Independent Study: A Photodiode Detector Michael Grogan 9th May, 2006

Contents 1

Goals

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Optical Atom Traps

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3

How A Photodiode Works

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4 4.1 4.2 4.3

Circuit A simple photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . A photodiode with gain . . . . . . . . . . . . . . . . . . . . . . . . A modified photodiode circuit . . . . . . . . . . . . . . . . . . . . .

5.1 5.2 5.3

Circuit Testing and Response 9 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Voltage and Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Phase-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusions 16

1

Goals

The aim of this independent study is to arrive at a working photodiode in preparation for making a noise eater to clean up a laser light. Instability in the intensity of lasers can be caused by many things, such as power supply noise, small thermal changes, or even something as simple as dust floating in front of the beam. Even these small changes are undesirable when the laser is being used in a precision scientific experiment, such as an atomic trap. The fluctuations which are important in this case are limited; above frequencies of ∼100kHz there is no problem with the fluctuations because the detector can have only limited response at these frequencies. The fluctuations due to these effects are small, so we require a large gain to be able to use the output. This means that we will need a high Signal:Noise ratio.

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Optical Atom Traps

The simplest way to understand an optical trap in terms of a sub-wavelength particle is to think of an induced dipole moment within the particle, caused by the light’s E and B fields. The dipole 1

will move towards the largest field, which is at the focus of the beam, but if it passes over a local maximum then its motion will be altered at this point, and it may even get trapped if there is enough of a force gradient. Pairs of optical traps can be generated in the same way from a single beam using a hologram, even a simple diffraction grating can cause optical trapping at each of the diffraction maxima. A couple of interesting ideas that have arisen from this are Optical tweezers and Holographic Optical Traps [1], the latter of which has tremendous applications for the quick manipulation of an atomic lattice. Computer generated holograms are used to control the location of the optical traps, and assuming that the traps are strong enough, the atoms are confined to their sites until the computer makes a ‘new’ hologram. Several methods for computing the required hologram are proposed, including one interesting method which creates a random model and uses something similar to the Ising algorithm to converge on a solution giving the confinement that is required. Quasi-crystalline structures have already been demonstrated using the holographic technique [2], using silica spheres immersed in water. Condensed bosons represent an interesting field, slightly different from the above example due to the difference of their interactions and statistics [3]. In particular, it would be desirable to be able to withdraw single atoms from the trap and then study their interactions. To do this requires a very accurately controlled potential, with high stablilty, so that the atoms can tunnel from the large BEC reservoir into the quantum ‘pipette’.

3

How A Photodiode Works

In this section I plan to cover the very basics of how a photodiode works, (which is where I started learning) up to something a little more advanced, but all of the circuit design will be covered in other sections. This section is just the physics of the actual photodiode, and its advantages and disadvantages over other types of light transducers, and it is largely sourced from ‘Fundamentals of Photonics’ [4]. The general class of light transducers are called photo-detectors, and there are two common mechanisms by which they work. The first kind, thermal detectors, absorb the energy from photons and convert it into heat. This is inefficient and slow, and not a useful form because it is unsuitable for electronic circuits. The other kind of photo-detectors use the photoelectric effect. On the absorption of a photon an electron is promoted into a more excited state and the conductivity of the material is changed. It is the photoelectric effect which is the more useful of the two, especially in this situation, and it can be broken down into two subcatagories. Photoelectric emission occurs when a photon with enough energy strikes a metal and dislodges an electron. This is similar to the set-up by Millikan, who verified the photoelectric effect after Einsteins proposal. In this experiment, electrons travel through a vacuum from the cathode to the anode, and this current flow can generate a voltage. After undergoing photoelectric emission electrons have kinetic energies: (1) k.e = hv − W where W is the work function of the metal, typically > 2eV . By choosing a semiconductor instead of a metal, the kinetic energy of a conduction electron is instead given by: k.e = hv − (Eg + χ) where the combination Eg + χ can be as low as 1.4eV . 2

(2)

In a semiconductor, by the same mechanism we can get the internal photoelectric effect, where the photocurrent carriers remain within the semiconductor. This is how most photo-detectors work today, including the photodiode which is what I am specifically looking at. Photodiodes have the advantage that since neither the electrons nor holes leave the material, there is no energy lost in the work function. Additionally, both the electrons and the holes can be used as current carriers, so there is increased conductivity. An incident photon with enough energy causes an electron to be promoted from the valence band into the conduction band as shown in Figure 1. This also generates a hole in the valence band, which upon an applied electric field leads to the production of an electric current.

Figure 1: An electron and hole pair are formed when a photon with enough energy is absorbed by a semiconductor. This style of conversion from light into energy is used in two important classes of devices today, the phototransistor and the photodiode. The phototransistor has an associated gain because of the recombination of holes and electrons, which have different velocities across the sample. It also places a limit on the response time of the transistor and thus is not well suited to our application. One of the main advantages of the transistor in other applications is the ability to make it sensitive to long wavelengths by using different doping elements, which changes the size of the bandgap between the conduction and valence band. We will be using a P-N type photodiode for this detector because of its linear response and fast speed. The photodiode works by the formation of electrons and holes in a depletion layer formed at the junction of a p- and an n-type semiconductor (shown in Figure 2). Electron and hole pairs formed in the depletion region are forced apart by an electric field, with electrons heading towards the n-type region. The electron-hole pairs formed in the n- and p- type regions which cannot support an electric field, travel about randomly until they are annihilated by part of another pair. If they are close enough to drift into the depletion layer before they annhilate, and they drift in from the right side, then they will also contribute to conduction. For example, an electron drifting into the depletion layer from the p-type region will be a conductor whereas an electron trying to drift into the depletion region from the n-type region will not be able to. Photodiodes can be operated under reverse bias conditions to increase the speed of the current carriers and increase the range of the depletion region. We actually will use a P-I-N diode, with an insulating layer in between the p- and n- type layers. This makes the depletion region a lot larger, which has several advantages over a regular p-n junction.

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Figure 2: All of the regions in a p-n junction can support the formation of electron-hole pairs, but only the depletion region can support an electric field. Electrons and holes which drift into this region after formation can also be used as charge carriers, hence the drift regions are approximately the size of the mean free path of the electrons and holes. • The capacitance associated with the depletion layer is reduced because of the larger depletion area. • The drift velocity of electrons and holes in the depletion region is increased. • The larger surface area means that there is more area for the light to hit, causeing a larger photocurrent. • The larger depletion region results in a higher proportion of carriers coming from within the region as opposed to the drift region, thus increasing the speed of the photodiode. Now that we have a fair idea of how the conduction actually happens in a photodiode we can start thinking about what it will do in an electrical circuit, especially in terms of noise. I am going to examine briefly several different potential sources of noise which are caused by the photodiode. These are: 1. Dark Current 2. Terminal Capacitance 3. Photon Noise 4. Photoelectron Noise (Efficiency noise) Dark Current: The name given to the spontaneous formation of electron/hole pairs which causes a small current to flow even with no initial light. The dark current is increased by applying a reverse bias to the photodiode, and ranges from a typical value of 0.1nA up to a maximum of 10nA [5] for the S1223 series photodiode. 4

Terminal Capacitance: Even with the insulating junction there is still a small amount of terminal capacitance caused by the depletion layer, which is reduced by applying a reverse bias. The typical value of terminal capacitance is 10 − 20pF for a 2.4mm2 junction. Photon Noise: Due to the random arrival of photons in the material, and the actual discrete formation of electron-hole pairs, there is noise associated with random arrivals. The random arrival of photons is described by the Poisson Distribution. Photoelectron Noise: Sometimes the photon can be absorbed but not lead to the formation of an electron-hole pair. This depends on the Quantum efficiency of the device (a number η which ranges from 0-1), and leads to a random source of error in the photodiode if η < 1.

4

Circuit Design Process

In the circuit design process I went through steps trying to understand three different circuits theoretically before picking one to use, and then I tried to totally analyse the final circuit in terms of transient behavior and noise. The noise in a photodiode circuit has some contributions from the electronic components as well as from the photodiode noise that we have looked at previously. The three main things that need to be kept in mind when designing a circuit with amplifiers are: 1. The divergence of the amplifier from an ideal amplifier 2. The noise of the amplifier 3. The noise of other components such as resistors and wires An ideal op amp has an infinite open loop gain and infinite input impedance, leading to no current going into the op amp. It also has a very high Common Mode Rejection Ratio (CMRR), which means that signals common to both leads do not affect the op amp output. In reality the open loop gain is just very large, and there is a small amount of leakage current into the terminals. The open loop gain typically varies from 104 − 106 , input currents are of the order of µA and the CMRR ranges from 60−140dB.√The amplifier has an input voltage noise of around 10nV , an input current thermal noise of around 1.5pA/ Hz and a bandwidth of 43M Hz. Resistors experience random √ excitations which cause noise as a function of temperature. This noise, in units of V / Hz is given by the equation p VN = 4kB T R (3) where kB is the Boltzmann Constant, T is the temperature in Kelvin, and R is the value of the resistor.

4.1

A simple photodiode

A photodiode is a current producing device, so to turn it into a voltage device we just need a simple voltage converter. The simplest photodiode circuits consist of a resistor -the simplest voltage converter- in parallel with the photodiode, as shown in Figure 3, where the photodiode is shown in an equivalent circuit as a current source in parallel with an internal capacitance and a resistance. There are several problems with this circuit for the application that we are looking at. The output voltage is set by the value of the load resistor RL for small loads, since most of the current will take the path of least resistance as opposed to travelling through the large internal resistance RD . However, we cannot just increase the voltage output by increasing the load resistance,the internal 5

Figure 3: A simple photodiode circuit, where the current is converted into a voltage by a resistor in series with the photodiode. Typically, Rd is very large and Cd is small. resistance and the maximum voltage that a photodiode can supply is are both factors. This means that we need to use an amplifier [6]. Another thing to think about is charging over the internal capacitance. For the bandwidth which we require, there is a limitation on the maximum resistor value that we can attach to the photodiode. We are aiming for a flat response over the 100kHz range, which dictates the value of the load resistance to be RL < 0.5M Ω. This does not apply to this circuit but it is worth thinking about for further circuit design.

4.2

A photodiode with gain

This and the following circuit design are based largely on the application notes regarding monitoring photodiodes with op-amps [7]. The simplest circuit is shown first, and then some modification are made based on the real components to get a flat response and improve the sensitivity. It is worth noting that in many situations a photodiode just needs to measure the level of light, whereas we are trying to measure the fluctuations on top of this, hence we need to achieve higher stability over a large bandwidth. The simplest circuit for a photodiode with gain is shown in Figure 4, it is just a basic current to voltage converter and I will go through the ideal operation here. • The non-inverting input is connected to ground through a resistor and a capacitor, which means that it is at 0V • Since V− = V+ , V− = 0V also. • The photodiode produces a current Ip (Or ID when there is no light incident). • No current flows into the op amp, so all the current must go through the feedback resistor. • Thus, the total voltage output of the circuit is Vout = IR. • Hence the feedback resistance controls the gain of the circuit. • The RC network on the non-inverting input is chosen to give maximum stability in return for minimum noise. 6

Figure 4: A simple photodiode with gain provided by an operational amplifier. This circuit is useful for monitoring levels of light but is susceptible to noise from a lot of different sources. The non-inverting input requires an extra resistance because of the temperature sensitivity of the op amp and feedback resistor. This needs to be the same value as the feedback resistor to maintain the stability. The capacitor is chosen such that it causes almost all AC signal to short through it, removing almost all of the Johnson noise caused by the resistor. Johnson noise, we will see later, has significant contributions to the noise output. Hence, the main sources of noise in this amplifier circuit are linear combinations of the op amp input current noise (by a factor of R), the feedback resistor Johnson noise, the op amp input voltage noise, the dark current noise of the photodiode and any electromagnetic coupling introduced by the photodiode leads. Additionally there may be noise on the power supply and op amp drift with time. This is not a good circuit to use for precision measurements of the kind we are after because photodiodes are very suscptable to EM coupling.

4.3

A modified photodiode circuit

To reduce the amount of noise in the circuit I added several modifications which can be broadly grouped as either removing internal noise or extraneous noise. My first modification was a capacitive filter on the power supply and the choice of battery power. This removed a lot of the low frequency AC components from the supply, and high frequencies 7

shouldn’t have much effect on the op-amp output. Then I modified the circuit layout so that the photodiode was connected between the inverting and the non-inverting inputs. Choosing an op-amp with a high CMRR ensures that noise generated in the photodiode that is common to both leads does not get amplified by the op amp. I selected an op amp with a much higher bandwidth so that there was no cut off from the open loop gain, making the behavior closer to that of an ideal op-amp. Finally I added in a high cut filter to the feedback loop of the op amp and a similar one on the non-inverting input. This should have the effect of cutting the unwanted high frequencies including the high frequency noise out of the signal. This circuit is shown in Figure 5.

Figure 5: The final circuit used by my detector. This circuit has a 3dB cut off point at ∼ 100kHz and provides flat response with a gain of 10.8V/mW of light modulated at frequencies below this. The analysis of this circuit is rather more tricky, since the non-inverting input is no longer at ground. However, by treating the setup as an ideal amplifier I will derive an expression for the gain and the phase shift of the circuit. • There is a current Ip flowing through the whole circuit as the photodiode is the only current 8

source. • The RC networks are both equal and have a complex impedence of Z which can be found from Z1 = R1 + iωC. • Hence the voltage at V+ = Ip Z = • Since V− = V+ , V− =

Ip R 1+iωRC .

Ip R 1+iωRC .

• There is also a current of Ip flowing through the feedback loop so there is the same voltage Ip R drop there, Vf = 1+iωRC . • Hence the total output voltage is Vout = • This is equivalent to Vout =

2Ip R 1+iωRC .

2Ip R 1+iωτ .

• For this circuit τ is set so that ωτ ∼ 1 when ω = 100kHz. • More generally, the expression for the output voltage for different values of resistors and R1 R2 capacitors is Vout = Ip ( 1+iωR + 1+iωR ). 1 C1 2 C2 The main source of noise in this circuit is the Johnson noise from the resistor on the noninverting input, which is easy to see by looking at the dark circuit, or indeed at an open circuit as they are very close to the same thing. When there is no light on the photodiode, there is only a small dark current produced, around 0.1nA. The resistance is large, so it acts like an open circuit between the op amp terminals, and similarly the capacitance is small so at low frequencies it behaves like an open circuit too. Hence the circuit shown in Figure 6 is approximately equivalent, with the different noise sources marked in. In a circuit with no noise, the output of the op-amp would be zero since the non-inverting input is tied to ground and there is no voltage applied to the inverting input. The addition of the feedback resistor noise contributes directly to the output and is directly the Johnson noise. The voltage noise on the inverting input also contributes directly, while the dark current of the photodiode gets amplified by the feedback resistor. The resistor noise on the non-inverting input causes a difference between the two inputs which is then acted on by the op amp open loop gain. Since the open loop gain is large the majority of the noise on the output terminal arises from this. For an op-amp with an open loop gain of A0 ≈ 105 this gives us a noise of: √ p VN = 4kb T R ∗ A0 ≈ 1mV / Hz (4) As always, the only way to reduce the noise is to filter out the johnson noise with a larger capacitor between the non-inverting input and ground. However this leads to an extra phase shift, and different gain, as will be seen in the ideal circuit simulation.

5

Circuit Testing and Response

Having selected a final design I then built the circuit on bread-board and proto-board, and tested its noise, gain, bandwidth and phase-shift to make sure it was behaving as expected. To obtain information about phase shift I compared my signal to the signal from a High Speed Photodiode. 9

Figure 6: The noise sources in the photodiode circuit, which is largely affected by the Johnson noise of the resistor on the non-inverting input and the op-amp open loop gain. The light levels were measured with a light meter, the noise with a network analyser, and the bandwidth, phase-shift and output voltages on a precision oscilloscope. The simulations were run using a C program which calculates the voltage output and the phase shift following the conventions that were examined before, and the code is included as an appendix to this report. Most measurements were taken in situ, using the actual laser that needs to be stabilised and modulating with an accousto-optic modulator. About 0.1mW of light was delivered to the photodiode for these experiments.

5.1

Noise

In an attempt to identify as many sources of noise as possible I have looked at the following using the spectrum analyser or scope: the spectrum analyser noise, BNC lead pickup, the ambient lights, RF crosstalk, the circuit noise with no power, the circuit noise as compared with the high speed photodiode, and the circuit noise as measured by the digital scope. I have tried to explain each type of noise in the captions of the images to aviod a lot of broken text. All of the graphs were drawn in Gnuplot and the images from the scope were taken directly.

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-100 The spectrum analyser with no input -105 -110 -115

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Figure 7: The noise floor of the spectrum analyser machine, which was taken before any other measurements were made.This is a theoretical base limit for the amount of noise I can measure, although in reality there is some capacitive coupling from the air which is why this is non-zero.

-40 BNC Lead Noise(lower) BNC Lead with Clip Attached -60

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Figure 8: The BNC cables also provide some coupling, especially for frequencies in multiples of 16kHz. The clips that are attached to the BNC also add a large amount of noise from their unshielded pickups. 11

-70 The spectrum of ambient light -80

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Figure 9: The roomlight has some characteristic effect on the photodiode when it is being used on the optical table, since it is not being used in the dark. This has been categorised here.

Figure 10: Very high frequency RF that was used in modulating the light was leaking into the circuit, either internally through the scope or externally via BNC cables crossing. On turning off the light modulation, this signal went away (Figure 13).

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-90 The noise of the circuit with no power -100

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Figure 11: The circuit has some pickup even when it is not switched on, and this noise is shown here. It has only a small contribution to the overall noise level and could probably be improved considerably by putting a box around the diode.

-90 High Speed Photodiode Noise (lower) My Photodiode Noise -100

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Figure 12: This is a comparison between a High Speed photodiode (HSPD) and the photodiode which I made. Both were taken with no laser light shining on them. The spikes likely arise from the room lights, as they have the same characteristic shape as Figure 9. The HSPD is unaffected as it was pointing perpendicular to the direction of the room lights. 13

Figure 13: The actual noise output measured on the scope. High frequencies can be cut out because they will not be used in the accousto-optic modulator, so the overall noise is then left at about 5mV, which is in good agreement with the theoretical noise derived before.

5.2

Voltage and Bandwidth

Using the theoretical program I was able to compare the predicted and actual results for the voltage output, and from this calculate the 3dB bandwidth of the photodiode. 3dB is a convenient √ measure of bandwidth as it shows a drop in power by a factor of 2, and hence a voltage drop of 1/ 2. The first time I ran the test, I used 100kΩ resistors and 220pF capacitor. Figure 14 shows the theoretical

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Figure 14: The theoretical voltage output and the actual voltage observed of a circuit with a 100kΩ resistor and a 220pF capacitor. The 3dB point of this circuit is 10kHz, which is much lower than required.

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and actual output voltages. The real output voltage is probably a bit low because of poor alignment in the light source, which caused less current to be produced than theoretically predicted. Given the large size of R, even a small difference in Ip can have a signigicant effect on Vout . It may also be because the photodiode is not supplying enough current or the departure from the ideal op amp behaviour. The next figure, Figure 15, shows the exact circuit from Figure 5 with simulated and actual results. The difference between the two results is significantly smaller which suggests more careful alignment and a good operating current being produced by the photodiode. In this case, the small difference is likely to result from the op amps departure from ideal performance. It may also be due to the meaurement device, since the probe has a finite capacitance and resistance, possibly causing a loop to ground. To overcome this as much as possible I have buffered the output with an RC network, filtering out any high frequency feedback through the scope.

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Figure 15: The largest source of error in this circuit is likely to come from the departure from ideal behaveior for the op amps. There may also be a slight mismatch in resistors which could account for slight changes in output voltage. The 3dB point for this circuit is ∼ 170kHz.

5.3

Phase-shift

The expression for voltage has some imaginary components, corresponds to a phase shift brought about by the capacitors in the circuit. I have simulated and measured this phase shift, and the results for the final circuit are shown in Figure 16. The phase shift should be around 90◦ at the 3dB point, so from this we can estimate the 3dB point for our data fairly accurately and check that it corresponds with the value we get from the voltage gain. This doesnt happen in the theoretical limit, because the model doesn’t take into acount the finite speed of the op amp.

15

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Figure 16: As the frequency of modulated light on the photodiode is increased, there is a complex shift in the phase of Vout . A 90◦ phase shift corresponds to a frequency of ∼ 200kHz, which is in good agreement with the voltage gain 3dB cut off from 15

6

Conclusions

This report serves to analyse theortically and practically the behavior of a photodiode circuit comprising of an op amp, a photodiode, resistors and capacitors. The theoretical simulation has been carried out by a C program, with results in good agreement with what is actually seen. The noise analysis gives a reasonable value, close to what is measured, and some of the other extrinsic sources of noise are discussed. The computer program hopefully provides some room for exploration of different parameters, although it should be extended to include the noise of the componenets and the non-ideal nature of op amps. The response of the photodiode with the current values of resistors and capacitors can be summarised as: Vout = Ilight ∗ 10.8V where Ilight is the power of the incident light in mW. The signal:noise ratio in this case is 2160/mW , which is deciptively high as we would never expect to have even 1mW of light incident.

Acknowledgements I would like to thank Jeramy Hughes, John Burke, Ofir Garcia and Ben Deissler for all the help and explanations which they offered whenever I had questions. I would also especially like to thank Cass Sackett for guiding me through a subject in which I have great difficulty, namely electronics.

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This has been a hard project for me to undertake but I feel I have learned significant lessons about design processes as well as the material covered.

References [1] D. Grier and Y. Roichman, “Holographic Optical Trapping,” arXiv:cond-mat/0506284 v1, 13/06/2005 [2] D. Grier and Y. Roichman, “Holographic assembly of quasicrystalline photonic heterosctructures,” arXiv:cond-mat/0506283 v1, 13/06/2005 [3] E. Kolomeisky, J. Straley and R. Kalas, “Ground-state properties of artificial bosonic atoms, Bose interaction blockade and the single atom pipette,” arXiv:cond-mat/0307771 v2 05/08/2006 [4] B. Saleh and M. Teich, “Fundamentals of Photonics,” 1991, Wiley and Sons, New York. [5] Si PIN Photodiode S1223 Series, Hamamatsu Solid State Devision. Japan. [6] P. Horrowitz and W. Hill, “The Art of Electronics” 1989, Cambridge University Press. [7] “Photodiode Monitoring with Op Amps” 1995, Burr-Brown Corporation, USA.

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