Apodizing functions are used in Fourier transform spectroscopy (FTS) to reduce the magnitude of the sidelobes in the instrumental line shape (ILS), which are a direct result of the finite maximum optical path difference in the measured interferogram. Three apodizing functions, which are considered optimal in the sense of producing the smallest loss in spectral resolution for a given reduction in the magnitude of the largest sidelobe, find frequent use in FTS [J. Opt. Soc. Am. 66, 259 (1976)
]. We extend this series to include optimal apodizing functions corresponding to increases in the width of the ILS ranging from factors of 1.1 to 2.0 compared with its unapodized value, and we compare the results with other commonly used apodizing functions.
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.
Coefficients of the Original Norton–Beer Apodizing Functions
FWHM
1.0
1
0
0
0
1.2
0.384093
0.703484
0
1.4
0.152442
0.983734
0
1.6
0.045335
0
0.554883
0.399782
Table 2
Coefficients, , of the Extended Norton–Beer Apodizing Functions
FWHM
1.1
0.701551
0.937693
0
0
0
1.2
0.396430
0.754472
0
0
0
1.3
0.237413
0.827872
0
0
0
1.4
0.153945
0.987820
0
0
0
1.5
0.077112
0
0.703371
0.219517
0
0
1.6
0.039234
0
0.630268
0.234934
0.095563
0
1.7
0.020078
0
0.480667
0.386409
0.112845
0
1.8
0.010172
0
0.344429
0.451817
0.193580
0
1.9
0.004773
0
0.232473
0.464562
0.298191
0
2.0
0.002267
0
0.140412
0.487172
0.256200
0.113948
Table 3
FWHM, Relative Height with Respect to the Peak of the ILS,aand Position in Units of bof the First Five Minima of the Apodizing Functions Presented in This Paper
Relative FWHM
FWHM
1.0
0.60364
0.715525
1.736325
2.742149
3.745112
4.747042
1.1
0.66420
0.750012
1.713789
2.726237
3.733150
4.737500
1.2
0.72424
0.815562
1.734427
2.737289
3.740819
4.743405
1.3
0.78458
0.893129
1.747848
2.741476
3.743089
4.744947
1.4
0.84468
0.990710
1.744366
2.731459
3.734965
4.738350
1.5
0.90512
1.093694
1.794395
2.755126
3.750099
4.749515
1.6
0.96542
1.201594
1.885519
2.770838
3.754891
4.751677
1.7
1.02550
1.324419
1.974661
2.788058
3.760063
4.754043
1.8
1.08598
1.482119
2.061705
2.819646
3.767627
4.757247
1.9
1.14610
1.644506
2.098222
2.903101
3.783034
4.762789
2.0
1.20662
2.348144
3.776461
4.763836
5.758335
6.755960
Upper value in each pair of rows.
Lower value in each pair of rows.
Table 4
FWHM, Relative Height with Respect to the Peak of the ILS, aand Position in Units of bof the First Five Maxima of the Apodizing Functions Presented in this Paper
Relative FWHM
FWHM
1.0
0.60364
0.128375
0.070914
0.049029
0.037473
0.030332
1.230154
2.239839
3.243821
4.246163
5.247804
1.1
0.66420
0.096291
0.061050
0.043331
0.033434
0.027178
1.206513
2.220976
3.230148
4.235539
5.239143
1.2
0.72424
0.054493
0.037163
0.026814
0.020821
0.016972
1.243239
2.235294
3.239185
4.242212
5.244445
1.3
0.78458
0.027323
0.022755
0.017081
0.013447
0.011037
1.280246
2.242023
3.242146
4.244048
5.245779
1.4
0.84468
0.008161
0.013794
0.011295
0.009135
0.007587
1.328441
2.231925
3.233072
4.236753
5.239763
1.5
0.90512
0.003573
0.006695
0.005802
0.004800
0.004033
1.405000
2.264633
3.251529
4.249592
5.249655
1.6
0.96542
0.002738
0.002413
0.002696
0.002407
0.002089
1.505329
2.301476
3.259629
4.252705
5.251229
1.7
1.02550
0.001252
0.000782
0.001217
0.001189
0.001071
1.615708
2.343501
3.268347
4.256090
5.252965
1.8
1.08598
0.000555
0.000212
0.000501
0.000561
0.000532
1.691538
2.419983
3.282272
4.260803
5.255272
1.9
1.14610
0.000102
0.000151
0.000223
0.000232
1.830878
2.539345
3.315750
4.269465
5.259122
2.0
1.20662
0.000058
0.000097
0.000106
0.000104
0.000098
3.284766
4.268922
5.260508
6.256902
7.255372
Upper value in each pair of rows.
Lower value in each pair of rows.
Table 5
FWHM of ILS Relative to the Sinc and the Magnitude of the Largest Secondary Lobe Relative to the Maximum, Expressed as a Percentage, for the Extended Norton–Beer Apodizing Functions
FWHM
Magnitude of Largest Secondary Lobe
1.0
21.723%
1.1
9.631%
1.2
5.504%
1.3
2.732%
1.4
1.389%
1.5
0.674%
1.6
0.276%
1.7
0.130%
1.8
0.056%
1.9
0.028%
2.0
0.011%
Tables (5)
Table 1
Coefficients of the Original Norton–Beer Apodizing Functions
FWHM
1.0
1
0
0
0
1.2
0.384093
0.703484
0
1.4
0.152442
0.983734
0
1.6
0.045335
0
0.554883
0.399782
Table 2
Coefficients, , of the Extended Norton–Beer Apodizing Functions
FWHM
1.1
0.701551
0.937693
0
0
0
1.2
0.396430
0.754472
0
0
0
1.3
0.237413
0.827872
0
0
0
1.4
0.153945
0.987820
0
0
0
1.5
0.077112
0
0.703371
0.219517
0
0
1.6
0.039234
0
0.630268
0.234934
0.095563
0
1.7
0.020078
0
0.480667
0.386409
0.112845
0
1.8
0.010172
0
0.344429
0.451817
0.193580
0
1.9
0.004773
0
0.232473
0.464562
0.298191
0
2.0
0.002267
0
0.140412
0.487172
0.256200
0.113948
Table 3
FWHM, Relative Height with Respect to the Peak of the ILS,aand Position in Units of bof the First Five Minima of the Apodizing Functions Presented in This Paper
Relative FWHM
FWHM
1.0
0.60364
0.715525
1.736325
2.742149
3.745112
4.747042
1.1
0.66420
0.750012
1.713789
2.726237
3.733150
4.737500
1.2
0.72424
0.815562
1.734427
2.737289
3.740819
4.743405
1.3
0.78458
0.893129
1.747848
2.741476
3.743089
4.744947
1.4
0.84468
0.990710
1.744366
2.731459
3.734965
4.738350
1.5
0.90512
1.093694
1.794395
2.755126
3.750099
4.749515
1.6
0.96542
1.201594
1.885519
2.770838
3.754891
4.751677
1.7
1.02550
1.324419
1.974661
2.788058
3.760063
4.754043
1.8
1.08598
1.482119
2.061705
2.819646
3.767627
4.757247
1.9
1.14610
1.644506
2.098222
2.903101
3.783034
4.762789
2.0
1.20662
2.348144
3.776461
4.763836
5.758335
6.755960
Upper value in each pair of rows.
Lower value in each pair of rows.
Table 4
FWHM, Relative Height with Respect to the Peak of the ILS, aand Position in Units of bof the First Five Maxima of the Apodizing Functions Presented in this Paper
Relative FWHM
FWHM
1.0
0.60364
0.128375
0.070914
0.049029
0.037473
0.030332
1.230154
2.239839
3.243821
4.246163
5.247804
1.1
0.66420
0.096291
0.061050
0.043331
0.033434
0.027178
1.206513
2.220976
3.230148
4.235539
5.239143
1.2
0.72424
0.054493
0.037163
0.026814
0.020821
0.016972
1.243239
2.235294
3.239185
4.242212
5.244445
1.3
0.78458
0.027323
0.022755
0.017081
0.013447
0.011037
1.280246
2.242023
3.242146
4.244048
5.245779
1.4
0.84468
0.008161
0.013794
0.011295
0.009135
0.007587
1.328441
2.231925
3.233072
4.236753
5.239763
1.5
0.90512
0.003573
0.006695
0.005802
0.004800
0.004033
1.405000
2.264633
3.251529
4.249592
5.249655
1.6
0.96542
0.002738
0.002413
0.002696
0.002407
0.002089
1.505329
2.301476
3.259629
4.252705
5.251229
1.7
1.02550
0.001252
0.000782
0.001217
0.001189
0.001071
1.615708
2.343501
3.268347
4.256090
5.252965
1.8
1.08598
0.000555
0.000212
0.000501
0.000561
0.000532
1.691538
2.419983
3.282272
4.260803
5.255272
1.9
1.14610
0.000102
0.000151
0.000223
0.000232
1.830878
2.539345
3.315750
4.269465
5.259122
2.0
1.20662
0.000058
0.000097
0.000106
0.000104
0.000098
3.284766
4.268922
5.260508
6.256902
7.255372
Upper value in each pair of rows.
Lower value in each pair of rows.
Table 5
FWHM of ILS Relative to the Sinc and the Magnitude of the Largest Secondary Lobe Relative to the Maximum, Expressed as a Percentage, for the Extended Norton–Beer Apodizing Functions