- Email: [email protected]
Focal shift in Hermite–Gaussian beams based on the encircled-power criterion
Available online at www.sciencedirect.com
Optics & Laser Technology 35 (2003) 435 – 440 www.elsevier.com/locate/optlastec
Focal shift in Hermite–Gaussian beams based on the encircled-power criterion Baida L)ua; b;∗ , Runwu Penga a Physics
b State
Department, Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China Key Laboratory of Laser Technology, Huazhong University of Science & Technology, Wuhan 430074, China Received 19 September 2002; received in revised form 7 February 2003; accepted 17 February 2003
Abstract The focal shift in Hermite–Gaussian beams, whose axial irradiance vanishes, is studied based on the encircled-power criterion, and the results are illustrated numerically. The related problems are discussed. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Focal shift; Hermite–Gaussian beam; Encircled-power criterion
1. Introduction
2. Irradiance distribution of focused H–G beams
To date the beam focusing and the associated focal shift have been studied extensively because of the theoretical and experimental interest [1–9]. It is known that, when a Gaussian beam is focused by a lens the point of maximum irradiance along the z-axis is not positioned at the geometrical focus, but somewhat closer to the lens due to the focal shift [2,7]. Usually, the focal shift is determined by the axial maximum irradiance. For some types of beam, whose axial irradiance is equal to zero, the conventional approach to the focal shift fails. More recently, in examining three-dimensional focused irradiance distributions of vector Bessel–Gaussian beams, an encircled-power criterion was introduced to de=ne the focal plane and focal shift of vector Bessel–Gaussian beams [10]. The present paper is aimed at studying the focal shift in Hermite–Gaussian (H–G) beams. The main attention is focused on the case where the axial irradiance of H–G beams vanishes. First, the expression for focused H–G beams is derived starting from the generalized Huygens– Fresnel di>raction integral [11]. Then, the focal shift of H–G beams is studied and illustrated with detailed numerical examples. Finally, there is a discussion about the method used here and the conventional approach to the focal shift, and a brief summary concludes the paper.
As shown in Fig. 1, an H–G beam with initial =eld distribution [12] √ x0 √ y0 Emn (x0 ; y0 ; 0) = Hm Hn 2 2 w0 w0 ik 2 ×exp − (x + y02 ) (1) 2q0 0
∗
Corresponding author. Physics Department, Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China. Tel.: +8628-541-2819; fax: +86-28-541-0844. E-mail address: [email protected] (B. L)u).
is incident on an aperture lens of focal length f(f ¿ 0) at the plane z =0, where Hm ; Hn are Hermite polynomials of orders m and n, respectively, m and n also denote mode indices of the H–G beam. q0 = ikw02 =2 and w0 are the complex q parameter and waist width of the corresponding fundamental Gaussian beam, respectively, k = 2= is the wave number with being the wavelength. The full-width of the aperture is 2a. Within the framework of the paraxial approximation, the propagation of H–G beams through the aperture lens system is characterized by the well-known generalized Huygens– Fresnel di>raction integral of the form [11] a a ik Emn (x; y; z) = exp(−ikz) Emn (x0 ; y0 ; 0) 2B −a −a ik ×exp − [A(x02 + y02 ) − 2(x0 x + y0 y) 2B + D(x2 + y2 )] d x0 dy0 ; (2)
0030-3992/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0030-3992(03)00040-9
436
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
√ √ +4 PQv + 4Q2 v2 ) × (−1 + exp[4 PQv]) √ √ + 8 PQv] − Ty Qv ×(6 − 3P + 4Q2 v2 )};
(8)
where P=1−
(3)
On substituting from Eq. (1) into Eq. (2), and recalling the formulae of Hermite polynomials
(12)
√ √ Ty = erf ( P − Qv) + erf ( P + Qv);
(13)
Hm (−x) = (−1) Hm (x);
=
Hm+1 (x) = 2xHm (x) − 2mHm−1 (x);
Nw =
w02 (Fresnel number associated with the f
dHm (x) = 2mHm−1 (x); dx after some integral calculations we obtain E00 (u; v; z) = Tx Ty S=4P; E11 (u; v; z) =
E22 (u; v; z) =
√ 2S {exp[ − ( P + Qu)2 ] P2 √ √ − exp[ − ( P − Qu)2 ] + QTx u} √ ×{exp[ − ( P + Qv)2 ] √ √ − exp[ − ( P − Qv)2 ] + QTy v};
(4)
(11)
(truncation parameter);
corresponding Gaussian beam);
(14)
(15)
and the normalizedcoordinates (5)
u = x=w0 ;
(16)
v = y=w0 ;
(17)
erf denotes the error function. The irradiance of focused H–G beams is expressed as Imn (u; v; z) = |Emn (u; v; z)|2 : (6)
√ √ S {−4( P − Qu) exp[ − ( P + Qu)2 ] 3 P √ √ − 4( P + Qu) exp[ − ( P − Qu)2 ] √ √ + Tx (2 − P + 4Q2 u2 )}{−4( P − Qv) √ ×exp[ − ( P + Qv)2 ] √ √ −4( P + Qv) exp[ − ( P − Qv)2 ] √ + Ty (2 − P + 4Q2 v2 )}; (7)
√ 8S E33 (u; v; z) = − 4 {exp[ − ( P + Qu)2 ] P √ ×[(4 + 4 P − 3P + 4 PQu + 4Q2 u2 ) √ √ ×(−1 + exp[4 PQu]) + 8 PQu] √ − Tx Qu(6 − 3P + 4Q2 u2 )} √ ×{exp[ − ( P + Qv)2 ][(4 + 4 P − 3P
(10)
2
a w0
m
(9)
iNw ; P 1=2 z=f iNw iNw (u2 + v2 ) ; S= exp Q2 − z=f z=f √ √ Tx = erf ( P − Qu) + erf ( P + Qu);
Q=
Fig. 1. Schematic illustration of an aperture lens system.
where the ABCD transfer matrix reads as A B 1 − z=f z M= = : C D −1=f 1
iNw (z − f)=f ; z=f
(18)
It is worth noting that, from Eqs. (2), (5)–(8) a recurrence propagation equation of H–G beams can be derived, by means of which it is possible to obtain the propagation expression for H–G beams of any order. Details are found in Ref. [13], and are omitted here. 3. Focal shift in H–G beams In the conventional treatment, the point zmax of maximum irradiance along the z-axis is determined by [1,2] dImn (0; 0; z)=d z = 0
(19)
and the relative focal shift is expressed as Izf =
zmax − f : f
(20)
For H–G beams with m; n = odd numbers, Imn (0; 0; z) = |Emn (0; 0; z)|2 = 0, Eq. (19) is not applicable. For this case, the more general de=nition of the focal plane can be applied based on the encircled-power criterion [10]. For the
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
437
Cartesian coordinate system it reads as
W=w0 W=w0 Imn (u; v; z) du dv −W=w0
∞
−∞
−W=w0
∞
−∞
Imn (u; v; z) du dv = 0:8;
(21)
where W denotes the width of H–G beams. From Eq. (21) the smallest width, i.e. the waist width Wmin of H–G beams and its position, namely, the position of the focal plane zmax , can be found numerically. As a result, the relative focal shift Izf is obtained. 4. Numerical results and analysis Numerical calculations were performed using Eqs. (6), (8), (18) and (21) to study the focusing properties and focal shift of the TEM11 - and TEM33 -mode H–G beams. Fig. 2 gives the isophotes (contour lines of the irradiance) of the beams at the focal plane zmax . The calculation parameters are = 632:8 nm, w0 = 1 mm, f = 400 mm, a = 2 mm. Thus, from Eqs. (14) and (15) we have = 4 and Nw = 4. The structure of the beams is plotted in Fig. 3, which shows irradiance distributions of the beams along the direction of white lines in Fig. 2. As can be seen, the contour lines of the irradiance exhibit a symmetry about the axes u=0 and v=0, and the on-axis irradiance of the TEM11 - and TEM33 -mode H–G beams vanishes, so that the approach based on Eq. (19) is not applicable [10]. However, the irradiance distributions in Fig. 3 become asymmetric about the axis z = f and the power is more concentrated at a point just before z = f. It means that the focal plane shifts slightly toward the lens, which is further con=rmed in Fig. 4, where the beam width calculated by the use of Eq. (21) it plotted against Iz=f. The dependence of relative focal shift Izf on the Fresnel number Nw and truncation parameter is depicted in Fig. 5, from which we see that |Izf | increases with decreasing and increasing mode index, and |Izf | decreases as Nw increases, if the mode indices m and n are kept =xed. 5. Discussion and concluding remarks In this paper, the encircled-power criterion has been used to study the focal shift in H–G beams whose axial irradiance vanishes. Numerical calculation results have shown that the focal shift in H–G beams depends on the truncation parameter, mode index and Fresnel number. Although the encircled-power criterion provides a more general de=nition of the focal plane and the associated focal shift, two questions arise. First, why do you use a factor 0.8? The choice of the factor is related to the de=nition of the beam width, or to the fraction of power in the bucket [14,15]. Because there are di>erent de=nitions of the beam width, an appropriate choice of the de=nition is necessary. Second, what
Fig. 2. Contour lines of the irradiance of H–G beams at the focal plane: (a) TEM11 mode; (b) TEM33 mode. The calculation parameters are = 632:8 nm, w0 = 1 mm, f = 400 mm and a = 2 mm.
are the real focal plane and focal shift of beams whose axial irradiance does not vanish, but the transversal maximum irradiance is not achieved on the axis? A typical example is the TEM22 -mode H–G beam, whose contour lines are shown in Figs. 6(a) and (b). The other calculation parameters are the same as those in Fig. 2. For such a type of beam both Eqs. (19) and (21) can be applied to calculate the focal
438
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
Fig. 4. Beam width W of TEM11 - and TEM33 -mode H–G beams varies with Iz=f.
Fig. 3. Irradiance distribution of H–G beams along the direction of the white lines in Fig. 2. (a) TEM11 mode; (b) TEM33 mode.
shift, and the results are compiled in Figs. 7(a) and (b). The variation of Izf with and Nw is similar by using Eqs. (19) and (21), whereas the quantitative di>erence for both cases is apparent, and |Izf | calculated by the use of Eq. (19) is larger than that by the use of Eq. (21). The value of |Izf | for the latter case is several percent, but for the former case can reach 10% or more. From Fig. 6 it turns out that |Izf | ≈ 0:06, if only the axial irradiance is used to
Fig. 5. Relative focal shift Izf of TEM11 - and TEM33 -mode H–G beams is plotted against (a) Fresnel number Nw , (b) truncation parameter .
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
439
Fig. 7. Relative focal shift Izf of a TEM22 -mode H–G beam is plotted against (a) Fresnel number Nw , (b) truncation parameter .
dealt with. In principle, although the real focal plane or the factor in Eq. (21) can be chosen according to the practical application purpose and beams used, further study seems to be necessary.
Acknowledgements Fig. 6. (a) Contour lines of the irradiance of a TEM22 mode H–G beam at the focal plane. (b) Irradiance distribution of a TEM22 -mode H–G beam along the direction of the white line in Fig. 6(a). The other calculation parameters are the same as those in Fig. 2.
calculate the focal shift of the TEM22 -mode H–G beam. However, |Izf | becomes smaller as the encircled-power criterion is applied. Physically, for the beams with large irradiance modulation it is unsuitable to determine the real focal plane if only the on-axis irradiance is taken into consideration. However, what is the real focal plane for such a type of beam? A similar question as the =rst one has to be
This work was supported by the Foundations of State Key Laboratory of Laser Technology and National Hi-Tech Project of China.
References [1] Li Y, Wolf E. Focal shifts di>racted converging spherical waves. Opt Commun 1981;39:211–5. [2] Li Y, Wolf E. Focal shift in focused truncated Gaussian beams. Opt Commun 1982;42:151–6. [3] Carter WH. Focal shift and concept of e>ective Fresnel number for a Gaussian laser beam. Appl Opt 1982;21:1989–94.
440
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
[4] Li Y. Variations of the axial intensity pattern formed by a focused truncated Gaussian beams. Opt Commun 1988;68:324–8. [5] Carter WH, Aburdene MF. Focal shift in Laguerre–Gaussian beams. J Opt Soc Am A 1987;4:1949–52. [6] L)u B, Huang W, Zhang B, Cai B. Focal shift in apertured Gaussian beams and relation with the lens focus. Optik 1995;99: 8–12. [7] Li Y. Propagation of focal shift through axisymmetrical optical systems. Opt Commun 1993;95:13–7. [8] Borghi R, Santarsiero M, Vicalvi S. Focal shift of focused Sat-topped beams. Opt Commun 1998;154:243–8. [9] L)u B, Zhang B. Focusing of a Gaussian Schell-model beam through a circular lens. J Mod Opt 1995;42:289–98.
[10] Greene PL, Hall DG. Focal shift in vector beams. Opt Express 1999;4:411–9. [11] Collins Jr SA. Lens-system di>raction integral written in terms of matrix. J Opt Soc Am 1970;60:1168–77. [12] Siegman AE. Lasers. California: University Science Books Mill Valley, 1986. p. 644. [13] Tao X, Zhou N, L)u B. Recurrence propagation equation of Hermite– Gaussian beams through a paraxial optical ABCD system with hard-edge aperture, to be published. [14] Siegmen AE. How to (maybe) measure laser beam quality. OSA TOPS 1998;17:184–98. [15] Garay A. Continuous wave deuterium Suoride laser beam diagnostic system. SPIE 1988;888:17–22.
Optics & Laser Technology 35 (2003) 435 – 440 www.elsevier.com/locate/optlastec
Focal shift in Hermite–Gaussian beams based on the encircled-power criterion Baida L)ua; b;∗ , Runwu Penga a Physics
b State
Department, Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China Key Laboratory of Laser Technology, Huazhong University of Science & Technology, Wuhan 430074, China Received 19 September 2002; received in revised form 7 February 2003; accepted 17 February 2003
Abstract The focal shift in Hermite–Gaussian beams, whose axial irradiance vanishes, is studied based on the encircled-power criterion, and the results are illustrated numerically. The related problems are discussed. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Focal shift; Hermite–Gaussian beam; Encircled-power criterion
1. Introduction
2. Irradiance distribution of focused H–G beams
To date the beam focusing and the associated focal shift have been studied extensively because of the theoretical and experimental interest [1–9]. It is known that, when a Gaussian beam is focused by a lens the point of maximum irradiance along the z-axis is not positioned at the geometrical focus, but somewhat closer to the lens due to the focal shift [2,7]. Usually, the focal shift is determined by the axial maximum irradiance. For some types of beam, whose axial irradiance is equal to zero, the conventional approach to the focal shift fails. More recently, in examining three-dimensional focused irradiance distributions of vector Bessel–Gaussian beams, an encircled-power criterion was introduced to de=ne the focal plane and focal shift of vector Bessel–Gaussian beams [10]. The present paper is aimed at studying the focal shift in Hermite–Gaussian (H–G) beams. The main attention is focused on the case where the axial irradiance of H–G beams vanishes. First, the expression for focused H–G beams is derived starting from the generalized Huygens– Fresnel di>raction integral [11]. Then, the focal shift of H–G beams is studied and illustrated with detailed numerical examples. Finally, there is a discussion about the method used here and the conventional approach to the focal shift, and a brief summary concludes the paper.
As shown in Fig. 1, an H–G beam with initial =eld distribution [12] √ x0 √ y0 Emn (x0 ; y0 ; 0) = Hm Hn 2 2 w0 w0 ik 2 ×exp − (x + y02 ) (1) 2q0 0
∗
Corresponding author. Physics Department, Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China. Tel.: +8628-541-2819; fax: +86-28-541-0844. E-mail address: [email protected] (B. L)u).
is incident on an aperture lens of focal length f(f ¿ 0) at the plane z =0, where Hm ; Hn are Hermite polynomials of orders m and n, respectively, m and n also denote mode indices of the H–G beam. q0 = ikw02 =2 and w0 are the complex q parameter and waist width of the corresponding fundamental Gaussian beam, respectively, k = 2= is the wave number with being the wavelength. The full-width of the aperture is 2a. Within the framework of the paraxial approximation, the propagation of H–G beams through the aperture lens system is characterized by the well-known generalized Huygens– Fresnel di>raction integral of the form [11] a a ik Emn (x; y; z) = exp(−ikz) Emn (x0 ; y0 ; 0) 2B −a −a ik ×exp − [A(x02 + y02 ) − 2(x0 x + y0 y) 2B + D(x2 + y2 )] d x0 dy0 ; (2)
0030-3992/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0030-3992(03)00040-9
436
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
√ √ +4 PQv + 4Q2 v2 ) × (−1 + exp[4 PQv]) √ √ + 8 PQv] − Ty Qv ×(6 − 3P + 4Q2 v2 )};
(8)
where P=1−
(3)
On substituting from Eq. (1) into Eq. (2), and recalling the formulae of Hermite polynomials
(12)
√ √ Ty = erf ( P − Qv) + erf ( P + Qv);
(13)
Hm (−x) = (−1) Hm (x);
=
Hm+1 (x) = 2xHm (x) − 2mHm−1 (x);
Nw =
w02 (Fresnel number associated with the f
dHm (x) = 2mHm−1 (x); dx after some integral calculations we obtain E00 (u; v; z) = Tx Ty S=4P; E11 (u; v; z) =
E22 (u; v; z) =
√ 2S {exp[ − ( P + Qu)2 ] P2 √ √ − exp[ − ( P − Qu)2 ] + QTx u} √ ×{exp[ − ( P + Qv)2 ] √ √ − exp[ − ( P − Qv)2 ] + QTy v};
(4)
(11)
(truncation parameter);
corresponding Gaussian beam);
(14)
(15)
and the normalizedcoordinates (5)
u = x=w0 ;
(16)
v = y=w0 ;
(17)
erf denotes the error function. The irradiance of focused H–G beams is expressed as Imn (u; v; z) = |Emn (u; v; z)|2 : (6)
√ √ S {−4( P − Qu) exp[ − ( P + Qu)2 ] 3 P √ √ − 4( P + Qu) exp[ − ( P − Qu)2 ] √ √ + Tx (2 − P + 4Q2 u2 )}{−4( P − Qv) √ ×exp[ − ( P + Qv)2 ] √ √ −4( P + Qv) exp[ − ( P − Qv)2 ] √ + Ty (2 − P + 4Q2 v2 )}; (7)
√ 8S E33 (u; v; z) = − 4 {exp[ − ( P + Qu)2 ] P √ ×[(4 + 4 P − 3P + 4 PQu + 4Q2 u2 ) √ √ ×(−1 + exp[4 PQu]) + 8 PQu] √ − Tx Qu(6 − 3P + 4Q2 u2 )} √ ×{exp[ − ( P + Qv)2 ][(4 + 4 P − 3P
(10)
2
a w0
m
(9)
iNw ; P 1=2 z=f iNw iNw (u2 + v2 ) ; S= exp Q2 − z=f z=f √ √ Tx = erf ( P − Qu) + erf ( P + Qu);
Q=
Fig. 1. Schematic illustration of an aperture lens system.
where the ABCD transfer matrix reads as A B 1 − z=f z M= = : C D −1=f 1
iNw (z − f)=f ; z=f
(18)
It is worth noting that, from Eqs. (2), (5)–(8) a recurrence propagation equation of H–G beams can be derived, by means of which it is possible to obtain the propagation expression for H–G beams of any order. Details are found in Ref. [13], and are omitted here. 3. Focal shift in H–G beams In the conventional treatment, the point zmax of maximum irradiance along the z-axis is determined by [1,2] dImn (0; 0; z)=d z = 0
(19)
and the relative focal shift is expressed as Izf =
zmax − f : f
(20)
For H–G beams with m; n = odd numbers, Imn (0; 0; z) = |Emn (0; 0; z)|2 = 0, Eq. (19) is not applicable. For this case, the more general de=nition of the focal plane can be applied based on the encircled-power criterion [10]. For the
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
437
Cartesian coordinate system it reads as
W=w0 W=w0 Imn (u; v; z) du dv −W=w0
∞
−∞
−W=w0
∞
−∞
Imn (u; v; z) du dv = 0:8;
(21)
where W denotes the width of H–G beams. From Eq. (21) the smallest width, i.e. the waist width Wmin of H–G beams and its position, namely, the position of the focal plane zmax , can be found numerically. As a result, the relative focal shift Izf is obtained. 4. Numerical results and analysis Numerical calculations were performed using Eqs. (6), (8), (18) and (21) to study the focusing properties and focal shift of the TEM11 - and TEM33 -mode H–G beams. Fig. 2 gives the isophotes (contour lines of the irradiance) of the beams at the focal plane zmax . The calculation parameters are = 632:8 nm, w0 = 1 mm, f = 400 mm, a = 2 mm. Thus, from Eqs. (14) and (15) we have = 4 and Nw = 4. The structure of the beams is plotted in Fig. 3, which shows irradiance distributions of the beams along the direction of white lines in Fig. 2. As can be seen, the contour lines of the irradiance exhibit a symmetry about the axes u=0 and v=0, and the on-axis irradiance of the TEM11 - and TEM33 -mode H–G beams vanishes, so that the approach based on Eq. (19) is not applicable [10]. However, the irradiance distributions in Fig. 3 become asymmetric about the axis z = f and the power is more concentrated at a point just before z = f. It means that the focal plane shifts slightly toward the lens, which is further con=rmed in Fig. 4, where the beam width calculated by the use of Eq. (21) it plotted against Iz=f. The dependence of relative focal shift Izf on the Fresnel number Nw and truncation parameter is depicted in Fig. 5, from which we see that |Izf | increases with decreasing and increasing mode index, and |Izf | decreases as Nw increases, if the mode indices m and n are kept =xed. 5. Discussion and concluding remarks In this paper, the encircled-power criterion has been used to study the focal shift in H–G beams whose axial irradiance vanishes. Numerical calculation results have shown that the focal shift in H–G beams depends on the truncation parameter, mode index and Fresnel number. Although the encircled-power criterion provides a more general de=nition of the focal plane and the associated focal shift, two questions arise. First, why do you use a factor 0.8? The choice of the factor is related to the de=nition of the beam width, or to the fraction of power in the bucket [14,15]. Because there are di>erent de=nitions of the beam width, an appropriate choice of the de=nition is necessary. Second, what
Fig. 2. Contour lines of the irradiance of H–G beams at the focal plane: (a) TEM11 mode; (b) TEM33 mode. The calculation parameters are = 632:8 nm, w0 = 1 mm, f = 400 mm and a = 2 mm.
are the real focal plane and focal shift of beams whose axial irradiance does not vanish, but the transversal maximum irradiance is not achieved on the axis? A typical example is the TEM22 -mode H–G beam, whose contour lines are shown in Figs. 6(a) and (b). The other calculation parameters are the same as those in Fig. 2. For such a type of beam both Eqs. (19) and (21) can be applied to calculate the focal
438
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
Fig. 4. Beam width W of TEM11 - and TEM33 -mode H–G beams varies with Iz=f.
Fig. 3. Irradiance distribution of H–G beams along the direction of the white lines in Fig. 2. (a) TEM11 mode; (b) TEM33 mode.
shift, and the results are compiled in Figs. 7(a) and (b). The variation of Izf with and Nw is similar by using Eqs. (19) and (21), whereas the quantitative di>erence for both cases is apparent, and |Izf | calculated by the use of Eq. (19) is larger than that by the use of Eq. (21). The value of |Izf | for the latter case is several percent, but for the former case can reach 10% or more. From Fig. 6 it turns out that |Izf | ≈ 0:06, if only the axial irradiance is used to
Fig. 5. Relative focal shift Izf of TEM11 - and TEM33 -mode H–G beams is plotted against (a) Fresnel number Nw , (b) truncation parameter .
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
439
Fig. 7. Relative focal shift Izf of a TEM22 -mode H–G beam is plotted against (a) Fresnel number Nw , (b) truncation parameter .
dealt with. In principle, although the real focal plane or the factor in Eq. (21) can be chosen according to the practical application purpose and beams used, further study seems to be necessary.
Acknowledgements Fig. 6. (a) Contour lines of the irradiance of a TEM22 mode H–G beam at the focal plane. (b) Irradiance distribution of a TEM22 -mode H–G beam along the direction of the white line in Fig. 6(a). The other calculation parameters are the same as those in Fig. 2.
calculate the focal shift of the TEM22 -mode H–G beam. However, |Izf | becomes smaller as the encircled-power criterion is applied. Physically, for the beams with large irradiance modulation it is unsuitable to determine the real focal plane if only the on-axis irradiance is taken into consideration. However, what is the real focal plane for such a type of beam? A similar question as the =rst one has to be
This work was supported by the Foundations of State Key Laboratory of Laser Technology and National Hi-Tech Project of China.
References [1] Li Y, Wolf E. Focal shifts di>racted converging spherical waves. Opt Commun 1981;39:211–5. [2] Li Y, Wolf E. Focal shift in focused truncated Gaussian beams. Opt Commun 1982;42:151–6. [3] Carter WH. Focal shift and concept of e>ective Fresnel number for a Gaussian laser beam. Appl Opt 1982;21:1989–94.
440
B. L/u, R. Peng / Optics & Laser Technology 35 (2003) 435 – 440
[4] Li Y. Variations of the axial intensity pattern formed by a focused truncated Gaussian beams. Opt Commun 1988;68:324–8. [5] Carter WH, Aburdene MF. Focal shift in Laguerre–Gaussian beams. J Opt Soc Am A 1987;4:1949–52. [6] L)u B, Huang W, Zhang B, Cai B. Focal shift in apertured Gaussian beams and relation with the lens focus. Optik 1995;99: 8–12. [7] Li Y. Propagation of focal shift through axisymmetrical optical systems. Opt Commun 1993;95:13–7. [8] Borghi R, Santarsiero M, Vicalvi S. Focal shift of focused Sat-topped beams. Opt Commun 1998;154:243–8. [9] L)u B, Zhang B. Focusing of a Gaussian Schell-model beam through a circular lens. J Mod Opt 1995;42:289–98.
[10] Greene PL, Hall DG. Focal shift in vector beams. Opt Express 1999;4:411–9. [11] Collins Jr SA. Lens-system di>raction integral written in terms of matrix. J Opt Soc Am 1970;60:1168–77. [12] Siegman AE. Lasers. California: University Science Books Mill Valley, 1986. p. 644. [13] Tao X, Zhou N, L)u B. Recurrence propagation equation of Hermite– Gaussian beams through a paraxial optical ABCD system with hard-edge aperture, to be published. [14] Siegmen AE. How to (maybe) measure laser beam quality. OSA TOPS 1998;17:184–98. [15] Garay A. Continuous wave deuterium Suoride laser beam diagnostic system. SPIE 1988;888:17–22.