandwidth |1 [pic] 10 MHz |

| |(ωH) | |

| |Filter Bandwidth ((f)Amplifier Input Noise (en) |16 Hz10 pV |

|Stages |Physical Variables |Values |

| |Active Area |2 mm2 |

|Photodiode | | |

| |Operating Voltage |1 Volts |

| |Saturation Current (Isat) |10-11 A |

| |Quantum Efficiency |0.95 |

|Amplifier | | |

| |Parallel Resistance (Rd) |10 GΩ |

| |Capacitance (Cd) |10-9 F |

| |Input Impedance |100 MΩ |

| |Input Capacitance |10-12 F |

| |Feedback Resistor |15 kΩ |

| |Feedback Capacitance |10-9 F |

| |Open loop Gain |100 V/A |

|Bandpass Filter | | |

| | ωH |10 MHz |

| |en |10 pV |

| |Bandwidth |1 Hz |

Table 2: Values used for simulation.

3.10 7. Summary

This chapter offers an overview of the physical models model that governs the photodetector, behavior of the Integrated Heterodyne Optical System (IHOS)electronics and the MOEMS modulator, which constitute the IHOS, thereby providing the foundations for the design chapter (Chapter 4). As we formulated all the equations that define the system, this chapter can be considered as the basisfoundations of for a the development of a design CAD tool. First, we introduced all the noise mechanisms that limit the sensitivity of silicon photodiodes. Flicker noise is the most dominant source of noise, as it is characterized by its 1/f noise distribution. Hence, the motivation for modulation is to overcome the low-frequency noise content that limits the sensitivity of the photodiodes.

Furthermore, we introduced the noise mechanism of the transresistance amplifier integrated with a photodiode. The model was based on Hamstra and Wendland [8967], which neglects the series resistance of the photodiode. This chapter has been complemented with Appendix I. In addition, the noise due to the power supply variations was also neglected since the system will be connected to batteries. We have simulated the total noise of the system in order to estimate the minimum detectable signal.

We provided the noise spectral distribution at the output of the system. ThereafterFinally, we presented the model of the MOEMS modulator, which is the solution tois the heart of the increase the sensitivity of photodetectors IHOS, and can be implemented as a simple add-on device. We introduced all the equations of motions that define the comb-drive actuator, including the electro-static and mechanical stages. In addition, we explain how the transducer modulator works in the as a mechanical-electrical-optical domain transducer in order to achieve optical modulation at different frequency scales, which are electro-mechanically decoupled from the excitation signal of the modulator (the reader is referred to the experimental chapter, Chapter 56, for a description of the Double Harmonic Modulation Technique).

Furthermore, we introduced the noise mechanism of the transresistance amplifier integrated with photodiode. We describe in detail the transfer functions that describe the noise spectral distribution at the output of the system. Finally, we simulated the noise mechanism of the photodiode and photodetector in order to calculate the power spectral density of the noise, and the minimum detectable signal. In order to realize the integration of the IHOS with light-emitting whole-cell sensors, simulations for the noise figures were required to estimate a baseline for the minimum detectable signal (MDS)integrated . This chapter provides the basis for the Design Chapter.

[pic]

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4(e

Optical Response

Electrical Excitation

Cin

Amplitude

(Arbitrary Units)

2(e

(e

2(e

(e

vout

ao

Point ‘P’ for equating currents

Inoise

In_f

Cf

a0

MOEMS

Modulator

in

in_f

Cf

Rf

en

Cin

Rf

Rd

[pic]

Rin

Rd

ipnpd_n

ipd

In_f

en

Cf

Thermal Noise

Rf

en

Rin

[pic]

[pic]

[pic]

Cin

i1/f

ishot

Shot Noise

Flicker

Noise

Optical Signal

Bandpass Filter

Transresistance

Amplifier

Photodiode

MOEMS

Modulator

Point ‘P’ for equating currents

ip

i1/f

rRd

ishot

Cd

ithermal

Rsrs

Venth

Point ‘P’ for equating currents

If

Cf

Rf

en

Cin

Rin

In_in

Rd

Inoise

Shot

Noise

Rin

Thermal Noise

Mechanical Response

Flicker

Noise

Optical Signal

a0

Bandpass Filter

Transresistance

Amplifier

ip

Rd

in

in_f

vout

Cf

Rf

en

Cin

Rin

Rd

inpd

ip

InoiseIpd_n

Frequency [Hz]

Output Voltage [V]

2nd harmonic

1st harmonic

2nd harmonic

1st harmonic

1st harmonic

PHOTODIODE ACTIVE AREA

X

INPUT FLUX

Y

OUTPUT FLUX

Shutters

Substrate

Ip

PHOTODIODE ACTIVE AREA

Substrate

Shutters

OUTPUT FLUX

INPUT FLUX

Y

Ip

Thermal Amplifier Noise

Photodiode

X

Vth

Rs

rd

Cd

Inoise

ip

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