Kevin J. Coakley, Jolene Splett, and Thomas Gerrits, "Mixture model analysis of transition edge sensor pulse height spectra," J. Opt. Soc. Am. B 39, 137-144 (2022)
To calibrate an optical transition edge sensor, for each pulse of the light source (e.g., pulsed laser), one must determine the ratio of the expected number of photons that deposit energy and the expected number of photons created by the laser. Based on the estimated pulse height generated by each energy deposit, we form a pulse height spectrum with features corresponding to different numbers of deposited photons. We model the number of photons that deposit energy per laser pulse as a realization of a Poisson process, and the observed pulse height spectrum with a mixture model method. For each candidate feature set, we determine the expected number of photons that deposit energy per pulse and its associated uncertainty based on the mixture model weights corresponding to that candidate feature set. From training data, we select the optimal feature set according to an uncertainty minimization criterion. We then determine the expected number of photons that deposit energy per pulse and its associated uncertainty for test data that are independent of the training data. Our uncertainty budget accounts for random measurement errors, systematic effects due to mismodeling feature shapes in our mixture model, and possible imperfections in our feature set selection method.
Zachary H. Levine, Boris L. Glebov, Alan L. Migdall, Thomas Gerrits, Brice Calkins, Adriana E. Lita, and Sae Woo Nam J. Opt. Soc. Am. B 31(10) B20-B24 (2014)
Zachary H. Levine, Thomas Gerrits, Alan L. Migdall, Daniel V. Samarov, Brice Calkins, Adriana E. Lita, and Sae Woo Nam J. Opt. Soc. Am. B 29(8) 2066-2073 (2012)
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Cited By
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Normal Mixture Model Results for Filtered Experiment A Dataa
Feature
(mV)
(mV)
Weight
Training data
0-photon
19.90(03)
10.87(02)
0.22006(45)
1-photon
89.79(03)
15.02(03)
0.33812(53)
2-photon
165.71(04)
14.89(03)
0.25308(51)
3-photon
233.14(06)
14.50(05)
0.12652(41)
4-photon
292.32(10)
14.24(11)
0.04899(29)
5-photon
343.62(17)
11.28(10)
0.01323(15)
Test data
0-photon
19.88(03)
10.91(02)
0.22080(49)
1-photon
89.73(03)
15.03(03)
0.33802(53)
2-photon
165.74(04)
14.95(03)
0.25204(49)
3-photon
233.26(06)
14.46(05)
0.12720(40)
4-photon
292.71(10)
14.45(12)
0.04885(29)
5-photon
343.92(18)
11.12(11)
0.01308(15)
Bootstrap standard errors (components of uncertainty due to random measurement errors) are shown in parentheses for each estimate. For instance, 0.22006(45) means that the estimate and its associated bootstrap standard error are 0.22006 and 0.00045, respectively.
Table 2.
Six-Feature Normal Mixture Model Results for Filtered Experiment A Dataa
Feature Set Analyzed
Normal Model
Gamma Model
Training data
2-5
1.5048
0.0039
1.5047
0.0001
0.0039
2-4
1.5108
0.0044
1.5083
0.0014
0.0046
3-5
1.5132
0.0076
1.5090
0.0024
0.0080
1-5
1.5001
0.0022
1.5146
0.0083
0.0086
1-4
1.5018
0.0023
1.5163
0.0083
0.0086
0-5
1.5101
0.0015
1.4953
0.0086
0.0087
0-4
1.5114
0.0016
1.4953
0.0093
0.0094
0-3
1.5107
0.0018
1.4924
0.0106
0.0107
1-3
1.4978
0.0028
1.5159
0.0104
0.0108
0-2
1.5153
0.0024
1.4859
0.0170
0.0172
Test data
2-5
1.5114
0.0037
1.5141
0.0016
0.0040
2-4
1.5191
0.0042
1.5182
0.0005
0.0042
Here, we estimate ${\theta _{\text{dep}}}$ for different subsets of the features. The nomenclature for the feature set denotes the range of features from which we determine ${\theta _{\text{dep}}}$. For instance, the feature set “2-4” corresponds to results based on analysis of the weights associated with 2-photon, 3-photon, and 4-photon features. We determine ${\theta _{\text{dep}}}$ from the test data for the feature set that yields the lowest value of ${u_{\text{subtot}}}$ for the training data. From the test data, we also estimate ${\theta _{\text{dep}}}$ from the feature set that yields the second lowest value of the combined uncertainty ${u_{\text{subtot}}}$ for the training data. From the two estimates of ${\theta _{\text{dep}}}$ for the test data, we determine a component of uncertainty due to imperfect performance of our feature selection method. Uncertainty due to imperfect feature selection, ${u_{\text{feature}}}$, for the training and test data are 0.0017 and 0.0022, respectively.
Table 3.
Five-Feature Normal Mixture Model Results for Unfiltered Pulse Height Data Corresponding to Experiment B Dataa
Feature
(mV)
(mV)
Weight
Training data
0-photon
4.629(11)
2.337(06)
0.29460(94)
1-photon
13.945(11)
3.353(12)
0.44021(112)
2-photon
25.776(11)
2.622(11)
0.17815(66)
3-photon
35.399(23)
3.215(39)
0.07259(69)
4-photon
44.364(66)
2.528(29)
0.01444(31)
Test data
0-photon
4.611(11)
2.332(05)
0.29387(92)
1-photon
13.927(11)
3.379(12)
0.44125(112)
2-photon
25.767(12)
2.607(11)
0.17667(70)
3-photon
35.344(25)
3.288(43)
0.07408(74)
4-photon
44.435(73)
2.495(31)
0.01413(33)
Bootstrap standard errors are shown in parentheses.
Table 4.
Five-Feature Normal Mixture Model Results for Unfiltered Experiment B Dataa
Feature Set Analyzed
Normal Model
Gamma Model
Training data
2-4
1.1473
0.0083
1.1327
0.0084
0.0118
1-3
0.8759
0.0031
0.9078
0.0184
0.0187
1-4
0.8814
0.0028
0.9161
0.0200
0.0202
0-4
1.1369
0.0015
1.0852
0.0299
0.0299
0-3
1.1465
0.0017
1.0895
0.0329
0.0330
2-3
1.2225
0.0150
1.1686
0.0311
0.0345
0-2
1.1700
0.0022
1.1018
0.0394
0.0395
3-4
0.7958
0.0233
0.9567
0.0929
0.0958
Test data
2-4
1.1682
0.0088
1.1423
0.0150
0.0174
1-3
0.8737
0.0031
0.9102
0.0211
0.0213
Here, we estimate ${\theta _{\text{dep}}}$ for different subsets of photon features. For the test data, ${u_{\text{feature}}}$ is 0.0850.
Tables (4)
Table 1.
Normal Mixture Model Results for Filtered Experiment A Dataa
Feature
(mV)
(mV)
Weight
Training data
0-photon
19.90(03)
10.87(02)
0.22006(45)
1-photon
89.79(03)
15.02(03)
0.33812(53)
2-photon
165.71(04)
14.89(03)
0.25308(51)
3-photon
233.14(06)
14.50(05)
0.12652(41)
4-photon
292.32(10)
14.24(11)
0.04899(29)
5-photon
343.62(17)
11.28(10)
0.01323(15)
Test data
0-photon
19.88(03)
10.91(02)
0.22080(49)
1-photon
89.73(03)
15.03(03)
0.33802(53)
2-photon
165.74(04)
14.95(03)
0.25204(49)
3-photon
233.26(06)
14.46(05)
0.12720(40)
4-photon
292.71(10)
14.45(12)
0.04885(29)
5-photon
343.92(18)
11.12(11)
0.01308(15)
Bootstrap standard errors (components of uncertainty due to random measurement errors) are shown in parentheses for each estimate. For instance, 0.22006(45) means that the estimate and its associated bootstrap standard error are 0.22006 and 0.00045, respectively.
Table 2.
Six-Feature Normal Mixture Model Results for Filtered Experiment A Dataa
Feature Set Analyzed
Normal Model
Gamma Model
Training data
2-5
1.5048
0.0039
1.5047
0.0001
0.0039
2-4
1.5108
0.0044
1.5083
0.0014
0.0046
3-5
1.5132
0.0076
1.5090
0.0024
0.0080
1-5
1.5001
0.0022
1.5146
0.0083
0.0086
1-4
1.5018
0.0023
1.5163
0.0083
0.0086
0-5
1.5101
0.0015
1.4953
0.0086
0.0087
0-4
1.5114
0.0016
1.4953
0.0093
0.0094
0-3
1.5107
0.0018
1.4924
0.0106
0.0107
1-3
1.4978
0.0028
1.5159
0.0104
0.0108
0-2
1.5153
0.0024
1.4859
0.0170
0.0172
Test data
2-5
1.5114
0.0037
1.5141
0.0016
0.0040
2-4
1.5191
0.0042
1.5182
0.0005
0.0042
Here, we estimate ${\theta _{\text{dep}}}$ for different subsets of the features. The nomenclature for the feature set denotes the range of features from which we determine ${\theta _{\text{dep}}}$. For instance, the feature set “2-4” corresponds to results based on analysis of the weights associated with 2-photon, 3-photon, and 4-photon features. We determine ${\theta _{\text{dep}}}$ from the test data for the feature set that yields the lowest value of ${u_{\text{subtot}}}$ for the training data. From the test data, we also estimate ${\theta _{\text{dep}}}$ from the feature set that yields the second lowest value of the combined uncertainty ${u_{\text{subtot}}}$ for the training data. From the two estimates of ${\theta _{\text{dep}}}$ for the test data, we determine a component of uncertainty due to imperfect performance of our feature selection method. Uncertainty due to imperfect feature selection, ${u_{\text{feature}}}$, for the training and test data are 0.0017 and 0.0022, respectively.
Table 3.
Five-Feature Normal Mixture Model Results for Unfiltered Pulse Height Data Corresponding to Experiment B Dataa
Feature
(mV)
(mV)
Weight
Training data
0-photon
4.629(11)
2.337(06)
0.29460(94)
1-photon
13.945(11)
3.353(12)
0.44021(112)
2-photon
25.776(11)
2.622(11)
0.17815(66)
3-photon
35.399(23)
3.215(39)
0.07259(69)
4-photon
44.364(66)
2.528(29)
0.01444(31)
Test data
0-photon
4.611(11)
2.332(05)
0.29387(92)
1-photon
13.927(11)
3.379(12)
0.44125(112)
2-photon
25.767(12)
2.607(11)
0.17667(70)
3-photon
35.344(25)
3.288(43)
0.07408(74)
4-photon
44.435(73)
2.495(31)
0.01413(33)
Bootstrap standard errors are shown in parentheses.
Table 4.
Five-Feature Normal Mixture Model Results for Unfiltered Experiment B Dataa
Feature Set Analyzed
Normal Model
Gamma Model
Training data
2-4
1.1473
0.0083
1.1327
0.0084
0.0118
1-3
0.8759
0.0031
0.9078
0.0184
0.0187
1-4
0.8814
0.0028
0.9161
0.0200
0.0202
0-4
1.1369
0.0015
1.0852
0.0299
0.0299
0-3
1.1465
0.0017
1.0895
0.0329
0.0330
2-3
1.2225
0.0150
1.1686
0.0311
0.0345
0-2
1.1700
0.0022
1.1018
0.0394
0.0395
3-4
0.7958
0.0233
0.9567
0.0929
0.0958
Test data
2-4
1.1682
0.0088
1.1423
0.0150
0.0174
1-3
0.8737
0.0031
0.9102
0.0211
0.0213
Here, we estimate ${\theta _{\text{dep}}}$ for different subsets of photon features. For the test data, ${u_{\text{feature}}}$ is 0.0850.
Add to BibSonomy
