Abstract

In this paper, mid-wave infrared (MWIR) sensor optimization is provided as a function of the parameter ${F}\lambda /{d}$, where ${F}$ is the ${f}$-number, $\lambda $ is the effective wavelength, and $d $ is the detector pitch. For diffraction limited systems, acquisition range is related to the instantaneous field of view (detector limited operation) when ${F}\lambda /{d} \lt {1}$, and to the optical properties (optics limited operation) when ${F}\lambda /{d} \gt {2}.{0}$. Range performance is a combination of detector and optics resolution limits when ${F}\lambda /{d}$ is in between. When the system is not strictly diffraction or sampling limited, the optimal ${F}\lambda /{d}$ depends on other system component characteristics and conditions. Optical system aberrations affect system resolution and decrease range performance. As background shot noise, dark current shot noise, and read noise increase, range decreases. In the infrared spectral region, atmospheric absorption leads to reemission of thermal energy. The detected reemission creates additional shot noise. Atmospheric attenuation greatly affects MWIR sensor range performance. Next-generation MWIR sensors will have smaller detectors, larger arrays, and better sensitivity to enable ${F}\lambda /{d}$-based optimization. Previous studies ($\Delta T = {4}\;{\rm K}$ for tracked vehicles) suggest that an initial design point is ${F}\lambda /{d} \approx {2}.{0}$. When detecting low contrast targets ($\Delta T \sim {0}.{1}\;{\rm K}$), sensor gain is used to increase the signal for a desired displayed contrast. This gain increases displayed noise and reduces acquisition range. This is typically not an issue for long-wave infrared sensors due to the excess number of photons in the 8–12 µm band but poses a problem for MWIR sensors, which are photon starved. Under such scenarios, the optimum ${F}\lambda /{d}$ appears to be about 1.5 for MWIR sensors. The results here provide reasonable strategies for MWIR system optimization and a direction associated with future MWIR focal plane development.

© 2020 Optical Society of America

1. INTRODUCTION

Numerous infrared system performance analyses have focused on long-wave infrared (LWIR) detector pitch optimization [1,2]. Although the work is similar, those analyses do not address the mid-wave infrared (MWIR) band. The MWIR band is more difficult to understand because it tends to be photon starved for many terrestrial background applications (the LWIR band has plenty of photons). Detailed analysis allows a knowledgeable selection of detector sizes and system implementations.

LWIR has long been a historical band that works well for many applications, and in particular, ground combat applications [1,2]. The U.S. Army still uses LWIR in primary targeting sights due to the abundance of photons under different conditions including cold weather, high turbulence, degraded atmosphere, smoke, and dirty battlefield conditions [35]. For an aperture-limited system, a MWIR sensor provides a greater acquisition range than a comparable LWIR sensor under good target and atmospheric conditions (i.e., high altitude). As a result, MWIR has become common for U.S. Air Force and U.S. Navy applications since their fighting environments are not as severe as ground combat. Considering the benefits of MWIR, the U.S. Army is pursuing dual-band (MWIR + LWIR) systems, where the primary benefit of the MWIR channel is that of long-range target identification under good atmospheric conditions.

Small pitch (5–10 µm) LWIR sensors with high pixel count arrays allow imaging of two fields of view without switching optics [610]. The benefit of such systems is the speed in detecting and identifying a target. This approach provides all of the LWIR benefits associated with classical switched optical systems but still does not compete with MWIR targeting range under good conditions. The MWIR large format small pitch approach has largely been ignored by the armed services for tactical applications, but has been studied in the applications of missile launch detection, infrared search and track, infrared countermeasures, and unmanned aerial vehicle detection.

Large format MWIR detectors have been under development for quite a few years for the applications listed above. The “standard” MWIR detector in production and/or under development has 12 µm pitch (Fig. 1). Many of the government laboratories are pushing to an 8 µm pitch, and it is likely that this will be the new standard. Changing the pitch will have significant implications on system design and optimization. Next-generation MWIR sensors will have even smaller detectors and better sensitivity (easily detect $\Delta T = {0}.{1}\;{\rm K}$ targets).

 figure: Fig. 1.

Fig. 1. Scanning microscope image of 12 µm pitch MCT MWIR detectors (Courtesy of J. Devitt, Raytheon).

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There have been significant technical advances in MWIR detectors in the past 7–10 years. The advances have been primarily motivated by the US-funded Tri-Service Vital Infrared Sensor Technology Acceleration (VISTA) programs. The first program emphasized the development of antimony (Sb)-based focal plane technology, and the second program focused on high-operating temperature (HOT) detectors and multiband detectors [11]. The second program also established an industrial base for Sb-based MWIR HOT focal plane manufacturing with large format and small pitch. The process improvement effort of the program provided for lower-cost focal planes with higher yield and operability. The program resulted in two material substrate foundries, two epitaxial material growers, and five infrared focal plane producers. The results of the VISTA program as well as the manufacturing technology follow-on programs have provided the MWIR focal plane development emphasis that will allow extremely high performance MWIR sensors in the near future. The applications that are impacted by VISTA are targeting, infrared search and track, threat warning, and intelligence–surveillance–reconnaissance (ISR).

The impact of the VISTA work is highly likely to replace InSb MWIR detectors with quantum structures such as type II superlattice (T2SL) and nBn detectors. Both of these newer detector technologies provide InSb-like performance, but with much higher operating temperatures. HgCdTe is also a candidate for the displacement of InSb, as it also has a higher operating temperature (130–150 K). There is a rule of thumb for detector coolers that states that an increase in operating temperature of 10 K provides for a cooler power reduction of one half. This rule emphasizes the importance of high operating temperature MWIR detectors since the cooler power impacts size, weight, power, and cost.

In this paper, night vision integrated performance model (NVIPM) predictions provide a path for system optimization. While NVIPM is valid for sensors sensitive anywhere in the spectral region from 0.4–14 µm, this paper focuses on next-generation MWIR sensor design. LWIR sensors have been addressed in previous work [610]. Important system parameters and conditions modeled in NVIPM are used as inputs in this study. Each input (background temperature, optical aberrations, noise, atmospheric transmission, notch filter, viewing distance, and well capacity) is individually analyzed to determine those that are most deleterious to system range performance. Previous studies [11] with large target $\Delta T$ (NVIPM recommends $\Delta T = {4}\;{\rm K}$ for tracked vehicles) suggested that a starting design point is ${F}\lambda /{d} \approx {2}.{0}$, where ${F}$ is the ${f}$-number, $\lambda $ is the effective wavelength averaged over the spectral photon irradiance and sensor spectral responsivity, and $d$ is the detector pitch.

With low $\Delta T$s, noise affects acquisition range, and the optimum ${F}\lambda /{d}$ for MWIR systems appears to be about 1.5. The acquisition ranges presented here are unique to the values selected (e.g., spectral quantum efficiency, atmospheric spectral transmission) but the shapes of the curves and conclusions are representative of all MWIR sensors. Any size detector with the appropriate f-number optics can achieve a desired ${F}\lambda /{d}$ [12,13] and corresponding range performance. The limitations are how small detectors can be manufactured, physical space constraints that limit the focal length and aperture diameter, and minimum practical ${f}$-number (sometimes taken as 1.2). Fiete [14] labeled the detector pitch as $\rho $ and provided imagery as a function of ${F}\lambda /\rho $ (sometimes called ${Q}$). This paper assumes 100% fill factor so that detector size and pitch are the same; we will refer to this as $d$ (${ \rho } = d$) in the analysis and discussion in this paper. We note that the results for a somewhat less than 100% closely match the results reported here.

2. NVIPM THEORY

NVIPM was developed for the detection, recognition, and identification of ground targets. It consists of an eye (observer) term and a noise term. The system contrast threshold function is

$$\begin{split}{\rm CTF}_{{\rm SYS}}=\sqrt{\underbrace{\left(\frac{{\rm CTF}_{{\rm NAKED\;EYE}}}{{\rm MTF}_{{\rm SYS}}}\right)^{2}+}_{\rm EYE\;TERM} \underbrace{\left(\frac{{\rm CTF}_{{\rm NAKED\;EYE}}}{{\rm MTF}_{{\rm SYS}}}\right)^{2}\left(\frac{\alpha \sigma_{P}}{L_{{\rm AVE}}}\right)^{2}}_{\rm NOISE\;TERM}}.\end{split}$$
${{\rm CTF}_{\rm NAKED\,EYE}}$ is Barten’s human visual system (HVS) contrast model and the system modulation transfer function $({{\rm MTF}_{\rm SYS}})$ is the product of all subsystem MTFs (limited here to optics, detector, and display). Usually there is a one-to-one mapping from a detector pixel to a display pixel so that ${{\rm MTF}_{\rm DISPLAY}} = {{\rm MTF}_{\rm DETECTOR}}$. The HVS provides tremendous spatial and temporal integration such that perceived SNR is much greater than the measured SNR. This integration is part of the complex noise term and is represented by the variable ${\sigma _P}$. ${{L}_{\rm AVE}}$ is the display luminance. The experimentally derived constant $\alpha $ matches predicted values to experimental results.

For mathematical convenience, the MTFs are considered separable. The targeting task performance (TTP) is

$${{\rm TTP}_u} =\int \sqrt {\frac{{{C_{\rm DISPLAYED}}}}{{{{\rm CTF}_{\rm SYS}}( u )}}} {\rm d}u,{{\rm TTP}_v} = \int \sqrt {\frac{{{C_{\rm DISPLAYED}}}}{{{{\rm CTF}_{\rm SYS}}( v )}}} {\rm d}v.$$
The constant, ${{C}_{\rm DISPLAYED}}$, is the observer’s desired displayed contrast. It is not the target contrast, ${{C}_{\rm TARGET}}$. The displayed contrast can be maintained constant by adjusting the sensor gain as a function of target contrast. This gain also modifies the displayed noise affecting the noise term in Eq. (1). Here we use ${{C}_{\rm DISPLAYED}} = {0}.{2}$ as the default suggested by the NVIPM help manual. The variables $u $ and $ v$ are, respectively, the horizontal and vertical spatial frequencies in cycles/milliradian. The range, ${R}$, is predicted from a ${V}$ parameter and target size ${W}$, where the variable ${W}$ is the square root of the target area in meters:
$${V_{\rm RESOLVED}}( R ) = \frac{W}{R}\sqrt {{{\rm TTP}_u}{{\rm TTP}_v}} .$$
The range of target discrimination at a 50% probability is related to
$$R = \frac{W}{{{V_{50}}}}\sqrt {{{\rm TTP}_u}{{\rm TTP}_v}} = \frac{W}{{{V_{50}}}}{\rm TTP},$$
where ${{V}_{\rm RESOLVED}}$ is set to ${{V}_{50}}$ (50% probability metric) in Eq. (3). The $V$ value for a given probability of task performance is obtained through an empirically derived targeting transfer probability function (TTPF) conversion function.

Figures 2 and 3 illustrate the effects of MTF, noise, and low contrast targets on the TTP calculation.

 figure: Fig. 2.

Fig. 2. Screen shot of NVIPM contrast illustrating a low noise system. The TTP is the area between the system CTF and the displayed contrast (line added). This area is also called excess contrast. As ${{\rm MTF}_{\rm SYS}}$ decreases, system ${{\rm CTF}_{\rm SYS}}$ increases and range decreases. This MWIR sensor is MTF limited. $\Delta {T} = {1}.{0}\;{\rm K}$.

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 figure: Fig. 3.

Fig. 3. Screen shot of NVIPM contrast illustrating effects of noise and low target contrast. This MWIR sensor is noise limited [noise term in Eq. (1) dominates]. $\Delta {T} = {0}.{1}\;{\rm K}$.

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Considering a simplified system, ${{\rm MTF}_{\rm SYS}} = {{\rm MTF}_{\rm DIFFRACTION}}{{\rm MTF}_{\rm DETECTOR}}{{\rm MTF}_{\rm DISPLAY}}$. Using the monochromatic diffraction limited optical MTF as an approximation to the polychromatic MTF,

$${{\rm MTF}_{\rm DIFFRACTION}}( u ) = \frac{2}{\pi }\!\left[\! {{\cos^{ - 1}}( u ) - ( u )\sqrt {1 - {{( u )}^2}} } \right]u \le 1,$$
where the spatial frequency is normalized with respect to the optics diffraction cutoff. The detector MTF, relative to diffraction MTF, is
$${{\rm MTF}_{\rm DETECTOR}}( u ) = {\rm sinc}\left( {\pi \frac{d}{{F\lambda }}u} \right).$$
In Eq. (6), $d$ is the detector linear dimension. In the frequency domain, $F\lambda /d$ is the ratio of the detector cutoff (first zero) to the optics cutoff. In physical space, 2.44 $F\lambda /d$ is the ratio of the Airy disk diameter (to the first zero) to the detector linear dimension. NVIPM calculates an effective wavelength (averaged over the photon spectral irradiance and detector spectral response). Since the spectral irradiance is a function of the background temperature, the effective wavelength is a function of the background temperature. While $F$ and $d$ are single valued system parameters $\lambda $ is not. Using a single value for a broadband spectral response sensor is a reasonable approximation for trade studies and optimization. For this study, $\lambda = {4}.{46}\; \unicode{x00B5} {\rm m}$ for the MWIR at an environmental temperature of 300 K. Ignoring ${{\rm MTF}_{\rm DISPLAY}}$, as ${F}\lambda /{d} \to \infty $, ${{\rm MTF}_{\rm SYS}} \to {{\rm MTF}_{\rm DIFFRACTION}}$. For small ${F}\lambda /{d}$ values, ${{\rm MTF}_{\rm SYS}} \to {{\rm MTF}_{\rm DETECTOR}}$.

For preliminary analysis, we set all noise except photoelectron shot noise to zero. This requires a transparent atmosphere ($\tau = 1/{\rm km}$ with Beer’s law) to avoid atmospheric thermal emission shot noise. With low noise, the eye term in Eq. (1) dominates, and Eq. (4) becomes

$$\begin{split}R \approx \frac{W}{{{V_{50}}}}\sqrt {{C_{\rm DISPLAY}}} \smallint \sqrt {\frac{{{{\rm MTF}_{\rm SYS}}( u )}}{{{{\rm CTF}_{\rm NAKED\,EYE}}( u ) \times {\rm NF}\big( {{{C}_{{\rm TARGET}}}} \big)}}} {\rm d}u,\end{split}$$
where the noise terms have been grouped in ${\rm NF}( {{{C}_{{\rm TARGET}}}})$, and NF represents a target contrast dependent noise factor term. In a system with only background limited noise, to maintain a constant ${C_{\rm DISPLAYED}}$ as mentioned above, a large target contrast results in ${\rm NF} \approx 1$, and a small target contrast results in NF greater than one, and that reduces range performance.

For this analysis we use a typical InSb MWIR sensor sensitive from 3.34–4.85 µm and includes a ${{\rm CO}_2}$ notch filter at 4.2 µm. To explore the eye and displayed noise term properties only in the presence of shot noise, the sensor has zero dark current, zero downstream noise, and a well capacity of ${{10}^{10}}$ electrons (considered an infinite well). The aperture diameter is 50 mm. The display element is 0.2 mm, and the viewing distance is 30 cm. The detection criterion is ${{V}_{50}} = {2}$. From Eq. (7), range is linearly proportional to target size so that selecting 1.0 m is representative of range performance. As $\Delta {T}$ decreases, the sensor gain increases to maintain a constant ${{C}_{\rm DISPLAY}}$. But the value of NF increases with gain, and the noise term in Eq. (1) can dominate for low $\Delta {T}$s. With only background limited noise and $\Delta {T} = {1}\;{\rm K}$, maximum range occurs at high ${F}\lambda /{d}$ (Fig. 4). This curve is based upon MTFs only and its shape is generic to all systems. The optimum ${F}\lambda /{d}$ decreases as displayed noise (due to gain) increases. Figure 4 has three distinct regions: detector limited, transition, and optics limited. Selecting ${F}\lambda /{d} = {0}.{5}$ and 1.5 as boundaries is arbitrary.

 figure: Fig. 4.

Fig. 4. NVIPM predicted relative range as a function of ${F}\lambda /{d}$. The focal length was varied to produce a variable ${F}\lambda /{d}$. With a transparent atmosphere, only background shot noise exists. Background temperature is 300 K. For low $\Delta {T}$ targets, the sensor increases gain to reach the desired displayed contrast. As the gain increases, the displayed noise increases and response moves from eye limited to noise limited operation near ${F}\lambda /{d} \approx {1}.{5}$.

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It was previously shown [12] that the integral in Eq. (7) can be expressed as a sixth-order polynomial valid up to ${F}\lambda /{d} = {4}$:

$$\begin{split}{\rm TTP}& \approx \left[ {a_6}{{\left( {\frac{{F\lambda }}{d}} \right)}^6} + {a_5}{{\left( {\frac{{F\lambda }}{d}} \right)}^5} + {a_4}{{\left( {\frac{{F\lambda }}{d}} \right)}^4}\right.\\&\quad +\left. {a_3}{{\left( {\frac{{F\lambda }}{d}} \right)}^3} + {a_2}{{\left( {\frac{{F\lambda }}{d}} \right)}^2} + {a_1}{{\left( {\frac{{F\lambda }}{d}} \right)}} \right]\frac{D}{\lambda },\end{split}$$
where $D$ is the optical aperture diameter. This equation fits the $\Delta {T} \gt {1}$ curve in Fig. 4. In the detector limited region (${{\rm MTF}_{\rm DETECTOR}}$ dominates ${{\rm MTF}_{\rm SYS}}$), the last term of the polynomial in Eq. (8) is retained. Then the 50% probability range is approximately
$$R \approx {k_{\rm DETECTOR}}\frac{W}{{{V_{50}}}}\frac{f}{d} = {k_{\rm DETECTOR}}\frac{W}{{{V_{50}}}}\frac{1}{{\rm IFOV}}.$$
Here, range increases with focal length (smaller instantaneous field of view or IFOV) leading to the popular saying: “A smaller IFOV increases range.” In the optics limited region, ${{\rm MTF}_{\rm OPTICS}}$ dominates, and the polynomial in Eq. (8) approaches a constant:
$$R \approx {k_{\rm OPTICS}}\frac{W}{{{V_{50}}}}\frac{D}{\lambda },$$
leading to the popular saying: “Range increases with aperture size.” With the wavelength in the denominator, a MWIR sensor is desirable over a LWIR sensor when operating in the optics limited region. Figure 5 includes detector sizes for MWIR with realistic optics of ${F} = {1}.{2}$. These ${F}\lambda /{d}$ values are different from those provided in Ref. [4] where the average wavelength was 4 µm.
 figure: Fig. 5.

Fig. 5. Detector sizes for a low noise MWIR sensor and transparent atmosphere. Background temperature is 300 K. A different f-number and/or wavelength provide(s) different ${F}\lambda /{d}$ values for each detector size.

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The above equations and figures are generic to all sensors.

In the following sections, we introduce system component and condition parameters that affect system performance deviating from the detector limited or optics limited regions.

3. BACKGROUND TEMPERATURE

The differential signal between the resolved target at temperature ${{T}_T}$ and the background at temperature ${{T}_B}$ is

$$\begin{split}\Delta {n_{\rm PE}}& \approx \int_0^\infty \frac{{\eta ( \lambda )\left[ {{M_q}( {\lambda ,{T_T}} ) - {M_q}( {\lambda ,{T_B}} )} \right]{A_D}{t_{\rm INT}}}}{{4{F^2}}}\\&\quad \times {\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda ,\end{split}$$
where $\eta (\lambda )$ is the spectral quantum efficiency, ${{M}_q}(\lambda ,{T})$ is the photon spectral exitance (Planck’s blackbody expression), ${{t}_{\rm INT}}$ is the integration time (for an infinite well, it is 33.3 ms for 30 Hz frame rate), ${\tau _{\rm OPTICS}}(\lambda )$ is the optical spectral transmittance, and ${{ A}_D} = {{d}^2}$. Let ${{T}_T} \approx {{T}_B} + \Delta {T}$. Then, ${{ M}_q}(\lambda ,\Delta {T}) = {{ M}_q}\;(\lambda ,{{T}_B} + \Delta {T}) - {{ M}_q}\;(\lambda ,{{T}_B})$. Expanding in a Taylor series, keeping the first term (small $\Delta {T}$), provides the camera formula as a function of $\Delta {T}$:
$$\begin{split}\Delta {n_{\rm PE}} \approx \int _{{\lambda _1}}^{{\lambda _2}} \frac{{\eta ( \lambda ){A_D}{t_{\rm INT}}}}{{4{F^2}}}\frac{{\partial {M_q}( {\lambda ,{T_B}} ){\Delta }T}}{{\partial T}}{\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda .\end{split}$$
Since the thermal derivative varies with the background temperature, the number of photoelectrons and acquisition range vary with background temperature. Figure 6 illustrates detection range (${{ V}_{50}} = {2})$ for a 1 m target whose $\Delta {T}$ is 1 K. The acquisition range of hot targets ($\Delta {T} \gt {1}\;{\rm K}$) is nearly independent of the background temperature. In Fig. 7, $\Delta {T} = {0}.{1}\;{\rm K}$. As the target $\Delta {T}$ decreases, the system gain increases, producing more perceptible noise. The acquisition range of low $\Delta {T}$ targets varies significantly with background temperature. The optimum ${F}\lambda /{d}$ depends upon the anticipated target $\Delta {T}$ and background temperature. As will be shown in Sections 6 and 7, the atmospheric transmission significantly affects acquisition range.
 figure: Fig. 6.

Fig. 6. Detection range as a function of ${F}\lambda /{d}$ for background temperatures of 240, 260, 280, 300, 320, and 340 K. 1 m target with $\Delta {T} = {1}.{0}\;{\rm K}$ and transparent atmosphere. The 300 K curve is identical to that in Fig. 4.

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 figure: Fig. 7.

Fig. 7. Detection range as a function of ${F}\lambda /{d}$ for background temperatures of 240, 260, 280, 300, 320, and 340 K. 1 m target with $\Delta {T} = {0}.{1}\;{\rm K}$ and transparent atmosphere. The 300 K curve is identical to that in Fig. 4.

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4. OPTICAL ABERRATION

Aberration is a generic term applied to any lens “defect” that degrades the image. It is usually characterized by an MTF. In the detector limited region, optical aberrations have minimal effect on acquisition range. In the optics limited region, aberrations reduce the optics MTF and thereby reduce range. For convenience, ${{\rm MTF}_{\rm OPTICS}}$ is approximated by

$${{\rm MTF}_{\rm OPTICS}} \approx {{\rm MTF}_{\rm DIFFRACTION}}{{\rm MTF}_{\rm ABERRATION}}.$$
${{\rm MTF}_{\rm ABERRATION}}$ is a fictitious MTF such that the combination approximates the measured sensor optical MTF. For a rms wave front error of ${{\rm W}_{\rm RMS}}$, the fictitious aberration MTF is
$$\begin{split}{{\rm MTF}_{\rm ABERRATION}}( u ) \approx 1 - {\left( {\frac{{{W_{\rm RMS}}}}{{0.18}}} \right)^2}\left[ {1 - 4{{\left( {u - \frac{1}{2}} \right)}^2}} \right],\end{split}$$
where ${{\rm W}_{\rm RMS}} \approx {{W}_{\rm PP}}/{3}.{5}$. For peak-to-peak wave front error of $\lambda /{4}$, the rms error is $ \approx {0}.{\rm 25/3}.{5} = {0}.{0714}$. Figures 8 and 9 illustrate detection ranges for $\Delta {T} = {1}.{0}$ and 0.1 K targets, respectively. High-quality optics is required in the optics limited region.
 figure: Fig. 8.

Fig. 8. Acquisition range as a function of ${F}\lambda /{d}$ and peak-to-peak wave front error (0, $\lambda /{4}$, and $\lambda /{2}$) for ${{T}_B} = {240}$ (dashed lines) and 300 K (solid lines). Zero wave front error curve at 300 K is identical to those in Fig. 6. $\Delta {T} = {1}.{0}\;{\rm K}$. As $\Delta {T}$ increases, aberrations have a smaller effect.

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 figure: Fig. 9.

Fig. 9. Acquisition range as a function of ${F}\lambda /{d}$ and peak-to-peak wave front error (0, $\lambda /{4}$, and $\lambda /{2}$) for ${{T}_B} = {240}$ (dashed lines) and 300 K (solid lines). Zero wave front error curves are identical to those in Fig. 7. $\Delta {T} = {0}.{1}\;{\rm K}$.

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5. DARK CURRENT DENSITY

Dark current is the flow of thermally generated electrons. Usually the manufacturer provides the dark current density, ${{J}_{D}}$. For 100% fill factor arrays, the number of dark current electrons is

$${n_{\rm DARK}} = {J_D}{d^2}\frac{{{t_{\rm INT}}}}{q},$$
where ${q}$ has the usual meaning of electronic charge, and the dark current flows during an entire frame integration time, ${t_{\rm INT}}$. The number of generated background photoelectrons is
$${n_{\rm PE}} \approx \int_0^\infty \frac{{\eta ( \lambda ){M_q}( {\lambda ,{T_B}} ){d^2}{t_{\rm INT}}}}{{4{F^2}}}{\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda .$$
Due to the discrete nature of electrons, electron shot noise appears to follow Poisson statistics and the variance is equal to the mean:
$$\langle n_{\rm SHOT}^2\rangle = \langle n_{\rm PE}^2 \rangle+ \langle n_{\rm DARK}^2\rangle = {n_{\rm PE}} + {n_{\rm DARK}}.$$
Dark current shot noise can be ignored when ${n_{\rm DARK}} \lt {n_{\rm PE}}/10$. Equivalently,
$${J_D} \lt \frac{q}{{40{F^2}}} \int _0^\infty \eta ( \lambda ){M_q}( {\lambda ,{T_B}} ){\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda .$$
Note that ${{M}_q}(\lambda {{T}_B})$ and hence ${{n}_{\rm PE}}$ decrease with decreasing temperature. Therefore, dark current shot noise will have a bigger effect at low ambient temperatures. For the representative MWIR sensor with ${\tau _{\rm OPTICS}}(\lambda ) = {1}$, the integral is ${{3.03 \times 10}^{14}}\;{\rm photons}/{{\rm cm}^2}$ at ${{T}_B} = {240}\;{\rm K}$ and ${{6.43 \times 10}^{15}}\;{\rm photons}/{{\rm cm}^2}$ at ${{T}_B} = {300}\;{\rm K}$. Then the dark current density should be less than about ${{1210/F}^2}\;{\rm nA}/{{\rm cm}^2}$ (Fig. 10). Table 1 lists the maximum dark current density for ${{T}_B} = {240} $ and 300 K. Generally, dark current does not appreciably affect range performance when ${{T}_B} = {300}\;{\rm K}$.
 figure: Fig. 10.

Fig. 10. MWIR acquisition range as a function of ${F}\lambda /{d}$ and dark current density (0, 25, 50, 75, ${100}\;{{\rm nA/cm}^2}$). Note the vertical scale. The zero curve is identical to that in Fig. 9 (dashed curve for zero aberrations). Maximum range occurs when ${F}\lambda /{d}\;\sim\;{1}.{5}$. It is assumed that the dark current density is independent of the ambient temperature.

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Tables Icon

Table 1. Maximum Acceptable Current Density for a Typical MWIR Sensor $({\rm nA}/{{\rm cm}^2})$

6. DOWNSTREAM NOISE

Downstream noise is the root-sum-of-the-squares combination of read noise, analog-to-digital noise, and clock noise. Since downstream noise is entered as rms electrons, it should be less than 10% of the background shot noise:

$$\begin{split}&\langle {n_{\rm DOWNSTREAM}}\rangle\\&\quad \lt \frac{1}{{10}}\sqrt {\int _0^\infty \frac{{\eta ( \lambda ){M_q}( {\lambda ,{T_B}} ){d^2}{t_{\rm INT}}}}{{4{F^2}}}{\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda } .\end{split}$$
With ${\tau _{\rm OPTICS}}(\lambda ) = {1}$,
$$\begin{split}\langle {n_{\rm DOWNSTREAM}}\rangle \lt \frac{{d\sqrt {{t_{\rm INT}}} }}{{20F}}\sqrt {\int_0^\infty \eta ( \lambda ){M_q}( {\lambda ,{T_B}} ){\rm d}\lambda } .\end{split}$$
For the representative MWIR sensor, the integral is ${3}.{03} \times {{10}^{18}}\;{\rm photons}/{{\rm m}^2}$ at ${{T}_B} = {240}\;{\rm K}$ and ${6}.{43} \times {{10}^{19}}\;{\rm photons}/{{\rm m}^2}$ at ${{T}_B} = {300}\;{\rm K}$. Figure 11 illustrates detection range for ${{T}_B} = {240}\;{\rm K}$. Values for 30 and 60 Hz frame rates at 240 and 300 K are listed in Table 2.
 figure: Fig. 11.

Fig. 11. Relative range in the MWIR region as a function of downstream noise (0, 50, 100, 150, and 200 electrons rms). Note vertical axis scale. As noise increases, range decreases in the optics limited region. Maximum range occurs when ${F}\lambda /{d}\;\sim\;{1}.{5}$.

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Tables Icon

Table 2. Maximum Acceptable Downstream rms Electrons for a Typical MWIR Sensor

7. ATMOSPHERIC TRANSMISSION

In the IR region, there is little scattering, and the atmosphere mostly absorbs radiation. By Kirchhoff’s law, all absorbed radiation is reemitted. When detected, the reemission contributes to photoelectron shot noise. Over long distances, reemission dominates, producing significant shot noise, which is incorporated into the noise component of Eq. (1). Figure 12 illustrates detection range for $\Delta {T} = {0}.{1}\;{\rm K}$ and 1 m target. Targets with $\Delta {T} = {1}\;{\rm K}$ provide similar results. While MODTRAN can be used for specific spectral cases, we use Beer’s law to generalize. For targets that have spectral features embedded in spectrally varying atmosphere, the results may differ, but Beer’s law provides good generalization for most ground targets of interest.

 figure: Fig. 12.

Fig. 12. MWIR acquisition range as a function of ${F}\lambda /{d}$ and atmospheric transmission (Beer’s law) $\Delta {T} = {0}.{1}\;{\rm K}$ and ${W} = {1}\;{\rm m}$. Solid curves represent ${{T}_B} = {300}\;{\rm K}$, and dashed represent ${{T}_B} = {240}\;{\rm K}$. The $\tau = 1/{\rm km}$ curves are identical to those in Fig. 9. On the average, maximum range occurs when ${F}\lambda /{d} \sim {1}.{5}$.

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8. NOTCH FILTER

In Section 6, the atmospheric transmission was modeled by Beer’s law. While Beer’s law provides generalized results, MODTRAN provides a unique spectral transmission for each scenario. The results can be quite different for each of MODTRAN’s standard environment models (e.g., mid-latitude summer, mid-latitude winter, 1962 U.S. standard, etc.) and aerosol models (e.g., rural, maritime, etc.). The transmission at the ${{\rm CO}_2}$ absorption band (4.2 µm) is essentially zero, and only reemission exists. Therefore, it is worthwhile to place a blocking filter (aka a notch filter or twin-peak filter) at this wavelength. The advantage of the blocking filter becomes obvious when detecting target features that have a low $\Delta {T}$ (Fig. 13).

 figure: Fig. 13.

Fig. 13. MWIR acquisition range as a function of ${F}\lambda /{d}$ and mid-latitude summer with rural visibility (5 km). $\Delta {T} = {0}.{1}\;{\rm K}$ and ${W} = {1}\;{\rm m}$. Beer’s law (Fig. 12) acquisition ranges cannot be directly compared with MODTRAN results. When considering the ${{\rm CO}_2}$ absorption band, a notch filter will always provide better range performance.

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9. VIEWING DISTANCE

Reducing the visual angle defined by the display element size and observer viewing distance minimizes perceived noise. There is always an optimum [13] viewing distance (Fig. 14). Most observers will move closer to the display to discern detail but rarely move away from the display to reduce perceived noise. As the viewing distance increases, ${{\rm CTF}_{\rm NAKED\,EYE}}$ increases relative to ${{\rm MTF}_{\rm SYS}}$ and thereby reduces the perceived noise. Maximum range becomes a function of the noise level and visual angle. The noise level depends upon the sensor and atmospheric transmission, which is a function of range.

 figure: Fig. 14.

Fig. 14. Acquisition range as a function of ${F}\lambda /{d}$ and viewing distance. The 30 cm viewing distance curve is identical to that in Fig. 12 for $\tau = {0.8} / {\rm km}$. While the viewing distance affects acquisition range, it may not be possible for the observer to alter this distance (e.g., pilot strapped in a seat). Changing the display is an alternative.

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10. NEDT AND SNR

The noise equivalent differential temperature (NEDT) is a historical noise metric, and early target acquisition models were based upon the minimum discernible signal-to-noise ratio (SNR). Traditionally, SNR is expressed by

$${{\rm SNR}_{\rm BLIP}} = \frac{{{\tau ^R}\Delta T}}{{{{\rm NEDT}_{\rm BLIP}}}},$$
where the target $\Delta {T}$ is modified by the atmospheric transmittance. Only background shot noise (background limited performance or BLIP) was considered in the original NEDT expression. NVIPM includes all thermal noise sources as part of its predicted NEDT (labeled as ${{\rm NEDT}_{\rm NVIPM}}$, not discussed here). These include dark current shot noise, atmospheric thermal emission shot noise, optical thermal emission shot noise, and downstream noise. The incorporation of noise into Eq. (1) is complex, and its effects are illustrated in the numerous figures in this paper. As indicated in the last section, perceived noise depends upon the viewing distance.

11. FINITE WELL CAPACITY

With the infinite well discussed in the previous sections, the number of photoelectrons was limited by the frame time of 33.3 ms (30 Hz operation). Generally, the desired well capacity is twice that of the background photoelectrons. This allows detection of targets that are either warmer or cooler than the ${{T}_B}$. Equation (16) becomes

$$\begin{split}\frac{{{n_{\rm WELL}}}}{2} \approx \int_0^\infty \frac{{\eta ( \lambda ){M_q}( {\lambda ,{T_B}} ){t_{\rm INT}}}}{{4{{\left( {\frac{{F\lambda }}{d}} \right)}^2}}}{\lambda ^2}{\tau _{\rm OPTICS}}( \lambda ){\rm d}\lambda .\end{split}$$
When ${{t}_{\rm INT}} \lt {{t}_{\rm FRAME}}$, ${{t}_{\rm INT}}$ is inversely proportional to ${{M}_q}(\lambda ,{{T}_B})$ to maintain the 50% well fill. Figure 15 illustrates typical integration times as a function of ${F}\lambda /{d}$ for a 300 K background. As the background temperature decreases, there are fewer photoelectrons and the integration time increases reaching ${{t}_{\rm FRAME}}$ at lower ${F}\lambda /{d}$ values. The integration time primarily affects the noise term in Eq. (1). Recall that low $\Delta {T}$ targets cause the sensor gain to increase, thereby increasing the noise factor term (Fig. 16).
 figure: Fig. 15.

Fig. 15. Integration time as a function of ${F}\lambda /{d}$ and charge well capacity. The curves are a function of background temperature only.

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 figure: Fig. 16.

Fig. 16. Detection range (km) as a function of ${F}\lambda /{d}$ and charge well capacity (${1}.{0}\;{{\rm Me}^ - }$, ${3}.{0}\;{{\rm Me}^ - }$, and ${10}.{0}\;{{\rm Me}^ - }$). Transparent atmosphere.

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For $\Delta {T} \gt {1}$, detection ranges are not significantly affected by the well capacity. On the other hand, image processing algorithms perform better when more photoelectrons are available. While Eq. (22) appears complex, an estimate of desired well capacity is

$$\!\!\!{n_{\rm WELL}} \approx 2\frac{{\eta ( {{\lambda _O}} ){M_q}( {{\lambda _O},{T_B}} ){t_{\rm INT}}}}{{4{{\left( {\frac{{F{\lambda _O}}}{d}} \right)}^2}}}\lambda _O^2{\tau _{\rm OPTICS}}( {{\lambda _O}} ){\Delta }\lambda .\!$$
Let $\eta ({\lambda _O}) = {0}.{8}$ (band average), ${\tau _{\rm OPTICS}}({\lambda _O}) = {0}.{8}$ (band average), ${\lambda _{O\:}} = {4}.{46}\;\unicode{x00B5} {\rm m}$ (effective wavelength), $\Delta \lambda = {1}.{4}\;\unicode{x00B5} {\rm m}$, and ${{t}_{\rm INT}} = {33}.{3}\;{\rm ms}$ (30 Hz). Figure 17 illustrates the estimated desired well capacity as a function of background temperature for 30 Hz frame rate and ${F}\lambda /{d} = {1}$. For any other frame rate, multiply by 30 and divide by the new frame rate. For other ${F}\lambda /{d}$ values, simply divide by ${({F}\lambda /{d})^2}$. The actual number of photoelectrons is one-half of the desire well capacity. Figure 17 represents the charge well required when ${{t}_{\rm INT}} = {{t}_{\rm FRAME}}$.
 figure: Fig. 17.

Fig. 17. Rough estimate of the desired charge well capacity as a function of ${{T}_B}$ and 50% well fill. Equation (22) should be reevaluated for a specific sensor design.

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12. DISCUSSION

${F}\lambda /{d}$ links focal length, detector size, aperture diameter, and wavelength to acquisition range. As previously stated, the next-generation thermal imagers will have smaller detectors, larger arrays, and better sensitivity. Previous studies suggested an initial design point for LWIR sensors is ${F}\lambda /{d} \approx {2}.{0}$. For MWIR sensors that are photon starved, when detecting low contrast targets ($\Delta {T} \sim {0}.{1}\;{\rm K}$), sensor gain is used to achieve a desired displayed contrast. This also increases the perceived noise, and that reduces range performance. When considering noise aberrations and atmospheric transmission, the optimum ${F}\lambda /{d}$ for the photon-starved MWIR band appears to be on average, about 1.5. Any combination of f-number and detector size that provides 1.5 is acceptable (Fig. 18).

 figure: Fig. 18.

Fig. 18. Detector size as a function of f-number for ${F}\lambda /{d} = {1}.{5}$ and $\lambda = {4}.{46}\;\unicode{x00B5} {\rm m}$.

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The background temperatures (240–350 K) in this paper are consistent with ground-to-ground scenarios. The acquisition ranges presented here are unique to the values selected (e.g., spectral quantum efficiency, atmospheric spectral transmission) but the shapes of the curves and conclusions are general.

All figures are functions of ${F}\lambda /{d}$ with the aperture diameter fixed at 50 mm and the focal length varied. Changing the aperture size, D, to, say, 150 mm changes ${F}\lambda /{d}$, and the range increases linearly [Eq. (10)]. Normalizing the range (dividing by D) provides identical curves. Simply put, the curves presented here are generic to all MWIR sensors. Likewise, according to Eq. (4), as the target size increases, the range increases linearly, suggesting that selection of a 1 m target is representative. The idealized sensor presented here considers monochromatic diffraction limited optics, 100% fill factor array (detector dimension = pitch), no crosstalk [15], and no other MTF degrading issues.

The HVS affords tremendous temporal and spatial integration. This integration is included in NVIPM. As a contrast model, neither SNR nor NEDT is used to predict acquisition range. Because of the HVS spatial frequency dependence, perceived noise can be reduced by increasing the viewing distance. However, it is rare that an observer will change his/her viewing distance.

When ${F}\lambda /{d} \gt {1}.{5}$, the sensor is operating in the optics limited region. Here, range is sensitive to any optical degrading MTF. Even small changes such as $\lambda /{4}$ optical wave front error affect acquisition range. This puts additional burden on the optical designer. Spatial noise (also called three-dimensional or 3D noise) was not considered here and was assumed to be small compared to temporal noise.

Considering all the figures and analyses in this paper, it appears that ${F}\lambda /{d}\;\sim\;{1}.{5}$ is optimum for a MWIR sensor when viewed by a human observer. This is a general observation that applies to MWIR systems in a wide variety of conditions. To minimize size, weight, and cost (SWAP), a detector pitch of 3.6 µm is appropriate for an ${F}/{1}.{2}$ system ($\lambda = {4}.{46}\;\unicode{x00B5} {\rm m}$), and a detector pitch of 4.2 µm is appropriate for an ${F}/{1}.{4}$ system. Larger pitches (future 8 µm or current 12 µm) require larger chip size resulting in more weight and power consumption.

With a small detector pitch, it is possible to create arrays with ${5}\;{\rm K} \times {5}\;{\rm K}$ detectors. This poses a challenge due to the extremely high data rates required. One solution is a smart readout integrated circuit (ROIC) design where the full frame is not read out but regions of interest can be read out at conventional rates. Detector outputs can be combined for search and then addressed individually for target identification. There are clever ROIC approaches that essentially trade the ROIC functionality for optical switching.

Optical switching is when lenses are moved into the optical system to create a different magnification. An example of this is when an afocal system is moved into the optical train to convert a wide-field-of-view sensor to a narrow field of view. An afocal of ${5\times }$ can narrow the field of view by a factor of five. Optical switching is common when a smaller format detector array (e.g., ${640} \times {480}$) is used in a wide field of view, and then an optical switch causes the detector array to image a narrower field of view. The wide field of view is used for searching, and the narrow field of view is used for target identification and object interrogation. Optical switching is time consuming, where the switching can take a large fraction of a second to execute, so searching and target identification are a long series of wide-field searching and narrow-field interrogation. Once a potential target is found in the wide field, the imager has to be centered on the target (which also takes time) before the optical switch is invoked to provide the narrow-field interrogation. Small formats require optical switches that are expensive and time consuming. Large format, small pitch focal planes have the potential to replace optical switches with electronic magnification.

There are a number of ROIC programs currently underway that are looking at advanced ROIC functionality that can achieve the necessary technology improvements to realize these suggested advanced MWIR systems. With the large formats, optical switching can be accomplished by using the entire array for the wide field of view and decimation/interpolation in the ROIC. For a narrow field of view, a subsection of the ROIC can be displayed anywhere on the focal plane so that optical centering is not required, and interpolation and electronic zoom can provide the desired magnification. There are many ROIC functions such as time-delay-integration, electronics stabilization, and super-resolution that can be integrated with the ROIC or off-board interactions in concert with the ROIC. These ROIC enhancements continue to be the subject of many development programs.

As a final comment, NVIPM was released in May 2013. A major improvement over previous models is its loop capability (aka batch mode operation). Depending upon on the number of variables and loops, a single push of the run button exercises NVIPM up to several thousand times. While not counted, it is estimated that NVIPM was run well over 100,000 times to create the figures in this paper.

13. CONCLUSION

The ${F}\lambda /{d}$ parameter is generic to all imaging systems. In a photon-rich environment (typical of the LWIR band), it uniquely predicts acquisition range for low noise systems; ${F}\lambda /{d} \approx {2}$ provides optimum range performance. For the MWIR band (sometimes referred to as photon starved) ${F}\lambda /{d} \approx {1}.{5}$ is appropriate over a wide variety of conditions. This conclusion supports focal plane development efforts for smaller detectors. With a minimum practical f-number of 1.2 to 1.4, MWIR detector pitch ranges from 3.6–4.2 µm, respectively. However, as suggested in this paper, current and near future detector sizes are still larger than they need to be for a high-performance, compact, low-cost system. Cooled detector selection is increasing as detector operating temperature continues to rise (aka HOT) [1618]. Rapid advances in size reduction and in HOT technology will provide for extremely high-performance systems in the near future.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. J. N. Vizgaitis, “Dual f/number optics for third generation FLIR systems,” Proc. SPIE 5783, 875–886 (2005). [CrossRef]  

2. J. Vizgaitis, “Third generation infrared optics,” Proc. SPIE 6940, 69400S (2008). [CrossRef]  

3. D. A. Reago, S. B. Horn, J. Campbell Jr., and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999). [CrossRef]  

4. P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006). [CrossRef]  

5. A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006). [CrossRef]  

6. R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017). [CrossRef]  

7. G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012). [CrossRef]  

8. R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019). [CrossRef]  

9. R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019). [CrossRef]  

10. R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012). [CrossRef]  

11. “High Operating Temperature and Multi-Band Focal Plane Arrays Manufacturing Technology,” https://www.armymantech.com/SucessStories/Hot_MBFPA.pdf.

12. G. C. Holst, “Imaging system performance based upon Fλ/d,” Opt. Eng. 46, 103204 (2007). [CrossRef]  

13. G. C. Holst, “Optimum viewing distance for target acquisition,” Proc. SPIE 9452, 94520K (2015). [CrossRef]  

14. R. D. Fiete, “Image quality and λFN/ρ for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999). [CrossRef]  

15. T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017). [CrossRef]  

16. A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018). [CrossRef]  

17. M. Z. Tidrow and D. A. Reago Jr., “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017). [CrossRef]  

18. E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019). [CrossRef]  

References

  • View by:

  1. J. N. Vizgaitis, “Dual f/number optics for third generation FLIR systems,” Proc. SPIE 5783, 875–886 (2005).
    [Crossref]
  2. J. Vizgaitis, “Third generation infrared optics,” Proc. SPIE 6940, 69400S (2008).
    [Crossref]
  3. D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
    [Crossref]
  4. P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006).
    [Crossref]
  5. A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
    [Crossref]
  6. R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017).
    [Crossref]
  7. G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012).
    [Crossref]
  8. R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
    [Crossref]
  9. R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
    [Crossref]
  10. R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
    [Crossref]
  11. “High Operating Temperature and Multi-Band Focal Plane Arrays Manufacturing Technology,” https://www.armymantech.com/SucessStories/Hot_MBFPA.pdf .
  12. G. C. Holst, “Imaging system performance based upon Fλ/d,” Opt. Eng. 46, 103204 (2007).
    [Crossref]
  13. G. C. Holst, “Optimum viewing distance for target acquisition,” Proc. SPIE 9452, 94520K (2015).
    [Crossref]
  14. R. D. Fiete, “Image quality and λFN/ρ for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
    [Crossref]
  15. T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017).
    [Crossref]
  16. A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018).
    [Crossref]
  17. M. Z. Tidrow and D. A. Reago, “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017).
    [Crossref]
  18. E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
    [Crossref]

2019 (3)

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
[Crossref]

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

2018 (1)

A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018).
[Crossref]

2017 (3)

M. Z. Tidrow and D. A. Reago, “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017).
[Crossref]

T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017).
[Crossref]

R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017).
[Crossref]

2015 (1)

G. C. Holst, “Optimum viewing distance for target acquisition,” Proc. SPIE 9452, 94520K (2015).
[Crossref]

2012 (2)

G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

2008 (1)

J. Vizgaitis, “Third generation infrared optics,” Proc. SPIE 6940, 69400S (2008).
[Crossref]

2007 (1)

G. C. Holst, “Imaging system performance based upon Fλ/d,” Opt. Eng. 46, 103204 (2007).
[Crossref]

2006 (2)

P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006).
[Crossref]

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

2005 (1)

J. N. Vizgaitis, “Dual f/number optics for third generation FLIR systems,” Proc. SPIE 5783, 875–886 (2005).
[Crossref]

1999 (2)

R. D. Fiete, “Image quality and λFN/ρ for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[Crossref]

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

Campbell, J.

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

Corbin, T.

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Driggers, R.

R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
[Crossref]

R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017).
[Crossref]

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Driggers, R. G.

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006).
[Crossref]

Fanning, J. D.

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

Fiete, R. D.

R. D. Fiete, “Image quality and λFN/ρ for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[Crossref]

Goss, T. M.

T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017).
[Crossref]

Hodgkin, A.

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Holst, G. C.

G. C. Holst, “Optimum viewing distance for target acquisition,” Proc. SPIE 9452, 94520K (2015).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012).
[Crossref]

G. C. Holst, “Imaging system performance based upon Fλ/d,” Opt. Eng. 46, 103204 (2007).
[Crossref]

Horn, S. B.

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

Jenkins, G. D.

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

Kowalewski, B.

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Littlejohn, D.

R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
[Crossref]

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

Logan, J. V.

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

Maestas, D.

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

Morath, C. P.

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

Reago, D. A.

M. Z. Tidrow and D. A. Reago, “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017).
[Crossref]

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

Reynolds, J. P.

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

Richardson, P.

P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006).
[Crossref]

Rivera-Ortiz, C. J.

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

Scholten, M. J.

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

Short, R.

R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
[Crossref]

Short, R. E.

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

Steenbergen, E. H.

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

Teaney, B. P.

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Tidrow, M.

A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018).
[Crossref]

Tidrow, M. Z.

M. Z. Tidrow and D. A. Reago, “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017).
[Crossref]

Tomkinson, D.

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

Vizgaitis, J.

J. Vizgaitis, “Third generation infrared optics,” Proc. SPIE 6940, 69400S (2008).
[Crossref]

Vizgaitis, J. N.

J. N. Vizgaitis, “Dual f/number optics for third generation FLIR systems,” Proc. SPIE 5783, 875–886 (2005).
[Crossref]

Vollmerhausen, R.

R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017).
[Crossref]

Vollmerhausen, R. H.

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

Willers, C. J.

T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017).
[Crossref]

Williams, A.

A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018).
[Crossref]

Opt. Eng. (5)

G. C. Holst and R. G. Driggers, “Small detectors in infrared system design,” Opt. Eng. 51, 096401 (2012).
[Crossref]

R. E. Short, D. Littlejohn, M. J. Scholten, C. J. Rivera-Ortiz, R. H. Vollmerhausen, and R. G. Driggers, “Holistic approach to high-performance long-wave infrared system design,” Opt. Eng. 58, 023113 (2019).
[Crossref]

R. G. Driggers, R. H. Vollmerhausen, J. P. Reynolds, J. D. Fanning, and G. C. Holst, “Infrared detector size: how low should you go?” Opt. Eng. 51, 063202 (2012).
[Crossref]

R. D. Fiete, “Image quality and λFN/ρ for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[Crossref]

G. C. Holst, “Imaging system performance based upon Fλ/d,” Opt. Eng. 46, 103204 (2007).
[Crossref]

Proc. SPIE (12)

G. C. Holst, “Optimum viewing distance for target acquisition,” Proc. SPIE 9452, 94520K (2015).
[Crossref]

T. M. Goss and C. J. Willers, “Small pixel cross-talk MTF and its impact on MWIR sensor performance,” Proc. SPIE 10178, 101780L (2017).
[Crossref]

A. Williams and M. Tidrow, “III-V infrared focal plane array development in US (Conference Presentation),” Proc. SPIE 10624, 106240P (2018).
[Crossref]

M. Z. Tidrow and D. A. Reago, “VISTA video and overview (Conference Presentation),” Proc. SPIE 10177, 101770M (2017).
[Crossref]

E. H. Steenbergen, C. P. Morath, D. Maestas, G. D. Jenkins, and J. V. Logan, “Comparing II-VI and III-V infrared detectors for space applications,” Proc. SPIE 11002, 110021B (2019).
[Crossref]

R. Short, D. Littlejohn, and R. Driggers, “An update on PWP enhancement for LWIR target acquisition sensors,” Proc. SPIE 11001, 110010M (2019).
[Crossref]

J. N. Vizgaitis, “Dual f/number optics for third generation FLIR systems,” Proc. SPIE 5783, 875–886 (2005).
[Crossref]

J. Vizgaitis, “Third generation infrared optics,” Proc. SPIE 6940, 69400S (2008).
[Crossref]

D. A. Reago, S. B. Horn, J. Campbell, and R. H. Vollmerhausen, “Third-generation imaging sensor system concepts,” Proc. SPIE 3701, 108–117 (1999).
[Crossref]

P. Richardson and R. G. Driggers, “Atmospheric turbulence effects on 3rd generation FLIR performance,” Proc. SPIE 6207, 620706 (2006).
[Crossref]

A. Hodgkin, B. Kowalewski, D. Tomkinson, B. P. Teaney, T. Corbin, and R. Driggers, “Modeling of IR sensor performance in cold weather,” Proc. SPIE 6207, 620708 (2006).
[Crossref]

R. Vollmerhausen and R. Driggers, “Extending the range performance of diffraction limited imagers,” Proc. SPIE 10178, 101780F (2017).
[Crossref]

Other (1)

“High Operating Temperature and Multi-Band Focal Plane Arrays Manufacturing Technology,” https://www.armymantech.com/SucessStories/Hot_MBFPA.pdf .

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Figures (18)

Fig. 1.
Fig. 1. Scanning microscope image of 12 µm pitch MCT MWIR detectors (Courtesy of J. Devitt, Raytheon).
Fig. 2.
Fig. 2. Screen shot of NVIPM contrast illustrating a low noise system. The TTP is the area between the system CTF and the displayed contrast (line added). This area is also called excess contrast. As ${{\rm MTF}_{\rm SYS}}$ decreases, system ${{\rm CTF}_{\rm SYS}}$ increases and range decreases. This MWIR sensor is MTF limited. $\Delta {T} = {1}.{0}\;{\rm K}$.
Fig. 3.
Fig. 3. Screen shot of NVIPM contrast illustrating effects of noise and low target contrast. This MWIR sensor is noise limited [noise term in Eq. (1) dominates]. $\Delta {T} = {0}.{1}\;{\rm K}$.
Fig. 4.
Fig. 4. NVIPM predicted relative range as a function of ${F}\lambda /{d}$. The focal length was varied to produce a variable ${F}\lambda /{d}$. With a transparent atmosphere, only background shot noise exists. Background temperature is 300 K. For low $\Delta {T}$ targets, the sensor increases gain to reach the desired displayed contrast. As the gain increases, the displayed noise increases and response moves from eye limited to noise limited operation near ${F}\lambda /{d} \approx {1}.{5}$.
Fig. 5.
Fig. 5. Detector sizes for a low noise MWIR sensor and transparent atmosphere. Background temperature is 300 K. A different f-number and/or wavelength provide(s) different ${F}\lambda /{d}$ values for each detector size.
Fig. 6.
Fig. 6. Detection range as a function of ${F}\lambda /{d}$ for background temperatures of 240, 260, 280, 300, 320, and 340 K. 1 m target with $\Delta {T} = {1}.{0}\;{\rm K}$ and transparent atmosphere. The 300 K curve is identical to that in Fig. 4.
Fig. 7.
Fig. 7. Detection range as a function of ${F}\lambda /{d}$ for background temperatures of 240, 260, 280, 300, 320, and 340 K. 1 m target with $\Delta {T} = {0}.{1}\;{\rm K}$ and transparent atmosphere. The 300 K curve is identical to that in Fig. 4.
Fig. 8.
Fig. 8. Acquisition range as a function of ${F}\lambda /{d}$ and peak-to-peak wave front error (0, $\lambda /{4}$, and $\lambda /{2}$) for ${{T}_B} = {240}$ (dashed lines) and 300 K (solid lines). Zero wave front error curve at 300 K is identical to those in Fig. 6. $\Delta {T} = {1}.{0}\;{\rm K}$. As $\Delta {T}$ increases, aberrations have a smaller effect.
Fig. 9.
Fig. 9. Acquisition range as a function of ${F}\lambda /{d}$ and peak-to-peak wave front error (0, $\lambda /{4}$, and $\lambda /{2}$) for ${{T}_B} = {240}$ (dashed lines) and 300 K (solid lines). Zero wave front error curves are identical to those in Fig. 7. $\Delta {T} = {0}.{1}\;{\rm K}$.
Fig. 10.
Fig. 10. MWIR acquisition range as a function of ${F}\lambda /{d}$ and dark current density (0, 25, 50, 75, ${100}\;{{\rm nA/cm}^2}$). Note the vertical scale. The zero curve is identical to that in Fig. 9 (dashed curve for zero aberrations). Maximum range occurs when ${F}\lambda /{d}\;\sim\;{1}.{5}$. It is assumed that the dark current density is independent of the ambient temperature.
Fig. 11.
Fig. 11. Relative range in the MWIR region as a function of downstream noise (0, 50, 100, 150, and 200 electrons rms). Note vertical axis scale. As noise increases, range decreases in the optics limited region. Maximum range occurs when ${F}\lambda /{d}\;\sim\;{1}.{5}$.
Fig. 12.
Fig. 12. MWIR acquisition range as a function of ${F}\lambda /{d}$ and atmospheric transmission (Beer’s law) $\Delta {T} = {0}.{1}\;{\rm K}$ and ${W} = {1}\;{\rm m}$. Solid curves represent ${{T}_B} = {300}\;{\rm K}$, and dashed represent ${{T}_B} = {240}\;{\rm K}$. The $\tau = 1/{\rm km}$ curves are identical to those in Fig. 9. On the average, maximum range occurs when ${F}\lambda /{d} \sim {1}.{5}$.
Fig. 13.
Fig. 13. MWIR acquisition range as a function of ${F}\lambda /{d}$ and mid-latitude summer with rural visibility (5 km). $\Delta {T} = {0}.{1}\;{\rm K}$ and ${W} = {1}\;{\rm m}$. Beer’s law (Fig. 12) acquisition ranges cannot be directly compared with MODTRAN results. When considering the ${{\rm CO}_2}$ absorption band, a notch filter will always provide better range performance.
Fig. 14.
Fig. 14. Acquisition range as a function of ${F}\lambda /{d}$ and viewing distance. The 30 cm viewing distance curve is identical to that in Fig. 12 for $\tau = {0.8} / {\rm km}$. While the viewing distance affects acquisition range, it may not be possible for the observer to alter this distance (e.g., pilot strapped in a seat). Changing the display is an alternative.
Fig. 15.
Fig. 15. Integration time as a function of ${F}\lambda /{d}$ and charge well capacity. The curves are a function of background temperature only.
Fig. 16.
Fig. 16. Detection range (km) as a function of ${F}\lambda /{d}$ and charge well capacity (${1}.{0}\;{{\rm Me}^ - }$, ${3}.{0}\;{{\rm Me}^ - }$, and ${10}.{0}\;{{\rm Me}^ - }$). Transparent atmosphere.
Fig. 17.
Fig. 17. Rough estimate of the desired charge well capacity as a function of ${{T}_B}$ and 50% well fill. Equation (22) should be reevaluated for a specific sensor design.
Fig. 18.
Fig. 18. Detector size as a function of f-number for ${F}\lambda /{d} = {1}.{5}$ and $\lambda = {4}.{46}\;\unicode{x00B5} {\rm m}$.

Tables (2)

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Table 1. Maximum Acceptable Current Density for a Typical MWIR Sensor ( n A / c m 2 )

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Table 2. Maximum Acceptable Downstream rms Electrons for a Typical MWIR Sensor

Equations (23)

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C T F S Y S = ( C T F N A K E D E Y E M T F S Y S ) 2 + E Y E T E R M ( C T F N A K E D E Y E M T F S Y S ) 2 ( α σ P L A V E ) 2 N O I S E T E R M .
T T P u = C D I S P L A Y E D C T F S Y S ( u ) d u , T T P v = C D I S P L A Y E D C T F S Y S ( v ) d v .
V R E S O L V E D ( R ) = W R T T P u T T P v .
R = W V 50 T T P u T T P v = W V 50 T T P ,
M T F D I F F R A C T I O N ( u ) = 2 π [ cos 1 ( u ) ( u ) 1 ( u ) 2 ] u 1 ,
M T F D E T E C T O R ( u ) = s i n c ( π d F λ u ) .
R W V 50 C D I S P L A Y M T F S Y S ( u ) C T F N A K E D E Y E ( u ) × N F ( C T A R G E T ) d u ,
T T P [ a 6 ( F λ d ) 6 + a 5 ( F λ d ) 5 + a 4 ( F λ d ) 4 + a 3 ( F λ d ) 3 + a 2 ( F λ d ) 2 + a 1 ( F λ d ) ] D λ ,
R k D E T E C T O R W V 50 f d = k D E T E C T O R W V 50 1 I F O V .
R k O P T I C S W V 50 D λ ,
Δ n P E 0 η ( λ ) [ M q ( λ , T T ) M q ( λ , T B ) ] A D t I N T 4 F 2 × τ O P T I C S ( λ ) d λ ,
Δ n P E λ 1 λ 2 η ( λ ) A D t I N T 4 F 2 M q ( λ , T B ) Δ T T τ O P T I C S ( λ ) d λ .
M T F O P T I C S M T F D I F F R A C T I O N M T F A B E R R A T I O N .
M T F A B E R R A T I O N ( u ) 1 ( W R M S 0.18 ) 2 [ 1 4 ( u 1 2 ) 2 ] ,
n D A R K = J D d 2 t I N T q ,
n P E 0 η ( λ ) M q ( λ , T B ) d 2 t I N T 4 F 2 τ O P T I C S ( λ ) d λ .
n S H O T 2 = n P E 2 + n D A R K 2 = n P E + n D A R K .
J D < q 40 F 2 0 η ( λ ) M q ( λ , T B ) τ O P T I C S ( λ ) d λ .
n D O W N S T R E A M < 1 10 0 η ( λ ) M q ( λ , T B ) d 2 t I N T 4 F 2 τ O P T I C S ( λ ) d λ .
n D O W N S T R E A M < d t I N T 20 F 0 η ( λ ) M q ( λ , T B ) d λ .
S N R B L I P = τ R Δ T N E D T B L I P ,
n W E L L 2 0 η ( λ ) M q ( λ , T B ) t I N T 4 ( F λ d ) 2 λ 2 τ O P T I C S ( λ ) d λ .
n W E L L 2 η ( λ O ) M q ( λ O , T B ) t I N T 4 ( F λ O d ) 2 λ O 2 τ O P T I C S ( λ O ) Δ λ .

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