The beam propagation factor M2 of truncated Standard and Elegant-Hermite-Gaussian beams

Optics and Laser Technology xxx (2017) xxx–xxx

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Research Note

The beam propagation factor M2 of truncated Standard and Elegant-Hermite-Gaussian beams K. Mihoubi b, A. Bencheikh a,b,⇑, Ai. Manallah b a b

Département électromécanique, faculté des sciences et de la technologie, Université BBA, Al Anassar, Bordj Bou Arréridj 34000, Algeria Laboratoire d’Optique Appliquée, Institut d’Optique et Mécanique de Précision, Université Sétif 1, Sétif 19000, Algeria

a r t i c l e

i n f o

Article history: Received 1 March 2017 Received in revised form 26 August 2017 Accepted 7 September 2017 Available online xxxx Keywords: Beam propagation factor Standard Hermite-Gaussian beam Elegant-Hermite-Gaussian beam Truncated second-order moments

a b s t r a c t Based on the truncated second-order-moment method, exact analytical expressions of the beam propagation factors M2 of truncated Standard and Elegant-Hermite-Gaussian beam are derived for the first time. According to the derived expressions, the beam propagation factors are illustrated and analyzed with numerical examples, and the influence of the truncation parameter and the order of Hermite polynomials on the beam propagation factors are also discussed in detail. To clarify the main physical results, only M2 of truncated 1D standard and Elegant- Hermite-Gaussian beams have been treated in this paper, but further extension to the 2D space is straightforward. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The research on laser beam quality and laser beam propagation rule has been active field of laser subject, which has pushed the development of laser science and technology [1,2]. So much attention has been paid to the parametric characterization of the laser beam based on the intensity moments. It is known that the second-order-moment based M2 factor is useful parameter describing the laser beam quality of propagation [3–9]. The concept of Elegant-Hermite-Gaussian (EHG) modes was introduced by Siegman [10]. They are also solutions of the paraxial wave equation, but are not orthogonal in the usual sense and the argument of the Hermite part is complex. Recently, an increasing interest has developed in such beams having complex arguments, because they describe a number of beams whose field distribution may vary upon propagation [11–14]. Moreover, all aspects of their propagation and transformation through optical systems were given in detail [15–19]. In the other side near and far field distributions, beam propagation factor M2 of Elegant-Hermite-Gaussian beams (EHG) have been given and compared with those of StandardHermite-Gaussian beams (SHG) [10]. Other many works were devoted to studying more general forms of standard and ElegantHermite-Gaussian beams [20–30].

In the present paper, and in the same manner we follow the studies presented for Standard and Elegant-Laguerre-Gaussian beams to give for the first time analytical expressions for M2 factor of both truncated 1-D Standard and Elegant-Hermite-Gaussian, after numerical results will be presented to investigate the behavior of M2. 2. Basic theory We start by reviewing the essential element of theory about the two types of Standard-Hermite-Gauss beam and Elegant-HermiteGauss, and then we move to look at the different aspects of the laser beam quality factor M2. The optical field distribution of Standard-Hermite-Gaussian beam, at the z = 0 plane in Cartesian coordinate domains is given by [10]

pffiffiffi !  2 x 2 x exp w0 w20

Eðx; 0Þ ¼ Hm

ð1Þ

In the other side the optical field distribution of 1-D ElegantHermite-Gaussian beams at the plane z = 0 in Cartesian coordinate is given by [10]

 Eðx; 0Þ ¼ Hm ⇑ Corresponding author at: Département électromécanique, faculté des sciences et de la technologie, Université BBA, Al Anassar, Bordj Bou Arréridj 34000, Algeria. E-mail address: [email protected] (A. Bencheikh).

  2 x x exp w0 w20

ð2Þ

where Hm denotes Hermite polynomial with mode order m, and in the whole work and for shake of simplicity we take w0 = 1 mm

https://doi.org/10.1016/j.optlastec.2017.09.002 0030-3992/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: K. Mihoubi et al., The beam propagation factor M2 of truncated Standard and Elegant-Hermite-Gaussian beams, Opt. Laser Technol. (2017), https://doi.org/10.1016/j.optlastec.2017.09.002

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M2x

And hu2 i is the second-order moment irradiance in the spatialfrequency domain which corresponds to the square of the divergence, it reads

1 3

hu2 i ¼

Table 1 pffiffiffi Standard-Hermite-Gauss polynomials (SHG) Hm ð 2x=w0 Þ of order m. pffiffiffi Standard-Hermite polynomials (SHG) Hm ð 2x=w0 Þ

m

Beam

0 1

SHG00 SHG10

1

2

SHG20

5

3

SHG30

8xðx=w0 Þ2  2 pffiffiffi pffiffiffi 16 2ðx=w0 Þ3  12 2ðw0 Þ

4

SHG40

9

5

SHG50

64ðx=w0 Þ4  96ðx=w0 Þ2 þ 12 pffiffiffi pffiffiffi pffiffiffi 128 2ðx=w0 Þ5  320 2ðx=w0 Þ3 þ 120 2ðx=w0 Þ

pffiffiffi 2 2ðx=w0 Þ

7 11

m

beam

Elegant-Hermite polynomials Hm ðx=w0 Þ

M2x

0 1 2

EHG00 EHG10 EHG20

1 2ðx=w0 Þ 4ðx=w0 Þ  2

1 3 3.87

3

EHG30

8ðx=w0 Þ3  12ðx=w0 Þ

3.92

4

EHG40

16ðx=w0 Þ4  48ðx=w0 Þ2 þ 12

4.39

5

EHG50

32ðx=w0 Þ5  160ðx=w0 Þ3 þ 120ðx=w0 Þ

4.82

2

which presents the waist width. Tables 1 and 2 respectively give the first five polynomials of Standard and Elegant Hermite Gaussian beams and the associate beam propagation factor M2 [12]. The field distributions in the waist for the standard and Elegant Hermite-Gaussian beams for m = 0.1.2.3 are plotted in Fig. 1, for this latter we easily distinguish the well known symmetric and anti-symmetric structures of the field distributions. For the beam propagation factor M2, the theory is well known, the calculation is based on the definitions of the near and the far field widths. It is worth to note that when the intensity distribution exhibits a complicated form it is not easy to determine the near and the far field widths, so in the beginning of 90th years researchers proposed a statistical technique to overcome this problem, this technique is based on the determination of second-order irradiance moments. We have hx2 i presents the second-order moment irradiance in the spatial domain which corresponds to the square of the width, and is defined by

1 I0

Z

a

x2 jEðxÞj2 dx

ð3Þ

a

k I0

a

jE0 ðxÞj2 dx þ

a

4ðjEðaÞj2 þ jEðaÞj2

SHG20

3

SHG10

Z

a

jEðxÞj2 dx

ð5Þ

a

is the total power entering through the aperture. It has been implicitly assumed that the first order moments are zero. As an example and based on the second order irradiance moments the beam propagation factor for standard-Hermite Gaussian beam is given by M2x ¼ ð2m þ 1Þ and for Elegant-Hermite Gaussian beam HGm0 of order m was calculated by shaghafi et al. [12], it take the form M2x ¼ ðð4m  1Þð2m þ 1Þ=ð2m  1ÞÞ1=2 , using these formulas the values of M2x are given in Tables 1 and 2 for the first five orders of standard and Elegant-Hermite Gaussian beams The cross second-order irradiance moment is given by

hxui ¼

1 2ikI0

Z

a

fxE0 ðxÞ EðxÞ  xE0 ðxÞE ðxÞgdx

Transverse coordinate x

ð6Þ

a

where the asterisk denotes complex conjugation. Note that hxui vanishes since the Standard and ElegantHermite-Gaussian field distributions, are real valued in the waist plane. By grouping Eqs. (3)–(6) the generalized beam propagation factor M2 takes the form [3–6]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 ¼ 2k hx2 ihu2 i  hxui2

ð7Þ

3. Mathematical development of the generalized M2 for truncated Standard and Elegant-Hermite-Gaussian beams 3.1. Standard-Hermite-Gaussian beams (SHG) We start these development by the Standard-Hermite-Gaussian beam (SHG), so by substituting from Eqs. (2)–(7) into Eq. (1) and recalling the integral formulae [31]

EHG30

EHG20

EHG10

EHG00

SHG00

0

ð4Þ

2

k I0 a

SHG30

6

Field distribution [a.u]

2

Field distribution [a.u]

hx2 i ¼

Z

where k is the wave number and the prime denotes derivation with respect to x.with

I0 ¼ Table 2 Elegant-Hermite-Gauss polynomials (EHG) Hm ðx=w0 Þ of order m.

1

Transverse coordinate

Fig. 1. Intensity distributions for Standard (SHGm0) and Elegant (EHGm0) Hermite Gaussian beams for m = 0, 1, 2, 3.

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The series expansion of Hermite function is given by

12

m=5

11

pffiffiffi ! m=2 pffiffiffi!m2s X ð1Þs m! 2 2 2 x ¼ xm2s s!ðm  2sÞ! w0 w 0 s¼0

Hm

10

m=4

9

m=3

After tedious integral calculations, we can write the quality factor M2 of truncated 1-D Standard-Hermite-Gaussian as function of the beam truncation parameter as follows

m=2

M ¼ 24

8 7

M

2

2

6

2

5

m=2 X m=2 X f ðm;s1 ;s2 Þ s1¼0 s2¼0

4

ð11Þ

m=1

m=2 X m=2 X f ðm;s1 ;s2 Þ



3

ms1 s2 þpþ0:5 1 X ð1Þp ð2d2 Þ

1

!1

2ms1 s2 þ1:5 p¼0 p!ðms1 s2 þ0:5þpÞ ms1 s2 þ1:5þp 1 X ð1Þp ð2d2 Þ

1

!12

2ms1 s2 þ2:5 p¼0 p!ðms1 s2 þ1:5þpÞ ( ms1 s2 0:5þp 1 ðm2s1 Þðm2s2 Þ X ð1Þp ð2d2 Þ f ðm;s1 ;s2 Þ ms1 s2 þ0:5 p!ðms1 s2 0:5þpÞ 2 p¼0 ¼0

s1¼0 s2¼0

2

m=2 X m=2 X



m=0

1

s1¼0 s2

ms1 s2 þ0:5þp

0

0.0

0.5

1.0

1.5

2.0

2.5

1 ððm2s1 Þþðm2s2 ÞÞ X ð1Þp ð2d2 Þ  ms1 s2 þ0:5 p!ðms1 s2 þ0:5þpÞ 2 p¼0

3.0

δ Fig. 2. Variation of M2 the beam propagation factor for truncated StandardHermite-Gaussian beams (SHGm0) as function of the beam truncation parameter d.

þ

ms1 s2 þ1:5þp 1 X ð1Þp ð2d2 Þ

1

2ms1 s2 þ0:5 p¼0 p!ðms1 s2 þ1:5þpÞ

þ4ðdÞ2ðm2s1 2s2 Þ1 expð2d2 Þ

)!12 3 5

ð12Þ where

6

f ðm; s1 ; s2 Þ ¼ m=5 m=4

5

d¼

M²

m=2 m=1

3

m=0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ð14Þ

incomplete gamma function which reduces to the gamma function ðaÞ for x ! 1, so Eq. (12) simplifies to

δ Fig. 3. Variation of M2 the beam propagation factor for truncated Elegant-HermiteGaussian (EHGm0) beams as function of the beam truncation parameter d.

Z

a w0

is the beam truncation parameter. As can be seen from Eq. (12) the M2 factor for truncated Standard-Hermite-Gaussian beams is dependent on the mode order m of Hermite polynomial and also dependent on the beam truncation parameter. An important particular case has to be considered, is the nontruncated Standard-Hermite-Gaussian beams yielding by putting P ð1Þp xaþp d ! 1 in Eq. (12), we recall that cða; xÞ ¼ 1 is the p¼0 p!ðaþpÞ

2 1

2

M2d!1

¼ 24

m=2 X m=2 X f ðm; s1 ; s2 Þ s1¼0 s2¼0

u

n

xm ebx dx ¼

0

cðv ; bun Þ

ð8Þ

nbv



mþ1 n

ð9Þ

f ðm; s1 ; s2 Þ

1 X ð1Þp xaþp p¼0



m=2 X m=2 X

!1

1 2ms1 s2 þ1:5

Cðm  s1  s2 þ 0:5Þ !12

1

2

f ðm; s1 ; s2 Þ

ms1 s2 þ0:5 s1¼0 s2 ¼0 2

Cðm  s1  s2 þ 1:5Þ ms1 s2 þ2:5

fððm  2s1 Þðm  2s2 Þ

ð2m  2s1  2s2 ÞÞðm  s1  s2  0:5Þ #

and

cða; xÞ ¼

m=2 X m=2 X s1¼0 s2¼0

where

v¼

ð13Þ

and

m=3

4

ð1Þs1 þs2 ðm!Þ2 ð8Þms1 s2 s1 !s2 !ðm  2s1 Þ!ðm  2s2 Þ!

1

ð10Þ

p!ða þ pÞ

ð15Þ

þCðm  s1  s2 þ 1:5ÞgÞ2

Table 3 pffiffiffi Roots of Standard-Hermite polynomials (SHG) Hm ð 2x=w0 Þ. m

beam

ðx=w0 Þ

1 2 3 4 5

HG10 HG20 HG30 HG40 HG50

0 0.5 0.866 1.167 1.429

0.5 0 0.371 0.678

0.866 0.371 0

1.167 0.678

1.429

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K. Mihoubi et al. / Optics and Laser Technology xxx (2017) xxx–xxx

6.0

6

SHG20

5.5

SHG10

5.0

EHG10

SHG20

EHG20

5

4

EHG20

M²

M²

4.5 4.0

3

3.5

2

3.0 1 2.5 0.0

0.5

1.0

1.5

2.0

2.5

0.0

3.0

1.0

1.5

2.0

2.5

Truncation parameter δ

(a)

(b)

8

3.0

10

SHG30

SHG30

7

SHG40

SHG40 EHG40

8

EHG30 6

6

EHG40

M²

M²

0.5

Trancation parameter δ

5

4

EHG30

4

2 3 0.0

0.5

1.0

1.5

2.0

2.5

0

3.0

0.0

0.5

Truncation parameter δ

1.0

1.5

2.0

2.5

3.0

Truncation parameter

(c)

(d) 12

SHG50 SHG50

10

EHG50

M²

8

6

EHG50 4

2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

truncation parameter

(e) Fig. 4. The beam propagation factor M2as function of the beam truncation parameter d; a comparison between truncated Standard and Elegant-Hermite-Gaussian beams. (a) HG10, (b) HG20, (c) HG30, (d) HG40, (e) HG50.

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K. Mihoubi et al. / Optics and Laser Technology xxx (2017) xxx–xxx

beam, we implement Eq. (18) in Mathematica 10, wecalculate the M2 for beam order m = 2 the M2 = 3.87, for m = 4 the M2 = 4.39, so obtained expressions od M2 are very consistent.

Table 4 Roots of Elegant-Hermite polynomials (EHG) Hm ðx=w0 Þ. m

beam

ðx=w0 Þ

1 2 3 4 5

HG10 HG20 HG30 HG40 HG50

0 0.707 1.225 1.651 2.020

0.707 0 0.526 0.958

4. Numerical results and analyses

1.225 0.526 0

1.651 0.958

2.020

It is obvious from Eq. (15) that the M2 for the non-truncated Standard-Hermite-Gaussian beam depends only on the order of the beam. To verify that the expression of M2 given by Eq. (15) is consistent, we implement Eq. (15) in the software Mathematica.10 for some values of beam order m and the results are perfect and correspond to M2 = 2m + 1, as predicted by the theory (for m = 2, M2 = 5, for m = 4, M2 = 9). 3.2. Elegant-Hermite-Gaussian beams (EHG) Using a similar approach as for the Standard-Hermite-Gaussian beams, the analytical expression for the M2 factor of truncated beams is derived as follows 2 M 2 ¼ 24

m=2 X m=2 X

gðm;s1 ; s2 Þ

s1¼0 s2¼0 m=2 X m=2 X



s1¼0 s2¼0

gðm;s1 ;s2 Þ

ms1 s2 þ0:5þp p 1 X ð1Þ ð2d2 Þ

1 2

ms1 s2 þ1:5

1

p¼0

!1

p!ðm  s1  s2 þ 0:5 þ pÞ

ms1 s2 þ1:5þp 1 X ð1Þp ð2d2 Þ

!12

2ms1 s2 þ2:5 p¼0 p!ðm  s1  s2 þ 1:5 þ pÞ ( ms1 s2 0:5þp 1 ðm  2s1 Þðm  2s2 Þ X ð1Þp ð2d2 Þ gðm;s1 ;s2 Þ ms1 s2 þ0:5 p!ðm  s  s2  0:5 þ pÞ 1 2 p¼0 ¼0

m=2 X m=2 X



s1¼0 s2

ms1 s2 þ0:5þp

1 ððm  2s1 Þ þ ðm  2s2 ÞÞ X ð1Þp ð2d2 Þ  ms1 s2 þ0:5 p!ðm  s1  s2 þ 0:5 þ pÞ 2 p¼0

þ

ms1 s2 þ1:5þp p 1 X ð1Þ ð2d2 Þ

1

2ms1 s2 þ0:5 p¼0 p!ðm  s1  s2 þ 1:5 þ pÞ

þ 4ðdÞ

2ðm2s1 2s2 Þ1

)!12 3 5 expð2d Þ

ð16Þ

2

where

gðm; s1 ; s2 Þ ¼

ð1Þs1 þs2 ðm!Þ2 ð4Þms1 s2 s1 !s2 !ðm  2s1 Þ!ðm  2s2 Þ!

ð17Þ

In the same manner to the Standard-Hermite-Gaussian beams, the M2 factor for truncated Elegant-Hermite-Gaussian beams is also dependent on the mode order m of Hermite polynomial and on the beam truncation parameter d. Using a similar approach as for the Standard-Hermite-Gaussian beam, the M2 for the non- truncated Elegant-Hermite-Gaussian beam is expressed as

2 M ¼ 24 2

m=2 X m=2 X gðm; s1 ; s2 Þ s1¼0 s2¼0

1 2ms1 s2 þ1:5

m=2 X m=2 X gðm; s1 ; s2 Þ 

!1

Cðm  s1  s2 þ 0:5Þ !12

1

Cðm  s1  s2 þ 1:5Þ 2ms1 s2 þ2:5  m=2 X m=2 X ðm  2s1 Þðm  2s2 Þ  gðm; s1 ; s2 Þ Cðm  s1  s2  0:5Þ 2ms1 s2 þ0:5 s1¼0 s2 ¼0 s1¼0 s2¼0

 þ

ððm  2s1 Þ þ ðm  2s2 ÞÞ 2ms1 s2 þ0:5 1 2ms1 s2 þ0:5

Cðm  s1  s2 þ 0:5Þ

Cðm  s1  s2 þ 1:5Þ

12 #

5

In this section, and based on the two novel analytical expressions Eqs. (12) and (16) we illustrate numerically the behavior of the beam propagation M2 as function of the truncation parameter d for two types of Hermite-Gaussian beams, the standard and the Elegant one. As shown in Fig. 2 where the beam propagation factor M2 for the first six orders of Standard-Hermite-Gaussian beams (SHG) (SHG00, SHG10, SHG20, SHG30, SHG40 and SHG50) is plotted as function of the truncation parameter d, one can notice that when d = 0 we distinguish two distinct departure points, the first at M2 = 2.4 for symmetric beams (even orders); m = 0, m = 2 and m = 4, and the second departure point is at M2 = 6 for antisymmetric ones (odd orders); m = 1, m = 3, m = 5. Fig. 3 shows in the same way the variation of M2 as function of truncation parameter d for the first five orders of Elegant-HermiteGaussian beams (EHG), as for (SHG) we can split the curves in two families, the first which takes as departure point M2=2.4 when d = 0 which corresponds to symmetric beams (even orders), and the second family takes as departure point M2=6 when d = 0 which corresponds antisymmetric beams (odd orders). Fig. 4 joins the curves of truncated M2 as function of d for all five orders of both Standard and Elegant-Hermite-Gaussian beams, where we present in five subfigures a comparison between standard and Elegant Hermite Gaussian beams. The variation of M2 as function of d for EHGm0 presents always only one minima for each order, where we have only one minima at M2 = 1 for all even orders (m = 2, 4,. . ..) and we have also only one minima at M2=3 for all odd orders (m = 1, 3, 5,. . .). In the other side, the number of minima which correspond to the variation of M2 as function of d for SHGm0 is function of the number and the structure of the order (symmetric or antisymmetric), as an example for SHG40 we have two minima (at d = 0.371, M2=1 and at d = 1.167, M2=5), where it is worth to note that when we truncate SHG40 at its first zero intensity (Tables 3 and 4) we obtain M2=1 which corresponds to M2 of the non-truncated SHG00 (M2=2 m + 1) and when we truncate SHG40 at its second zero intensity we obtain M2=5 which corresponds to M2 of the non-truncated SHG20 (M2=2 m + 1), and we have observed the same thinks for odd orders, where, as an example when we truncate SHG50 at its first zero intensity we obtain M2=3 which corresponds to M2 of the non-truncated SHG10 (M2=2 m + 1) and when we truncate SHG50 at its second zero intensity we obtain M2=7 which corresponds to M2 of the non-truncated SHG30 (M2=2 m + 1). It is clear that a simple diaphragm allows obtaining from higher order SHGm0 beam of order m a new lower order SHGp0 beam. We notice that when the aperture of the diaphragm exceeds the width of the SHG or EHG beam, the M2 become a constant where we reach the non-truncated case. Finally, we note that Hermite Gaussian beams in their standard and elegant forms behave in two different manners, where all symmetric modes SHG and EHG (even orders) behaves in the same manner under diffraction effects, and all anti-symmetric modes SHG and EHG (odd orders) behaves also in the same manner under diffraction effects.

ð18Þ

It is clear from Eq. (18) that the M2 for the non-truncated Elegant-Hermite-Gaussian beam depends only on the order of the beam. In the same manner as for Standard-Hermite-Gaussian

5. Conclusions In this paper, new analytical expressions of the beam propagation factor M2 for truncated Hermite-Gaussian beams in their stan-

Please cite this article in press as: K. Mihoubi et al., The beam propagation factor M2 of truncated Standard and Elegant-Hermite-Gaussian beams, Opt. Laser Technol. (2017), https://doi.org/10.1016/j.optlastec.2017.09.002

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Please cite this article in press as: K. Mihoubi et al., The beam propagation factor M2 of truncated Standard and Elegant-Hermite-Gaussian beams, Opt. Laser Technol. (2017), https://doi.org/10.1016/j.optlastec.2017.09.002