Modeling of the nonlinear response of the intrinsic HgCdTe photoconductor by a two-level rate equation with a finite number of carriers available for photoexcitation Rene´ S. Hansen
A description of the nonlinear response of a HgCdTe photoconductor on the incident optical radiation is derived from a two-level rate equation for the photoexcited carriers. The model accounts for a limited number of electrons available for photoexcitation, and the possibility of photostimulated relaxation of an already excited photoelectron. The detector response as well as the level of incident optical power where saturation occurs is directly related to the physical constants of the detector material. The saturation of the detector is the same phenomenon as is known for the saturable absorbers—the detector material becomes transparent when the incident optical power exceeds the saturation power. Even though the description given here is applied to the HgCdTe detector, the general model can be applied to other detector materials as well. The parameters for the model are fitted to a thermoelectrically cooled HgCdTe detector and afterwards employed to predict the detector noise performance in a heterodyne setup. Unlike for liquid-nitrogen-cooled detectors, shot-noise-limited detection cannot be obtained with use of only thermoelectrical cooling 共226 K兲 of the detector. However, the detector performance can be optimized to be able to perform Doppler measurements from aerosol backscatter by use of the model presented to optimize the applied optical power in the reference wave. © 2003 Optical Society of America OCIS codes: 040.5160, 040.5150, 040.2840, 040.3060, 040.3780, 280.3340.
1. Introduction
The mercury-cadmium-telluride, Hg1-xCdxTe, semiconductor is a very important material for the detection of infrared light in the mid- and long-wavelength range 共3–30 m兲. It is widely used in a range of applications, such as Fourier transform IR spectroscopy, pyrometers,1 and in heterodyne CO2 laser lidars and laser anemometers.2– 4 In almost all of these applications, a very low level of light is of concern, and the detector material is usually cooled to liquidnitrogen temperature, minimizing the noise and the dark current. In the heterodyne laser anemometer, the weak 共10⫺11 ⬃ 10⫺14 W兲 signal light, backscattered from the airborne aerosols, is mixed locally with
R. S. Hansen 共
[email protected]兲 is with the Department of Optics and Fluid Dynamics, Risø National Laboratory, Frederiksborgvej 399, DK-4000 Roskilde, Denmark. Received 18 February 2003; revised manuscript received 7 May 2003. 0003-6935兾03兾244819-08$15.00兾0 © 2003 Optical Society of America
a reference beam, extracted from the same laser as is used for illuminating the aerosols. The signal strength of the Doppler beat signal is proportional to the applied power of the reference beam. Unfortunately, the noise generated also increases with reference power. Quantum limited detection of the backscattered light is obtainable at relatively low power levels of the reference beam when liquidnitrogen cooling of the detector is used. Saturation of the detector is not of concern in many of the above pyrometer and spectrometer applications involving direct detection, and a linear model for the detector response can safely be applied.3,4 However, when the temperature of hot objects is measured or a room temperature or a TE-cooled 共226-K兲 detector in the heterodyne laser anemometer is employed, detector saturation becomes a limitation for the equipment.5 When the reference power in the heterodyne anemometer is increased in order to make the shot noise become the dominating noise of a TE-cooled detector, saturation of the detector will occur before shot-noiselimited detection can be obtained.5,6 The TE-cooled detectors are attractive because of the opportunity to 20 August 2003 兾 Vol. 42, No. 24 兾 APPLIED OPTICS
4819
Fig. 2. Photoconductor with dimensions lx, ly and thickness lz is illuminated with photons of energy ប, where is the angular frequency of the incident light. The bias voltage is applied to the electrodes at each end of the photoconducting material and the current i flows through the detector material. Fig. 1. Bias circuit for the detector. Owing to the photogeneration of free electrons, the resistance of the photoconductor, Zd, varies as a function of the incident optical power.
avoid the costly cryogenic cooling system. Detectors operating at “room temperature” or being TE cooled have been presented and discussed.2,4,7,8 These detectors are especially good candidates for low-altitude laser anemometry because of a backscattering of the laser light being high enough to circumvent the need for shot-noise-limited detection.9 To achieve the optimum signal to noise for the Doppler signal— equal to obtaining the best possible sensitivity for the instrument—the optimum operating conditions for the detector must be found. In the present case of a TE-cooled photoconductor, the optimum operating condition involves a power level of the reference beam that partly saturates the detector material. Therefore the full response curve of the detector must be known, which includes the effect of the full saturation of the detector. The nonlinear response of the photoconductor has been modeled by fitting a quadratic equation to the measured values for the detector response.10 In the present paper, a model for the full response of the photoconductor is developed from a rate equation for the number density of photoexcited charge carriers. The rate equation accounts for a limited number of carriers available for photoexcitation. A change of the carrier lifetime due to the illumination has been discussed in the literature.2,7,11 In the rate equation discussed here, the variable lifetime of the photocarriers is included as the possibility of photostimulated relaxation of the carriers back to the valence band. In Section 2, the results of the rate equation are discussed. Measurements on the detector, employed in the laser anemometer build at the department,9 are fitted with the above model in Section 3. Section 4 is devoted to the conclusions. 2. Two-Level Rate Equation for the Photoconductor
The bias circuit for the photoconductor consists of a bias voltage Vb and a bias resistor Rb as shown in Fig. 1. A load resistance can be included in the model as 4820
APPLIED OPTICS 兾 Vol. 42, No. 24 兾 20 August 2003
an extra conductance of the detector material. The output voltage is given by Vo ⫽
Vb , 1 ⫹ Rb Sd
(1)
where Sd ⫽ 1兾Zd is the conductance of the photoconductor. The conductance of the photoconductor is separated in two contributions: the photoconductance due to the photoexcitation of intrinsic carriers, Sdp, and the extrinsic conductance Sdo due to impurities and dislocations in the crystal. In the literature, the extrinsic conductance of the Hg1-xCdxTe crystal is found to be due to a low carrier concentration of p-type12,13 or of n-type11,12,14 dependent on crystal growth conditions, impurities, and annealing. The total conductance is written as Sd ⫽
1 ⫽ S dp ⫹ S do . Zd
(2)
The photoconductance is assumed only to be due to free electrons excited into the conduction band and to the corresponding holes they leave behind in the valence band. Because the number of electrons and of matching holes are equal, the photoconductance needs only to be given by the number density of photoexcited carriers into the conduction band, N2: S dp ⫽ q e
lx lz N2 , ly
(3)
where qe is the charge of an electron, the mobility is equal to the sum of the nobilities of the major current carrying charges 共holes, h, and兾or electrons, e兲. The dimensions and layout of the photoconducting material used in the current model are shown in Fig. 2. The photoconductor is sandwiched between two electrodes. Improved performance of the detector, owing to an extended lifetime of the photocarriers, is obtained if the weak n- or p-type photon-absorbing photoconductor is connected to the electrodes through highly n- or p-doped contact areas.2,7,8,14 In the discussion to follow, the highly doped end con-
where the rate at which electrons are photoexcited up to the conduction band, W12 is equal to the rate of photostimulated relaxation of the photocarriers back to the valence band W21. From the harmonic oscillator model, the excitation rates are given by17 W 12共t兲 ⫽ W 21共t兲 ⫽
␥ rad n 4 2 ប⌬ a Ac
P opt共t兲 3 2共 opt ⫺ a兲 1⫹ ⌬ a
冋
册
2
,
(5)
Fig. 3. Valence band and the conduction band with energies E1 and E2 and number densities for electrons N1 and N2, respectively. W12 and W21 are the rates of photoexcitation and photostimulated relaxation, respectively, and the quantities w12 and w21 are the rates of thermal excitation and relaxation, respectively.
tacts and the metallic electrodes are treated in common as the contact electrodes. The gap between the highest lying valence band and the lowest lying empty conduction band is found to be a direct band gap.15 The size of the energy gap Eg can be tailored by the stoichiometry factor x. For the present detector with sensitivity at the 10.59-m wavelength for the CO2 laser, x ⬇ 0.2 and Eg ⬇ 0.1 eV.15,16 The system under consideration is shown in Fig. 3. At time t the valence band with energy E1 contains N1共t兲 electrons ready to be photoexcited into the conduction band, which contains N2共t兲 freecurrent carrying electrons with energy E2. The density of states for the respective bands gives the widths of the respective bands. Especially the density of states in the valence band is very localized. The number densities N1共t兲 and N2共t兲 are counting only the photocarriers contributing to the photoconduction Sdp. These densities are dependent on the rate of photoexcitation and relaxation of electrons and of course dependent on the temperature of the material. The extrinsic conductance Sdo is solely due to acceptors and兾or donors close to the valence and conduction bands, respectively. Their contribution to the total number of current carriers is treated separately. The mechanisms for photoexciting the electrons into the conduction band and the relaxation mechanisms are outlined in Fig. 3. The number density of photoexcited current carrying electrons is found by adopting the rate equation description of the number of carriers in a two-level system exposed to a flux of photons.17 The rates at which the electrons are excited into the conduction band and relax back to the valence band are given by the rate equation for the number densities
where ␥rad is the radiative decay rate for the transition, the quantities ⌬a and a are the linewidth and the transition frequency of the two-level transition, respectively; ប is the Planck constant divided by 2; and the photoconductor with the area, A ⫽ lx ly, is illuminated with light having a total optical power, Popt共t兲, and with the wavelength and the corresponding angular frequency opt. The photodetector has the refractive index n, and c is the speed of light in vacuum. The factor is equal to 3 when the atoms in the detector material are fully aligned and the optical field is optimally polarized. In various other situations, values in the range 0 ⱕ ⱕ 3 are appropriate.17 For convenience, the optical excitation efficiency is defined as ⫽
␥ rad n 4 2 ប⌬ a Ac
3 2共 opt ⫺ a兲 1⫹ ⌬ a
冋
⫽ ⫺关W 12 ⫹ w 12兴 N 1共t兲 ⫹ 关W 21 ⫹ w 21兴 N 2共t兲,
(4)
,
(6)
with units in inverse joules, so W 12共t兲 ⫽ W 21共t兲 ⫽ P opt共t兲.
(7)
The quantities w12 and w21 are the thermal excitation and the relaxation rate of the electrons, respectively. Their ratio is determined by the Boltzmann ratio, w 12 ⫽ exp ⫺ 共E g兾k B T兲, w 21
(8)
where Eg ⫽ E2 ⫺ E1 is the energy gab of the photoconductor material, kB is the Boltzmann constant, and T is the temperature of the photoconductor material. As in Ref. 17, Eq. 共4兲 is rewritten by introducing the total number density of electrons available for photoexcitation, N ⫽ N1 ⫹ N2, and the population difference ⌬N ⫽ N1 ⫺ N2, such that N2 ⫽ 1兾2 共N ⫺ ⌬N兲, and N1 ⫽ 1兾2 共N ⫹ ⌬N兲, ⌬N共t兲 ⫺ ⌬N 0 d⌬N共t兲 , ⫽ ⫺2W 12共t兲⌬N共t兲 ⫺ dt T1 where
冋 册
c B ⫽ tanh dN 2 dN 1 ⫽⫺ dt dt
册
2
Eg , 2k B T
(9)
(10)
⌬No ⫽ cB N is the population difference occurring when the photoconductor is at thermal equilibrium with the surroundings, and T1 ⫽ 共w12 ⫹ w21兲⫺1 is the 20 August 2003 兾 Vol. 42, No. 24 兾 APPLIED OPTICS
4821
effective relaxation time of the electronic population difference. T1 also includes the effects from recombination processes at the material surfaces.1 As discussed in Ref. 17, Eq. 共9兲 expresses the use of the absorbed photons for either exciting a photocarrier to the conducting band or acting on an already excited carrier to perform a photostimulated relaxation. When half of the available carriers are photoexcited 共N2 ⫽ 1⁄2N兲, ⌬N becomes zero and, on average, the distribution of carriers will not change from N1 ⫽ N2, even for higher power of the incident illumination, i.e. the detector is saturated. This phenomenon is well known from saturable absorbers.17 The steadystate solution to Eq. 共9兲 is readily found with Popt ⫽ PDC, ⌬N 0 ⌬N ⫽ . 1 ⫹ 2T 1P DC
dratic term in PDC gives the quadratic equation employed in the literature: V o共P DC兲 ⫽ V b兵G o ⫺ G o2c B R b S d,max p t P DC ⫹ G o3c B R b S d,max
Go ⫽
冋
册
1 lx lz N, q e 2 ly
⌬N共t兲 ⫽
(12)
冋 册
(14)
(15)
Q共t兲dt dt ,
(19)
Q共t兲dt
1 . T1
(16)
APPLIED OPTICS 兾 Vol. 42, No. 24 兾 20 August 2003
(20)
The optical power incident on the detector, Popt is often consisting of a large constant 共DC兲 power overlaid by a tiny sinusoidal signal, as for example in heterodyne laser Doppler systems19 where
冋
2 冑P ref P sig cos共 D t兲 P ref ⫹ P sig
⫽ P DC关1 ⫹ M cos共 D t兲兴,
册
(21)
where Pref is the power of the 共internal兲 local oscillator, is the heterodyne mixing efficiency, Psig is the optical power, backscattered from the target 共i.e., from airborne aerosols兲, and D is the beat frequency between the local oscillator and the backscattered 共Doppler-shifted兲 light. The constant detector illumination and the modulation depth of the signal are PDC ⫽ Pref ⫹ Psig and M ⫽ 关2共Pref Psig兲1兾2兴兾关Pref ⫹ Psig兴, 0 ⱕ M ⱕ 1, 0 ⱕ ⱕ 1, respectively. In many situations, the backscattered signal is very weak, so M ⬍⬍ 1. Inserting the expression for the optical power, Eq. 共21兲 into Eq. 共19兲, the integral 兰Q共t兲dt becomes
兰
Q共t兲dt ⫽ 0 t ⫹
2P DCM sin共 D t兲, D
(22)
where T1 1 ⫽ 0 ⫽ 0 1 ⫹ p t P DC
and Eex is the ionization energy of the acceptors or donors in the material. Using Sdp,ss in Eq. 共1兲 for the voltage output and calculating the linear and qua4822
冋兰 册 冋兰 册
⌬N 0exp
P opt共t兲 ⫽ 共P ref ⫹ P sig兲 1 ⫹
where E ex c Bex ⫽ tanh , 2k B T
兰
Q共t兲 ⫽ 2W 12共t兲 ⫹
(13)
with units in inverse watts has been introduced. pt is the generation rate of current-carrying electrons per unit incident optical power. As discussed above, Eq. 共12兲 shows the saturation of the conductance of the detector material as the incident optical power is increased. Equation 共14兲 gives the increase of the detector efficiency with the carrier lifetime T1. A longer carrier lifetime improves the detector sensitivity; however, the detector will then saturate at a lower level of the incident optical power. The extrinsic conductance is assumed not to be due to any photon absorption processes, and the temperature dependency is thus expressed as S do ⫽ S do,max关1 ⫺ c Bex兴,
1 T1
exp
with units in inverse ohms, where the photosensitivity p t ⫽ 2T 1,
(18)
where
where PDC is the steady-state power of the incident light and the maximum obtainable conductance of the photoconductor is given by S d,max ⫽
1 . 1 ⫹ R b关S d,max共1 ⫺ c B兲 ⫹ S do兴
The full solution of Eq. 共9兲 is found from standard textbooks18
Solving for N2 and substituting in Eq. 共3兲 yields the steady-state conductance, S dp,ss ⫽ S d,max
(17)
where
(11)
cB 1⫺ , 1 ⫹ p t P DC
⫻ 关1 ⫹ R b共S d,max ⫹ S do兲兴 p t P DC2其,
(23)
is the time constant of redistributing the electronic distribution under influence of the incident optical radiation. The frequency 0 is the cutoff frequency for the detector response. Note that 0 decreases with increasing DC optical power. Equation 共19兲 is
further evaluated by assuming that the signal is small, i.e. 共2⌿PDCM兲兾D ⬍⬍ 1, and by using the Taylor expansion of ex to the first order. The electronic population difference is then found to be ⌬N ⫽
⌬N 0 1 ⫹ 2T 1P DC
冋
⫻ 1⫹
V d共t兲 ⯝
册
2MP DC cos共 D t ⫺ 兲 , 共w 02 ⫹ D2 兲 1兾2
再
冋
⫻ 1⫺
cB 1 ⫹ p t P DC
2P signal cos共 D t ⫺ 兲 共w 02 ⫹ D2 兲 1兾2
册冎
,
(25)
where the amplitude of the optical signal is Psignal ⫽ MPDC, the maximum obtainable conductance of the photoconductor is given by Eq. 共13兲, and the photosensitivity is given by Eq. 共14兲. Equations 共12兲, 共17兲, and 共25兲 are the main results of this paper and express the nonlinear response of the photoconductance due to the limited number of electrons available for photoexcitation, and the possibility of photostimulated recombination of the exited electrons. Sd,max is the maximum conductance that can be obtained at high incident optical power and兾or high detector temperature. The lowest conductance of the photoconductor with no incident optical radiation, but having a temperature above absolute zero, is Sd,min ⫽ Sd,max共1 ⫺ cB兲, which will become zero with decreasing temperature. The conductance is following the sinusoidal variation of the incident light delayed by the phase . The cutoff frequency for the detector response, 0, is determined by the relaxation time of the electronic population difference, T1, and the average optical power impinging on the detector, PDC. All the values appearing in Eq. 共25兲 adhere to physical values for the detector material in use, and the equations presented can thus be analyzed in order to optimize the parameters for the detector material during the process of fabrication. If the incident optical power is constant, the conductance of the photoconductor is found by letting Psignal ⫽ 0; Eq. 共25兲 then becomes equal to the steadystate solution, Eq. 共12兲. Equation 共12兲 is also valid if the optical field is a slowly varying function of time, i.e. for D ⬍⬍ 0. Substituting Eq. 共25兲 into Eq. 共1兲 will yield the voltage output of the photoconductor circuit as a function of time and applied optical power. Again, assuming a small signal and calculating the Taylor expansion of Eq. 共1兲 around the
Vb 1 ⫹ R b S d,ss
冦
⫻ 1⫺
(24)
where ⫽ arctan共D兾0兲 is the phase lag between the applied sinusoidal signal and the sinusoidal response of the electronic energy distribution in the detector material. Calculating the number density of electrons in the conduction band N2 and inserting in Eq. 共3兲 yields the photoconductance of the illuminated photoconductor S dp共t兲 ⫽ S d,max 1 ⫺
steady-state conductance of the photoconductor, Sd,ss ⫽ Sdp,ss ⫹ Sdo, will give an analytical expression of the amplitude of the voltage signal corresponding to the incident optical signal,
⫻
R b S d,max c B p t P signal 1 ⫹ R b S d,ss 共1 ⫹ p t P DC兲 2
1 D 1⫹ 0
冋 冉 冊册
2 1兾2
冧
cos共 D t ⫺ 兲 .
(26)
The nonlinear response of the detector is clearly seen from Eq. 共26兲. Increasing the power of the reference on the detector, Pref, will increase the signal power, Psignal. However, increasing the reference power above an optimum value severely decreases the amplitude of the voltage response from the detector. The decrease is seen to be of the fourth order in PDC. The detector responsitivity, Rd 共in volts per watt兲 that determines the amplitude of the detector output signal is readily seen to be Rd ⫽
V b R b S d,maxc B p t 共1 ⫹ R b S d,ss兲 2共1 ⫹ p t P DC兲 2 ⫻
1 D 1⫹ 0
冋 冉 冊册
2 1兾2
共V兾W兲.
(27)
Again, the saturation effects appear as a reduction of Rd. When knowing the physical parameters for the detector material, the largest amplitude of the signal is achieved for a reference power equal to P DC,optimum ⫽
冢
1 1 ⫺ cB 3p t
1 1 1⫹ R b S d,max
冣
.
(28)
The noise is not considered in this expression but will be discussed in Section 3. 3. Experiments A.
Determining the Detector Parameters
The quantities, pt, Sdp,max, Sdo,max, and Eex are found by fitting measurements of the detector responsitivity and the temperature dependency of the DC conductance to Eqs. 共12兲 and 共27兲. The detector under consideration is a three-stage TE-cooled HgCdTe detector from VIGO detectors, Poland. The operating temperature of the detector is 226 K. The area of the detector material is 100 ⫻ 100 m which, by an immersion lens, is turned into an effective area of 1 ⫻ 1 mm.8 The time constant T1 is 10 ns. The detectivity is D* ⫽ 1*109 cm公Hz兾W. During the measurements, the bias circuit is configured with Rb ⫽ 1 k⍀ and Vb ⫽ ⫺12 V. At room temperature 共T ⫽ 298 20 August 2003 兾 Vol. 42, No. 24 兾 APPLIED OPTICS
4823
Fig. 4. Responsitivity Rd for the detector is decreasing with the increase in incident optical power. The dots are the three measured values of Rd.
K兲 the measured detector resistance is Sd⫺1 ⫽ 45 ⍀, and the resistance at the operating temperature 共T ⫽ 226 K兲 Sd⫺1 ⫽ 77 ⍀. The detector responsitivity is measured in a Michelson-like Doppler heterodyne detection setup. The reference wave is established as one of the arms of the Michelson interferometer. A low reflectivity target is assembled by an antireflex-coated ZnSe window, which partly reflects the output beam from the second arm. Further reduction of the target reflection coefficient is obtained with a noncoated ZnSe window placed in front of the antireflex-coated ZnSe target. A telescope focuses the output beam on the target in order to obtain a plane wave front at the detector plane, matching the reference wave front. The target is mounted on a piezocontrolled translation stage to introduce a low-frequency Doppler shift, D ⬍⬍ 0, of the backreflected wave. The power of the backreflected wave, impinging on the detector, is Psig ⫽ 0.4 mW with a 3-W waveguide CO2 laser as the laser source. Values for the detector responsitivity are measured by measuring the amplitude of the detector signal at different power levels of the reference wave PDC. The measurements and the fitting of Eq. 共27兲 are shown in Fig. 4. As discussed above, the detector responsitivity Rd decreases when the incident optical power PDC is increased. The above fitting procedure yields the parameters: Eg ⫽ 0.10378 eV, pt ⫽ 367.109 W⫺1, and Sd,max ⫽ 0.00279845 ⍀⫺1. Figure 5 is the voltage across the detector calculated from Eq. 共1兲 with the above model and data for the detector inserted and fitted with Sdo,max ⫽ 0.08303 ⍀⫺1 and Eex ⫽ 0.0481 eV. In addition, Fig. 5 shows the three measured detector responsitivities, which is the slope of the PDC–Vo curve, at the three values of the incident optical power. B.
Noise and Heterodyne Detection
The nonlinearity of the detector response places a limit to the maximum intensity of the reference wave and thereby for the sensitivity obtainable in the heterodyne detection scheme by simply increasing the 4824
APPLIED OPTICS 兾 Vol. 42, No. 24 兾 20 August 2003
Fig. 5. Voltage Vo across the detector as a function of the incident optical power. The detector is seen to be saturating at incident powers above 10 mW. The three dashed curves are the measured detector responsitivities, Rd ⫽ dVo兾dPDC, at three levels of the incident optical power.
intensity of the reference wave. The best obtainable sensitivity is achieved when the optical power in the reference wave is high enough to have the detector shot noise dominating the other sources of noise. The noise voltage output of the detector and bias circuit shown in Fig. 1 is a sum of the Johnson noise from the bias detector, the detector generationrecombination noise, and the shot noise from the donor–acceptor dark current, N2 ⫽ R 2
冉
冊
4k b T a ⫹ 4Gq e V o S dp,ss ⫹ 2q e V o S do ⌬f, Rb (29)
where Ta is the ambient temperature, ⌬f is the bandwidth of the following amplifier, R is the impedance of the circuit, R⫽
Rb , R b S d,ss ⫹ 1
(30)
and G is the photoconductive gain given by G⫽
o o e V o ⫽ . tp l y2
(31)
o is given by Eq. 共23兲, tp is the time for a electron to pass the photoconductor, and e ⫽ 10 m2兾共Vs兲.1 The three contributions and the total noise voltage is plotted in Fig. 6 as a function of the incident optical power on the detector operating at T ⫽ 226 K. It is observed that the shot noise cannot dominate the other noise sources for the TE-cooled detector. Cooling the same detector material to liquid nitrogen temperature enables shot-noise-limited detection, as seen in Fig. 7. However, the shot noise as well as the signal level quickly drops at higher incident optical power owing to the detector nonlinearity.
Fig. 6. Three noise contributions and the total detector noise plotted as a function of incident optical power. The plot is for the TE-cooled detector.
Fig. 8. Detector noise equivalent power plotted as a function of incident optical power. The plot is given for the detector at room temperature, TE-cooled, and cooled further to liquid nitrogen temperature. The plots are for ⌬f ⫽ 1 Hz.
The detector saturation can also be seen as an increase of the detector noise equivalent power, P NEP ⫽
冑N2 Rd
,
(32)
which is plotted in Fig. 8. The model correctly predicts the detectivity within the measurements accuracy, D* ⫽
冑lx ly⌬f P NEP
冏
PDC30
⫽ 1 ⫻ 10 9
cm 冑Hz , W
(33)
for the present TE-cooled detector. From the three plots with the detector at room temperature, TEcooled 共the actual design兲, and at liquid nitrogen temperature, it is seen that cooling down to 77 K strongly increases the sensitivity of the detector. However, the performance of the liquid-nitrogen-cooled detector decreases strongly with the incident power. The TE-cooled detector does not have the same performance as the former, but its performance decreases less with increasing incident power, making it a possible choice for heterodyne detection systems, which
Fig. 9. Heterodyne noise equivalent power plotted as a function of incident optical power. The plot is given for three situations: the detector at room temperature, the present TE-cooled detector, and the same detector material cooled to liquid nitrogen temperature. The plots are for ⌬f ⫽ 1 Hz.
do not need the sensitivity of the liquid-nitrogencooled detector. The optimum reference power is found from minimizing the noise equivalent power, PNEP, for the heterodyne detection. The heterodyne PNEP is the power of the backscattered light from the aerosols that produce unity signal-to-noise ratio of the detected Doppler signal: P NEP ⫽
冑N2 4R d2P ref
,
(34)
which is plotted in Fig. 9. Even though shot-noise-limited detection cannot be achieved, the heterodyne PNEP obtainable by optimizing the reference power to the present TE-cooled detector, is sufficient for performing wind speed measurements at low altitudes where the aerosol backscatter coefficient is high 共 ⫽ 1 ⫻ 10⫺7 m⫺1 sr⫺1兲.20 Fig. 7. Three noise contributions and the total detector noise plotted as a function of incident optical power. This plot is for the case when the detector is cooled to liquid-nitrogen temperature.
4. Conclusion
A formulation of the nonlinear electrical response of the photoconductor on the incident optical radiation 20 August 2003 兾 Vol. 42, No. 24 兾 APPLIED OPTICS
4825
is derived from the two-level rate equation. The detector response and the optical power where saturation occurs are related to the physical constants of the detector material. The saturation level of the detector is found to be closely connected to the saturation phenomenon known for saturable absorbers. The detector material becomes transparent when the incident optical power exceeds the saturation power. The current description is also able to give the correct noise characteristics for the detector. In the present article, only the HgCdTe detector material is considered, but the model can be applied to other detector materials as well. The parameters for the model are fitted to a thermoelectrically cooled HgCdTe detector and afterwards are used to predict the detector noise performance in a heterodyne setup. Unlike for the liquid-nitrogen-cooled detector, shot-noise-limited detection cannot be obtained when TE cooling the detector to 226 K. However, the heterodyne noise equivalent power obtainable with the model for optimizing the reference power for the present TE-cooled detector is sufficient for performing wind speed measurements at low altitudes, where the aerosol backscattering coefficient is high. The research is partly funded by the Danish Technical Research Council.
7.
8.
9.
10.
11.
12.
13. 14.
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