Abstract:
Electro-thermal feedback is utilized for reducing the effective thermal conductance between the detector stage of a bolometer pixel in a thermal radiation sensor assembly and the environment through its mechanical support structure and electrical interconnects, thereby coming closer to achieving thermal conductance limited primarily through photon radiation. Minimization of the effective thermal conductance associated with the mechanical support structure and electrical interconnects is achieved by electro-thermal feedback that adjusts the temperature of an intermediate stage and the mechanical support structure and electrical interconnects, connecting it to the detector stage, to equal the temperature of the bolometer pixel&#39;s detector stage (i.e., by active thermal isolation). Increased temperature sensitivity is preferably achieved through temperature sensing with reverse biased Schottky diodes connected in series.

Description:
GOVERNMENT RIGHTS NOTICE 
     This invention was made with government support under U.S. Government Contract No. W911QX-04-C0117. The Government has certain rights to this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Technical Field 
     This invention relates generally to ultra sensitive silicon sensors (USSS), and more particularly to an electro-thermal feedback circuit providing active thermal isolation using a series temperature-sensing diode configuration. 
     2. Description of the Background Art 
     Bolometers are used in many imaging applications because they provide spectral versatility. Bolometers are most commonly used in low cost and reduced sensitivity long-wavelength infrared (LWIR) cameras. Bolometers have also been used, at cryogenic temperatures, for microwave imaging. Bolometers inherently respond over a broad band and this makes them viable for imaging in different spectral bands. 
     Quantum detectors, operating above liquid Helium temperatures, offer better sensitivity and faster response than bolometers but are not spectrally versatile. The lack of spectral versatility with quantum detectors is particularly evident at mm-waves where no viable quantum detector technology exists. By contrast, in the visible spectrum, high performance solid state quantum detector cameras are commercially available. These imaging cameras are made in silicon and provide high performance e at low cost. Generally, cameras based on quantum detectors provide very impressive imagery at higher photon energies (0.5-2 eV). At lower photon energies (&lt;0.5 eV), or longer wavelengths, quantum detectors require more expensive crystalline materials and complicated integrated circuit (IC) processing. Moreover, at mid-wavelength infrared (MWIR), or longer wavelengths, cryogenic cooling is needed to realize their full performance potential. Hence, quantum detectors have been limited to imaging at photon energies greater than about 0.1 eV, i.e., LWIR. Additionally, the requirement for cryogenic cooling of IR quantum detector is a constant detractor in terms of cost, weight and reliability. 
     The aforementioned issues with quantum LWIR cameras are the main impetus for searching for alternative approaches. In principle, bolometers with theoretically limited sensitivity are a commercially attractive competitor to quantum based LWIR cameras. Hence, large development efforts were started to improve the sensitivity of bolometers. The expectation is that with theoretically limited sensitivity, there are many applications where LWIR bolometers will offer a viable replacement to LWIR quantum detectors; with the advantages of room temperature operation, adequate sensitivity, good reliability, low cost, and low weight. Additionally, bolometers with theoretically limited sensitivity open the opportunities for room temperature mm-wave imaging cameras. 
     A bolometer&#39;s focal plane is made up of pixels assembled into an x-y array with associated readout circuits. A conventional bolometer pixel  9  and its thermal equivalent circuit are shown in  FIGS. 1 and 2 , respectively. A bolometer pixel  9 , as shown in  FIG. 1 , includes an absorber element (i.e., detector)  10 , represented by a rectangle with area A D  that is mechanically supported by several low thermal conductance bridges  12 , which sits atop of and are anchored to a thermal bath member  14  having a temperature T HB . The detector  10  in each pixel acts as a photon absorber and has a temperature T 1 . From a scene  8 , at temperature Ts, photons (i.e., radiation power) hν are incident on detector  10 , of pixel  9 , and are absorbed and cause a temperature change δT 1  from T 1 . As shown in  FIG. 2 , the detector&#39;s heat capacity is C 1 , and the thermal conductance of the bridges  12  between the detector  10  and the heat bath  14  is G 2 . The scene, at temperature Ts, is radiating energy hν at the detector  10 , and this is represented in  FIG. 2  as a thermal current Q R . The detector  10 , in addition to being mechanically attached to the heat bath  14  by the bridge&#39;s thermal conductance G 2 , is radiating power Q D1  and receiving Q S1  from radiation shields (not shown) attached to heat bath  14 . The performance of pixel  9  is analyzed by computing the signal and noise in terms of the small signal equivalent circuit of  FIG. 2 . 
     The analysis is based on small signal analysis technique developed for electronic circuits. Translation from a thermal to an electrical domain is made by treating temperatures as voltages; thermal conductances as reciprocal electrical resistances; and thermal currents as electrical currents. Specifically, temperatures T S , T 1 , T HB  are treated as voltages. Thermal conductance between the heat bath  14  and absorber element  10  is modeled as a reciprocal resistance with conductance G 2 . The heat capacity of absorber  10  is represented by capacitor C 1 . The thermal currents are: from the scene Q R =σT S   4 A D /4F 2 ; from the absorber  10  to the outside world Q D1 =σT 1   4 A D ; and from the radiation shields to the absorber  10  Q S1 =σT HB   4 A D [1-1/4F 2 ], where σ=5.6697×10-8 W-m −2 -° K −4  and F is the optic&#39;s f-number. Functional differences between expression for Q R , Q D1 , and Q S1  are because photon flux arrives through different solid angles, accounted for by the lens&#39;s f-number. Additionally, the detector  10  conducts thermal energy, through conductances G 2 , to the heat bath  14 , at temperature T HB . Analytically, the thermal balance conditions at the bolometer are represented as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Q 
                             R 
                           
                           - 
                           
                             Q 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           + 
                           
                             Q 
                             
                               S 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∫ 
                               
                                 T 
                                 HB 
                               
                               
                                 T 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   G 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
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                                   ) 
                                 
                               
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                                 1 
                               
                               
                                 
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                                   1 
                                 
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                                   δ 
                                   ⁢ 
                                   
                                       
                                   
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                                     1 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               jω 
                               ⁢ 
                               
                                   
                               
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                                   1 
                                 
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                                   ( 
                                   T 
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                                       ) 
                                     
                                   
                                 
                                 
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                               ⁢ 
                               
                                 ❘ 
                                 
                                   T 
                                   HB 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           T 
                                           1 
                                         
                                         - 
                                         
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                                           HB 
                                         
                                       
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                                   n 
                                 
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                                     1 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     T 
                                     ) 
                                   
                                 
                               
                               
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                                   T 
                                   n 
                                 
                               
                             
                           
                           ⁢ 
                           
                             ❘ 
                             
                               T 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     1 
                                   
                                 
                                 ) 
                               
                               
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                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The temperature dependence of G 2 (T) and C 1 (T) are included in Equation (1). 
     Using small signal analysis on a conventional bolometer, shown in  FIG. 1 , it is reasonable to assume that the derivatives of G 1 (T) and C 1 (T) are a weak function of temperature. And for simplicity, only first order terms are retained. At equilibrium, or constant flux conditions, the bolometer&#39;s equilibrium temperature T 1   0  is obtained from Equation (1), for ω=0, and is given by: 
     
       
         
           
             
               
                 
                   
                     T 
                     1 
                     0 
                   
                   = 
                   
                     
                       T 
                       HB 
                     
                     + 
                     
                       
                         
                           Q 
                           R 
                         
                         + 
                         
                           Q 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         - 
                         
                           Q 
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       
                         
                           G 
                           2 
                         
                         + 
                         
                           0.5 
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 G 
                                 2 
                               
                             
                             
                               ∂ 
                               T 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 T 
                                 1 
                                 0 
                               
                               - 
                               
                                 T 
                                 HB 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Thus, at equilibrium, the temperature of absorber  10  is different from the heat bath temperature by the net bridge heat current divided by conductance G 2 *. As expected, the more power the absorber  10  receives, which is equal to (Q R +Q S1 −Q D1 ), the higher will be its operating temperature. Since the radiation shields are attached to the heat bath  14 , their temperature [and Q S1 ] is fixed to the heat bath&#39;s temperature. Thus the temperature of absorber  10  will change monotonically with changes in scene temperature T S , and these temperature changes are maximized with minimum conductance G 2 *. 
     Under dynamic conditions, changes in the scene&#39;s temperature by δT S  will produce changes in the absorber&#39;s temperature equal to δT 1 , about the thermal equilibrium temperature T 1   0 . Assuming the radiation shield is at a constant temperature (i.e., δT HB =δT S1 =0), the expression relating δT 1  to δT S  is computed by taking the differentials of Equation (1) and Equation (2) and combining them. The relationship between δT 1  and δT S  is given by: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       1 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           G 
                           R 
                         
                         
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           + 
                           
                             G 
                             2 
                             * 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           S 
                         
                       
                       
                         1 
                         + 
                         
                           jω 
                           ⁡ 
                           
                             [ 
                             
                               
                                 C 
                                 1 
                                 * 
                               
                               / 
                               
                                 ( 
                                 
                                   
                                     G 
                                     
                                       D 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   + 
                                   
                                     G 
                                     2 
                                     * 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     For brevity, G 2 *=G 2 +(T 1   0 -T HB )∂G 2 /∂T 1  is used to account for the temperature dependence of G 2 (T). G R =∂Q R /∂T S =σT S   3 A D /F 2  is the thermal conductance between the scene and absorber, and G D1 =∂Q D1 /∂T 1 =4σT 1   3 A D  is the conductance between the absorber and its surroundings. 
     According to Equation (3), a δT S  change in the scene&#39;s temperature causes a δT 1  change in the absorber&#39;s temperature. Although signal δT 1  is monotonically related to δT S , it is attenuated by the factor G R /[G D1 +G 2 *], and the changes are retarded by thermal time constant or radial frequency ω TM =[G D1 +G 2 *]/C 1 *. The signal is attenuated because: (a) the absorber reradiates at a wider solid angle that it received radiation from the scene [G D1 &gt;G R ], and (b) a portion of the energy received from the scene is shunted to the heat bath  14  through conductance G 2 *. For maximum signal, conductances G D1  and G 2 * should be made as small as possible, and approach G R . 
     Hence, much effort has been invested into minimizing the thermal conductance G 2 *. Bridge materials with poor thermal conductivity have been developed, but there are practical limits The time thermal constant responsible for AC attenuation is minimized by minimizing C 1 */[G D1 +G 2 *], and this is accomplished by making C 1 * the absorber&#39;s mass very small. However, C 1 * cannot be made arbitrarily small because it inversely impacts: absorption, pixel fill factor, and thermal noise temperature level. Below, we develop expressions for maximum signal and minimum noise which are given in terms of G D1 , G 2 *, and C 1 *. 
     Signal analysis is performed in the temperature domain because LWIR bolometers&#39; signals are based on temperature changes. Correspondingly, noise is also represented as the temperature variance at the absorber  10 . Several noise sources contribute to the total temperature variance at the absorber  10 , and they are: (1) variance in the scene&#39;s photon power absorbed by the absorber  10 , δQ 2   R , (2) variance in the photon power emitted by the absorber  10 , δQ D1   2 , (3) variance in the radiation shield&#39;s photon power absorbed by the absorber  10 , δQ S1   2 , (4) variance in the thermal bath temperature, δT HB   2 , and (5) variance in the absorber&#39;s temperature produced by noise in readout electronics, δT EL   2 . 
     Scene&#39;s photon flux fluctuations increase the absorber&#39;s temperature variance. Ultimately, the bolometers sensitivity, expressed in noise equivalent temperature resolution (NEΔT), is limited by this temperature variance. The minimum noise level in the scene&#39;s signal Q R  is obtained from the variance in the signal&#39;s noise power spectral density δQ 2   R  arriving at the absorber element  10 , and it is given by: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       R 
                       2 
                     
                   
                   = 
                   
                     
                       8 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         A 
                         D 
                       
                       ⁢ 
                       σ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         k 
                         B 
                       
                       ⁢ 
                       
                         T 
                         S 
                         5 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       f 
                     
                     
                       4 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         F 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where, Δf represents the absorber&#39;s electrical frequency bandwidth, k B  is Boltzmann&#39;s constant, and σ is the Stefan-Boltzmann constant. The 4F 2  factor attenuates the signal flux at the absorber in accordance to the solid angle define by the optics F#. The signal&#39;s noise power spectral density δQ 2   R  is readily translated into a temperature variance at the absorber  10 , δT S   2 , by combining Equations (3) and (4) and integrating over frequency. The temperature variance δT S   2  at the absorber  10  from the scene&#39;s variance δQ R   2  is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             S 
                             2 
                           
                         
                         = 
                         
                           
                             2 
                             
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 F 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     A 
                                     D 
                                   
                                   ⁢ 
                                   σ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     B 
                                   
                                   ⁢ 
                                   
                                     T 
                                     S 
                                     5 
                                   
                                 
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           G 
                                           
                                             D 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                         
                                         + 
                                         
                                           G 
                                           2 
                                           * 
                                         
                                       
                                       ] 
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                       
                                         ω 
                                         2 
                                       
                                       ⁡ 
                                       
                                         [ 
                                         
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                                           1 
                                           * 
                                         
                                         ] 
                                       
                                     
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 ω 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               G 
                               R 
                             
                             
                               ( 
                               
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   G 
                                   2 
                                   * 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               
                                 k 
                                 B 
                               
                               ⁢ 
                               
                                 T 
                                 S 
                                 2 
                               
                             
                             
                               C 
                               1 
                               * 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Equation (5) reveals that the temperature variance, induced by the scene  8  on the absorber element  10 , is a product of two factors. The first factor is a conductance ratio: the conductance through free space between the scene  8  and the absorber  10 , divided by the conductance between the absorber  10  and its surroundings G R /[G D1 +G 2 *]. The second factor is the thermodynamic temperature variance for any object at temperature T S  and with a heat capacity C 1 *. For best performance, the scene&#39;s noise should dominate over all other noise sources. This is facilitated with fast optics (small F#) and minimum conductance [G D1 +G 2 *], or excellent thermal isolation. The scene&#39;s temperature variance at the absorber  10  is increased by additional sources and these are considered next. 
     The temperature variance [δT 1   2 ] at the absorber has several thermal contributors, which include the absorber  10  itself, the heat bath  14 , and radiative shields (not shown). In addition, electronic noise indirectly increases the absorber&#39;s temperature variance. The largest direct thermal contributor to the absorber&#39;s thermal variance is the absorber and this contribution is given as an integral of the power spectral temperature variance density [δT 1   2 (ω)] at the absorber by, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             1 
                             2 
                           
                         
                         = 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   T 
                                   1 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 ω 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   4 
                                   ⁢ 
                                   
                                     k 
                                     B 
                                   
                                   ⁢ 
                                   
                                     T 
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     G 
                                     2 
                                     * 
                                   
                                 
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           G 
                                           2 
                                           * 
                                         
                                         + 
                                         
                                           G 
                                           
                                             D 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                         
                                       
                                       ] 
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                       [ 
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                       ] 
                                     
                                     2 
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 ω 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               G 
                               2 
                               * 
                             
                             
                               
                                 G 
                                 2 
                                 * 
                               
                               + 
                               
                                 G 
                                 
                                   D 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               
                                 k 
                                 B 
                               
                               ⁢ 
                               
                                 T 
                                 1 
                                 2 
                               
                             
                             
                               C 
                               1 
                               * 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The absorber temperature variance, δT 1   2 , is a product of two factors: a leading factor [G 2 *]/[G D1 +G 2 *] is equal almost to unity. The second factor represents the theoretical temperature variance of an object at temperature T 1  and with a heat capacity of C 1 *. When fluctuations from the lens and radiation shields are included, the leading factor becomes unity. The contributions from the heat bath are made small by increasing the heat bath&#39;s heat capacity. Hence, the only remaining contributor to the temperature variance at the absorber is electronic noise. 
     Electronic noise degrades the bolometer&#39;s sensitivity. The electronic noise analysis assumes the most mature and most widely used bolometers: resistive VOx bolometers. Thus, the electronic noise includes the readout voltage noise and noise voltage from the resistive bolometer, biased with a current I CR . The total electronic noise voltage is converted to a temperature variance [δT EL   2 ] by dividing it by [I CR ∂R CB /∂T S ] 2 , where ∂R CB /∂T S  is the resistive temperature coefficient, I CR  is the bias current during readout, and R CB  the absorber&#39;s resistance. The total electrical noise is a sum of the absorber&#39;s and amplifier&#39;s noise power spectral densities, and these are represented by white and 1/f noise terms dE 2   N /df and dE 2   NA /df, respectively. Integrating these noise terms over radial frequency provides an expression for the equivalent temperature variance produced by the electronic noise voltage given by: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       EL 
                       2 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             I 
                             CR 
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 R 
                                 CB 
                               
                             
                             
                               ∂ 
                               
                                 T 
                                 S 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             ⅆ 
                             
                               E 
                               N 
                               2 
                             
                           
                           / 
                           
                             ⅆ 
                             f 
                           
                         
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             E 
                           
                           ⁢ 
                           
                             R 
                             CB 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             ω 
                             LO 
                           
                           
                             ω 
                             EL 
                           
                         
                         ⁢ 
                         
                           
                             
                               ⅆ 
                               
                                 E 
                                 NA 
                                 2 
                               
                             
                             
                               ⅆ 
                               f 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ⅆ 
                               ω 
                             
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The temperature variance at the absorber due to electronic noise is a sum of two noise terms, multiplied by a factor [I CR ∂R CB /∂T S ] −2 . The two terms are the integrated white noise and 1/f noise integral from the cut-on radial frequency ω LO  to the cut-off radial frequency ω EL =1/C E R CB . Minimum noise requires that the factor [I CR ∂R CB /∂T S ] −2  is minimized. Selection of absorbers with a high temperature coefficient ∂R CB /∂T S  is important. Second, use of a larger readout current I CR  helps, but it creates problems. Larger I CR  currents inject I 2 R power into the absorber that is larger (&gt;1000X) that the signal power resolution (in the pWatts range). The work around large I 2 R power is to use a pulsed readout to minimize the current heating duty cycle. However, pulsed operation requires wider signal bandwidth, accompanied with more noise. In the analysis it was assumed that the readout amplifiers voltage noise is less than the absorber&#39;s electrical noise. Such is facilitated with absorbers that have a large resistance, typically greater than 10KΩ. The ideal white noise power spectral density voltage equals to 12.9 nV/Hz 1/2 . The 1/f noise power contributions diverge logarithmically with the low cut-on radial frequency ω LO . For the purpose of simplification, 1/f noise contribution is included by increasing the white noise value from 12.9 nV/Hz 1/2  to 0.1 μV/Hz 1/2 . The absorber&#39;s total temperature variance δT T   2  is approximated by the sum of Equations (5), (6), and (7), and is given by, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       T 
                       2 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             k 
                             B 
                           
                           ⁢ 
                           
                             T 
                             S 
                             2 
                           
                         
                         
                           C 
                           1 
                           * 
                         
                       
                       ) 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             G 
                             R 
                           
                           
                             
                               G 
                               
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             + 
                             
                               G 
                               2 
                               * 
                             
                           
                         
                         + 
                         
                           
                             T 
                             1 
                             2 
                           
                           
                             T 
                             S 
                             2 
                           
                         
                         + 
                         
                           
                             
                               C 
                               1 
                               * 
                             
                             
                               
                                 k 
                                 B 
                               
                               ⁢ 
                               
                                 T 
                                 S 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   I 
                                   CR 
                                 
                                 ⁢ 
                                 
                                   
                                     ∂ 
                                     
                                       R 
                                       CB 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       T 
                                       S 
                                     
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               2 
                             
                           
                           ⁢ 
                           
                             0.25 
                             
                               
                                 C 
                                 E 
                               
                               ⁢ 
                               
                                 R 
                                 CB 
                               
                             
                           
                           ⁢ 
                           
                             
                               ⅆ 
                               
                                 E 
                                 N 
                                 2 
                               
                             
                             
                               ⅆ 
                               f 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     The total temperature variance δT T   2  reveals how the bolometers sensitivity is affected. The leading factor is the minimum possible thermodynamic temperature variance, governed by scene temperature and the absorber&#39;s heat capacity. The second factor shows how this variance is increased by the thermal conductivity between the absorber and its environment relative to thermal conductivity with the scene. This factor reveals that poor thermal isolation lead to sensitivity degradation that cannot be overcome by better electronics or responsivity. This factor is a sum of there terms representing contributions, respectively, from the scene&#39;s thermal noise, the absorber&#39;s thermal noise, and the electronics&#39; thermal noise. The values of second term is significantly greater than the scene&#39;s radiation noise and electronics noise can only further degrade sensitivity, and this is calculated next in terms of NEΔT. 
     The bolometer&#39;s sensitivity (minimum resolvable temperature: NEΔT) is defined at unity signal to noise ratio. The equation for NEΔT is obtained equating the change in absorber&#39;s temperature (given by Equation (3)) to the total noise temperature fluctuations at the absorber (given by the square root of Equation (8)). At unity signal to noise ratio, NEΔT equals δT S , making this substitution and solving for NEΔT the following is obtained: 
     
       
         
           
             
               
                 
                   
                     NE 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     T 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 k 
                                 B 
                               
                               ⁢ 
                               
                                 T 
                                 S 
                                 2 
                               
                             
                             
                               C 
                               1 
                               * 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   G 
                                   2 
                                   * 
                                 
                                 + 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               
                                 G 
                                 R 
                               
                             
                             + 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         G 
                                         2 
                                         * 
                                       
                                       + 
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                     
                                     
                                       G 
                                       R 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       T 
                                       1 
                                       2 
                                     
                                     
                                       T 
                                       S 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         C 
                                         1 
                                         * 
                                       
                                       
                                         
                                           k 
                                           B 
                                         
                                         ⁢ 
                                         
                                           T 
                                           S 
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           
                                             ω 
                                             EL 
                                           
                                           4 
                                         
                                         ⁢ 
                                         
                                           
                                             ⅆ 
                                             
                                               E 
                                               N 
                                               2 
                                             
                                           
                                           / 
                                           
                                             ⅆ 
                                             f 
                                           
                                         
                                       
                                       
                                         
                                           ( 
                                           
                                             
                                               I 
                                               CR 
                                             
                                             ⁢ 
                                             
                                               
                                                 ∂ 
                                                 
                                                   R 
                                                   CB 
                                                 
                                               
                                               / 
                                               
                                                 ∂ 
                                                 
                                                   T 
                                                   S 
                                                 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                       
                         1 
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               ω 
                               2 
                             
                             
                               ω 
                               TM 
                               2 
                             
                           
                         
                         ) 
                       
                       
                         1 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     This Equation has been condensed by defining ω TM =[G D1 +G 2 *]/C 1 * and ω EL =1/R E C CB . NEΔT is given as a product of three factors. Maximum sensitivity (smallest NEΔT) is achieved by minimizing each one of these factors. 
     The first factor represents the thermodynamic sensitivity limit dictated by the absorber&#39;s heat capacity C 1 * and scene&#39;s temperature T S . The actual sensitivity can only get worse than this theoretical thermodynamic limit 
     The second in the square brackets is a sum of three terms beginning with [G 2 *+G D1 ]/G R , which represents the sensitivity degradation due to thermal loading [G D1 +G 2 *] on the absorber relative to the thermal conductance to the scene G R . This factor is typically greater than 10 and much effort has gone into minimizing it because the sensitivity degradation it produces cannot be undone. The degradation to thermal loading can be minimized with maximizing G R  [with fast optics], and minimizing [G D1 +G 2 *] by construing thermal isolation bridges with poor thermal conductivity. Thermal loading has an additional deleterious impact because it also decreases the signal thereby making things more susceptible to electronic readout noise. The second term, in the square brackets, incorporates all the thermal noise contributions from the absorber element [including radiation shields]. Heat bath noise is not included since it can be made arbitrarily small by increasing the heat bath&#39;s heat capacity. Thus the second term is much larger than the leading term. The last term is in the square bracket represents readout electronics&#39; noise. Electronics noise can only make things worse and it can be minimized by maximizing the responsivity ∂R CB /∂T S . Optimistically, let&#39;s assume that the noise from the resistive bolometer dominates over all electronic noises. For a 10 4 Ω resistive bolometer, the noise dE N   2 /df is about 2×10 −16 V 2 /Hz. 1/f noise will only makes things worse. The bolometer&#39;s TCR, equals to ∂R CB /∂T S ≈200Ω/° K and lets make G 2 */G D1 ≈10. For: T S ≈300° K, A D =0.25×10 −4  cm 2 , and F=1; the third term in the sum in Equation (9) equals to ω EL 3×10 −10 /I CR   2 . The electronic noise should be significantly less than the second term in the sum, [equals &lt;10]. If (ω EL )=30, the electronics noise becomes insignificant if the circuit current I CR &gt;&gt;0.01 mA, or I CR &gt;0.03 mA. The I 2 R power associated with I CR &gt;0.03 mA is about 10 μWatts, and this is much larger than the desired power resolution of about 10 pWatts. This large I 2 R readout power is about a million times larger than the desired sensitivity thereby imposing unreasonable temperature sensitivity. The I 2 R power can be minimized by minimizing the I CR  duty cycle. For a 400×500 elements staring array, the current on time [or I 2 R power] can be reduced by up to 2×10 5  times by minimizing the I CR  duty cycle. However, a minimum duty cycle maximizes the operating bandwidth given by ω EL , and this increases noise. The larger noise can be attenuated by increasing I CR  from 0.03 mA to about 12 mA. Unfortunately, this increases the I 2 R readout power by about 10 5 , an unacceptable situation. Including 1/f noise makes things even more difficult. Parallel readout can be used to reduce electronic bandwidth, but it will increase the power by the number of parallel channels. 
     The third factor reveals how sensitivity decreases with frequency ω TM =[G D1 +G 2 *]/C 1 *. The optimal solution for this factor is to minimize the frequency response ω TM , and this is achieved by making ω TM  equal the video frame rate of 30 Hz. These limitations addressed by the active thermal isolation USSS approach described next. 
     According to Equation (9), thermal loading reduces the NEΔT sensitivity by at least [G D1 +G 2 *]/G R . The responsivity is reduced by [G R /(G D1 +G 2 *)], see Equation (3). The values of G D1  and G R  are limited by optics, and the most direct way for improvement is by minimizing the thermal conductivity G 2   * ≈2×10 −7  Watts/° K to approach G D1 , and G R . Given inherent limitations with material and geometrical approaches, further reductions in G 2 * required a different approach. The approach, incorporated in the all silicon ultra sensitive silicon sensor (USSS), is to utilize electro-thermal feedback to substantially reduce the effective thermal conductivity of G 2 *. Electro-thermal feedback offers active, as opposed to passive, thermal isolation and thereby obtain a ten-fold improvement in the thermal isolation. Improved thermal isolation results in increases in photo-responsivity and corresponding improvements in NEΔT. The performance advantages of the USSS approach are detailed in the description of the operation and the structure of a single USSS pixel. 
     Active thermal isolation, mechanized with electro-thermal feedback, can significantly improve thermal isolation between the absorber element  10  and heat bath  14  in  FIG. 1 . Thermal isolation improvements with active techniques versus passive techniques directly lead to corresponding improvement in responsivity, and sensitivity. These improvements are shown by analysis of the electro-thermal feedback. 
     Mechanizing electro-thermal feedback in each USSS pixel requires incorporation of special circuits and components. Specifically, electro-thermal feedback requires: (1) temperature difference sensors, (2) temperature difference amplifier, (3) a heater whose output power dependent on temperature difference, and (4) an approach that incorporates all these into a single pixel. 
     The USSS pixel approach in  FIG. 3  shows a three-temperature-tier design for maximizing area efficiency. The three tier structure includes: an absorber  10  at the top tier at temperature T 1 , an intermediate stage  16  in the middle at temperature T 2 , and at the bottom a heat bath  14  at temperature T HB . These tiers are mechanically and electrically connected by thermal isolation bridges G 1 * and G 2 *. 
     Electro-thermal feedback requires two temperature sensors to control the power delivered into the intermediate stage  16  so that T 2  converges to T 1 . Conventionally, the temperature difference signal is obtained from two silicon diodes  34  and  40 , connected back-to-back (see  FIG. 4 ). Temperature sensing diode  34  is built into the absorber  10  and a temperature sensing diode  40  is built into the intermediate stage  16 . Conventionally, temperature sensing diodes  34  and  40  are biased with a constant current and the voltage is a function of temperature. The voltage signal from diodes  34  and  40 , biased with a constant current, is about −2 m° K per degree Kelvin. 
     The voltage signal from the back-to-back diodes  34  and  40  is amplified and controls the temperature T 2  of the intermediate stage  16 . In each pixel, a single voltage amplifier  18  functions as USSS pixel readout and as a heater with variable output power Q H . Combining readout and heater functions facilitates pixel miniaturization. The power Q H  delivered into the intermediate stage  16  depends on the temperature difference between T 1  and T 2 . If T 2 &gt;T 1  (T 2 &lt;T 1 ) the quiescent power Q H  delivered decreases (increases) thereby converging T 2  towards T 1 . The potential across the back-to-back diodes  34  and  40  is amplified and produces an output voltage signal V S . Relative to a dc level, the amplifier output V S  will be positive if T 2 &gt;T 1  and negative if T 2 &lt;T 1 . As V S  increases, the quiescent power Q H  delivered by the amplifier to the intermediate stage  16  decreases. Conversely, as V S  decreases, the quiescent power Q H  delivered by the amplifier to the intermediate stage  16  increases. The power Q H  delivered by the amplifier to the intermediate stage  16  is given by,
 
 Q   H   =I   H   B   +   −I   H   V   s   =I   H   B   +   −I   H   A   G [ΔΦ D1 ( T   1 )−ΔΦ D2 ( T   2 )]  (10)
 
where A G  is the amplifier&#39;s voltage gain and I H  is the dc bias current flowing between B+ and ground.
 
     The amplifier&#39;s voltage gain is typically greater than 10 4 , I H  is about 1 μA, and B+ is about 10 volts. Since B+ and I H  are constants, the amplifier&#39;s quiescent power Q H  heating the intermediate stage varies linearly with the output signal V S , and it becomes a measure of the power received by the absorber. Specifically, as V S  decreases (increases) the power delivered by the amplifier increases (decreases). The differential power delivered by the amplifier depends on temperature difference and is readily computed from Equation (10) and given by, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Q 
                       H 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           I 
                           H 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           G 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   ∂ 
                                   
                                     ( 
                                     
                                       
                                         ΔΦ 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           T 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     T 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 1 
                               
                             
                             - 
                             
                               
                                 
                                   ∂ 
                                   
                                     ( 
                                     
                                       
                                         ΔΦ 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           T 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     T 
                                     2 
                                   
                                 
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                     ≈ 
                     
                       
                         - 
                         
                           I 
                           H 
                         
                       
                       ⁢ 
                       
                         A 
                         G 
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             
                               ( 
                               
                                 
                                   ΔΦ 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     T 
                                     1 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           
                             ∂ 
                             
                               T 
                               1 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 1 
                               
                             
                             - 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where ∂[ΔΦ D1 (T 1 )]/∂T 1 ≈∂[ΔΦ D2 (T 2 )]/∂T 2  represents the sensitivity of the temperature sensing diodes (for current biased p/n diodes it is about −2mV/° K). 
     The electro-thermal coefficient “A” depends on the USSS heater power in  FIG. 3  and is given by, 
     
       
         
           
             
               
                 
                   A 
                   ≈ 
                   
                     
                       - 
                       
                         I 
                         H 
                       
                     
                     ⁢ 
                     
                       
                         A 
                         G 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ∂ 
                             
                               ( 
                               
                                 
                                   ΔΦ 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     T 
                                     1 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           
                             ∂ 
                             
                               T 
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                   ≈ 
                   
                     
                       - 
                       
                         I 
                         H 
                       
                     
                     ⁢ 
                     
                       
                         A 
                         G 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ∂ 
                             
                               ( 
                               
                                 
                                   ΔΦ 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     T 
                                     2 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           
                             ∂ 
                             
                               T 
                               2 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The value of “A” depends on the voltage amplifier&#39;s gain and frequency response. The amplifier&#39;s voltage gain is A G =A 0 /(1+jω/ω 0 ) with ω 0 ≈10 4 rad/sec and A 0 ≈10 4 . For ∂[ΔΦ D2 (T 2 )]/∂T 2 −2 mV/° K, and I H ≈2 μA, the value of the electro-thermal coefficient is given by A≈4×10 −5 /(1+jω/ω 0 ) W/° K. For proper operation, analysis shows that the electro-thermal coefficient “A” must be large relative to conductances G 1 * and G 2 *, which are about 10 −7  W/° K. 
     The USSS (see  FIG. 3 ) photoresponse is computed with thermal analysis of the USSS thermal equivalent circuit shown in  FIG. 5 . The USSS analysis follows the approach presented for conventional bolometers. For bidirectional temperature tracking, T HB  is always less than T 1  and T 2 . As anticipated, thermal analysis reveals that the conductance of G 1 * is severely reduced by electro-thermal feedback. This reduction produces greater than a ten-fold improvement in thermal isolation and increase in photoresponse. The thermal balance conditions at the absorber  10  and intermediate stages  16  are expressed in terms of two integral/differential equations. At the absorber stage  10 , the thermal balance equation is given by: 
     
       
         
           
             
               
                 
                   
                     
                       Q 
                       R 
                     
                     - 
                     
                       Q 
                       
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     + 
                     
                       Q 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             ∫ 
                             
                               T 
                               1 
                             
                             
                               T 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               
                                 G 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 T 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               T 
                             
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             T 
                             1 
                           
                           
                             
                               T 
                               1 
                             
                             + 
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 1 
                               
                             
                           
                         
                         ⁢ 
                         
                           jω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               C 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               T 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             T 
                           
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         ∞ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 
                                     
                                   
                                     
                                       
                                         ∂ 
                                         n 
                                       
                                       ⁢ 
                                       
                                         
                                           G 
                                           1 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           T 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         T 
                                         n 
                                       
                                     
                                   
                                    
                                 
                                 
                                   T 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       2 
                                     
                                     - 
                                     
                                       T 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ! 
                               
                             
                           
                           + 
                           
                             
                               
                                   
                                 
                                   jω 
                                   ⁢ 
                                   
                                     
                                       
                                         ∂ 
                                         n 
                                       
                                       ⁢ 
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           T 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       
                                         T 
                                         n 
                                       
                                     
                                   
                                 
                                  
                               
                               
                                 T 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       T 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ! 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     For small temperature signals G 1 (T) and C 1 (T), the integrals in Equation (13) can be approximated by the first two terms of the Taylor series, and the temperature differential of Equation (13) yields an expression given by:
 
 G   R   δT   S   =└G   1   *+G   D1il +   jωC   1   *┘δT   1   −G   1   *δT   2   (14)
 
where, G 1 *=G 1 +(T 2 −T 1 )∂G 1 /∂T 1 .
 
     Similarly, thermal balance conditions at the intermediate stage  16  give rise to a differential/integral equation given by: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         Q 
                         
                           D 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                     + 
                     
                       Q 
                       H 
                     
                     + 
                     
                       Q 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             ∫ 
                             
                               T 
                               2 
                             
                             
                               T 
                               HB 
                             
                           
                           ⁢ 
                           
                             
                               
                                 G 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 T 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               T 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 
                                   T 
                                   2 
                                 
                                 
                                   
                                     T 
                                     2 
                                   
                                   + 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       T 
                                       2 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 jω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     T 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   T 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             T 
                             1 
                           
                           
                             T 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             
                               G 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               T 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             T 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           ∞ 
                         
                         ⁢ 
                         
                           [ 
                           
                             ( 
                             
                               
                                 
                                   - 
                                   
                                     
                                         
                                       
                                         
                                           
                                             ∂ 
                                             n 
                                           
                                           ⁢ 
                                           
                                             
                                               G 
                                               2 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               T 
                                               ) 
                                             
                                           
                                         
                                         
                                           ∂ 
                                           
                                             T 
                                             n 
                                           
                                         
                                       
                                        
                                     
                                     
                                       T 
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           T 
                                           HB 
                                         
                                         - 
                                         
                                           T 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                     ! 
                                   
                                 
                               
                               + 
                               
                                 jω 
                                 ⁢ 
                                 
                                   
                                       
                                     
                                       
                                         
                                           ∂ 
                                           n 
                                         
                                         ⁢ 
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             T 
                                             ) 
                                           
                                         
                                       
                                       
                                         ∂ 
                                         
                                           T 
                                           n 
                                         
                                       
                                     
                                      
                                   
                                   
                                     T 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ( 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           T 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                   
                                     
                                       ( 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                     ! 
                                   
                                 
                               
                             
                             ) 
                           
                           ] 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           ∞ 
                         
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                   
                                 
                                   
                                     
                                       ∂ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         G 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         T 
                                         ) 
                                       
                                     
                                   
                                   
                                     ∂ 
                                     
                                       T 
                                       n 
                                     
                                   
                                 
                                  
                               
                               
                                 T 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       T 
                                       2 
                                     
                                     - 
                                     
                                       T 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ! 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   15 
                   ) 
                 
               
             
           
         
       
     
     For small temperature signal G 1 (T) and C 1 (T) dependent weakly on temperature, the integrals are approximated by the first two Taylor series terms. Taking the temperature differential of Equation (15) and combining with Equations (11) and (12), we obtained:
 
└ G   1   *A┘δT   1 =└ G   1   *+G   2   *+G   D2   +A+jωC   2   *┘δT   2   (16)
 
     Variable G 2 * is defined as: G 2 *=G 2 +(T HB -T 2 )∂G 2 /∂T 2 . For sufficiently large A&gt;&gt;[G 1 *, G 2 *, G* D2 ], Equation (16) shows that electro-thermal feedback forces δT 1 ≈δT 2 . This condition minimizes change in the thermal current through G: with changes in absorber&#39;s temperature T 1 . Having the intermediate stage  16  track the absorber&#39;s temperature effectively makes G 1 * behave as a much smaller thermal conductance. In the limit as A goes to infinity δT 1 =δT 2 , and the absorber&#39;s thermal isolation should approach the ideal radiative limit G D1 . 
     Responsivity also improves because electro-thermal feedback greatly improves the effective thermal isolation. This improvement is revealed by combining Equations (14) with (16) to eliminate T 2 , and obtaining an expression for T 1  as a function of T S , specifically: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       1 
                     
                   
                   = 
                   
                     
                       
                         
                           G 
                           R 
                         
                         
                           [ 
                           
                             
                               G 
                               
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             + 
                             
                               jω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 1 
                                 * 
                               
                             
                             + 
                             
                               
                                 
                                   G 
                                   1 
                                   * 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       G 
                                       2 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                     + 
                                     
                                       jω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         C 
                                         2 
                                         * 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               
                                 [ 
                                 
                                   
                                     G 
                                     1 
                                     * 
                                   
                                   + 
                                   
                                     G 
                                     2 
                                     * 
                                   
                                   + 
                                   
                                     G 
                                     
                                       D 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   + 
                                   A 
                                   + 
                                   
                                     jω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       C 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         S 
                       
                     
                     ≅ 
                     
                       
                         
                           
                             G 
                             R 
                           
                           / 
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                         
                           [ 
                           
                             1 
                             + 
                             
                               
                                 jω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                   * 
                                 
                               
                               
                                 G 
                                 
                                   D 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         S 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Comparing Equations (17) and (3) reveals that with large electro-thermal values A&gt;&gt;[G 1 *, G 2 *, G D2 ] the responsivity increases. The electro-thermal coefficient A≈4×10 −5 /(1+ω/ω 0 )W/° K is much larger than typical values for G 1 *≈G 2* ≈2×10 −7 W/° K, and the result is that electro-thermal feedback reduces conductance G 1 * to about G D1 . This reduction represents better than a 40 fold decrease in the effective thermal conductance of G D1 . The improved responsivity is limited by the optical solid angle, and, according to Equation (17), is approximated by: δT 1 /δT S =1/(4F 2 ). With greatly improved photoresponsivity (X40), susceptibility to absorber stage noise and readout circuit noise are reduced thereby facilitating an improved NEΔT. 
     The USSS voltage photoresponse is calculated in several steps, starting by substituting Equation (10) for Q H  into Equation (15) and taking the differential. Under the constant bias current I H , the differential power delivered by the amplifier is simply δQ H =−δV S I H . Incorporating this into the differential of Equation (15), and after some rearrangements, a new equation is obtained given by,
 
δ V   S   I   H   =G   1   *δT   1   −└G   1   *+G   2   *+G   D2   +jωC   2   ┘δT   2   (18)
 
     The voltage responsivity in terms of changes in scene temperature δT S  is obtained by first using Equation (16) to factor δT 2  from Equation (18) and from the result factoring δT 1  with Equation (17). Performing these substitutions, and after some rearrangements, the USSS&#39;s voltage responsivity is given by, 
     
       
         
           
             
               
                 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           V 
                           S 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           T 
                           S 
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             AG 
                             R 
                           
                         
                         / 
                         
                           I 
                           H 
                         
                       
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     1 
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             G 
                                             1 
                                             * 
                                           
                                           + 
                                           A 
                                         
                                         ) 
                                       
                                       / 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         2 
                                         * 
                                       
                                       + 
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       + 
                                       
                                         jω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 G 
                                 
                                   D 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               + 
                               
                                 jω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           G 
                           1 
                           * 
                         
                       
                     
                     ≈ 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               
                                 G 
                                 2 
                                 * 
                               
                               + 
                               
                                 G 
                                 
                                   D 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             
                               I 
                               H 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             G 
                             R 
                           
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             1 
                             + 
                             
                               jω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   C 
                                   2 
                                 
                                 / 
                                 
                                   ( 
                                   
                                     
                                       G 
                                       2 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         
                           [ 
                           
                             1 
                             + 
                             
                               jω 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   C 
                                   1 
                                 
                                 / 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Volts 
                       ⁢ 
                       
                         / 
                       
                       ⁢ 
                       Kelvin 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     The approximations to the right side of Equation (19) are obtained with A&gt;&gt;[G 1 *, G 2 *, G D2 ], and A is defined by Equation (12). The responsivity approximation equals about (0.25/F 2 )(G 2 */I H )≈0.025V/° K: when I H =2 μA, G D1 /G R =4F 2 =0.25 and G 2 *≈2×10 −7  W/° K. Thus, by having made A 0  very large, the absorber element&#39;s dc response greatly increases and produces a signal sufficiently large to diminish the impact of electronics noise. There are two ac time constants, the first one is from the absorber element and is given as C 1 /G D1 , and the second is for the intermediate stage and it is given by C 2 /(G 2 *+G D1 ). For television frame rates, the bolometer time constant is required to be consistent with 30Hz frame rate, or C 1 /G D1 =1/(60π). The thermal time constant is minimized by using concentrators such as microlenses or microantennas. The concentrators will reduce further the value of G D1  to provide even better responsivity. The projected sensitivity improvements with the USSS are examined next. 
     The USSS model includes conventional thermal noise source and the electronic/thermal noise from electro-thermal feedback. The thermal noise sources include fluctuations in: the scene&#39;s flux, Q S , the radiation shields, Q S1  and Q S2 , the heat bath coupled through conductance G 1 * and G 2 *, the absorber element, Q D1 , the intermediate stage, Q D2 , and electro-thermal feedback noise is categorized as electronic readout noise. These noise sources manifest themselves as a fluctuation in the output voltage signal V S . Specifically, noise voltage fluctuations, produced by temperature fluctuations include: (1) the scene, δV S (T S ), (2) the heat bath, δV S (T HB ), (3) the absorber stage, δV S (T 1 ), (4) the intermediate stage, δV S (T 2 ), and electro-thermal/readout circuits noise voltage, δV S (EL). These noise sources are addressed next. 
     Spectral fluctuations in the scene temperature, δT S   2 (ω)/dω=4kT S   2 G R , contributes RMS noise to the USSS output node δV S (T S ). This contribution is computed by integrating the scene fluctuations times the squared absolute value of the transfer function given by Equation (19). Approximating Equation (19) at high frequency, the RMS voltage noise due to the spectral temperature fluctuations in the scene temperature T S (ω) is: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         
                           T 
                           S 
                         
                         ) 
                       
                     
                   
                   ≅ 
                   
                     
                       [ 
                       
                         
                           
                             G 
                             2 
                             * 
                           
                           + 
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         
                           I 
                           H 
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 kT 
                                 S 
                                 2 
                               
                               ⁢ 
                               
                                 G 
                                 R 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 ω 
                               
                             
                             
                               
                                  
                                 
                                   
                                     G 
                                     
                                       D 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   + 
                                   
                                     jω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       C 
                                       1 
                                       * 
                                     
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                     
                   
                   ≅ 
                   
                     
                       [ 
                       
                         
                           
                             G 
                             2 
                             * 
                           
                           + 
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         
                           I 
                           H 
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             G 
                             R 
                           
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                         ) 
                       
                       
                         1 
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             kT 
                             S 
                             2 
                           
                           
                             C 
                             1 
                             * 
                           
                         
                         ) 
                       
                       
                         1 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Contributions from fluctuations in the heat bath&#39;s temperature to the output node voltage noise δV S (T HB ) is calculated in terms of temperature fluctuations induced on the absorber&#39;s temperature T 1  and the intermediate stage&#39;s temperature T 2 . These relationships are obtained by taking the differentials of Equations (13) and (15) under the conditions that δT HB ≠0 and δT S =0. Taking differentials of Equations (13), and (15), the following is obtained:
 
└ G   S1   +jωC   HB   ┘δT   HB   (ω)+└G   1   *┘δT   2   (ω)=└G   1   *+G   D1   +jωC   1   *┘δT   1 (ω)  (21)
 
└ G   2   *+G   S2   +jωC   HB   ┘δT   HB   (ω)+└G   1   *+A┘δT   1 (ω)=└ G   1   *+G   2   *+G   D2   +A+jωC   2   *┘δT   2 (ω)  (22)
 
     An expression for δV S (ω) in terms of δT HB (ω) is obtained by combining Equations (21), (22), (10), and (12).This expression is simplified with approximations that are G S2 ≈G S1 , and G D2 ≈G D1 =G S2 ≈G S1 , and C 2 *≈C 1 . Combing all these yields a transfer function relating the output voltage to changes in the heat bath temperature, and it is given by, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         ω 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           G 
                           2 
                           * 
                         
                         + 
                         
                           G 
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       
                         I 
                         H 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             - 
                             A 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           jω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             C 
                             HB 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               T 
                               HB 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                         ) 
                       
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         1 
                                         * 
                                       
                                       + 
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       + 
                                       
                                         jω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       G 
                                       2 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                     + 
                                     
                                       jω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         C 
                                         2 
                                         * 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       G 
                                       1 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     + 
                                     
                                       jω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         C 
                                         1 
                                         * 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The RMS noise level is computed with the definition of the noise power spectral density, and of the transfer function given by Equation (23). Converting Equation (23) into a power spectral density, the RMS noise voltage due to thermal fluctuations in the heat bath temperature become, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         
                           T 
                           HB 
                         
                         ) 
                       
                     
                   
                   ≅ 
                   
                     
                       
                         
                           G 
                           2 
                           * 
                         
                         + 
                         
                           G 
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       
                         I 
                         H 
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             ⅆ 
                                             ω 
                                           
                                           / 
                                           2 
                                         
                                         ⁢ 
                                         π 
                                       
                                       ] 
                                     
                                     ⁢ 
                                     
                                       ω 
                                       2 
                                     
                                     ⁢ 
                                     
                                       C 
                                       HB 
                                       2 
                                     
                                     ⁢ 
                                     4 
                                     ⁢ 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         kT 
                                         HB 
                                         2 
                                       
                                       / 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                         2 
                                       
                                       + 
                                       
                                         
                                           ω 
                                           2 
                                         
                                         ⁢ 
                                         
                                           C 
                                           HB 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 G 
                                                 2 
                                                 * 
                                               
                                               + 
                                               
                                                 G 
                                                 
                                                   D 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 ω 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   C 
                                                   2 
                                                   * 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 G 
                                                 1 
                                                 * 
                                               
                                               + 
                                               
                                                 G 
                                                 
                                                   D 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   1 
                                                 
                                               
                                               + 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 ω 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   C 
                                                   1 
                                                   * 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     A 
                                   
                                 
                                  
                               
                               2 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         
                           G 
                           2 
                           * 
                         
                         + 
                         
                           G 
                           
                             D 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       
                         I 
                         H 
                       
                     
                     ⁢ 
                     
                       
                         
                           kT 
                           HB 
                           2 
                         
                         
                           C 
                           1 
                           * 
                         
                       
                     
                   
                 
               
               
                 
                   24 
                   ) 
                 
               
             
           
         
       
     
     We&#39;ve simplified Equation (24) by recognizing that at higher frequencies the integral is simplified because G D1 &lt;&lt;ω 2 C 2   HB , and the electro-thermal coefficient A&gt;&gt;[G 1 *+G 2 *+G D2 ]. The right side of Equation (24) is obtained by including the fact that (A+G 1 *+G 2 *+G D2 )C 1 &gt;&gt;G D1 C 2 . 
     The noise voltage δV S (T 1 ) due to fluctuations in the absorber&#39;s temperature δT 1 (ω) is calculated using the equivalent circuit in  FIG. 6 , when δQ 2 =δQ H =0. Summing the flow of all the thermal power at node T 1 , we obtain: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               Q 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   G 
                                   1 
                                   * 
                                 
                                 + 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     1 
                                     * 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 T 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               G 
                               1 
                               * 
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 T 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 ω 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 G 
                                 
                                   D 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               + 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   C 
                                   1 
                                   * 
                                 
                               
                               + 
                               
                                 
                                   
                                     G 
                                     1 
                                     * 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         2 
                                         * 
                                       
                                       + 
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       + 
                                       
                                         jω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           2 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   ( 
                                   
                                     
                                       G 
                                       1 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       2 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                     + 
                                     A 
                                     + 
                                     
                                       j 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         C 
                                         2 
                                         * 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               T 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     δQ 1  squared equals to the squared power spectral density and it is given by dQ 2   1 /df=4G D1 k B (T 1 ) 2 . Using Equation (16), δT 2 (ω) is eliminated to yield the right side of Equation (25). Combining the right side of Equation (11) with Equation (12) and using Equation (16) to eliminate δT 2 (ω), the spectral noise voltage δV S [T 1 (ω)] produced by temperature fluctuations δT 1 (ω) is obtained and is given by, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             - 
                             δ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               V 
                               S 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   T 
                                   1 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   ω 
                                   ) 
                                 
                               
                               ] 
                             
                           
                           ⁢ 
                           
                             I 
                             H 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             A 
                             ⁡ 
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         T 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         ω 
                                         ) 
                                       
                                     
                                   
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         T 
                                         1 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         ω 
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               T 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           A 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   G 
                                   2 
                                   * 
                                 
                                 + 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 + 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                     * 
                                   
                                 
                               
                               
                                 
                                   G 
                                   1 
                                   * 
                                 
                                 + 
                                 
                                   G 
                                   2 
                                   * 
                                 
                                 + 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 + 
                                 A 
                                 + 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     2 
                                     * 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               T 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               ω 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Solving Equation (25) for δT 1 (ω) and using it to eliminate δT 1 ( 107  ) from Equation (26), an analytical solution is obtained relating the spectral noise voltage δV S [T 1 ( 107  )] to fluctuations in the noise power δQ 1 . Squaring the absolute value of this solution, replacing δQ 1  by dQ 2   1 /df=4G D1 k B (T 1 ) 2 , integrating [0≦ω≦∞], and taking the square root, the RMS voltage fluctuations δV S (T 1 ), produced by δT 1 (ω), is obtained and it is given by: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         
                           T 
                           1 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           I 
                           H 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   AG 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   k 
                                   B 
                                 
                                 ⁢ 
                                 
                                   T 
                                   1 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     ω 
                                   
                                   / 
                                   π 
                                 
                               
                               
                                 
                                    
                                   
                                     
                                       
                                         
                                           
                                             G 
                                             1 
                                             * 
                                           
                                           + 
                                           
                                             G 
                                             
                                               D 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               1 
                                             
                                           
                                           + 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             ω 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               C 
                                               1 
                                               * 
                                             
                                           
                                           + 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             [ 
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     1 
                                                     + 
                                                     A 
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                   G 
                                                   
                                                     D 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     1 
                                                   
                                                 
                                               
                                               + 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 ω 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   C 
                                                   1 
                                                   * 
                                                 
                                               
                                             
                                             ] 
                                           
                                           / 
                                           
                                             [ 
                                             
                                               
                                                 G 
                                                 2 
                                                 * 
                                               
                                               + 
                                               
                                                 G 
                                                 
                                                   D 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   2 
                                                 
                                               
                                               + 
                                               
                                                 j 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 ω 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   C 
                                                   2 
                                                   * 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               G 
                               2 
                               * 
                             
                             + 
                             
                               G 
                               
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                           
                             I 
                             H 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 k 
                                 B 
                               
                               ⁢ 
                               
                                 T 
                                 1 
                                 2 
                               
                             
                             
                               C 
                               1 
                               * 
                             
                           
                           ] 
                         
                       
                       
                         1 
                         / 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Temperature fluctuations in the intermediate stage δT 2 (ω) produce noise voltage at the output node and this contribution is calculated as contributions from the absorber stage δT 1 ( 107  ). Using the equivalent circuit in  FIG. 6 , the power at node T 1  is summed when δQ 1 =δQ H =0, and a relationship between δT 1 (ω) and δT 2 (ω) is obtained and given by:
 
δ T   1 (ω)=└ G   1 */( G   1   *+G   D2   +jωC   1 *)┘δ T   2 (ω)  (28)
 
     The intermediate stage noise source δQ 2 , produces temperature fluctuations δT 2 (ω), hence |T 2 |&gt;|T 1 |, and this is given by Equation (28). Summing the power at node T 2  in  FIG. 6 , a spectral power relationship given by the following is obtained:
 
δ Q   2 (ω)+δ Q   H (ω)=└ G   1   *+G   2   *+G   D2   +jωC   2   *┘δT   2 (ω)− G   1   *δT   1 (ω)  (29)
 
     The variables δQ H and δT 1  in Equation (29) are eliminated with δQ H =A[δT 1 -δT 2 ], and Equation (28), respectively, to obtain:
 
δ Q   2 (ω)=└( G   2   *+G   D2   +jωC   2 *)+( A+G   1 *)( G   D1   +jωC   1 *)/ G   1   *+G   D1   +jωC   1 *)┘δ T   2 ( 107 )   (30)
 
     Using Equations (11) and (28), an expression is obtained for the spectral fluctuations in the output voltage V S [T 2 (ω)] produced by thermal fluctuations from the intermediate stage and it is given by:
 
−δ V   S   [T   2 (ω)] I   H   =A[ 1 −δT   1 (ω)]δ T   2 (ω)=− A ( G   D1   +jωC   1 *)( G   1 *+G D1   +jωC   1 *)δ T   2 (ω)  (31)
 
     Fluctuations δQ 2  squared is given in terms of black body radiation by dQ 2   2 /df=4G D2 k B (T 2 ) 2 . Combining Equations (30) and (31), to eliminate δT 2 (ω), and taking the absolute value, squared, integrating [0≦ω≦∞], and taking the square root yields the voltage fluctuations at node T 2  in terms of the intermediate stage&#39;s temperature given by, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         
                           T 
                           2 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           I 
                           H 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   G 
                                   
                                     D 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   k 
                                   B 
                                 
                                 ⁢ 
                                 
                                   T 
                                   2 
                                   2 
                                 
                                 ⁢ 
                                 
                                   
                                     ⅆ 
                                     ω 
                                   
                                   / 
                                   π 
                                 
                               
                               
                                 
                                    
                                   
                                     1 
                                     + 
                                     
                                       
                                         G 
                                         1 
                                         * 
                                       
                                       A 
                                     
                                     + 
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   G 
                                                   2 
                                                   * 
                                                 
                                                 + 
                                                 
                                                   G 
                                                   
                                                     D 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     2 
                                                   
                                                 
                                                 + 
                                                 
                                                   j 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   ω 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     C 
                                                     2 
                                                     * 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   G 
                                                   1 
                                                   * 
                                                 
                                                 + 
                                                 
                                                   G 
                                                   
                                                     D 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     1 
                                                   
                                                 
                                                 + 
                                                 
                                                   j 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   ω 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     C 
                                                     1 
                                                     * 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         A 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               G 
                                               
                                                 D 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 1 
                                               
                                             
                                             + 
                                             
                                               j 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               ω 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 C 
                                                 1 
                                                 * 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             G 
                             2 
                             * 
                           
                           + 
                           
                             G 
                             
                               D 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                         
                           I 
                           H 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               k 
                               B 
                             
                             ⁢ 
                             
                               T 
                               2 
                               2 
                             
                           
                           
                             C 
                             2 
                             * 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Equation (32) is simplified by recognizing that for large “A” the approximation on the right is obtained and it reveals that the voltage fluctuations are proportional to the classical temperature fluctuations of a black body at temperature T 2 . The factor in front is the total thermal conductivity from the intermediate stage divided by the amplifier&#39;s bias current I H . The total fluctuation in voltage V S  from fluctuations in temperatures is the RMS sum of fluctuations from T HB , T S , T 1  and T 2 . 
     Electrical noise from the R/O and electro-thermal feedback circuit corrupts the USSS output signal. The level of corruption has been previously computed, and is given in terms of the amplifier&#39;s noise current, the conductivity of the thermal bridges, and amplifier&#39;s bias current I H . The contribution of electrical noise took into account the affects of electro-thermal feedback and electrical feedback. In the analysis below, the amplifier&#39;s effective channel noise I N0  without any feedback is defined. Noise analysis is performed by representing noise as a Fourier series sum, and analyzing the contribution of each term. Including electrical and thermal feedback, the electrical voltage noise is given by, 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         V 
                         S 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             E 
                             L 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         Z 
                         S 
                       
                       ⁢ 
                       
                         
                           I 
                           
                             N 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             0 
                           
                         
                         ⁡ 
                         
                           ( 
                           ω 
                           ) 
                         
                       
                     
                     
                       1 
                       + 
                       
                         
                           
                             g 
                             m 
                           
                           ⁢ 
                           
                             Z 
                             S 
                           
                           ⁢ 
                           
                             Z 
                             D 
                           
                         
                         
                           ( 
                           
                             
                               Z 
                               S 
                             
                             + 
                             
                               Z 
                               G 
                             
                             + 
                             
                               Z 
                               D 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         g 
                                         m 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         S 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         G 
                                       
                                     
                                     
                                       
                                         Z 
                                         S 
                                       
                                       + 
                                       
                                         Z 
                                         G 
                                       
                                       + 
                                       
                                         Z 
                                         D 
                                       
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 
                                   
                                     I 
                                     H 
                                   
                                   ⁢ 
                                   
                                     
                                       ∂ 
                                       
                                         
                                           E 
                                           F 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           T 
                                           ) 
                                         
                                       
                                     
                                     
                                       ∂ 
                                       T 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       G 
                                       2 
                                       * 
                                     
                                     + 
                                     
                                       G 
                                       
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                     
                                     + 
                                     
                                       j 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         C 
                                         2 
                                         * 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         1 
                                         * 
                                       
                                       + 
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     G 
                                     1 
                                     * 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         G 
                                         
                                           D 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       + 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           C 
                                           1 
                                           * 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Affects of electrical feedback are represented by the first algebraic term in denominator, and affects of thermal feedback are represented by the second algebraic term in the denominator. The larger the denominator is the smaller is the noise. The negative sign in front of the thermal feedback term is cancelled when it is multiplied by the negative factor ∂E F (T)/∂T≈2 mV/° K. Noise attenuation produced by the electrical and thermal feedbacks terms is estimated in terms of thermal conductivities. Typically, Z G &gt;&gt;[Z D , Z S ], g m Z S &gt;&gt;1, G 2 *≈G 1 *&gt;[10 G D1 , 10 G D2 ]. Additionally, the USSS operating conditions are given by: I H =2 μA, ∂E F (T)/∂T≈2 mV/° K, [G D1 , G D2 ]≈1.5×10 −8 W/° K, g m Z S ≈10 4 , and [G 1 *, G 2 *]≈2×10 −7  W/° K. The value of the electro-thermal term A≈4×10 −5 W/° K dominates over the other thermal conductivities. Incorporating all these into Equation (33), a simplified expression is obtained, and it is given as, 
     
       
         
           
             
               
                 
                   
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                   34 
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     The right side of Equation (34) incorporates these approximations combined with A=−I H g m Z S ∂E F (T)/∂T, given by Equation (12). For large electro-thermal feedback (A&gt;&gt;[G 1 *, G 2 *, and G D1 ] the noise is attenuated. The spectral noise density is readily converted into a standard deviation by integrating over frequency. If 1/f noise is neglected, the integral of Equation (34) is simplified and the noise voltage, in terms of the electrical bandwidth, (Δ BW ) 1/2 , is given by: 
     
       
         
           
             
               
                 
                   
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                   35 
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     The total noise at the USSS output node is the RMS sum of contributions from: the scene [Equation (20)], the heat bath [Equation (24)], the bolometer stage T 1  [Equation (27)], the intermediate stage T 2  [Equation (32)], and the readout electronics [Equation (35)]. The USSS sensitivity, NEΔT USSS , or minimum resolvable temperature, is obtained by equating the total noise to the signal: unity signals to noise ratio. RMS summing Equations (20), (24), (27), (32), (35) and dividing it by the absolute value of right side of Equation (19) yields the expression for NEΔT USSS , and it is given by, 
     
       
         
           
             
               
                 
                   
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                   36 
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     The NEΔT USSS  sensitivity in Equation (36) has been factored into three factors for comparison with the NEΔT sensitivity of conventional bolometers given by Equation (9). Sensitivity comparisons are made on a term by terms basis. 
     The first factor (kT S   2 /C 1 *) 1/2  represents the theoretical thermal sensitivity limit for a body at temperature T S  and heat capacity C 1 *. As expected, this factor is the same in Equations (9) and (36). 
     The second factor in the square brackets in Equation (36) is a sum of five terms with the upper terms corresponding to the first term in the square brackets in Equation (9). These terms show how sensitivity is reduced by thermal loading. With active thermal isolation, thermal loading is the USSS is much smaller than thermal loading in conventional bolometers: [G D1 /G R ] 2 &lt;&lt;[(G D1 +G 2 *)/G R ] 2 . This reduction in thermal loading and this leads to greater than 10X sensitivity improvement. The other thermal noise terms on the first line inside the square brackets in Equation (36) include radiation shield noise, absorber noise, and heat bath noise. These noise sources are multiplied by the factors, respectively, [G D1 /G R ] 2  and [(G D1 +G 2 *)/G R ] 2  in Equations (36) and (9). Clearly, thermal loading amplifies the affects of thermal noise from the absorber and radiation shields, and the multiplication is much worse (at least 10X) in conventional bolometers. 
     This affective noise “amplification” occurs because thermal loading attenuates the signal and not the thermal noise. In the NEΔT equations this attenuation manifests itself as though the noise increases. This “noise amplification” also applies to the electronics&#39; readout noise and is apparent from comparing the second line inside the square brackets in Equation (36) with the last term inside the square brackets in Equation (9). The electronics&#39; readout noise is smaller in the USSS compared to electronics&#39; readout noise in conventional bolometers. This is readily confirmed by comparing the respective expressions for the readout noise in Equations (36) and (9). Comparison is facilitated by recognizing the relationship between the variable in Equations (36) and (9). The electro-thermal feedback term is given by A=I H Z S g M ∂E F /∂T, with ∂E F /∂T≈−2 mV/° K, and (I N0 /g M ) 2 =dE N   2 /df. Also, the thermal conductance G 2 *≈G 1 * and these are much larger G D1  or G R . The resistive bolometer&#39;s temperature coefficient ∂R CB /∂T S ≈200, and the bias current is about I CR ≈0.1 mA. Because the conventional bias current is large, it is pulsed and for a 400×500 element array, the value of ω EL ≈400×500 Δ BW . Inserting all these values reveals that for the same readout transistors the impact of the readout noise in the USSS is about 50 times smaller than in a conventional bolometer. The reduced impact of readout noise in the USSS over conventional bolometers indicates that the signal in the USSS is higher thereby making the readout noise less significant. 
     The third factor in Equation (36) represents the AC frequency response and is given as a ratio of a zero and pole. In conventional bolometers the AC factor in Equation (9) includes only a pole at ω TM . This pole is the bolometer&#39;s thermal time constant and pulsed readout makes this factor significantly greater than one and adversely degrades sensitivity. In the USSS the AC factor includes a pole and a zero. The pole is the absorber&#39;s time constant and the zero is the intermediate stage&#39;s time constant. Since pulsed readout is not used, the ratio of these as a function of frequency can be made equal to about one. 
     For an absorber element 10 heat capacity C 1 * equal to about C 1 *≅1.5×10 −1 J/° K, the thermodynamic sensitivity limit is about (kT S   2 /C 1 *) 1/2 ≈0.1 m° K. This is much more sensitive than the performance of conventional bolometers. The sensitivity degradation are due to thermal loading that decrease this by about 50X to about 5 m° K. Readout noise and bandwidth further reduces the sensitivity by about 4X, resulting in an NEΔT≈20 m° K. As a result, there is a need for improved sensitivity in conventional bolometers. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to overcome the disadvantages of the prior art by increasing sensitivity while reducing power consumption. 
     An aspect of the invention is a bolometer pixel. The bolometer pixel has a detector stage, an intermediate stage, a heat bath and an electro-thermal feedback circuit. The electro-thermal feedback circuit has first and second diodes, first and second circuit elements, an amplifier and a heater. The first diode is contained in the detector stage. The second diode is contained in the intermediate stage. The cathode of the second diode is connected to the anode of the first diode. The first and second circuit elements are contained in the heat bath. The first and second circuit elements are connected in series. The amplifier includes a first input connected to the cathode of the second diode and the anode of the first diode, a second input connected to node between the first and second circuit components, and an output. The heater is connected to the output of the amplifier and heats the intermediate stage in accordance the temperature difference between the detector stage and the intermediate stage. The first and second diodes can be Schottky diodes. The first circuit element can be a third diode, and the second circuit element can be a fourth diode. The third and fourth diodes can be connected in series by the cathode of the fourth diode being connected to the anode of the third diode, and the second input can be connected to the cathode of the fourth diode and the anode of the third diode. The third and fourth diodes can also be Schottky diodes. The Schottky diodes can have a reverse bias saturation current of about one nanoampere. The Schottky diodes can have a junction diameter of about 2.3 micrometers. The first circuit element can be a first capacitor, and the second circuit element can be a second capacitor. The bolometer pixel can include a switch one side of the switch being connected to the first and second capacitors and the other side of the switch being connected to the cathode of the second diode and to the anode of the first diode. The heater can operate at a constant current and can be formed by a bias network and a heater field effect transistor (FET). The amplifier can be a differential complimentary metal-oxide-semiconductor (CMOS) circuit. The bolometer pixel can also have a first thermal isolation bridge between the detector stage and the intermediate stage, and a second thermal isolation bridge between the intermediate stage and the heat bath. The bolometer pixel can also have an associated readout circuit and a pixel row address switch. The bolometer pixel can be incorporated into a bolometer having a plurality of bolometer pixels and a column address switch for each column of the array. 
     Another aspect of the invention is an electro-thermal feedback circuit for a bolometer pixel having a detector stage, an intermediate stage and a heat bath. The electro-thermal feedback circuit includes a first diode contained in the detector stage, a second diode connected to the first diode in series and contained in the intermediate stage, and a heater which heats the intermediate stage to minimize the temperature difference between the detector and intermediate stage based on the voltage between the first and second diode. The detector stage absorbs and is heated by incident photons. The electro-thermal feedback circuit includes an amplifier which receives as an input the voltage between the first and second diode and outputs a signal to control the heater the heater to minimize the temperature difference between the detector and intermediate stage. The cathode of the second diode can be connected to the anode of the first diode. The first and second diodes can be Schottky diodes. The Schottky diodes can act as their own generator with there being no need to add a current generator. 
     An additional aspect of the invention is an active thermal isolation method. The active thermal isolation method includes heating a second stage to minimize the temperature difference between the second stage and a first stage using a voltage between a first diode contained in the first stage and a second diode contained in the second stage and connected in series with the first diode. The first stage can absorb and be heated by energy from incident photons. The cathode of the second diode can connected to the anode of the first diode. The first and second diodes can be Schottky diodes. The active thermal isolation method can include comparing the voltage between the first diode and the second diode with a voltage between third and fourth diodes connected in series and contained in a heat bath. The active thermal isolation method can include comparing the voltage between the first diode and the second diode with a voltage between first and second capacitors connected in series and contained in a heat bath. 
     Another aspect of the invention is an active thermal isolation method for a bolometer pixel with a first stage containing a first diode and second stage containing a second diode connected in series to the first diode. The active thermal isolation method includes using temperature-change-induced variation in saturation currents of the first diode and the second diode to heat the second stage so as to minimize the temperature difference between the second stage and the first stage. The first stage can be heated with the energy from incident photons. The cathode of the second diode can be connected to the anode of the first diode. The first and second diodes can be Schottky diodes. The temperature-change-induced variation can be used by comparing a voltage between the first and second diodes to a constant voltage. The constant voltage can be a voltage between third and fourth diodes connected in series and maintained at a constant temperature. The third and fourth diodes can be contained in a heat bath. The constant voltage may be a voltage between first and second capacitors connected in series and maintained at a constant temperature. The first and second capacitors may be contained in a heat bath. The method may include performing calibration to remove voltage errors due to variations between the first and second diodes. The performing calibration may include recording a voltage error in the first and second capacitors. The voltage error may be recorded by momentarily connecting a switch connected to the first and second diodes on one end and to the first and second capacitors on the other. 
     Some advantages that may be obtained by practicing the present invention include improving sensitivity of the bolometer pixel and reduced power consumption. Other aspects, embodiments, features and advantages of the invention will become apparent to those of skill in the art upon reviewing the following detailed description of the preferred embodiments and the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a side view of a prior art bolometer with a detector attached to a thermal isolation bridge on top of a substrate including a heat bath; 
         FIG. 2  is a thermal equivalent circuit for the bolometer shown in  FIG. 1 ; 
         FIG. 3  is an electrical block diagram illustrating a prior art USSS bolometer with an electro-thermal feedback circuit incorporating three temperature tiers; 
         FIG. 4  is an electrical circuit diagram illustrating a prior art USSS bolometer with an electro-thermal feedback circuit configured with two back-to-back diodes; 
         FIG. 5  is a thermal equivalent circuit of the USSS bolometer shown in  FIG. 3 ; 
         FIG. 6  is a thermal equivalent circuit for noise sources associated with the USSS bolometer shown in  FIG. 3 ; 
         FIG. 7  is a simplified electrical circuit diagram illustrating a bolometer with an electro-thermal feedback circuit according to an embodiment of the present invention; 
         FIG. 8  is an electrical circuit diagram illustrating a bolometer with an electro-thermal feedback circuit according to an embodiment of the present invention; 
         FIG. 9  is a simplified electrical circuit diagram illustrating a bolometer with a normalized electro-thermal feedback circuit according to an embodiment of the present invention; 
         FIG. 10  is an electrical circuit diagram illustrating a bolometer with a normalized electro-thermal feedback circuit according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A diagram of a bolometer pixel  109  with an electro-thermal feedback circuit according to an embodiment of the present invention is shown in  FIG. 7 . A plurality of such pixels can be arranged in an array, such as in rows and columns, to create nil array of pixels to form a focal plane with bolometer pixels. Each bolometer pixel  109  includes a detector stage  110 , an intermediate stage  116 , and a heat bath stage  114 . Within each pixel, the detector, intermediate and heat bath stages are connected in spaced relation to on another by a low thermal conductance bridge structure, e.g., as described in U.S. Pat. No. 7,361,900 entitled “‘I’ Beam Bridge Interconnection for Ultra-Sensitive Silicon Sensor,” U.S. Pat. No. 7,064,328 entitled “Ultra Sensitive Silicon Sensor Millimeter Wave Passive Imager” or U.S. Pat. No. 6,489,615 entitled “Ultra Sensitive Silicon Sensor,” the disclosures of which are incorporated herein by reference. It will be appreciated that the bolometer pixel can be fabricated in accordance with any known design or method of manufacture. The heat bath stage fowls a common thermal reference to the plurality of such pixels. 
     The temperatures of detector stage  110 , intermediate stage  116  and heat bath  114  are T D , T IN  and TH B , respectively. The electro-thermal feedback circuit performs active thermal isolation by heating the intermediate stage  116  to minimize the temperature difference between the intermediate stage  116  and the detector stage  110 . The result is minimized thermal conductivity between the detector stage  110  and the intermediate stage  116 , effectively minimizing the thermal conductivity from the detector stage to the environment, including the intermediate stage, thermally isolated from the heat bath stage  114 . 
     The detector stage  110 , containing diode D 1 , is mechanically and electrically connected through thermal isolation bridges to the intermediate stage  116 . The intermediate stage, containing diode D 2 , is mechanically and electrically connected through thermal isolation bridges to the heat bath stage  114 . Diodes D 1  and D 2  are connected in series such that the cathode of diode D 2  is connected to the anode of diode D 1 . Diodes D 1  and D  2  are used as temperature sensing diodes. The heat bath stage  114  contains diodes D 3  and D 4 . Diodes D 3  and D 4  are connected in series such that the cathode of diode D 4  is connected to the anode of diode D 3 . Diodes D 3  and D 4  are thermally isolated from diodes D 1  and D 2 , are held at a constant temperature T HB , and provide a voltage reference. An amplifier  118  amplifies the temperature difference (T D -T IN ) between detector stage  110  and intermediate stage  116 . The inputs of amplifier  118  are connected to the node N 1 , where diodes D 1  and D 2  are connected, and to the node N 2 , where diodes D 3  and D 4  are connected, respectively. The output of amplifier  118  functions as the USSS pixel  109  readout signal and as a control signal for a heater formed by bias network T 2  and heater field-effect transistor (FET) T 1 . Because reverse bias Schottky diodes exhibit better temperature sensitivity than pin diodes forward biased with a constant current, diodes D 1 -D 4  are Schottky diodes in a preferred embodiment. The benefits of using Schottky diodes are discussed in a later section. 
     In operation, the detector stage  110  absorbs incident photons and is thereby heated. Changes in temperature change how the voltage divides between the two temperature sensing diodes D 1  and D 2 . The current flowing through the diodes is the reverse bias saturation current and is designed to be about one nano ampere. As the temperature changes, the saturation currents for diodes D 1  and D 2  change, and this produces a voltage change at the node N 1 , which is the point where diodes D 1  and D 2  are connected in series. The resulting voltage signal is a reflection of the relative temperatures of detector stage  110  and intermediate stage  116 . If the detector stage  110  and the intermediate stage  116  are at the same temperature (i.e., T D =T IN ), the voltage across each series diode will be equal, and the voltage at the node N 1  will be equal to the voltage at node N 2 , which is the point where diodes D 3  and D 4  are connected. On the other hand, if the detector stage  110  and the intermediate stage  116  are at the different temperatures (i.e., T D ≠T IN ), the voltage across diodes D 1  and D 2 , respectively, will not be equal, and the voltage at node N 1  will be different than the voltage at the node N 2 . Accordingly, an error voltage signal is produced and this voltage error signal is proportional to the temperature difference (T D −T IN ) between detector stage  110  and intermediate stage  116 . 
     The heater supplies a power to heat intermediate stage  116  in accordance with the error voltage signal produced by the temperature difference between the intermediate stage  116  and the detector stage  110 . The heater is designed to operate at a constant current. The heater output power into the intermediate stage is monotonically proportional to the output voltage signal applied to the drain of FET T 1 . The voltage applied to the drain of NET T 1  is produced by amplifier  118  in response to the input voltages; one input is from the node N 1  between diodes D 1  and D 2  connected in series, and a second input is from the node N 2  between diodes D 3  and D 4  connected in series. 
     As mentioned above, diodes D 1  and D 2  are preferably Schottky diodes. Schottky diodes exhibit a very large reverse bias dynamic resistance. Hence, temperature changes in the detector stage  110  relative to the intermediate stage  116  produce an imbalance in the reverse bias currents of Schottky diodes D 1  and D 2 . This current imbalance, translated into a voltage signal, results in a very large responsivity of about 100 mV/° K. This voltage responsivity is about 100 times larger than the voltage responsivity of forward biased p/n junctions, which operate at higher currents, and consume more power. This voltage responsivity can further be increased with larger reverse bias current, for example 50 nA. However, for many applications it has been found that the 1 nA reverse bias Schottky current design provides good performance while minimizing the (I ^2 )R power consumption in the Schottky diodes. The 1 nA nominal reverse bias Schottky current is obtained by selecting Schottky diode metal, the silicon doping concentration, and junction area in the manner described below. 
     In the embodiment shown in  FIG. 8 , the amplifier  118  is a differential complimentary metal-oxide-semiconductor (CMOS) circuit biased with a current produced by transistor T 16 . The current from the drain of transistor T 16  is divided between the input FETs T 13  and T 14  and mirrored within the amplifier by a first mirror circuit comprised of transistors T 10  and T 8 , a second mirror circuit comprised of transistors T 9  and T 6 , and a third mirror circuit comprised of transistors T 11  and T 12 . The difference between the channel currents in FETs T 13  and T 14  produces the output voltage signal. The two channel current difference is obtained by replicating twice the current flowing in channel FET T 14  with the first and third mirror circuits and replicating once the current flowing in channel of FET T 13  with the second mirror circuit. After two replications, channel current in FET T 14  equals the channel current in FET T 12  and currents flow in the same direction. After a single replication, the channel current in BET T 13  equals the channel current in FET T 6  and flow in opposite directions. A large voltage gain of greater than 40 dB is obtained by combining the counter flowing currents in channel FET T 12  and channel FET T 6 . The output voltage of the differential amplifier drives the heater FET T 1  through a source follower stage made with FET T 18 . The electro-thermal coefficient “A” for the combination of the Schottky diode temperature sensor, the heater current, and the 40 dB amplifier gain is about 10 −4  W/° K. This larger electro-thermal coefficient (A equals about 10 −4  W/° K) is much larger (100 times) than the thermal conductivity of the thermal isolation bridges. The large electro-thermal coefficient reduces the effective thermal conductivity of the bridges by about 100 times. 
     In the present invention, having the temperature sensing diodes D 1  and D 2  connected in series instead of the conventional back-to-back design increases the dynamic range of the circuit and simplifies the design. All the diodes can be the same size in the series design. In the back-to-back design, the load diodes need to be adjusted in area to accommodate the temperature range anticipated for the bolometer. In the series design, when the detector and intermediate stage temperatures are equal (i.e., T D =T IN ), the voltage at node N 1  where the series diodes connect is equal to B-12. With the back-to-back design, the common mode voltage to the CMOS amplifier will change with temperature even though the detector and intermediate stages are at the same temperature. With the series diode arrangement, the same common mode voltage will be presented to the CMOS amplifier as long as the detector and intermediate stages are at the same temperature. 
     At the end of each integration time, the amplifier  118  output is gated off by the reset switch T 7 . When the switch T 7  is turned on, the amplifier&#39;s voltage output minimizes the heater power delivered to the intermediate stage  116 . This causes the detector stage  110  and intermediate stage  116  to converge in temperature. When the switch T 7  is turned off, the electro-thermal feedback loop is reestablished, and, as the detector stage  110  starts heating up, its temperature change will be tracked by the intermediate stage  116 . The temperature tracking reduces the effective thermal conductance (or thermal loading) of the bridges between the detector and intermediate stages by about 100 times. With the reduced thermal loading by 100X, the responsivity improves, and the signal will correspondingly increase. Furthermore, the pulsed operation of the electro-thermal feedback loop avoids the problem of loop oscillation and transients, by establishing initial electro-thermal conditions with very small temperature differences between detector and intermediate stages. 
     Included in each bolometer pixel  109  are: a readout FET T 19  and pixel row address FET switch T 20 . A column address switch T 21  is shown in  FIG. 8  but is not part of the bolometer pixel  109  circuitry. Instead, column address switch T 21  is part of the bolometer pixels FPA array into which the column of bolometer pixels  109  is intended to be incorporated, and a single switch is used for each column. The output FET for the entire FPA is provided by the combination of the column enable FET T 21  and the output current shared for all the outputs. Multiple outputs FET can be utilized with a single bolometer focal plane array. Transistor T 15  globally controls the amplifier quiescent current and the heater current. In the present design, the heater and amplifier currents are about 1 μA each. Thus, the power consumed by each pixel is only about 10 μWatts. 
     In another embodiment, the bridge made up of four Schottky diodes D 1 -D 4 , which produces the input to amplifier circuit  118 , may alternatively be made up of only two diodes in series and a midpoint reference instead. 
     The improved temperature sensitivity achieved through use of the Schottky diodes connected in series is discussed below. 
     Schottky diode characteristics are given by an equation similar to the pin junction diode, and is,
 
 I ( V,T )= I   S ( V,T )[exp(qV/kT)−1]  (37)
 
     This equation is very similar in format to the equation for a pin junction diode. The difference is in the expression for the Schottky diode saturation current I S (V,T). The expressions for a Schottky and p/n junction diodes saturation currents are different, and the equation for a Schottky diode&#39;s saturation current is given by, 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       S 
                     
                     ⁡ 
                     
                       ( 
                       
                         V 
                         , 
                         T 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       AA 
                       ** 
                     
                     ⁢ 
                     
                       T 
                       2 
                     
                     ⁢ 
                     
                       exp 
                       ⁡ 
                       
                         [ 
                         
                           
                             q 
                             kT 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   Φ 
                                   MS 
                                 
                               
                               + 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     where A is the Schottky diode area, A** is the Richardson constant, Φ MS  is the Schottky barrier height and Δφ is the amount the Schottky barrier height is lowered with reverse bias. The expression for the amount of Schottky barrier lowering with reverse bias is given by, 
     
       
         
           
             
               
                 
                   Δϕ 
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               q 
                               0.75 
                             
                             ⁢ 
                             
                               N 
                               D 
                               0.25 
                             
                           
                           
                             
                               2 
                               0.75 
                             
                             ⁢ 
                             
                               π 
                               0.5 
                             
                             ⁢ 
                             
                               ɛ 
                               S 
                               0.75 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             Φ 
                             MS 
                           
                           - 
                           V 
                           - 
                           
                             
                               ( 
                               
                                 kT 
                                 / 
                                 q 
                               
                               ) 
                             
                             ⁢ 
                             
                               Ln 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     N 
                                     C 
                                   
                                   / 
                                   
                                     N 
                                     D 
                                   
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             ( 
                             
                               kT 
                               / 
                               q 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                     0.25 
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Where, ε S , N C , N D  and V are, respectively, is the dielectric constant for Silicon, the effective density of states in the conduction band, the donor doping level in the substrate, and the Schottky&#39;s reverse bias voltage. 
     From Equation (39) it is evident that the effective Schottky barrier Φ MS  will be reduced with higher silicon substrates doping N D  and larger reverse bias voltage V. Thus, the Schottky saturation current I S (V,T) will increase with reverse bias and temperature. 
     The saturation current varies differently in p/n junction diodes than in Schottky diodes. The saturation current depends on intrinsic and extrinsic parameters. Unlike invariant intrinsic parameters, extrinsic parameters (like process variations) contribute to device variations and result in temperature measurement errors. A p/n junction diode&#39;s saturation current will vary directly with the extrinsic parameters: junction area A and doping concentration as 1/N D . A Schottky diode&#39;s saturation current will vary directly with the extrinsic junction area parameter of A, but only very weakly with the extrinsic parameter of doping concentration N D . The very weak variation with N D  is because the numerical coefficient in Equation (39) for [N D ] 0.25  is 8.16×10 −8 , and this is much smaller than the coefficient in a p/n diode that for 1/N D , equal to about 5. Thus the extrinsic contributor to Schottky diode saturation current is the junction area, and the substrate doping variations are neglected. 
     The second advantage offered by a Schottky diode is an adjustable saturation current with the Schottky barrier Φ MS , something that cannot be done with conventional p/n junction diodes. 
     Adjusting the Schottky diodes saturation current with barrier Φ MS  offers two very important advantages. First there is no need for a adding a separate current generator circuit because the Schottky diode acts as its own current generator. Second, the sensitivity of temperature measurements is adjustable with the Schottky diode&#39;s saturation current. The temperature sensitivity at the reverse bias Schottky temperature sensor depends on the noise from the diode saturation current I S (V,T) and the current signal determined from changes in the diode&#39;s saturation current with temperature, ∂I S (V,T)/∂T. The Schottky diode noise temperature resolution ΔT Diode  is given as, 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       Diode 
                     
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           
                             qI 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             
                               V 
                               , 
                               T 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           Δ 
                           BW 
                         
                       
                     
                     
                       
                         ∂ 
                         
                           
                             I 
                             S 
                           
                           ⁡ 
                           
                             ( 
                             
                               V 
                               , 
                               T 
                             
                             ) 
                           
                         
                       
                       / 
                       
                         ∂ 
                         T 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     Temperature resolution at the Schottky temperature sensor depends directly on the variations in the saturation current and improves with minimum electrical bandwidth Δ BW  and inversely with the value of the saturation current. Specifically, the temperature resolution ΔT Diode  improves as the reciprocal of the saturation current&#39;s square root. This dependence is deduced by recognizing that in Equation (40) the numerator varies as square root of the Schottky diode area “A” and the denominator varied linearly with diode area “A.” Since the Schottky diode saturation current varies directly with area, the sensitivity improves with [A] −0.5 . 
     The value selected for the Schottky diode saturation current depends on the denominator in Equation (40). The temperature dependence of the Schottky saturation current is computed from the derivative of Equation (38) and this is given by, 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           I 
                           S 
                         
                         ⁡ 
                         
                           ( 
                           
                             V 
                             , 
                             T 
                           
                           ) 
                         
                       
                     
                     / 
                     
                       ∂ 
                       T 
                     
                   
                   = 
                   
                     
                       
                         I 
                         S 
                       
                       ⁡ 
                       
                         ( 
                         
                           V 
                           , 
                           T 
                         
                         ) 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           2 
                           T 
                         
                         + 
                         
                           
                             q 
                             
                               kT 
                               
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Φ 
                                 MS 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ϕ 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             
                               ∂ 
                               Δ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                           
                             ∂ 
                             T 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     The saturation current&#39;s temperature dependence scales directly with the saturation current&#39;s amplitude multiplied by three terms (see right side of Equation (41)). The expression for the last term in Equation (41) is obtained by differentiating Equation (41), and the following is obtained, 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         Δ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ϕ 
                     
                     
                       ∂ 
                       T 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             k 
                             
                               4 
                               ⁢ 
                               q 
                             
                           
                           [ 
                           
                             
                               
                                 q 
                                 0.75 
                               
                               ⁢ 
                               
                                 N 
                                 D 
                                 0.25 
                               
                             
                             
                               
                                 2 
                                 0.75 
                               
                               ⁢ 
                               
                                 π 
                                 0.5 
                               
                               ⁢ 
                               
                                 ɛ 
                                 S 
                                 0.75 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               Ln 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     N 
                                     C 
                                   
                                   / 
                                   
                                     N 
                                     D 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             1 
                           
                           ] 
                         
                         
                           
                             [ 
                             
                               
                                 Φ 
                                 MS 
                               
                               - 
                               V 
                               - 
                               
                                 
                                   ( 
                                   
                                     kT 
                                     / 
                                     q 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   Ln 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         N 
                                         C 
                                       
                                       / 
                                       
                                         N 
                                         D 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 ( 
                                 
                                   kT 
                                   / 
                                   q 
                                 
                                 ) 
                               
                             
                             ] 
                           
                           0.75 
                         
                       
                     
                     ≈ 
                     
                       3.4 
                       × 
                       
                         10 
                         
                           - 
                           6 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     An approximation for the Schottky diode saturation current&#39;s temperature dependence is obtained with a numerical estimate of the right most factor in Equation (41) leading to a simplified expression given by,
 
∂ I   S ( V,T )/∂ T≅ (0.07) I   S ( V,T )  (43)
 
     The noise limited temperature resolution of the Schottky diode sensors is readily estimated by combining Equations (43) and (42) to obtain a simplified expression for the temperature sensitivity given by, 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         Diode 
                       
                     
                     ≅ 
                     
                       
                         1 
                         0.07 
                       
                       ⁢ 
                       
                         
                           
                             2 
                             ⁢ 
                             q 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               Δ 
                               BW 
                             
                           
                           
                             
                               I 
                               S 
                             
                             ⁡ 
                             
                               ( 
                               
                                 V 
                                 , 
                                 T 
                               
                               ) 
                             
                           
                         
                       
                     
                     ≅ 
                     
                       2.5 
                       × 
                       
                         10 
                         
                           - 
                           4 
                         
                       
                       ⁢ 
                       
                         
                           Δ 
                           BW 
                         
                       
                     
                   
                   ⁢ 
                   
                     | 
                     
                       at 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       A 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Designing a Schottky diode with a saturation current of 10 −9  amps and operating in a 60 Hz bandwidth will provide a 2 m° K temperature resolution. The temperature resolution can be improved by narrowing the electrical bandwidth or increasing the saturation current (e.g., to about 50 nA). However, it is desirable to minimize the I 2 R power consumed by the Schottky diode. Accordingly, in the present invention, the saturation current is limited to about 10 −9  amps and the electrical bandwidth is reduced to obtain better temperature resolution. 
     The saturation current temperature signal is changed to a voltage with a large resistive load provided by diode D 2 . The resistive load is obtained by reverse biasing Schottky diode D 2 , with a back biased resistance greater than 10 9  Ohms. The current noise from the resistive load D 2  needs to be much less than the noise current in the temperature sensor D 1 . The expression for the noise current from D 1  is given by the numerator in Equation (44). For a saturation current of 10 −9  amps the noise current is equivalent to a 5×10 7  Ohms resistor. Thus the noise current from D 2  with a 10 9  Ohms dynamic resistance is negligible. 
     A large temperature responsivity voltage signal is obtained from the Schottky diodes D 1  and D 2  connected in series. The responsivity is readily computed with Equation (41). For a le amps saturation current and a D 2  load greater than 10 9  Ohms, the temperature responsivity voltage signal is greater than 70 mV/° K. This temperature responsivity represents almost a 100 times improvement over the temperature responsivity with a forward biased pin junction silicon diode. As a result, in the present invention, the pin diodes are replaced with temperature sensing Schottky diodes operating at a 10 −9  amps saturation current. 
     Schottky diodes D 1 -D 4  preferably have small areas and small saturation current of about 10 −9  Amps Design constrains limit the junction&#39;s diameter to about 2.3 μm. After careful examination of Equations (38) and (40), it was determined that, for 10 −9  Amps, the needed Schottky barrier height [Φ MS ] is equal to 0.53 eV. Such a barrier height is obtained for a TiSi 2  silicide formed over a p-type silicon tub. 
     A bolometer pixel  209  with a normalized electro-thermal feedback circuit according to an embodiment of the present invention is shown inn.  FIG. 9 . Bolometer pixel  209  is similar in structure and operation to the bolometer pixel  109  according to the embodiment shown in  FIG. 7 . However, bolometer pixel  209  is designed to remove error differences due to local diode variations. 
     In the embodiment shown in  FIG. 9 , heat bath stage  114  contains capacitors C 1  and C 2 . Capacitors C 1  and C 2  are connected in series. Capacitors C 1  and C 2  are thermally isolated from diodes D 1  and D 2 , and are held at a constant temperature MB, and provide a voltage reference. Details of the amplifier and electro-thermal feedback circuit are shown in  FIG. 10 . An amplifier  118  amplifies the temperature difference (T D -T IN ) between detector stage  110  and intermediate stage  116 . The inputs of amplifier  118  are connected to the node N 1 , where diodes D 1  and D 2  are connected, and to the node N 2 , where capacitors C 1  and C 2  are connected, respectively. The output of amplifier  118  functions as the USSS pixel  209  readout signal and as a control signal for a heater formed by bias network T 2  and heater field-effect transistor (FET) T 1 . A switch S 2  is connected between nodes N 1  and N 2 . 
     In operation, bolometer pixel  209  operates similarly to bolometer pixel  109  but with the reference voltage at node N 2  being provided by capacitors C 1  and C 2  instead of diodes D 3  and D 4 . In addition, removal of any systematic error due to differences in the diode characteristics is performed with the aid of the two capacitors C 1  and C 2  and the switch S 2 . 
     Ideally, when the temperatures T D  and T IN  of the detector and intermediate stages  110  and  116  are the same signals, the voltage across each of the diodes D 1  and D 2 , connected in series, should be equal. However, differences between the diodes D 1  and D 2  will result in a different voltage across diodes D 1  and D 2 . The operation of the electro-thermal feedback circuit will be compromised if this systematic voltage error, interpreted as a temperature error, is not removed. 
     Diodes D 1  and D 2 , to first order, depend primarily on the junction area and the Schottky barrier height. The Schottky barrier height is intrinsic and the process was adjusted for the uniform formation of TiSi 2 . Thus, variations in the Schottky diode characteristics are expected to be due to extrinsic variations in the diode junction areas. Variations in diode junction area produce errors which are temperature independent. Specifically, as long as the detector and intermediate stages  110  and  116  are at the same temperature, the voltage error does not change. The electro-thermal feedback equalizes the temperatures of the detector and intermediate stages  110  and  116 , and this equalization keeps the voltage error constant. The voltage error due to variations in diode junction areas is recorded with the capacitors C 1  and C 2  when switch S 2  is on. Switch S 2  is turned on momentarily to connect and record the voltage at the junction between the two Schottky diodes D 1  and D 2  on the capacitors C 1  and C 2 . 
     The calibration to remove the voltage errors due to variations in Schottky diode areas has to be performed when the detector and intermediate stages  110  and  116  are at the same temperature (i.e., T D =T IN ). This is mechanized by turning off the heater and shuttering the radiation on the pixel  209 . The heater is turned off when switch S 1  is turned on and the incident radiation can be shuttered with a chopper or an electrical switch to turn off the microantenna. The chopper may be a mechanical chopper or an electrical chopper. Thus when switch S 2  is turned on, the error voltage is recorded on the capacitors C 1  and C 2 . 
     The recording of the error voltage is performed as part of operating pixel  209 . Before each integration time, the calibration step described above is performed. The calibration is completed by opening switch S 2  first and then the heater switch S 1 . Next, the shutter is opened and the electro-thermal feedback loop resumes its operation. Any errors in the bridge formed by diodes D 1  and D 2  and capacitors C 1  and C 2  are removed by the time the feedback loop&#39;s operation is resumed. 
     It is noted that, because of the large thermal time constants, the electro-thermal feedback circuit never reaches steady state operation. Operation in a transient mode alleviates the issue of oscillation in the electro-thermal feedback loop. 
     In  FIG. 10 , a bolometer pixel  209  with a normalized electro-thermal feedback circuit according to an embodiment of the present invention is shown. Bolometer pixel  209  is similar in structure and operation to the bolometer pixel  109  shown in  FIG. 8 . However, the bolometer pixel  209  shown in  FIG. 10  is designed to remove error differences due to local diode variations in a manner similar to the bolometer pixel  209  shown in  FIG. 9 . As shown in  FIG. 10 , nodes N 1  and N 2  are connected by transistor T 0 , which acts as a switch controlled by the row calibration signal Row Cal. As a result, when the row calibration signal Row Cal turns transistor TO on, calibration to remove the voltage errors due to variations in Schottky diode areas of diodes D 1  and D 2  is performed by recording the error voltage on the capacitors C 1  and C 2 . 
     While this invention has been described in conjunction with specific embodiments thereof, many alternatives, modifications and variations will be apparent to those skilled in the art. For example, while the present invention is preferably fabricated in silicon using well-known semiconductor fabrication techniques, embodiments of the present invention may be realized in semiconductor materials other than silicon. In addition, the diodes of the present invention may be Schottky, pin or other known types Also, while the present invention has been described as a bolometer, the electro-thermal feedback circuit can be used in any device requiring a stage to be isolated from a heat bath. Accordingly, the preferred embodiments of the invention as set forth herein, are intended to be illustrative, not limiting. Various changes may be made without departing from the true spirit and full scope of the invention as set forth herein.