Metric for the measurement of the quality of complex beams: a theoretical study

Sergiy Kaim; Julien Lumeau; Vadim Smirnov; Boris Zeldovich; Leonid Glebov

doi:10.1364/JOSAA.32.000538

Journal of the Optical Society of America A
Vol. 32,
Issue 4,
pp. 538-548
(2015)
•https://doi.org/10.1364/JOSAA.32.000538

Metric for the measurement of the quality of complex beams: a theoretical study

Sergiy Kaim, Julien Lumeau, Vadim Smirnov, Boris Zeldovich, and Leonid Glebov

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Sergiy Kaim, Julien Lumeau, Vadim Smirnov, Boris Zeldovich, and Leonid Glebov, "Metric for the measurement of the quality of complex beams: a theoretical study," J. Opt. Soc. Am. A 32, 538-548 (2015)

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Abstract

We present a theoretical study of various definitions of laser beam width in a given cross section. Quality of the beam is characterized by dimensionless beam propagation products (BPPs) $Δ x \cdot Δ θ_{x} / λ$ , which are different for the 21 definitions presented, but are close to 1. Six particular beams are studied in detail. In the process, we had to review the properties for the Fourier transform of various modifications and the relationships between them: physical Fourier transform (PFT), mathematical Fourier transform (MFT), and discrete Fourier transform (DFT). We found an axially symmetric self-MFT function, which may be useful for descriptions of diffraction-quality beams. In the appendices, we illustrate the thesis “the Fourier transform lives on the singularities of the original.”

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	Gauss, $E (r) = \exp [- {(r / w)}^{2}]$	Super Gauss, $E (r) = \exp [- {(r / w)}^{4}]$	2D Sech $C_{00} (r / w = u)$	Fiber Mode ${LP}_{01} (r)$ , $V = 2.4$	Round Top Hat of Radius $w$	Factorized 1D Hyperbolic Secant $c_{0} (x / w = u)$
$Δ x$ HWHIM, [ $width / w$ ]	0.5887	0.7677	0.6930	0.6840	1	0.7032
$Δ x$ $HW e^{- 1} IM$ , [ $width / w$ ]	0.7070	0.8409	0.8476	0.8052	1	0.8657
$Δ x$ $HW e^{- 2} IM$ , [ $width / w$ ]	1	1	1.2685	1.0699	1	1.3225
$Δ x$ $HW 10^{- 2} IM$ , [ $width / w$ ]	1.5170	1.2318	2.1956	1.6923	1	2.3882
$θ_{x}$ HWHIM, $angle \cdot (w / λ)$	0.1874	0.2447	0.1099	0.1493	0.2572	0.1119
$θ_{x}$ $HW e^{- 1} IM$ , $angle \cdot (w / λ)$	0.2250	0.2913	0.1345	0.1825	0.3048	0.1378
$θ_{x}$ $HW e^{- 2} IM$ , $angle \cdot (w / λ)$	0.3183	0.3993	0.2012	0.2724	0.4112	0.2105
$θ_{x}$ $HW 10^{- 2} IM$ , $angle \cdot (w / λ)$	0.4829	0.5499	0.3483	0.4518	0.5442	0.3801
$r$ PIB $f = 0.5$ , [ $width / w$ ]	0.5887	0.5807	0.7830	0.6394	0.7070	0.7848
$r$ PIB $f = 0.75$ , [ $width / w$ ]	0.8326	0.7584	1.1551	0.8893	0.8660	1.1591
$r$ PIB $f = 0.865$ , [ $width / w$ ]	1.0006	0.8641	1.4370	1.0699	0.9299	1.4450
$r$ PIB $f = 0.9$ , [ $width / w$ ]	1.0730	0.9069	1.5677	1.1574	0.9487	1.5765
$r$ PIB $f = 0.95$ , [ $width / w$ ]	1.2239	0.99	1.8547	1.3573	0.9747	1.8686
$r$ PIB $f = 0.975$ , [ $width / w$ ]	1.3581	1.0586	2.1304	1.5559	0.9874	2.1509
$r$ PIB $f = 0.99$ , [ $width / w$ ]	1.5174	1.1349	2.4842	1.8161	0.9950	2.5158
$θ$ PIB $f = 0.5$ , $angle \cdot (w / λ)$	0.1874	0.2327	0.1246	0.1654	0.2654	0.1249
$θ$ PIB $f = 0.75$ , $angle \cdot (w / λ)$	0.2650	0.3237	0.1838	0.2414	0.3917	0.1845
$θ$ PIB $f = 0.865$ , $angle \cdot (w / λ)$	0.3185	0.3838	0.2287	0.2963	0.7766	0.2300
$θ$ PIB $f = 0.9$ , $angle \cdot (w / λ)$	0.3415	0.4095	0.2495	0.3208	0.9063	0.2509
$θ$ PIB $f = 0.95$ , $angle \cdot (w / λ)$	0.3896	0.4638	0.2952	0.3720	1.7802	0.2974
$θ$ PIB $f = 0.975$ , $angle \cdot (w / λ)$	0.4323	0.5203	0.3391	0.4175	2.9031	0.3423
$θ$ PIB $f = 0.99$ , $angle \cdot (w / λ)$	0.4830	0.8080	0.3954	0.4705	5.1887	0.4004
$s$ PIS $f = 0.5$ , [ $width / w$ ]	0.3372	0.3357	0.4488	0.3684	0.4040	0.4383
$s$ PIS $f = 0.75$ , [ $width / w$ ]	0.5752	0.5423	0.7864	0.6220	0.6347	0.7763
$s$ PIS $f = 0.865$ , [ $width / w$ ]	0.7467	0.6727	1.0498	0.8031	0.7607	1.0465
$s$ PIS $f = 0.9$ , [ $width / w$ ]	0.8224	0.7257	1.1724	0.8845	0.8054	1.1747
$s$ PIS $f = 0.95$ , [ $width / w$ ]	0.9800	0.8283	1.4425	1.0652	0.8783	1.4615
$s$ PIS $f = 0.975$ , [ $width / w$ ]	1.1207	0.9121	1.7024	1.2463	0.9237	1.7431
$s$ PIS $f = 0.99$ , [ $width / w$ ]	1.2880	1.0037	2.0371	1.4860	0.9587	2.1117
$θ_{s}$ PIS $f = 0.5$ , $angle \cdot (w / λ)$	0.1073	0.1335	0.0714	0.0946	0.1590	0.0697
$θ_{s}$ PIS $f = 0.75$ , $angle \cdot (w / λ)$	0.1831	0.2257	0.1252	0.1647	0.2838	0.1235
$θ_{s}$ PIS $f = 0.865$ , $angle \cdot (w / λ)$	0.2377	0.2907	0.1671	0.2179	0.4194	0.1665
$θ_{s}$ PIS $f = 0.9$ , $angle \cdot (w / λ)$	0.2618	0.3190	0.1866	0.2420	0.6093	0.1869
$θ_{s}$ PIS $f = 0.95$ , $angle \cdot (w / λ)$	0.3119	0.3782	0.2296	0.2935	1.1172	0.2326
$θ_{s}$ PIS $f = 0.975$ , $angle \cdot (w / λ)$	0.3567	0.4343	0.2709	0.3403	1.9249	0.2774
$θ_{s}$ PIS $f = 0.99$ , $angle \cdot (w / λ)$	0.4100	0.5308	0.3242	0.3966	3.7706	0.3361
${〈 x^{2} 〉}^{\frac{1}{2}}$ , $meter / w$	0.5	0.4465	0.7193	0.5509	0.5	0.7236
$〈 \| x \| 〉$ , $meter / w$	0.3989	0.3701	0.5564	0.4363	0.4244	0.5530
${(〈 \sqrt{\| x \|} 〉)}^{2}$ , $meter / w$	0.3380	0.3205	0.4641	0.3693	0.3723	0.4586
${〈 θ_{x}^{2} 〉}^{\frac{1}{2}}$ , $angle \cdot (w / λ)$	0.1591	0.2011	0.1145	0.1474	0.7182	0.1152
$〈 \| θ_{x} \| 〉$ , $angle \cdot (w / λ)$	0.1270	0.1581	0.0885	0.1154	0.3148	0.0880
${(〈 \sqrt{\| θ_{x} \|} 〉)}^{2}$ , $angle \cdot (w / λ)$	0.1076	0.1336	0.0739	0.0968	0.2093	0.0730

$r \cdot θ / λ$ , $s \cdot θ_{x} / λ$ , $M$	Gauss, $E (r) = \exp [- {(r / w)}^{2}]$	Super Gauss, $E (r) = \exp [- {(r / w)}^{4}]$	2D Sech $C_{00} (r)$	Fiber Mode ${LP}_{01} (r)$ , $V = 2.4$	Round Top Hat	Factorized 1D Hyperbolic Secant
HWHIM	0.1103	0.1878	0.0764	0.1021	0.2572	0.0787
$HW e^{- 1} IM$	0.1591	0.2450	0.1143	0.1469	0.3048	0.1193
$HW e^{- 2} IM$	0.3183	0.3993	0.2561	0.2915	0.4112	0.2783
$HW 10^{- 2} IM$	0.7329	0.6774	0.7672	0.7645	0.5443	0.9078
PIB $f = 0.5$	0.1103	0.1351	0.0976	0.1058	0.1877	0.0980
PIB $f = 0.75$	0.2206	0.2455	0.2123	0.2147	0.3392	0.2138
PIB $f = 0.865$	0.3183	0.3316	0.3286	0.3170	0.7221	0.3323
PIB $f = 0.9$	0.3665	0.3714	0.3911	0.3713	0.8598	0.3955
PIB $f = 0.95$	0.4768	0.4591	0.5475	0.505	1.7352	0.5557
PIB $f = 0.975$	0.5871	0.5508	0.7223	0.6496	2.8666	0.7363
PIB $f = 0.99$	0.7329	0.9169	0.9822	0.8544	5.1627	1.0073
PIS $f = 0.5$	0.0362	0.0448	0.0320	0.0348	0.0642	0.0306
PIS $f = 0.75$	0.1053	0.1224	0.0985	0.1025	0.1801	0.0959
PIS $f = 0.865$	0.1775	0.1955	0.1754	0.1750	0.3191	0.1743
PIS $f = 0.9$	0.2153	0.2315	0.2188	0.2141	0.4907	0.2196
PIS $f = 0.95$	0.3057	0.3133	0.3312	0.3126	0.9812	0.3400
PIS $f = 0.975$	0.3998	0.3962	0.4613	0.4241	1.7781	0.4836
PIS $f = 0.99$	0.5280	0.5327	0.6605	0.5893	3.6150	0.7097
$M_{x}^{2} = (4 π / λ) \cdot \sqrt{〈 x^{2} 〉〈 θ_{x}^{2} 〉}$	1	$2 / \sqrt{π} = 1.1281$	1.0349	1.0182	4.5771 ( $\infty$ )	1.0472
$M_{1} = (4 π / λ) \cdot 〈 \| x \| 〉〈 \| θ_{x} \| 〉$	0.6366	0.7349	0.6192	0.6326	1.6784	0.6117
$M_{0.5} = (4 π / λ) {〈 \sqrt{\| x \|} 〉}^{2} {〈 \sqrt{\| θ_{x} \|} 〉}^{2}$	0.4569	0.5381	0.4308	0.4493	0.9787	0.4206

$r \cdot θ / λ$ , $s \cdot θ_{x} / λ$ , $M$	Gauss, $E (r) = \exp [- {(r / w)}^{2}]$	Super Gauss, $E (r) = \exp [- {(r / w)}^{4}]$	2D Sech $C_{00} (r)$	Fiber Mode ${LP}_{01} (r)$ , $V = 2.4$	Round Top Hat	Factorized 1D Hyperbolic Secant
HWHIM	1	1.7026	0.6927	0.9257	2.3318	0.7135
$HW e^{- 1} IM$	1	1.5399	0.7184	0.9233	1.9158	0.7497
$HW e^{- 2} IM$	1	1.2545	0.8046	0.9158	1.2919	0.8744
$HW 10^{- 2} IM$	1	0.9243	1.0468	1.0431	0.7427	1.2392
PIB $f = 0.5$	1	1.2248	0.8849	0.9592	1.7017	0.8887
PIB $f = 0.75$	1	1.1129	0.9624	0.9733	1.5376	0.9693
PIB $f = 0.865$	1	1.0418	1.0324	0.9959	2.2686	1.0440
PIB $f = 0.9$	1	1.0134	1.0671	1.0131	2.346	1.0793
PIB $f = 0.95$	1	0.9629	1.1483	1.0591	3.6393	1.1655
PIB $f = 0.975$	1	0.9382	1.2303	1.1065	4.8826	1.2541
PIB $f = 0.99$	1	1.2511	1.3402	1.1658	7.0442	1.3744
PIS $f = 0.5$	1	1.2376	0.884	0.9613	1.7735	0.8445
PIS $f = 0.75$	1	1.1624	0.9354	0.9734	1.7104	0.9108
PIS $f = 0.865$	1	1.1014	0.9882	0.9859	1.7977	0.9820
PIS $f = 0.9$	1	1.0752	1.0163	0.9944	2.2791	1.0200
PIS $f = 0.95$	1	1.0249	1.0834	1.0226	3.2097	1.1121
PIS $f = 0.975$	1	0.991	1.1538	1.0608	4.4475	1.2095
PIS $f = 0.99$	1	1.0089	1.2509	1.1161	6.8466	1.3440
$M_{x}^{2} = (4 π / λ) \cdot \sqrt{〈 x^{2} 〉〈 θ_{x}^{2} 〉}$	1	1.1281	1.0349	1.0182	4.5771 ( $\infty$ )	1.0472
$M_{1} = (4 π / λ) \cdot 〈 \| x \| 〉〈 \| θ_{x} \| 〉$	1	1.1544	0.9727	0.9937	2.6365	0.9609
$M_{0.5} = (4 π / λ) {〈 \sqrt{\| x \|} 〉}^{2} {〈 \sqrt{\| θ_{x} \|} 〉}^{2}$	1	1.1777	0.9429	0.9834	2.142	0.9205

	Gauss, $E (r) = \exp [- {(r / w)}^{2}]$	Super Gauss, $E (r) = \exp [- {(r / w)}^{4}]$	2D Sech $C_{00} (r / w = u)$	Fiber Mode ${LP}_{01} (r)$ , $V = 2.4$	Round Top Hat of Radius $w$	Factorized 1D Hyperbolic Secant $c_{0} (x / w = u)$
$Δ x$ HWHIM, [ $width / w$ ]	0.5887	0.7677	0.6930	0.6840	1	0.7032
$Δ x$ $HW e^{- 1} IM$ , [ $width / w$ ]	0.7070	0.8409	0.8476	0.8052	1	0.8657
$Δ x$ $HW e^{- 2} IM$ , [ $width / w$ ]	1	1	1.2685	1.0699	1	1.3225
$Δ x$ $HW 10^{- 2} IM$ , [ $width / w$ ]	1.5170	1.2318	2.1956	1.6923	1	2.3882
$θ_{x}$ HWHIM, $angle \cdot (w / λ)$	0.1874	0.2447	0.1099	0.1493	0.2572	0.1119
$θ_{x}$ $HW e^{- 1} IM$ , $angle \cdot (w / λ)$	0.2250	0.2913	0.1345	0.1825	0.3048	0.1378
$θ_{x}$ $HW e^{- 2} IM$ , $angle \cdot (w / λ)$	0.3183	0.3993	0.2012	0.2724	0.4112	0.2105
$θ_{x}$ $HW 10^{- 2} IM$ , $angle \cdot (w / λ)$	0.4829	0.5499	0.3483	0.4518	0.5442	0.3801
$r$ PIB $f = 0.5$ , [ $width / w$ ]	0.5887	0.5807	0.7830	0.6394	0.7070	0.7848
$r$ PIB $f = 0.75$ , [ $width / w$ ]	0.8326	0.7584	1.1551	0.8893	0.8660	1.1591
$r$ PIB $f = 0.865$ , [ $width / w$ ]	1.0006	0.8641	1.4370	1.0699	0.9299	1.4450
$r$ PIB $f = 0.9$ , [ $width / w$ ]	1.0730	0.9069	1.5677	1.1574	0.9487	1.5765
$r$ PIB $f = 0.95$ , [ $width / w$ ]	1.2239	0.99	1.8547	1.3573	0.9747	1.8686
$r$ PIB $f = 0.975$ , [ $width / w$ ]	1.3581	1.0586	2.1304	1.5559	0.9874	2.1509
$r$ PIB $f = 0.99$ , [ $width / w$ ]	1.5174	1.1349	2.4842	1.8161	0.9950	2.5158
$θ$ PIB $f = 0.5$ , $angle \cdot (w / λ)$	0.1874	0.2327	0.1246	0.1654	0.2654	0.1249
$θ$ PIB $f = 0.75$ , $angle \cdot (w / λ)$	0.2650	0.3237	0.1838	0.2414	0.3917	0.1845
$θ$ PIB $f = 0.865$ , $angle \cdot (w / λ)$	0.3185	0.3838	0.2287	0.2963	0.7766	0.2300
$θ$ PIB $f = 0.9$ , $angle \cdot (w / λ)$	0.3415	0.4095	0.2495	0.3208	0.9063	0.2509
$θ$ PIB $f = 0.95$ , $angle \cdot (w / λ)$	0.3896	0.4638	0.2952	0.3720	1.7802	0.2974
$θ$ PIB $f = 0.975$ , $angle \cdot (w / λ)$	0.4323	0.5203	0.3391	0.4175	2.9031	0.3423
$θ$ PIB $f = 0.99$ , $angle \cdot (w / λ)$	0.4830	0.8080	0.3954	0.4705	5.1887	0.4004
$s$ PIS $f = 0.5$ , [ $width / w$ ]	0.3372	0.3357	0.4488	0.3684	0.4040	0.4383
$s$ PIS $f = 0.75$ , [ $width / w$ ]	0.5752	0.5423	0.7864	0.6220	0.6347	0.7763
$s$ PIS $f = 0.865$ , [ $width / w$ ]	0.7467	0.6727	1.0498	0.8031	0.7607	1.0465
$s$ PIS $f = 0.9$ , [ $width / w$ ]	0.8224	0.7257	1.1724	0.8845	0.8054	1.1747
$s$ PIS $f = 0.95$ , [ $width / w$ ]	0.9800	0.8283	1.4425	1.0652	0.8783	1.4615
$s$ PIS $f = 0.975$ , [ $width / w$ ]	1.1207	0.9121	1.7024	1.2463	0.9237	1.7431
$s$ PIS $f = 0.99$ , [ $width / w$ ]	1.2880	1.0037	2.0371	1.4860	0.9587	2.1117
$θ_{s}$ PIS $f = 0.5$ , $angle \cdot (w / λ)$	0.1073	0.1335	0.0714	0.0946	0.1590	0.0697
$θ_{s}$ PIS $f = 0.75$ , $angle \cdot (w / λ)$	0.1831	0.2257	0.1252	0.1647	0.2838	0.1235
$θ_{s}$ PIS $f = 0.865$ , $angle \cdot (w / λ)$	0.2377	0.2907	0.1671	0.2179	0.4194	0.1665
$θ_{s}$ PIS $f = 0.9$ , $angle \cdot (w / λ)$	0.2618	0.3190	0.1866	0.2420	0.6093	0.1869
$θ_{s}$ PIS $f = 0.95$ , $angle \cdot (w / λ)$	0.3119	0.3782	0.2296	0.2935	1.1172	0.2326
$θ_{s}$ PIS $f = 0.975$ , $angle \cdot (w / λ)$	0.3567	0.4343	0.2709	0.3403	1.9249	0.2774
$θ_{s}$ PIS $f = 0.99$ , $angle \cdot (w / λ)$	0.4100	0.5308	0.3242	0.3966	3.7706	0.3361
${〈 x^{2} 〉}^{\frac{1}{2}}$ , $meter / w$	0.5	0.4465	0.7193	0.5509	0.5	0.7236
$〈 \| x \| 〉$ , $meter / w$	0.3989	0.3701	0.5564	0.4363	0.4244	0.5530
${(〈 \sqrt{\| x \|} 〉)}^{2}$ , $meter / w$	0.3380	0.3205	0.4641	0.3693	0.3723	0.4586
${〈 θ_{x}^{2} 〉}^{\frac{1}{2}}$ , $angle \cdot (w / λ)$	0.1591	0.2011	0.1145	0.1474	0.7182	0.1152
$〈 \| θ_{x} \| 〉$ , $angle \cdot (w / λ)$	0.1270	0.1581	0.0885	0.1154	0.3148	0.0880
${(〈 \sqrt{\| θ_{x} \|} 〉)}^{2}$ , $angle \cdot (w / λ)$	0.1076	0.1336	0.0739	0.0968	0.2093	0.0730

$r \cdot θ / λ$ , $s \cdot θ_{x} / λ$ , $M$	Gauss, $E (r) = \exp [- {(r / w)}^{2}]$	Super Gauss, $E (r) = \exp [- {(r / w)}^{4}]$	2D Sech $C_{00} (r)$	Fiber Mode ${LP}_{01} (r)$ , $V = 2.4$	Round Top Hat	Factorized 1D Hyperbolic Secant
HWHIM	0.1103	0.1878	0.0764	0.1021	0.2572	0.0787
$HW e^{- 1} IM$	0.1591	0.2450	0.1143	0.1469	0.3048	0.1193
$HW e^{- 2} IM$	0.3183	0.3993	0.2561	0.2915	0.4112	0.2783
$HW 10^{- 2} IM$	0.7329	0.6774	0.7672	0.7645	0.5443	0.9078
PIB $f = 0.5$	0.1103	0.1351	0.0976	0.1058	0.1877	0.0980
PIB $f = 0.75$	0.2206	0.2455	0.2123	0.2147	0.3392	0.2138
PIB $f = 0.865$	0.3183	0.3316	0.3286	0.3170	0.7221	0.3323
PIB $f = 0.9$	0.3665	0.3714	0.3911	0.3713	0.8598	0.3955
PIB $f = 0.95$	0.4768	0.4591	0.5475	0.505	1.7352	0.5557
PIB $f = 0.975$	0.5871	0.5508	0.7223	0.6496	2.8666	0.7363
PIB $f = 0.99$	0.7329	0.9169	0.9822	0.8544	5.1627	1.0073
PIS $f = 0.5$	0.0362	0.0448	0.0320	0.0348	0.0642	0.0306
PIS $f = 0.75$	0.1053	0.1224	0.0985	0.1025	0.1801	0.0959
PIS $f = 0.865$	0.1775	0.1955	0.1754	0.1750	0.3191	0.1743
PIS $f = 0.9$	0.2153	0.2315	0.2188	0.2141	0.4907	0.2196
PIS $f = 0.95$	0.3057	0.3133	0.3312	0.3126	0.9812	0.3400
PIS $f = 0.975$	0.3998	0.3962	0.4613	0.4241	1.7781	0.4836
PIS $f = 0.99$	0.5280	0.5327	0.6605	0.5893	3.6150	0.7097
$M_{x}^{2} = (4 π / λ) \cdot \sqrt{〈 x^{2} 〉〈 θ_{x}^{2} 〉}$	1	$2 / \sqrt{π} = 1.1281$	1.0349	1.0182	4.5771 ( $\infty$ )	1.0472
$M_{1} = (4 π / λ) \cdot 〈 \| x \| 〉〈 \| θ_{x} \| 〉$	0.6366	0.7349	0.6192	0.6326	1.6784	0.6117
$M_{0.5} = (4 π / λ) {〈 \sqrt{\| x \|} 〉}^{2} {〈 \sqrt{\| θ_{x} \|} 〉}^{2}$	0.4569	0.5381	0.4308	0.4493	0.9787	0.4206

Abstract

Cited By

Figures (2)

Tables (3)

Equations (73)

Journal of the Optical Society of America A