Physics of Detectors

PHYSICS OF ELECTRO-OPTIC DETECTORS

copyright 1998, 1999, 2000, 2001, 2003, 2004, 2005  Everett Companies LLC

INTRODUCTION

An electro-optic detector absorbs electromagnetic radiation and outputs an electrical signal that is usually proportional to the irradiance (intensity of the incident electromagnetic radiation). Depending on the type of detector and the way in which it is operated, the output signal can be either a voltage or a current.

In order to understand how detectors work, we will first discuss the properties of the electromagnetic field.

ELECTROMAGNETIC RADIATION

The electromagnetic spectrum is rather arbitrarily divided into regions. The spectral region to which the human eye is sensitive is referred to as visible. It extends from wavelengths of about 400 nm to about 700 nm. Wavelengths shorter than visible are successively referred to as ultraviolet, x-rays and gamma rays. The infrared region is considered to extend from 700 nm to 1000 mm. Wavelengths longer than 1 mm are referred to as radio waves. There are no consistent definitions of the various infrared spectral bands, but the region from 700 nm to 1 mm is usually called the near infrared or NIR. The infrared region is further subdivided into short wave infrared or SWIR (1-3 mm), mid-wave infrared or MWIR (3-6 mm) and long wave infrared or LWIR (>6 mm). However, the MWIR region is often considered to be 3-5 μm and LWIR 8 μm and longer. The reason is that the atmosphere absorbs strongly in the 5-8 μm band, and these wavelengths are seldom used. 

Electromagnetic radiation has been understood since the time of Maxwell. An electromagnetic wave is emitted whenever an electrical charge is accelerated. Since matter is largely composed of electrically charged particles that are constantly in motion, electromagnetic radiation is continuously emitted by all objects. The intensity and spectral distribution of the radiation depend to a large degree on the temperature of the object. The higher the temperature the more vigorously the electrons in the material bounce around, the more they are accelerated and the more radiation they emit. Maxwell, however, did have a problem, the spectral density of the radiation predicted by his theory kept increasing with decreasing wavelength becoming infinite at zero wavelength. Since this would require an infinite source of energy, it did cause some conceptual problems, and the issue was referred to as the ultraviolet catastrophe. Observation, however, showed that there really was no catastrophe. The experimentally measured spectral density increased with decreasing wavelength as Maxwell predicted but then peaked and decreased toward zero at shorter wavelengths. The question then was why, and Max Planck, who came along not too much after Maxwell, came up with what he thought was a mathematical manipulation to solve the problem. He postulated that the energy was not really emitted as continuous waves as Maxwell has assumed but rather as little chunks called quanta. These quanta are actually little wave packets, and a bunch of them can add up to a pretty good wave. This solved the problem, and Planck's theory fit the observations essentially perfectly. This, of course, is the foundation on which modern physics and quantum mechanics is based. Planck, however, was a true classicist and never felt that his quanta were anything more than a mathematical device, rejecting the implications of his fundamental discovery. 

In any case, the spectral distribution of the radiation emitted by a perfect emitter or radiator (called a blackbody) is given by what is now known as Planck's Law

where h is Planck's constant (6.626x10^-34 Js), c is the speed of light (2.9979x10⁸ m/s), λ is the wavelength, k is Boltzmann's constant (1.381x10^-23 J/K) and T is the absolute temperature in K. The spectral radiant emittance, W_l , is the optical power radiated into a hemisphere per unit area of emitting surface per unit wavelength and is shown in Figure 1  for a blackbody at room temperature (20 ^oC). 

Figure 1  Electromagnetic emission from a 20 ^oC blackbody

The units of spectral radiant emittance are power per unit area per unit wavelength. While it would be appropriate to use standard SI units throughout, we will follow standard usage and give wavelengths in microns (mm) and areas in cm².  It is the reader's responsibility to keep the numbers straight. The emitted power distribution in Figure 1 peaks at about 10 μm and falls fairly rapidly toward zero. Nearly all of the electromagnetic emission from an object at a typical room temperature is in the infrared portion of the spectrum. As the temperature is increased the emission at all wavelengths increases, but it increases faster at shorter wavelengths causing the peak of the distribution to shift toward shorter wavelengths. This is illustrated in Figure 2.

Figure 2  Electromagnetic emission from 100 ^oC and 1000 ^oC blackbodies

We know that as an object is heated it starts to glow, first red hot and finally white hot. As the object is heated the distribution peak shifts to shorter wavelengths and the short wavelength tail starts to be pulled up. The emission at the long wavelength end of the visible spectrum, around 0.7 μm, rises to observable levels first, and the object appears to be red. As the temperature is raised further emission over the entire visible spectrum becomes significant, and the color moves from red toward white. However, even at very high temperatures, at least below the melting points of most materials, the distribution peak is still in the infrared. An incandescent light bulb casts a "warm" light that is weighted toward the red end of the spectrum compared to a fluorescent bulb that is more "white".¹ The majority of the emitted optical power from an incandescent light bulb is still in the infrared heating up a room as well as illuminating it. 

The total power emitted from an object (a perfect radiator or blackbody in this case), of course, is just the area under the Planck distribution, from zero to infinite wavelength. 

where W is called the radiant emittance and is the total emitted power per unit emitting area.  This is a fairly straightforward integral to do in closed form, and the result is

where σ = 5.6697x10^-12 Wcm^-2K^-4. This is known as the Stefan-Boltzmann Law, and σ is usually called Stefan's constant. The problem with the Stefan-Boltzmann Law is that it is hardly ever useful. The reason is that it includes all wavelengths. The law accurately represents the total optical power emitted by a blackbody.  However, it does not represent the optical power that reaches a detector or other object. This is because there is almost always something in the optical path that passes some wavelengths and absorbs or reflects others, such as the window on a detector package or a spectral filter. The atmosphere is a classic example where there is very strong absorption in the 5-8 μm region due primarily to the presence of water molecules. The Stefan-Boltzmann Law must be used only with great care, and it is almost always the case that the Planck equation must be integrated after inserting the appropriate spectral transmission function. 

The emission from most real objects is similar to that from a blackbody except the Planck equation is more or less scaled by a parameter called the emissivity, e. It is often assumed that e is independent of wavelength. Actually, this is not strictly true, and measurements have found that while the spectral dependence of the electromagnetic emission from common objects around us generally closely tracks the scaled Planck distribution for a blackbody, the correlation is not exact. It is, however, close enough that ignoring the spectral dependence of the emissivity does result in a description of the passive emission from most objects that is accurate for many if not most applications although absorption lines and emission spikes from strong individual quantum mechanical transitions may be superimposed on it.² The emission from a light bulb, for example, looks very much like the Planck distribution for an object at the temperature of the filament. 

However, if there is a need to be very precise we need to consider the spectral dependence of the emissivity. Then for a real emitting surface

The optical power emitted per unit surface area then is

If we ignore any spectral dependence of the emissivity, we can write  

where

where T, of course is the temperature of the emitting surface. If the emissivity does indeed exhibit a significant spectral dependence, this approach necessarily introduces a temperature dependence into ε_avg. When it comes to detection, it is often the case that the detected radiation is limited to a fairly narrow spectral bandwidth by optical elements that are placed in the radiation's path.  The value of ε_avg then must be averaged over the detection spectral bandwidth using the appropriate spectral weighting functions. The spectral emissivity itself can also exhibit a dependence on the temperature of the emitting surface although this dependence is usually quite small.  These issues will be discussed further below.

Most surfaces emit the radiation uniformly over the hemisphere in front of it, and such emission is referred to as Lambertian.  This makes it straightforward to determine the amount of radiation captured by an aperture. The fraction of the total emitted radiation that is incident on the aperture is just the solid angle subtended by the aperture divided by the solid angle of a hemisphere which, of course, is 2π steradians.

Radiation at specific wavelengths or within relatively narrow spectral bands can be generated electrically. Short wavelength radiation such as x-rays can be preferentially generated by accelerating electrons to high energy and having them rapidly decelerate by striking a target. Light emitting and laser diodes emit radiation at particular wavelengths by causing electron transitions between specific energy levels of the diode semiconducting material. These phenomena, of course, are not described by Planck's Law but rather by quantum mechanics.

Reflected Radiation

The radiation emanating from a surface includes not only that emitted by the surface but also that reflected by the surface. The objects surrounding the surface in question also emit radiation in spectral distributions appropriate to their respective temperatures. This radiation, or at least some portion of it, is incident on our surface of interest where it is partially absorbed and partially reflected. To an observer the reflected radiation is indistinguishable from the emitted radiation.

We will consider our radiating object of interest to be opaque. This means that the incident radiation at any wavelength is either absorbed or reflected.  We know that the reflectivities of the surfaces of most objects generally have spectral dependencies, particularly in the visible portion of the electromagnetic spectrum.  That is why one object appears to be red while other objects are blue, green or whatever. The radiation at some wavelength that is not reflected must be absorbed, and the absorbtivity or absorption coefficient must have the complementary spectral dependence. Then

where

a(λ) is the spectral absorption coefficient

r(λ) is the spectral reflection coefficient

We will consider an object at temperature T that is in thermal equilibrium with all of its surrounding objects which then are also at temperature T. Since everything is in thermal equilibrium whatever radiation is absorbed must be re-emitted. Also, the incident and emitted radiation have the same spectral distribution since they correspond to the same temperature.  

and then 

The emitted radiation and the reflected radiation from the object under consideration have the same spectral distribution as they should since they were emitted from objects at the same temperature. The emitted and reflected radiation then simply add to give the blackbody radiation, equivalent to that from a surface with unity emissivity and zero reflectivity. This leads us to the simple idea that the radiation within an enclosure where everything in the enclosed space including the enclosure walls is at the same temperature is identical to the radiation coming from a blackbody at that temperature. In fact, this is the best known method to approximate true blackbody radiation. A hollow sphere with a small "pin" hole is brought to a known uniform temperature. The radiation exiting the pin hole is the blackbody radiation corresponding to that temperature. 

We know that the emissivity, absorbtivity and reflectivity can exhibit a temperature dependence. Consider that the emitting surface has a multilayer antireflection coating. As the temperature of the surface changes the layers in the coating expand or contract and change the specular characteristics causing the reflection coefficient at a given wavelength to either increase or decrease while the opposite occurs at other wavelengths. This then introduces a temperature dependence into these parameters. To be more rigorous we should write

For the moment we will ignore this temperature dependence since it is small enough to do so in many if not most cases.  However, the issue should be revisited if the conditions involve a particularly broad range of emitting temperatures.  

We usually refer to the object or surface of interest as the target. The objects surrounding the target are normally referred to as the background. However, since we are considering reflected radiation, foreground might be a more appropriate term, but we will stick with convention. It is frequently the case that the objects in the background are at a more or less uniform temperature that we will call T_b. If this is not the case, we have to take into account the individual temperature of each object in the background, a laborious if not difficult task. If we assume that the surrounding objects are all at temperature T_b, the radiation incident on our target is the blackbody radiation field corresponding to T_b, assuming that the target is small enough that it does not disrupt the radiation field. The radiation reflected from the target then is

The total radiation coming from our target object, W_tot, then is the sum of the emitted and reflected components.

where the spectral distribution of the emitted radiation corresponds to the target temperature and the reflected radiation corresponds to the background temperature.

Radiation Reaching a Detector

Now we have to ask the question, how much radiation emitted by all sources reaches our detector. To keep things simple we will consider only planar detectors, detectors that have a single flat surface that is sensitive to the incident radiation.  This is not always the case.  We could, for example, have a spherical detector and sometimes do.  However, most of the time the detector is indeed planar, and such detectors have hemispherical fields-of-view.  Light, as they say, travels in a straight line.  Some portion of the radiation emitted by the surface of an object with an unimpeded view of the detector's optically sensitive surface will reach that surface and be detected.  All such objects must lie in the volume (or hemisphere) in front of the detector since any object behind the detector has no view of the detector's front surface at all.  While we only need to consider the front half when we are considering emission, we cannot ignore the objects behind the detector when it comes to reflected radiation. 

Image of Target Focused on Detector

The above expression for W_tot gives the amount of optical power emitted and reflected per unit surface area (usually given in units of W/cm²).  We are considering only diffuse or Lambertian surfaces so the radiation is uniformly distributed over the hemisphere in front of this surface. We typically set up an optical system with a lens that captures some fraction of this radiation and directs it onto a detector. We will first consider the situation where we adjust the lens-to-detector distance so that an image of the emitting surface is focused onto the detector. This is illustrated in Figure 3. 

Figure 3  Schematic diagram of detector with lens, spectral filter and window in adjustable focus sensor

Typically, the detector is mounted in an enclosure that keeps stray radiation from reaching the detector. Also, a spectral pass band filter is frequently placed at an appropriate location in front of the detector. The detector is frequently mounted in a package such as a TO-5 that has a window located in front of the detector that is transparent or at least partially transparent to the target radiation. All of these components should be at the same uniform temperature that we will call the ambient temperature, T_a. The lens can be larger than the field-of-view defined by the aperture of the detector package or other limiting stop within the enclosure, but it cannot be smaller. If it is smaller, stray radiation from other sources will enter the enclosure that must then be taken into account. Whatever the detector's field-of-view outside of the enclosure, the lens, filter and window are assumed to fill it.

From the thin lens formula we know that the distance from the lens to the detector (a) and the distance from the lens to the target or emitting surface (b) are related to the lens focal length (f) by

Also, simple proportion tells us that the area of the emitting surface that is focused onto the detector (A_target) is related to the size of the detector (A_det) by

Since the radiation is spread uniformly over the hemisphere in front of the target, the fraction of the emitted and reflected radiation captured by the lens and focused onto the detector is the ratio of the solid angle subtended by the lens as seen from the target surface to the 2π solid angle of a hemisphere. This fraction (F) is then given by

where A_l is the cross sectional area of the lens

The numerator in the above expression is actually the area of the sphere of radius b intercepted by the lens rather than the lens cross section. However, the difference is usually small enough that it can be neglected. We have assumed that the lens is the limiting stop in the optical train. If it is not, A_l should be replaced the area of the limiting stop projected onto the lens.

Notice that A_target increases as b² while the fraction of the radiation emitted by this surface and captured by the lens decreases as b^-2. The two cancel, and the amount of radiation from the target reaching the detector is independent of the distance to the target.

This radiation must pass through the lens, filter and window before it reaches the detector. Each of these components will have spectral transmission functions that will vary with wavelength, particularly the filter, and this transmission or lack of it must be taken into account. Then the amount of optical power from the target that reaches the detector is

where

and

t_l(λ) is the spectral transmission function of the lens

t_f(λ) is the spectral transmission function of the filter

t_w(λ) is the spectral transmission function of the enclosure window

If there are other components in the optical path, their transmission functions are simply included in the product for the net spectral transmission function. The above expression tells us how much optical power is incident on the detector that was emitted by the target and that emitted by the background and reflected by the target. We can replace this expression by 

where

and

where we have reintroduced the dependence of the emissivity on the temperature of the emitting surface just to remind ourselves of the issue.

Now if ε(λ) is sufficiently constant over the spectral detection band defined by the product of the transmission functions, the average emissivity can be removed from both integrals and replaced by a single number between 0-1. However, if this is not the case the two average emissivities defined above will not be equal since they will be averaged over different spectral distributions and will be functions of the temperatures to which those distributions correspond, the target and background temperatures, as well, of course, of the spectral transmission of the optical system. Whether or not this is an issue depends on the accuracy needed for the analysis. In some cases the whole thing can be ignored while in others it cannot.

In addition, the enclosure that surrounds the detector also emits radiation that must be included. We will assume that the enclosure walls, spectral filter and the detector package with its window are all at the ambient temperature, T_a. If everything were totally opaque at all wavelengths, the detector would be completely surrounded by objects at the ambient temperature and would be subjected to the blackbody radiation field corresponding to that temperature. In this case the optical power incident on the detector is given by 

This expression can be simplified for detectors with response that does not depend on wavelength by using the Stefan-Boltzmann Law.   However, semiconductor detectors such as photodiodes have responsivities that depend strongly on wavelength, and the responsivity must be included in the integration of the Planck function eliminating the use of the law. As we shall see below, P_bb subtracts out of the final result in the case of thermal detectors again obviating the usefulness of Stefan-Boltzmann.

The filter usually limits the transmission both in and out to a fairly narrow band. If the passband is narrow enough, the blackbody radiation field within the enclosure is negligibly disturbed, and we can use the above expression. It is, however, too large. The filter along with the lens and window do indeed act like opaque objects emitting and reflecting radiation like anything else except within their spectral passbands. Even in the passband some radiation is emitted and reflected since the transmission is not 100%, and we must subtract from P_bb  the ambient radiation that it is not emitted and reflected.  This correction is given by

where φ is the solid angle subtended by the limiting aperture stop as seen from the detector

The net optical power from the enclosure incident on the detector then is given by

or

The total optical power incident on the detector then is simply the sum of P_target and P_enclosure.

Fixed Focus

Frequently, we do not want to be bothered adjusting the focus every time we make a measurement. In this case we can place the detector at the focal point of the lens a distance f behind the lens. This is illustrated in Figure 4. In effect then we have focused the sensor to infinity.  The image of the target "focused" onto the detector will be relatively sharp when the distance to the target is large but fuzzy when this distance is small. 

Figure 4  Ray trace of emitted radiation reaching a detector placed at the lens focal point.  The enclosure with the filter and window are not shown.

The lens is assumed to define the aperture. All of the radiation emitted by the central violet region and captured by the aperture is focused onto the detector. However, the same is not true of the green ring or frame that surrounds the central region. Only a fraction of the radiation emitted from this outer region and incident on the lens is focused on the detector. This fraction decreases from 100% at the inner perimeter of the ring to zero at the outside edge. The area of the central region increases as b² while the area of the outer ring increases only linearly with b. The width of the ring is always equal to the diameter of the lens. At large distances then the radiation from the outer ring can be ignored and the above analysis used without modification or correction. A fixed focus sensor works well when the distance to the emitting surface is large compared to the diameter of the aperture. At closer separations, the power incident on the detector can still be calculated, but one must be careful to include the variable contribution from the green region in addition to the more directly calculated contribution from the violet region. The amount of radiation reaching the detector is independent of the distance from the lens to the source for large distances, but this is no longer true when the source is close to the lens. 

At large distances where we can ignore the green border, and the area of the violet region (the measurement spot) increases as the square of the distance from the lens to the emitting surface, and the solid angle subtended by the aperture decreases as the square of the inverse of this same distance.  The two cancel just as was the case with the adjustable focus, and the detector signal is independent of the distance to the target. The advantage of the adjustable focus is that there is no "fuzzy" band and the emitting surface does not need to be a substantial distance away.

The enclosure emitted radiation, of course, is not affected by whether the sensor has variable or fixed focus or no lens at all. 

"Pin Hole" Imaging

Now suppose we have a simple aperture without a lens. This is illustrated in Figure 5. 

Figure 5  Ray trace of emitted radiation reaching a detector through an aperture with no lens

The situation is similar to that with a lens. All of the radiation from the violet region passing through the aperture reaches the detector, and only a portion of the radiation from the green region finds the detector. The biggest difference is that the width of the green ring is not fixed and increases as the distance from the emitting surface to the aperture increases, and the contribution from the green region cannot be neglected at large separation. However, the area of the green ring also becomes proportional to the square of the distance to the emitting surface, at least for distances that are large enough. The fraction of the radiation emitted (per unit area) by the green ring, of course, varies with radius, and we must integrate over the ring for the specific geometry of the detector, aperture and their relative placement. Once done for that geometry the measurement is accurate and independent of distance to the emitting surface as long as that distance is large enough.

How Far Is Enough

The integration required to relate the optical power incident on a detector to the temperature of the emitting object goes from zero wavelength out to infinity. This integration is hardly ever done in any sort of closed form and is almost always done numerically on a computer.  Infinity is a long way, and the question is how far do you have to carry the integration to get reasonably good results. The answer, of course, depends on temperature as well as the spectral transmission and the accuracy with which you need to make the calculation. To get some idea of the temperature dependence we will set spectral transmission aside for the moment and do the integration with all of the spectral transmission functions set to unity. If the temperature of the emitting surface is 1000 K and we integrate from 0-100 μm, we will get all but 0.01% of the emitted power, accurate enough for most applications. However, if the temperature is 300 K, right around typical room temperature, we come up 0.5% short. If the temperature is 100 K, a bit above the boiling point of liquid nitrogen, the integration gives us less than 92% of the actual emitted power.

Quite often the spectral absorption of the elements in the optical path improves things, particularly so if those elements include a filter, since the longer wavelengths tend to be absorbed rather than passed by the materials used to fabricate lenses, filter substrates etc. Integration over a sufficiently wide spectral band is not really an issue with today's desktop computers. However, it does become an issue with less powerful microprocessors such as those that are typically imbedded in electro-optic sensors.

Waves and Particles

Electromagnetic radiation is usually well described in terms of waves traveling at the speed of light. However, this description is not complete and in many cases not sufficient. With the development of quantum mechanics in the first half of the last century, an important principle was discovered that is usually referred to as the wave-particle duality. All things exhibit both wave-like and particle-like properties, and one or the other of these descriptions, wave or particle, will be appropriate depending on the circumstances. The electron is usually thought of as a particle. However, the wave-like properties of the electron are dramatically exhibited in an electron microscope and in the tunneling of electrons through thin films. Concomitantly, things we think of as waves also exhibit particle-like properties. Electromagnetic radiation can also be described in terms of particles, and this description will be useful in discussing the operation of a number of electro-optic detectors. The description of radiation in terms of particles is usually referred to as second quantization, and for electromagnetic radiation the particle is called the photon.

The atoms that make up a crystalline material, like those used for many electro-optic detectors, vibrate about their nominal positions in the crystal lattice.  If the crystal is struck at a point, what we call a sound wave propagates though the material with the atomic vibrations increasing in amplitude and carrying the energy through the material. In the absence of a sound wave, the atoms still vibrate in response to the thermal energy or heat present in the material. The amplitude of these vibrations, of course, increases with temperature. Lattice vibrations also have particle-like properties, and the particle is called the phonon. This can be carried further, but we have enough to be able to describe how detectors work.

It is useful at this point to relate the particle and wave properties of electromagnetic radiation. The primary quantity of interest is as always energy. As we discussed above, the energy in an electromagnetic wave is condensed into packets of well defined amount. These packets are the photons, and the energy of a photon is given by

E = hc/λ

The number of photons per unit area per second incident on a surface then is given by

N = Hλ/hc

where H is the irradiance, sometimes called the intensity, of the incident radiation in units of power per unit area. Note that both N and H are spectrally dependent.

DETECTORS

An electro-optic detector is used to sense or measure the radiation emitted or reflected by objects within the detector's optical field-of-view. These detectors generally fall into two classes, thermal detectors and quantum detectors. 

Thermal Detectors

Thermal detectors simply absorb the incident radiation, and the detector's temperature increases or decreases until it comes into quasi-equilibrium with the radiation being absorbed. The temperature of the detector material will change until the rate at which energy is being radiated and thermally conducted away is equal to the rate at which it is being absorbed from the incident radiation. The detector is, of course, continuously radiating as well as absorbing. When the detector is in thermal equilibrium with the ambient, it is at the ambient temperature and it radiates at the same rate at which it absorbs. When the incident radiation rises above that of the ambient, the detector absorbs more energy than it radiates, and the vibration of its atoms and its temperature increase. The detector temperature continues to increase until the rate at which energy is being removed via thermal conduction equals the excess rate at which radiation is being absorbed. The increase in radiated energy due to the small increase in the detector temperature can usually be ignored.  Since thermal conduction is linear, the increase in detector temperature is proportional to the increase in absorbed power³. Of course, if the incident optical power drops below that of the ambient, the opposite occurs and the detector temperature drops below the ambient temperature. The detector material is usually thermally isolated to reduce the thermal conductivity and increase the temperature change as much as is practical. A property of the detector material is then measured to determine the temperature change and thus the incident optical power. The most commonly used thermal detectors are the thermopile, pyroelectric detector and the bolometer.

Thermopiles

The thermopile is a series combination of thermocouples. One set of the thermocouple junctions is heat sunk to the detector case which is maintained at the ambient temperature. The other set of junctions is attached to a membrane that is thermally isolated from the ambient. The incident radiation is absorbed by the membrane, and the temperature of the membrane with the set of junctions attached to it changes in correspondence. Multiple thermocouples are used simply to increase the size of the signal. Since the voltage generated across a thermocouple is nearly linear with the temperature difference between the two junctions, the thermopile voltage under open circuit conditions is essentially proportional to the difference in the optical power incident on the detector and that that would be incident if the thermopile were surrounded by the ambient, that is, the optical power the membrane radiates. The temporal response of the thermopile, however, is limited by the finite thermal mass of the exposed junctions and the substrate on which they are mounted. The energy is absorbed at the surface and must have time to spread through the material, be conducted to the ambient and allow the detector to come to a state of quasi-equilibrium.  The speed of response or time constant can be controlled to some degree by design. The thermopile, of course, generates its maximum response or signal at dc. As the incident radiation is modulated, the response of most thermopiles starts to fall off significantly around 5 Hz and to unusable levels by 10 Hz.  Thermopiles designed specifically for high speed operation can be used at somewhat higher frequencies or modulation rates.

Pyroelectric Detectors

Piezoelectric materials are insulators that generate a dipole electric field when strained, and these materials are widely used in strain gauges. A piece of the piezoelectric crystal is glued to something such as a steel beam. When the steel beam is bent or stretched, the piezoelectric crystal is stretched, and a voltage is generated across the crystal if it is oriented properly.  Of the twenty-one crystalline point groups that are piezoelectric, twenty are also pyroelectric. When a pyroelectric crystal is heated (or cooled) the expansion (or contraction) is anisotropic causing the material to be strained, and a voltage is generated across it due to the resulting dipole field. A pyroelectric detector is made by taking a thin slice of the material, thermally isolating it and exposing it to the incident radiation. In the same way as the thermopile, the temperature of the pyroelectric crystal comes to an equilibrium in concert with the rate at which energy is absorbed from the incident radiation. Unlike the thermopile the pyroelectric cannot be operated under static conditions.  This is because the charge generated on the crystal surfaces is eventually neutralized by ions in the surrounding atmosphere or by electrical conduction across the surface of the material.  This neutralization can be reduced by careful cleaning and evacuation of the surrounding atmosphere (and perhaps replacement by an inert gas).  However, it can never be completely eliminated, and the radiation incident on a pyroelectric detector must be modulated. A pyroelectric detector can be operated in either of two modes.

Bolometers

The bolometer was the first infrared detector. Basically, it is just a resistor that is thermally isolated and exposed to the incident radiation. The bolometer temperature changes with the net rate at which it absorbs the incident radiation, and, of course, the resistance of the device changes as well. This resistance change is measured by passing a small current through the bolometer and measuring the voltage across it or in extreme cases using a Wheatstone bridge. Bolometers are now finding fairly wide use in thermal imaging. Thermally isolated resistor elements, called microbolometers because of their small size, are deposited on each cell of an integrated circuit. The number of cells (or pixels for picture element) can be quite large in a typical imager, up to 100,000 and even larger. The megapixel resolution available today in visible cameras is certainly feasible for microbolometers as well. The circuitry in each cell senses the microbolometer resistance and outputs that information to the imaging electronics that generates the high resolution thermal image. Other types of detectors, particularly the photodiode, are used in thermal imaging application as well often with a higher level of performance, but the microbolometer is emerging as the cost effective approach for many applications primarily because cryogenic cooling is not required.  However, temperature stabilization of the microbolometer array usually is.

Quantum Detectors

In thermal detectors the incident radiation is absorbed by the detector material and is manifested as an increase in the vibrations of the atoms in that material. This is what is meant by an increase in temperature. The situation is different with quantum detectors where the detector is normally maintained at a constant temperature. A quantum detector is made from a semiconductor, and the incident radiation excites electrons from the semiconductor's valence band to the conduction band.  To understand what is going on we first have to understand something about semiconductor energy bands.

First, we have to discuss atoms for a bit. The electrons in an atom are quantized into energy levels (for those of you keeping track, this is first quantization). We are used to thinking of these energy levels in terms of Bohr orbits of the electrons around the nucleus. In quantum mechanical terms electrons are Fermions, and no two interacting Fermions can be in exactly the same energy state, or have the same set of quantum numbers. When atoms condense into a solid, the outermost or valence electrons interact strongly. Actually, the valence electrons bind the atoms together. When the atoms are far apart, corresponding electrons in the atoms have essentially the same energy. As the atoms come into proximity, the electrons push and pull on one another, and their energies are shifted. The discrete energy levels of the atoms are spread out into quasi-continuous bands with some 10²³ energy levels in each band. Sometimes the bands overlap and sometimes there is a gap between them where no energy levels exist.

In a metal the valence electrons disassociate from the individual atoms and become part of the material as a whole. They are free to move though the material and conduct an electrical current in response to an applied voltage. The energy band containing these electrons is not full. There is no gap between the states of the highest energy electrons, those at what is called the Fermi energy, and the adjacent sea of unoccupied states. An electron with the slightest provocation can move into another state and through the material.

In an insulator the valence electrons are actually locked tightly into the bonds between adjacent atoms. In terms of band theory, the valence electrons completely fill a band, and a large gap exists between the top of the valence band and the bottom the next band, called the conduction band. In an insulator the conduction band is nominally empty, has no electrons. In order to excite an electron to the conduction band, a substantial amount of energy must be given to a valence electron to break it loose from the bond which is exerting a strong attractive force on that electron. In an insulator very few electrons are in the conduction band, and the material is a poor conductor of electricity and heat.

In a semiconductor an energy gap between the valence and conduction bands exists as in an insulator but is smaller. A semiconductor such as silicon is held together by covalent bonds. In its cubic crystalline structure each silicon atom has four outermost or valence electrons and four other silicon atoms that are closest to it. Each silicon atom donates one of its valence electrons to the bonds with the adjacent atoms. A covalent bond between two atoms then consists of two electrons, one from each atom. The electrons are not as tightly bound as they are in insulators and can be broken loose by absorbing the energy of either photons or phonons, incident radiation or the indigenous thermal energy of the material. At a finite temperature a semiconductor always has some conduction electrons that have been thermally excited or broken loose from their bonds. As the temperature of the material is increased, more electrons are excited across the band gap to the conduction band. Unlike a metal, the electrical conductivity of a semiconductor increases with temperature. 

Conduction electrons can also be added to a semiconductor by doping. In doping, impurity atoms are used to replace the normal atoms in the crystal structure. In silicon, if another atom with four valence electrons, such as germanium, is used, nothing much happens. However, if an atom such as phosphorus or arsenic with five valence electrons is used as the dopant, the fifth valence electron is left without a bond. The extra electron is weakly attracted to its parent nucleus by electrostatic forces, but with any thermal agitation at all it breaks loose and becomes a conduction electron. These doping electrons provide a conductivity floor that is nominally independent of temperature except at very low temperatures where they tend to be "frozen out" (at these low temperatures there are no longer phonons with sufficient energy to overcome the electrostatic forces applied by the host nuclei). Semiconductors that are doped with extra valence electrons are called n-type.

Dopants with fewer valence electrons than the host may also be used. In the case of silicon this might be boron which has only three. In this case, there is a bond left short an electron. The covalent bond with only one electron is called a hole. It can also conduct electricity. The hole applies an attractive electrostatic force on the electrons attached to the complete bonds in its vicinity. If an electric field is applied to the material, the added inducement will cause an electron to jump from an adjacent bond to the hole completing that bond but leaving the hole behind on the bond it came from. The hole then moves in the opposite direction of the electron current (that is, in the direction of the electric field) and acts like a positive charge carrier. A hole is generally less effective in carrying an electrical current than an electron in the conduction band. In order for a hole to move it must pull an electron off of an adjacent bond that is trying to hold on to that electron while the conduction band electron is more or less free to move around. The conduction electron is said to have a higher mobility than the hole. Semiconductors with dopants that are short valence electrons are called p-type. Doping, both n-type and p-type, will be important when we discuss photoconductive detectors and photodiodes.  

Aside from doping and absorption of phonons, electrons can be excited to the conduction band by absorbing photons, which, of course, is the way radiation is detected. The band gap between the valence and conduction bands has a well defined energy. In order to be absorbed, the photon must have enough energy to break the electron loose from its bond and excite it to the conduction band. Since there is effectively a continuum of energy states above the band gap, those photons with energy equal to or greater than the band gap energy are able to excite valence electrons while those with energy less than this are not and do not cause an increase in the number of electrons in the conduction band. Only radiation with wavelength shorter than a critical value can generate conduction electrons. This critical wavelength is given by

λ_c = hc/E_g

where E_g is the energy across the bandgap. The bandgap energy varies from material to material. Silicon can detect radiation with wavelengths less than 1.1 μm.  It is the standard detector for visible radiation, and essentially all television cameras use silicon detector arrays. However, unless it is doped with specific impurities to insert localized states within the bandgap, silicon does not detect infrared radiation, at least at wavelengths longer than NIR. Lead sulfide detects radiation with wavelengths less than 3 mm and is a good SWIR detector. Lead selenide and indium antimonide detect radiation less than 5 mm and are good MWIR detectors. Silicon carbide, on the other hand, has a fairly wide bandgap and makes a good ultraviolet detector. We should note that when an electron is excited from the valence band to the conduction band, two charge carriers are generated, the electron in the conduction band and a hole in the valence band.   The combination is usually called an electron-hole pair.

As E_g is reduced, more and more electrons are excited to the conduction band by thermal agitation. This is a significant source of noise since these thermally generated charge carriers are dumped on top of the signal generated by the radiation. Wide bandgap detectors like silicon and even lead sulfide can be operated at room temperature without ill effect unless exceptionally low levels of radiation are being detected. The performance of narrower bandgap detectors such as those used for MWIR and LWIR detection are almost always improved by cooling. Thermoelectric coolers can often be used to refrigerate many but not all MWIR detectors to -40 ^oC or so with good effect. Some MWIR and nearly all LWIR detectors require cryogenic cooling to the vicinity of liquid nitrogen temperature (78 K). Wide band gap detectors such as those used for ultraviolet and x-rays usually do not need to be cooled except in cases where they are being used to detect very low levels of radiation. As a general rule, the best performance is achieved by using a detector with the widest bandgap that still allows detection of the radiation wavelengths of interest.

Now that we have an understanding of energy bands in semiconductors, we can discuss quantum detectors.  There are two common types of quantum detectors, photoconductors and photodiodes. 

Photoconductors

Photoconductors are made from semiconductors that have been heavily doped n-type or p-type and are frequently used for infrared detection. Detectors made from these materials have a finite electrical conductivity that increases with temperature. The resistance of an infrared photoconductive detector can be as little as 10 W or as much as 10 MW. When these detectors are exposed to radiation, additional conduction electrons and holes are generated and the detector resistance is reduced. However, the change is not much, usually less than 1% for normal radiation intensities.

Photoconductive detectors are usually operated by biasing them with a fixed voltage and measuring the current flowing in the biasing circuit. The current in the absence of radiation is called the dark current. The photo-signal is the increase in current when the detector is exposed to radiation. Since the bandgaps of infrared detectors are relatively small, a substantial number of electrons are thermally excited to the conduction band. A small increase in the detector temperature can in most cases excite an additional number of electrons that substantially exceeds those excited by the radiation. If the photo signal is obtained by simply subtracting the dark current from the total current, the detector temperature must be controlled to something on the order of 0.01 ^oC or less in order to keep the thermally excited electrons from burying the optical signal. In most cases, this is not practical. The way around it is to modulate or chop the incident radiation. Any drift in the detector temperature causes a signal change that is nearly dc while the optically generated signal appears at the modulation frequency. Detection at the modulation frequency then separates the optical signal from the background. Preamplifiers for Electro-optic Detectors describes electronics that can be used to extract the photo signal.

The materials used most commonly for infrared photoconductive detectors are PbS, PbSe and HgCdTe. As mentioned above, PbS has a cutoff wavelength something less than 3 mm and is the standard SWIR photoconductive detector. Lead selenide has a cutoff wavelength a bit less than 5 mm and is the most commonly used MWIR photoconductive detector. Mercury cadmium telluride is a tertiary compound with half of the lattice sites occupied by Te atoms. The other half is divided between Hg and Cd. Mercury telluride is a semimetal with an effectively zero bandgap between the valence and conduction bands while CdTe is a fairly wide bandgap semiconductor. The bandgap of HgCdTe can be adjusted continuously by varying the relative concentrations of Hg and Cd. If the Hg/Cd half of the material is divided to be 30% Cd and 70% Hg, the cutoff wavelength is around 5 mm making the detector sensitive to MWIR. If the concentrations are 20% Cd and 80% Hg, the cutoff wavelength is about 10 mm making an LWIR detector. Since PbS and PbSe are available and less expensive, they are usually used for shorter wavelength detection. However, cryogenically cooled HgCdTe has higher performance and is used for shorter wavelength detection when that performance is needed. Mercury cadmium telluride is the standard LWIR photoconductive detector. 

Photoconductors are frequently, but not always, polycrystalline materials. That is, instead of being one big single crystal, the detector is made up of a number of smaller crystals. In a crystal, the atoms have an orderly geometric arrangement. A grain boundary is the transition region between a crystal of one orientation of that geometric arrangement and an adjacent crystal with a different orientation. The orderliness of the atomic arrangement breaks down within the grain boundary. Since band theory requires the orderly arrangement of the atoms, the concept breaks down and the band gap goes away. However, the performance of a photoconductive detector is still well described by band theory as long as the grain boundaries make up the small part and good crystals the large part of the material. This will not be the case when we discuss photodiodes.

Photodiodes

A diode consists of a single crystal semiconductor where a portion has been doped n-type and the remainder p-type. The electrons in the conduction band on the n-side will tend to drift across the junction between the two sides and combine with the holes on the p-side.  That is, the conduction band electrons from the n-side will cross over to the p-side and be captured by the bonds that are short an electron. This will not continue indefinitely. Both sides of the junction are normally electrically neutral. When the conduction electrons cross over they leave the n-type material in the vicinity of the junction with a net positive charge and generate a net negative charge on the p-side. This residual charge keeps any more electrons on the n-side or holes on the p-side from moving toward the junction and stops the process. The region of n-type material in the immediate vicinity of the junction is depleted of conduction electrons, and a similar region in the p-type material is depleted of holes. The combined region from both sides is called the depletion region. A potential difference called the contact potential is generated across the junction.

Additional charge carriers are generated when a photodiode is exposed to radiation with wavelength short enough to bridge the bandgap. The increased density of charge carriers causes more electrons from the n region to cross over and combine with holes in the p region until electrostatic forces again stop the process. This causes the depletion region to widen and the potential across the junction to increase. If no current is allowed to flow through the diode, a voltage is generated across it that is logarithmically dependent on the radiation intensity. This is called the open circuit condition, and the voltage across the junction is called the open circuit voltage, V_oc. The detector can also be operated in the short circuit condition. Here the two sides of the detector are shorted together so that no voltage is allowed to develop across the diode. In order to keep the voltage from developing, the charge carriers being generated by the radiation and crossing the junction must be continuously removed from one side of the diode and sent back to the other. In other words, a current flows through the diode that is proportional to the incident intensity. This current is called the short circuit current, I_sc. The I-V curves for a photodiode are shown in Figure 6, and V_oc and I_sc are indicated. 

These parameters allow reasonably accurate prediction of the performance that will be achieved with a specific detector  

CHARACTERIZATION MEASUREMENTS

The detector output signal is usually either a voltage or a current. The responsivity is the ratio of the output signal to the input which is usually the incident optical power. The measurement of responsivity is fairly straightforward. A known amount of optical power is directed onto the detector, and the output signal is measured. The responsivity is simply the ratio of the detector signal to the incident power with the units of A/W or V/W. The background, of course, contributes to the total power on the detector and must be included.  This is sometimes hard to determine. To get around this ac detection is often used. A known amount of flux is generated with a blackbody at a known temperature and with a known aperture. Therefore, the amount of flux reaching the detector from the blackbody is known. A chopper is placed in front of the blackbody and operated at a reasonable frequency, frequently 1 kHz. The detector is then alternately exposed to the background flux and the background flux plus that from the blackbody  The difference signal from the detector then represents just the blackbody flux.

Noise measurements are not quite so obvious. There are usually three contributions to the noise. These are from the detector, the electronics used to make the measurement and the photon flux itself. The incident optical power actually consists of a current of photons with each photon individually striking and being absorbed by the detector. The photon noise is just the shot noise, the uncertainty in the number of photons striking the detector which is the square root of that number of photons. In any case, the three sources of noise are statistically independent and add in quadrature.

where

N_flux is the noise contribution from the photon flux

N_d is the noise contribution from the detector

N_elec is the noise contribution from the detection electronics

As mentioned above, noise is usually measured in A/√Hz or V/√Hz. What we are usually interested in is the detector noise. This, of course, is given by

In order to determine this we must know both N_elec and N_flux in addition to N_tot. The electronics noise can be measured directly by shorting the input leads at the detector.  The photon noise is more difficult since the detector sits between it and the measurement equipment. However, it can be calculated if the incident flux is known. Since the photon noise is determined indirectly, it is always best to make it as small as possible. This is usually done by enclosing the detector by a cold shield with no aperture. This is what is usually done with cryogenically cooled photodiodes. The cold shield is also cooled to cryogenic temperatures reducing the photon noise to, in most cases, a negligible level. This is not really practical for detectors operated at higher temperatures such as normal room temperature. In such a situation it is still best to enclose the detector with a zero aperture shield at what we have called the ambient temperature, a temperature that is usually well known. Then we know that the radiation incident on the detector is the blackbody radiation corresponding to the ambient temperature, and this radiation can be calculated accurately. 

When we know the detector noise and the responsivity, the noise equivalent power of the detector is simply

where

R is, as above, the detector's responsivity

As mentioned above, the units or N_d are usually A/√Hz or V/√Hz and NEP_d usually W/√Hz. The parameter most often used to characterize detector performance is D*, and D* is determined directly from NEP.  In order that D* characterize the properties of the detector and not the conditions under which it was measured, one needs to be certain that only the detector noise was included in the determination of the value for D*. This is often not the case, and the values quoted for D* are then are not really meaningful, particularly for low noise detectors.

The detector's time constant is usually measured with an oscilloscope. A radiation source significantly different from the ambient is placed in front of the detector with a fast shutter in the middle. The detector is exposed to the source radiation when the shutter is open and just the ambient radiation when it is closed. The rise (step increase in radiation) and fall (step decrease in radiation) times are observed on an oscilloscope. The rise and fall time constants are then determined directly from the oscilloscope traces. The rise and fall time constants may very well be different, and both should be measured. The oscilloscope trace, of course, incorporates the time constant of the electronics as well as that of the detector. If the electronics is not fast enough that their rise and fall times can be ignored, they will have to be independently measured and analytically extracted from the measurements of the detector's rise and fall times.

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¹The emission from a fluorescent light bulb results from direct stimulation of the electrons to higher energy levels and is not governed by Planck’s Law.

²Each atom has a large number of energy levels (really infinite). The electrons then have a large number of potential transitions they can undergo thereby emitting energy in all sorts of amounts. By the time we put 10²³ atoms together to make up an object that might radiate, the atoms push and pull on each other shifting these energy levels all over the map. We actually come up with what is called a quasi-continuum of energy levels. Each energy level is unique, but there are so many of them fit into a restricted range that they effectively appear as a quasi-continuous distribution. As long as the transitions from one level to another occur with roughly equal frequency, the radiation emitted as a result of these transitions is continuously distributed over wavelength. However, every now and then some of the transitions (for quantum mechanical reasons) occurs with substantially greater or lesser frequency producing a spike or dip here or there in the distribution. 

³Since the temperature of the detector material is raised by absorption of the incident radiation, radiation from the surface of this material is also increased. Thermal conduction is the dominant mechanism for removing energy from the detector and maintaining thermal equilibrium as long as the temperature difference between the detector and its surroundings is relatively small. If this is not the case, radiation from the detector surface must also be considered. In most cases the temperature difference is small enough that radiation can be ignored. This is important because the linearity of detector response is based on the linearity of heat capacity and the linearity of thermal conduction. Radiation is not linear with temperature, and detector responsivity is no longer a simple number if radiation becomes a significant thermal equilibrium mechanism for the detector.

⁴We are implicitly assuming that all of the incident radiation is absorbed by the detector. We know this is not the case. However, in most situations the fraction of the incident radiation that is absorbed is more or less constant. This fraction can then be absorbed into the overall amplitude factor reducing it somewhat. In effect, it can be absorbed into the detector responsivity that is normally measured in a calibration procedure. 

Infrared and Electro-optic Technology

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