Study and Measurement of the Main Parameters of a Laser quadrant Detector

Transcription

1 Cairo University National Institute of Laser Enhanced Sciences Laser Sciences and Interactions Study and Measurement of the Main Parameters of a Laser quadrant Detector By Eng. Mohamed Abd-Elfattah Abd-Elazim Ibrahim Under supervision of Assoc. Prof. Dr. Iftitan Elsayed Mounir National Institute of Laser Enhanced Sciences Dr. Eng. Adel Muharram Alnozahi National Institute of Laser Enhanced Sciences Dr. Eng. Wael Elsayed Swelam Egyptian Armed Forces This thesis is submitted as a partial fulfillment of the requirements for the degree of Master of Science. Cairo 2012

2 Approval Sheet Title of the Thesis "Study and Measurement of the Main Parameters of a Laser Quadrant Detector" Name of the Candidate Mohamed Abd-Elfattah Abd-Elazim Ibrahim Submitted to Laser Sciences and Interactions National Institute of Laser Enhanced Sciences Cairo University Supervision Committee: Assoc. Prof. Iftitan Elsayed Mohamed Mounir Azzoz Dr. Eng. Adel Muharram Alnozahi Dr. Eng. Wael Elsayed Ahmed Swelam

3 - 1 - Chapter 1 Introduction and Background Position Sensitive Detector (PSD) is used for obtaining the position information of the targets. It is a simple photodiode configuration capable of detecting the centroid position of a light spot projected on its surface and converts the incident laser spot to its corresponding photocurrent; then the electronic circuits convert the photocurrent to its corresponding voltage, and this can be Quadrant Detector (QD) or Lateral Effect Photodiode (LEP). The position-sensitive detector (PSD) is the major part in many applications such as optical communication, object detection, and laser tracking systems. For example, in Laser Tracking Systems (LTS) which are divided into active and semi-active laser tracking systems. In Active LTS all the components of the system are in the same missile, while in the semi-active LTS the illumination of the target is performed with a separate laser source (laser designator). Semi-Active Laser (SAL) guidance is utilized on the modern battlefield for multiple weapon systems ranging from rockets to missiles to guided bombs. The SAL guidance scheme relies on a laser designator, either ground-based or airborne, to illuminate the target with laser energy. The reflected light is then sensed by the seeker on the weapon system, typically containing a quadrant photo detector. This information is then processed, using the guidance system, to determine miss-angles and computes the required corrections. In this sense, SAL seekers provide a terminal homing capability, based upon laser energy being reflected from the target, in such geometry, the incoming radiation is detected and used for the guidance process [1]. These weapons are said to be semi-active because, the weapon system does not have a laser source installed but relies on an external laser designator [2]. 1.1 Position Sensitive Diode The position sensitive diode (PSD) is a core component in the laser tracker, since the displacement between the measured beam incident point on the target and center is measured by the sensor, it also has an effect on how fast the target can be tracked.

4 - 2 - A PSD is a continuous silicon photodetector used for optical position sensing and basically consists of a uniform resistive layer, which is formed on a silicon substrate. A pair of electrodes is formed at the ends of the resistive layer from which photocurrents are measured. These photo-currents are generated as a result of the incident radiation effect and their magnitude is relative to the distance of the electrode to the center of the beam spot on the sensor's active area. The photo-currents are typically amplified and converted to form the measured voltage signal used by the control system. PSDs can be divided into two general groups: segmented PSDs and lateral effect PSDs Segmented Position Sensitive Diodes Segmented PSDs are quadrature PSD each is divided into four segments, which are separated by a small gap, called the dead region. The sensor is used for positioning by measuring each segment's photocurrent. A symmetrical beam is positioned at the center of the PSD if all these currents are equal. Figure 1.1 shows schematic crosssectional view of a quadrant detector showing the two of four photodiode element with contact and gap. Figure 1.1: Schematic cross-sectional view of a quadrant detector showing the two of four photodiode element with contact and gap. Segmented PSDs have a higher accuracy level than that of the lateral effect PSDs, due to the superior responsitivity match between the elements. Higher signal to noise ratio (S/N) of these system enable them to detect very low light levels. This group of PSDs is limited though, since the beam spot must be bigger than the dead region and must also overlap all the segments at all times and a uniform beam spot intensity is also

5 - 3 - required. These restrict the beam spot displacement measurements to a small and narrow tracking range [3] Lateral Effect Position Sensitive Diodes The other group is the lateral Effect Photodiodes (LEP), which can be either one or two dimensional and they are continuous single element PSDs, and have no dead areas. The beam spot position can be calculated with their photocurrents over the entire active area. Lateral effect photodiodes have the main advantage of a wide dynamic range and that the measured position is independent of the light spot intensity distribution (unlike segmented PSDs), and the resolution however is detector (or circuit) signal to noise ratio dependent. Figure 1.2 shows the schematic crosssectional of a lateral effect photodiode with n-type substrate showing incident light creating electron hole pairs that divides between the two contacts in proportion to resistance R1 and R2. This sensor is also more expensive [3]. Figure 1.3 shows Structure of 2-axis LEPs: a) Wallmark b) Tetralateral c) Duolateral d) Pincushion. In the case of pincushion LEP, the radius of the curvature of the active area boundary should be equal to R/r, where R is the sheet resistance within the active area and r is the boundary resistance. The Duolateral type has the highest position detecting ability, but it is also the most expensive. The Pincushion type is an improved Tetralateral PSD, as it has a bigger high linearity region than the Tetralateral type PSD. Both of these though have a simpler bias scheme, smaller dark current and faster response time than the Duolateral type. 1.2 Thesis Objectives Performing the necessary performance measurements. Developing and implementing a suitable mathematical model for PSD. This model will be used to verify the photodiode performance using the proposed model. Designing and implementing an optical system for the laser spot detection and optimization. Implementing and testing a laser spot position determination system. 1.3 Thesis Contributions The main contributions of this thesis are summarized below:

6 - 4 - Measuring the Quadrant photodiode main parameters using the available setup. Implementing the Quadrant photodiode model using a Pspise CAD tool. Model verification by comparing the simulation results with the experimental measurements. Analysis study to improve the Q-Photodiode performance using the proposed model. Designing and implementing an optical system to collect and optimize the laser spot size (By using Zemax optical design program). Implementing the Laser Spot Position Determination circuit using P-CAD program. Experimental verification testing of a Laser Spot Position Determination System. Figure 1.2: Schematic cross-sectional of a lateral effect photodiode with n-type substrate showing incident light creating electron hole pairs that divides between the two contacts in proportion to resistance R1 and R2.

7 - 5 - Figure 1.3: Structure of 2-axis LEPs: a) Wallmark b) Tetralateral c) Duolateral d) Pincushion. 1.4 Thesis Organization This thesis consists of four chapters plus the conclusion chapter. Chapter 1 presents a historical background, and discusses the development progress of photodiodes and four quadrant detectors for position determination applications, the general overview of previous works regarding the photodiode model and the literature survey are parented. Chapter 2 describes the analytical photodiode modeling and implementation using Pspice ORCAD tool, and the experimental measurement setup of the Quadrant photodiode main parameters such as the photocurrent, the dark current, and the junction capacitance. Chapter 3 presents the photodiode model verification and analysis study. Chapter 4 presents the designing of an optical system for laser spot detection and optimization using Zemax program and implementing and testing a laser spot position determination system. Chapter 5 summarizes the conclusions and discusses the future work. 1.5 Photodetection with Semiconductors: Basic Phenomena Photodetection in semiconductors works on the general principle of the creation of electron-hole pairs under the action of light. When a semiconductor material is illuminated by photons of energy greater than or equal to its barrier, the absorbed photons promote electrons from the valence band into excited states in the conduction band, where they behave like free electrons able to travel long distances across the

8 - 6 - crystal structure under the influence of an intrinsic or externally-applied electric field. The positively-charged holes left in the valence band contribute to electrical conduction by moving from one atomic site to another under the effect of the electric field. In this way the separation of electron-hole pairs generated by the absorption of light gives rise to a photocurrent, which refers by definition to the fraction of the photogenerated free charge-carriers collected at the edges of the material by the electrodes of the photodetecting structure, and whose intensity at a given wavelength is an increasing function of the incident light intensity. On this level, distinguish between two main categories of photodetectors based on the nature of the electric field, which causes the charge separation of photogenerated electron-hole pairs [4]: photoconductors, which consist of a simple layer of semiconductor simply with two ohmic contacts, where the electric field leading to the collection of the charge-carriers is provided by applying a bias voltage between the contacts at either end, and photovoltaic photo detectors, which use the internal electric field of a p-n or Schottky (metal semiconductor) junction to achieve the charge separation. This last term covers p-n junction photodetectors (photovoltaic structures consisting of a simple p-n junction, and p-i-n photodetectors which include a thin layer of semiconductor material between the p and n region which is not deliberately doped), as well as all Schottky junction photo detectors (Schottky barrier photodiodes and metal semiconductor metal (MSM) photodiodes). Now, briefly introduce of the main physical concepts at the root of the operation of the different semiconductor photodetector families will be presented Semiconductor Devices Photoconductors represent the simplest conceivable type of photodetector: they consist of a finite-length semiconductor layer with an ohmic contact at each end. In Figure 1.4 [5, 6], A fixed voltage of magnitude V p is applied between the two end contacts, in such a way that a bias current I b flows through the semiconductor layer, simply following Ohm's law. The active optical surface is formed from the region between the two collection electrodes. When it is illuminated, the photogenerated charges produced under the effect of the applied electric field lead to a photocurrent I Ph which is added to the bias current, effectively increasing the conductivity of the device.

9 - 7 - The main point of interest in a photoconducting device is its increased gain, the response of photoconductors being typically several orders of magnitude greater than that of photo-voltaic detectors for a given material. On the other hand, its other operational parameters (bandwidth, UV/ visible contrast, infrared sensitivity) are generally below that of other types of photodetectors, which often greatly limits the scope of its potential applications (this is particularly the case for photoconductors based on III-V nitrides) p-n junctions and p-i-n Structures In p-n diodes, the metallurgical linkage of a region of a p-type doped semiconductor and a region of n-type doping forms a p-n junction, where the joining of the Fermi levels in equilibrium mostly occurs through a flow of charge between the n and p regions. In equilibrium we therefore find a region with no free charge carriers immediately around the junction, similar to a charged capacitor, on the n side, Figure 1.4: Diagram of a photoconducting device [5, 6].

10 - 8 - Figure 1.5: Curvature of the energy bands and mechanisms of photocurrent generated in a p-n junction [5]. positively ionized donors, and on the p side, negatively ionized acceptors (this zone is known as the space charge region (SCR), where ionized donors and acceptors provide fixed charges). The presence of charged donors and acceptors produces an electric field in that region which curves the energy bands and, in equilibrium, forms an energy barrier between the two regions: the bottom of the conduction band and the top of the valence band on the n side are below the corresponding levels on the p side Figure 1.5. The width of the SCR is a decreasing function of the level of doping in the material, while the height of the energy barrier is an increasing function of it. An electron-hole pair produced in this SCR (situation 2 in Figure 1.5) [6] is therefore separated by the effect of the internal electric field of the junction, and so does not recombine. These are the charge carriers which contribute to the photocurrent, to which we can add, to some extent, those generated at a distance from the junction less than or equal to the diffusion length (situations 1 and 3 in Figure 1.5) [6]. The band structure of the junction implies that the photocurrent will consist of minority charge carriers. For this reason, the photocurrent flows in the opposite direction to the bias on the diode, where the forward direction is defined as the direction of flow of the majority charge carriers (from the n to the p region in the case

11 - 9 - of electrons, and vice versa for holes). The application of an opposing external electric field (Vp-Vn < 0) allows us to increase the height of the energy barrier in the vicinity of the junction, and also increase the spatial extent of the SCR, which significantly improves the efficiency of the separation of electron-hole pairs by increasing the electric field within the junction. We note that when the doping level is moderate, the width of the SCR is important. This effect is beneficial in the case of p-n junction photodetectors, where in order to increase the photoresponse it is desirable to ensure that the mechanisms of electron-hole pair generation through incident light take place predominately inside the SCR. A simple way of increasing the spatial extent of the SCR is to introduce between the n and p regions a thin layer of intrinsic semiconductor material which is not intentionally doped: the structure is therefore referred to as p-i-n. Such a structure is interesting because it is possible to maintain high levels of doping in the n and p regions without significantly reducing the extent of the SCR, whose width is then largely determined by the thickness of the "i" layer. Additionally, increasing the width of the SCR reduces the capacitance of the structure, which makes p-i-n structures particularly well-suited for high-speed operation P-I-N Diodes The solution of the problem created by the extreme thickness of a p-n junction is to make it thicker (increase depletion width by i type material). The junction is extended by the addition of a very lightly doped layer called the intrinsic zone between the p and n doped zones. Thus the device is called a p-i-n diode rather than a p-n diode. This is illustrated in Figure 1.6 [6]. The wide intrinsic (i) layer has only a very small amount of dopant and acts as a very wide depletion layer. There are a number of improvements here: It increases the chances of an entering photon being absorbed because the volume of absorbent material is significantly increased. Since it makes the junction wider it reduces the capacitance across the junction. The lower the capacitance of the junction the faster the device response. There are two ways of current carriage across the junction: Diffusion and Drift current.

12 Figure 1.6: Typical Silicon P-I-N Diode Schematic [6]. Increasing the width of the depletion layer favors current carriage by the drift process which is faster than the diffusion process. The result is that the addition of the "i" layer increases the Responsivity and decreases the response time of the detector to around a few tens of picoseconds. The key of the operation of a PIN diode is that the energy of the absorbed photon must be sufficient to promote an electron across the bandgap, the material will absorb photons of any energy higher than its bandgap energy. Thus when discussing PIN diodes, it is a common to talk about the "cutoff wavelength". Typically PIN diodes will operate at any wavelength shorter than the cutoff wavelength. This suggests the idea of using a material with low bandgap energy for all PIN diodes regardless of the wavelength. The lower the bandgap energy the higher the "dark current" (thermal noise). For this characteristic germanium would be the material of choice for all PIN diodes. It is low in cost and has two useful bandgaps (an indirect bandgap at 0.67 ev and a direct bandgap at 0.81 ev) [6]. It has a relatively high dark current compared to other materials. This means that the materials used for PIN diode construction are different depending on the band of wavelengths for which it is to be used. However, this restriction is nowhere near as stringent as it is for lasers and LEDs where the characteristics of the material restrict the device to a very narrow range. The optimal way is to choose a material with bandgap energy slightly lower than the energy of the longest wavelength we want to detect. An interesting consequence to note here is that

13 these crystalline semiconductor materials are transparent at wavelengths longer than their cutoff. Thus, if we could "see" with 1500 nm eyes a crystal of pure silicon (which appears dark grey in visible light) would look like a piece of quartz or diamond. Typical materials used in the three communication wavelength "windows" are as follows: nm band [7]: Silicon PIN diodes operate over a range of 500 to 1120 nm as silicon has bandgap energy of 1.11 ev [6]. Since silicon technology is very low cost, silicon is the material of choice in this band. However, silicon is an indirect bandgap material (at the wavelengths we are interested in) and this makes it relatively inefficient. Silicon PIN diodes are not as sensitive as PIN diodes made from other materials in this band. This is the same characteristic that prevents the use of silicon for practical lasers nm (1250 nm to 1400 nm) band [7]: in this band indium gallium arsenide phosphide (InGaAsP) and germanium can be used. Germanium has lower bandgap energy (0.67 ev versus 0.89 ev for InGaAsP) and hence it can theoretically be used at longer wavelengths. However, other effects in Ge limit it to wavelengths below 1400 nm [6]. InGaAsP is significantly more expensive than Ge but it is also significantly more efficient (devices are more sensitive) nm band (1500 nm to 1600 nm) [7]: the material used here is usually InGaAs (indium gallium arsenide). InGaAs has bandgap energy of 0.77 ev [6]. Operation with reverse bias as described above (called "Photoconductive Mode") has a significant problem. At low light levels the random current produced by ambient heat is a source of noise. At higher light levels however this is not a problem and the device offers the advantages of much higher speeds (than the alternative mode of operation) and linear response characteristics over a wide range. The higher speed characteristics are the result of lowered capacitance caused by the widening of the depletion layer in the presence of reverse bias. If the device is operated without an externally applied current the natural potentials of the p-i-n junction will cause electrons and holes to migrate across the junction anyway. This is called "Photovoltaic Mode". Thus you get a small voltage (around 0.15 V) developed across the device. The advantage of this is that there is much less "dark current" caused by ambient heat. Thus in this mode the device is more sensitive but the output requires immediate amplification because of the low voltage levels produced.

14 The efficiency of p-i-n diodes at long wavelengths can be improved by the use of heterostructures [6]. For example using GaAlAs for the p-layer and GaAs for the i and n-layers. Incident light must pass through the p-layer before it enters the i-layer. If it is absorbed by the p-layer then the energy is lost and does not contribute to the current produced. By using a material with high bandgap energy for the p-layer (higher than the energy of the incident photons) we can prevent incident long wavelength light from being absorbed. This means that there is less light lost before it reaches the i- layer where we want it to be absorbed. Measures of Efficiency in PIN Diodes There are two common measures parameters when the efficiency of PIN photodetectors is discussed. Quantum Efficiency (QE). This is simply the ratio of the number of electrons collected at the junction to the number of incident photons. In an ideal situation 1 photon releases 1 electron (and this matching hole of course). Perfect Quantum Efficiency is therefore efficiency of "1". In real devices QE is different at each operating wavelength and so it should always be quoted in association with a wavelength. Responsivity. Quantum efficiency does not take into account the energy level of the incident pho- tons. Responsivity is a measure that does take photon energies into account. It is simply the output photocurrent of the device (in amperes) divided by the input optical power (in watts). Thus Responsivity is quoted in amperes per watt. A typical Responsivity for a silicon photodiode at a wavelength of 900 nm is Of course Responsivity is very closely linked to quantum efficiency. It is just quantum efficiency adjusted to account for the variation in energy level implied by different wavelengths Avalanche Effect in p-i-n Structures When the reverse-bias voltage established at the terminals of a p-i-n structure increases sufficiently that the electric field established in the junction reaches values close to the breakdown field (in structures of micron-scale thickness, this is generally the case when the bias voltage at the terminals reaches a few dozen volts), the photogenerated charge carriers in the SCR (which is effectively the region that is not intentionally doped) are accelerated enough to separate other secondary charge

15 carriers from the atoms in the lattice that they impact in the course of their motion: this is the avalanche effect which results in a multiplication of the charge carriers in the SCR [6, 8]. The gain is therefore greater than 1 for the generation of charge carriers by light, and this gain can even typically reach 10 or 20 under favorable conditions. This effect is exploited in what are called avalanche photodiodes where the levels of n- and p-type doping are generally adjusted to high values above cm -3 to maximize the intrinsic electric field of the junction Schottky Junction A Schottky junction is formed by bringing a metal and a semiconductor into contact. The basic phenomena which lead to the formation of a Schottky junction with an n-type semiconductor are summarized in Figure 1.7. In thermal equilibrium, when the Fermi levels of the metal and the semiconductor are equalized, a transfer of electronic charge occurs from the semiconductor to the metal in the case where the work function q. φ M of the metal (q being the elementary charge) is greater than the electron affinity X of the semiconductor and a SCR appears at the edge of the semiconductor of width x d next to the junction, where the only charges present are the positively-ionized donors. A curvature of the energy bands therefore occurs at the junction, which leads to the appearance of an energy barrier between the metal and the semiconductor, called a Schottky barrier, whose height is given to first approximation by the expression [6]: q. φ Bn = q. φ M χ (1.1) In equilibrium, therefore, we find an intrinsic electric field immediately next to the metal-semiconductor junction which is comparable in form to that found in a p-n junction. Consequently, it is the phenomenon of photogeneration of charge carriers inside and near to the SCR which is responsible for the appearance of a photocurrent, with the electron-hole pairs being separated by the effect of the electric field in the Schottky junction. It is possible, as in the case of the p-n junction, to modify the intensity of the internal electric field in the junction by applying a bias voltage V between the semiconductor and the metal of the Schottky contact (fig 1.8).

16 Figure 1.7: Formation of a Schottky junction (In an n-type semiconductor) [5]. In the case of the n-type semiconductor, the application of a negative voltage between the semiconductor and the metal electrode of the Schottky contact has the effect of reverse-biasing the Schottky junction, which leads to an increase in the height of the effective barrier, along with an increase in the width of the SCR. This last effect is of course favorable for photodetection. Indeed, it follows that the majority charge carriers (electrons) cannot flow towards the Schottky contact, and only the minority carriers (holes) generated by external excitation (in particular photogeneration) can reach the Schottky contact and hence produce an electric current: as in the case of the p-n junction, we therefore find that the current flows in reverse through the Schottky junction, that is, from the semiconductor towards the Schottky contact. The illumination of Schottky photodiodes can occur through the front or rear face (often this second option is chosen in the case where the substrate material is transparent to the light to be detected, as is the case for example with sapphire). In the case of illumination through the front face, we resort to a semi-transparent Schottky contact, characterized by a very small thickness of metal (in the order of 100 angstrom) selected to ensure sufficient optical transmission: while a thin layer of gold of 100 angstrom thickness transmits up to 95% of the incident light in the infrared, the percentage transmitted in the ultraviolet is around 30% in the range nm.

17 The gain of p-i-n photodiodes (other than the specific case of avalanche photodiodes) and Schottky photodiodes is at most 1, which would be the case if all the photogenerated charge carriers were collected by the electrodes at the ends of the device. Figure 1.8: Reverse-bias of a Schottky junction (n-type semiconductor material) [5] Metal-Semiconductor-Metal (MSM) Structures An MSM structure consists of two Schottky electrodes, often interlinked in the form of a comb structure, leaving a free semiconductor surface between the two contacts which forms the active region in which light will be absorbed [6, 9]. A bias voltage can be applied between the two electrodes, in order to break the initial electrical symmetry of the contacts: one of the Schottky junctions is reverse-biased, producing a SCR of increased width, and the other junction is forward-biased. The absorption of light near the reverse-biased junction creates electron-hole pairs which are separated under the effects of the electric field present in the SCR, thus creating the photocurrent. The other electrode, consisting of a forward-biased (and hence transmissive) Schottky junction, simply acts as a collection electrode. The band diagram of the device under increased bias voltage (V B ) is represented schematically in Figure (1.9), in which L is the distance between two adjacent contact fingers, φ is the height of the Schottky barrier and I ph is the photocurrent. MSM photodetectors normally use semiconductor materials which are not intentionally doped, are chemically very pure and electrically very resistive. The SCRs associated with Schottky junctions made of these materials are hence of significant width which, for a

18 given bias voltage, allows the electric field of the junction to extend more easily into semiconductor regions some way from the contact. It follows that photogenerated electron hole pairs are more easily separated and collected by the electrodes at either end. Figure 1.9: Energy band diagram for a MSM structure under electrical bias; effect of illumination [5]. 1.6 Optical Receiver Parameters with the Position Sensitive Detector Figure 1.10 shows the geometry of the laser transmitter and the receiving optics used with the position sensitive detector for laser spot position determination Background power The received background optical power at the input of the position sensitive optical receiver can be calculated from [10, 11]: P B = L λ GT R T F τ at (1.2) Where: P B is the background optical power that is received at the input of the position sensitive optical receiver (PSOR) L λ is the solar spectral radiance at the object site, T R is the receiving optics transmission coefficient, T F is the optical filter transmission coefficient, τ at is the transmission coefficient of the atmosphere, G is a geometrical factor.

19 Figure 1.10: Geometry of the laser transmitter and the receiving optics [10]. Where: The geometrical factor G is defined as a factor of radiation exchange of two small areas [10]: Where: A D G = A D cos θ A R cos θpo R M 2 (1.3) is the area of the detector footprint at the background, A R cosθ po is the effective area of the optical receiver, θ po R M Where: is the angle between the receiver surface normal and the line joining the object and receiver centers (θ po, because in the case of good positioning and tracking the receiving optics are always directed towards the object), is the distance between the object and the optical receiver. The solar spectral radiance L λ, induced by solar radiation of a diffuse reflector for a specific wavelength is given by [10, 12]:

20 L λ = E λρ B π Where: E λ is the solar spectral irradiance, (1.4) ρ B is the background reflectance. The atmosphere transmission coefficient τ at is given by [12]: τ at = e ςr M (1.5) Where: σ is the atmospheric extinction coefficient. The background power, which is given by equation (1.2), after combining with equation (1.3) to equation (1.5), it becomes: P B = π 16 E λ Δλ ρ B β 2 D 2 T R T F e ςr M (1.6) Where: Δλ is the optical spectral filter bandwidth, β is the receiving optics field-of-view, and D is the receiving optics aperture diameter Signal Power The received optical signal power P s of the reflected laser radiation from the illuminated object, in the case when the area of the laser beam is smaller than the area of the object, is given by [10, 11, and 12]: P S = L T A T Ω D T R T F e ςr Mcosθ (1.7) Where: L T is the spectral radiance of the reflected radiation from the object, A T is the area of the laser spot on the object, Ω D is the solid angle subtended by the optical receiver aperture. The spectral radiance L T is given by: Where: P L is the laser peak power, L T = 4P LT L η ρ T e ςr L cos θ L π 2 β T 2 R L 2 (1.8) T L is the transmission coefficient of transmitting optics,