The transverse modes of the lasers used in coherent anti-Stokes Raman spectroscopy (CARS) affect the signal strength, spatial resolution, and noise. A general formulation for CARS signal calculations with arbitrary TEM modes and 3-D phase matching is presented. Numerical calculations for useful beam geometries are reported and compared with calculations based on simple models. The variation of intensity and phase across a laser beam is shown to have significant effect on the CARS signal strength. The effects of the transverse modes on window damage are discussed. Estimates of CARS signal strengths are presented for nitrogen thermometry near windows.
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The geometric HG and plane wave HPW estimates of the signal are also compared with the true value HT. For the 25- and 50-μm beams (∑I)max is 3.1E5 and 7.6E4 cm−2, respectively.
Table II
Relative CARS Signal Strength HT for the Case Where Beams One and Three are TEM00 and Beam Two is TEM33a
R (μm)
θ2 (rad)
HT (cm−2)
HG/ Ht
HPW/ HT
L90 (cm)
25
0.10
5.56E4
14.4
117
0.043
25
0.05
3.01E5
10.7
86
0.091
25
0.03
2.79E6
3.0
26
0.094
25
0.01
1.02E6
35.4
636
0.21
50
0.10
4.61E3
43.2
352
0.077
50
0.05
5.55E4
14.4
117
0.17
50
0.03
2.17E5
10.2
83
0.31
50
0.01
4.45E6
3.8
36
0.87
The waist of beam two is half the waist of the other two beams so all three beams have the same nominal radii. The geometric HG and plane wave HPW estimates of the signal are also compared with the true value HT. For the 25- and 50-μm beams, (∑I)max is 2.5E5 and 6.3E4 cm−2, respectively.
Table III
Relative CARS Signal Strength HT for the Case Where All Three Beams are TEM33 (the Waists are Half the Nominal Radii Shown)a
R (μm)
θ2 (rad)
Ht (cm−2)
HG/ Ht
HPW/ HT
L90 (cm)
25
0.10
2.60E5
1.8
25
0.030
25
0.05
1.26E6
1.5
21
0.10
25
0.03
3.04E6
1.8
24
0.17
25
0.01
1.80E7
1.5
36
0.10
50
0.10
6.31E4
1.9
26
0.059
50
0.05
2.59E5
1.8
25
0.12
50
0.03
8.16E5
1.6
22
0.30
50
0.01
6.86E6
1.7
24
0.96
The geometric HG and plane wave HPW estimates of the signal are also compared to the true value HT. For the 25- and 50-μm beams, (∑I)max is 4.5E5 and 1.1E5 cm−2, respectively.
Table IV
Comparison of CARS Signal Strength HT for BOXCARS and USED CARS with Similar Spatial Resolution L90 and Intensity (∑I)maxa
HT (cm−2)
HG/ Ht
L90 (cm)
(∑I)max (cm−2)
BOXCARS
2.59E9
1.0
0.35
3.0E6
USED CARS a
1.79E9
1.4
0.29
3.2E6
USED CARS b
4.96E8
2.2
0.21
2.7E6
For the BOXCARS, all three beams had 8-μm waists, and θ2 was 0.005. For the USED CARS, beams 1 and 3 were each made of a sum of TEM00 modes with waists of 10 and 10.0001 μm. (The intensity profiles are given in Fig. 9). Beam 2 was TEM00 with a waist of 10 μm for case a and 20 μm for case b.
Table V
Base Line Nitrogen CARS Signals (Detector Counts) for Plane Waves; Parameters are Given in the Text
P (atm)
T (K)
1
10
20
40
60
100
300
2.41E5
3.61E7
1.44E8
6.69E8
1.78E9
6.41E9
800
1.39E4
9.39E5
3.00E6
1.05E7
2.26E7
6.27E7
1300
3.05E3
1.33E5
4.14E5
1.33E6
2.72E6
7.05E6
1800
9.24E2
2.99E4
9.23E4
2.83E5
5.61E5
1.38E6
2300
3.35E2
1.06E4
2.56E4
7.78E4
1.52E5
3.65E5
2800
1.40E2
5.45E3
8.63E3
2.49E4
4.82E4
1.15E5
Table VI
Peak Nitrogen CARS Signal (Detector Counts) Assuming TEM00 Beams with the Measurement Volume 1 cm from a Windowa
P (atm)
T (K)
1
10
20
40
60
100
300
1.23E4
1.04E5
1.75E5
3.43E5
5.52E5
1.05E6
800
2.42E3
9.20E3
1.24E4
1.83E4
2.37E4
3.48E4
1300
9.72E2
2.39E3
3.13E3
4.24E3
5.23E3
7.17E3
1800
4.42E2
8.06E2
1.05E3
1.35E3
1.62E3
2.10E3
2300
2.18E2
3.89E2
3.95E2
5.05E2
5.95E2
7.55E2
2800
1.18E2
2.55E2
1.70E2
2.07E2
2.41E2
3.04E2
All parameters are the same as for the base line case in Table V except the optical throughput is 0.1, and the beam waists and energies are given in Tables VII and VIII. The window fluence limit and the gas breakdown intensity are given in the text.
Table VII
Optimum Beam Waists in μm for Producing the CARS Signals of Table VI
P (atm)
T (K)
1
10
20
40
60
100
300
10.1
13.4
14.7
16.0
16.9
18.1
800
8.9
11.9
13.0
14.2
14.9
15.9
1300
8.4
11.2
12.2
13.3
14.0
15.0
1800
8.0
10.7
11.7
12.8
13.4
14.3
2300
7.8
10.4
11.4
12.4
13.0
13.9
2800
7.6
10.1
11.1
12.1
12.7
13.6
Table VIII
Optimum Beam Energies in Millijoules Associated with Table VI
The geometric HG and plane wave HPW estimates of the signal are also compared with the true value HT. For the 25- and 50-μm beams (∑I)max is 3.1E5 and 7.6E4 cm−2, respectively.
Table II
Relative CARS Signal Strength HT for the Case Where Beams One and Three are TEM00 and Beam Two is TEM33a
R (μm)
θ2 (rad)
HT (cm−2)
HG/ Ht
HPW/ HT
L90 (cm)
25
0.10
5.56E4
14.4
117
0.043
25
0.05
3.01E5
10.7
86
0.091
25
0.03
2.79E6
3.0
26
0.094
25
0.01
1.02E6
35.4
636
0.21
50
0.10
4.61E3
43.2
352
0.077
50
0.05
5.55E4
14.4
117
0.17
50
0.03
2.17E5
10.2
83
0.31
50
0.01
4.45E6
3.8
36
0.87
The waist of beam two is half the waist of the other two beams so all three beams have the same nominal radii. The geometric HG and plane wave HPW estimates of the signal are also compared with the true value HT. For the 25- and 50-μm beams, (∑I)max is 2.5E5 and 6.3E4 cm−2, respectively.
Table III
Relative CARS Signal Strength HT for the Case Where All Three Beams are TEM33 (the Waists are Half the Nominal Radii Shown)a
R (μm)
θ2 (rad)
Ht (cm−2)
HG/ Ht
HPW/ HT
L90 (cm)
25
0.10
2.60E5
1.8
25
0.030
25
0.05
1.26E6
1.5
21
0.10
25
0.03
3.04E6
1.8
24
0.17
25
0.01
1.80E7
1.5
36
0.10
50
0.10
6.31E4
1.9
26
0.059
50
0.05
2.59E5
1.8
25
0.12
50
0.03
8.16E5
1.6
22
0.30
50
0.01
6.86E6
1.7
24
0.96
The geometric HG and plane wave HPW estimates of the signal are also compared to the true value HT. For the 25- and 50-μm beams, (∑I)max is 4.5E5 and 1.1E5 cm−2, respectively.
Table IV
Comparison of CARS Signal Strength HT for BOXCARS and USED CARS with Similar Spatial Resolution L90 and Intensity (∑I)maxa
HT (cm−2)
HG/ Ht
L90 (cm)
(∑I)max (cm−2)
BOXCARS
2.59E9
1.0
0.35
3.0E6
USED CARS a
1.79E9
1.4
0.29
3.2E6
USED CARS b
4.96E8
2.2
0.21
2.7E6
For the BOXCARS, all three beams had 8-μm waists, and θ2 was 0.005. For the USED CARS, beams 1 and 3 were each made of a sum of TEM00 modes with waists of 10 and 10.0001 μm. (The intensity profiles are given in Fig. 9). Beam 2 was TEM00 with a waist of 10 μm for case a and 20 μm for case b.
Table V
Base Line Nitrogen CARS Signals (Detector Counts) for Plane Waves; Parameters are Given in the Text
P (atm)
T (K)
1
10
20
40
60
100
300
2.41E5
3.61E7
1.44E8
6.69E8
1.78E9
6.41E9
800
1.39E4
9.39E5
3.00E6
1.05E7
2.26E7
6.27E7
1300
3.05E3
1.33E5
4.14E5
1.33E6
2.72E6
7.05E6
1800
9.24E2
2.99E4
9.23E4
2.83E5
5.61E5
1.38E6
2300
3.35E2
1.06E4
2.56E4
7.78E4
1.52E5
3.65E5
2800
1.40E2
5.45E3
8.63E3
2.49E4
4.82E4
1.15E5
Table VI
Peak Nitrogen CARS Signal (Detector Counts) Assuming TEM00 Beams with the Measurement Volume 1 cm from a Windowa
P (atm)
T (K)
1
10
20
40
60
100
300
1.23E4
1.04E5
1.75E5
3.43E5
5.52E5
1.05E6
800
2.42E3
9.20E3
1.24E4
1.83E4
2.37E4
3.48E4
1300
9.72E2
2.39E3
3.13E3
4.24E3
5.23E3
7.17E3
1800
4.42E2
8.06E2
1.05E3
1.35E3
1.62E3
2.10E3
2300
2.18E2
3.89E2
3.95E2
5.05E2
5.95E2
7.55E2
2800
1.18E2
2.55E2
1.70E2
2.07E2
2.41E2
3.04E2
All parameters are the same as for the base line case in Table V except the optical throughput is 0.1, and the beam waists and energies are given in Tables VII and VIII. The window fluence limit and the gas breakdown intensity are given in the text.
Table VII
Optimum Beam Waists in μm for Producing the CARS Signals of Table VI
P (atm)
T (K)
1
10
20
40
60
100
300
10.1
13.4
14.7
16.0
16.9
18.1
800
8.9
11.9
13.0
14.2
14.9
15.9
1300
8.4
11.2
12.2
13.3
14.0
15.0
1800
8.0
10.7
11.7
12.8
13.4
14.3
2300
7.8
10.4
11.4
12.4
13.0
13.9
2800
7.6
10.1
11.1
12.1
12.7
13.6
Table VIII
Optimum Beam Energies in Millijoules Associated with Table VI