Optimizing the Energy Resolution of a Detector with Nonlinear Response and Non-Stationary Noise Goddard Space Flight Center DJ Fixsen (UMD) SH Moseley (GSFC) SE Busch (GSFC) SW Nam (NIST) T Gerrits (NIST) A Lita (NIST) B Calkins (NIST) A Migdall (NIST)

Why Nonlinear? Optimizing the Detector Goddard Space Flight Center Energy resolution dE=kT sqrt(C) Lower Temperature Lower Heat Capacity Good electronics Maximize absorption

Laser + DetectorExperiment Goddard Space Flight Center Laser + filters + Detector + Electronics (1.55 um) (few  ) (TES) (10 MHz)

“Typical Pulse” Goddard Space Flight Center Sharp Rise Exponential Decay Noise

Histogram of fit to “Average” Pulse Goddard Space Flight Center Energy resolution  E=.08 eV Hidden assumptions A: Noise is White B: Noise is Stationary C: Signal is linear E=.S

Noise has 1/f Component Goddard Space Flight Center Wiener Filter or fit in Fourier Domain Deals only with the non-white aspect of Noise

Weight is inverse of Covariance Matrix Goddard Space Flight Center E=P W S/(P W P)

Improved Results Goddard Space Flight Center Energy resolution  E=.09 eV Trivial to rescale to, Hz, J…

Extend to Higher Energy Goddard Space Flight Center Problem Signal is non-linear

Extend to Higher Energy Goddard Space Flight Center Pulses 1-16 Signal is non-linear

Nonlinear means Pulses Change Shape Goddard Space Flight Center Effect of 16 th photon is different from 1 st

Fit to  P at P=9 Goddard Space Flight Center Accounted for Pulse shape  E=.13 eV Photon But what about Noise? E=(P 9 -P 8 ).W.S

Covariances Change too Goddard Space Flight Center W 0 is no longer optimal at P=9 Use W 9 instead

Optimum Fit Goddard Space Flight Center Energy resolution  E=.13 eV 9 photons

Goddard Space Flight Center Energy is one dimension in a 256 dimensional space.

Goddard Space Flight Center But larger signal is not aligned with the low signal direction P2 P1 P3 P4 P5 P6 P7 P8

Goddard Space Flight Center Want to look at dE rather than total energy

Goddard Space Flight Center E is curve in 256 dimensional space. This is what Differential Geometry was designed for.

Differential Geometry Goddard Space Flight Center Fit is in Local Tangent Space Fit is closest Distance in sub-space General m dimensional model in n dimensional space. Metric is Covariance\Weight Matrix Model M(P) to fit D i Minimize [D i -M i (P  )] Minimize [D i -M i (P  )]W ij [D j -M j (P  )] Minimize [D i -M i,  P  ]W ij [D j -M j, P )] Solution P  =M i,  W ij D j G  =(M i,  W ij M j, ) -1

Differential Geometry Goddard Space Flight Center Fit is in Local Tangent Space Fit is closest Distance in sub-space D i D i =  2 +P  P  Warning: Metric is Covariance\Weight Matrix You need lots of clean covariance data Handles multi-dimensional data well

Conclusion Goddard Space Flight Center It also simplifies to well known weighted fit in the linear case. Differential Geometry provides convenient and correct framework to optimally fit in non- white, nonlinear signal and nonstationary noise Differential Geometry provides convenient “Picture” of model fitting.