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Effective radius of curvature of Hermite–Gaussian array beams
ARTICLE IN PRESS Optics & Laser Technology 42 (2010) 1054–1058
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Effective radius of curvature of Hermite–Gaussian array beams Guangming Ji a,, Xiaoling Ji b a b
College of Information Management, Chengdu University of Technology, Chengdu 610059, China Department of Physics, Sichuan Normal University, Chengdu 610068, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 October 2009 Received in revised form 16 January 2010 Accepted 19 January 2010 Available online 6 February 2010
Analytical expressions for the effective radius of curvature, R, of Hermite–Gaussian (H–G) array beams propagating in free space for both coherent and incoherent combinations are derived. It is shown that for the two types of beam combination a minimum of the effective radius of curvature, Rmin, appears as the propagation distance z increases. For the coherent combination, R is larger than that for the incoherent combination. The position zmin where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. For the two types of beam combination, R and zmin increase with increasing beam number, increasing beam separation distance, increasing waist width, and decreasing beam order and wavelength. In particular, the R of single H–G beams is always smaller than that of H–G array beams; the R of Gaussian array beams is always larger than that of H–G array beams. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Effective radius of curvature Hermite–Gaussian (H–G) array beam Coherent and incoherent beam combinations
1. Introduction The propagation of laser beams is a topic that has been of considerable theoretical and practical interest. Much work has been carried out concerning the propagation properties of different laser beams, such as the intensity, beam width, far-field divergence angle, coherence length, kurtosis parameter and beam propagation factor (i.e., M2-factor) [1–9]. However, until now few papers have dealt with the radius of curvature of laser beams. In 1992, the radius of curvature of non-Gaussian and nonspherical beams was defined in Ref. [10] by using the spherical wave front that best fits the actual wave front. However, based on the definition in Ref. [10], it is difficult to obtain an easy analytical expression for the radius of curvature of non-Gaussian and nonspherical beams, since the derivatives of the field are involved in the definition (see Eq. (16) in Ref. [10]). In 2002, the radius of curvature of Gaussian Schell-model (GSM) beams in turbulence was evaluated from a mutual coherence expression [11]. However, the method used in Ref. [11] is only applicable to laser beams expressed by a single Gaussian exponential term. Beam combination is a useful method for scaling diode arrays, CO2 lasers and hydrogen fluoride/deuterium fluoride chemical lasers up to high-power level [12,13]. Beam combination has been used widely in high-power system, inertial confinement fusion and high-energy weapons. Some propagation properties of laser array beams have been studied [14–22]. In practice, there are certain scenarios to minimize excitation of nonlinearities within Corresponding author.
E-mail address: [email protected] (G. Ji). 0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.01.030
the crystal of a laser, or when the received spot needs to have a multiple spot pattern. In addition, the conversion efficiency, power out/power in, is greater for higher-order modes. Higherorder solutions of the paraxial wave equation can take the form of the either Hermite–Gaussian (H–G) functions in a rectangular coordinate system or Laguerre–Gaussian (L–G) functions in a cylindrical coordinate system. In real lasers, the Brewster windows and other tilted surfaces or distorted elements usually provide a small but inherent rectangular symmetry to the laser cavity [23]. Real lasers, therefore, overwhelmingly elect to oscillate in near Hermite–Gaussian (H–G) modes. In this paper, based on the definition of effective radius of curvature of an arbitrary field [24], the easy analytical expressions for the effective radius of curvature of H–G array beams propagating in free space are derived, where coherent and incoherent beam combinations are considered. The effective radius of curvature of H–G array beams are studied both analytically and numerically, and some interesting results are obtained.
2. Theoretical formulae As shown in Fig. 1, a one-dimensional array beam in linear symmetry consists of N individual off-axis H–G beams positioned at the plane z = 0, the waist width of the corresponding TEM00 mode is w0, and the beam separation distance is xd. For the sake of simplicity, let N= odd number. It is clear Fig. 1 reduces to a single H–G beam centered at the origin when N= 1.
ARTICLE IN PRESS G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
1055
y x 0 xd Fig. 1. A schematic diagram of the one-dimensional array beam.
tedious integral calculations we obtain
2.1. Coherent combination For the phase-locked case, the N individual H–G beams combine coherently. The cross-spectral density function of the H–G array beam at the plane z =0 is expressed as "pffiffiffi # ðN1=2Þ ðN1=2Þ X X 2ðx01 i1 xd Þ Hm Wðx01 ; x02 ; 0Þ ¼ w0 i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ "pffiffiffi # ( ) ðx01 i1 xd Þ2 þðx02 i2 xd Þ2 2ðx02 i2 xd Þ exp ; Hm w0 w20 ð1Þ
/x2 S ¼ A þBz2 ;
ð10Þ
where A¼
ðN1=2Þ X
ðN1=2Þ X
2 2 fw20 ½Lm ðDÞ2Lð1Þ m ðDÞ þði1 þ i2 Þ xd Lm ðDÞg
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
expðD=2Þ=8C;
B¼
ðN1=2Þ X
ðN1=2Þ X
ð11Þ
ð1Þ fLm ðDÞ2Lð1Þ m ðDÞ½Lm ðDÞ4Lm ðDÞ
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
where * denotes the complex conjugate, Hm is the mth-order Hermite polynomial. Based on the Huygens–Fresnel principle, the intensity of the H–G array beam represented by Eq. (1) propagating in free space reads as [25] ZZ k dx01 dx02 Wðx01 ; x02 ; 0Þ Iðx; zÞ ¼ 2pz ik exp ð2Þ ½ðx0 21 x0 22 Þ2xðx01 x02 Þ ; 2z where k is the wave number related to the wavelength l by k= 2p/l. To obtain the analytical results, the new variables of integration are introduced as x02 þ x0 ; v ¼ x02 x01 ; ð3Þ 2 Eq. (2) then becomes "pffiffiffi # "pffiffiffi # ZZ k 1 1 2 2 du dvHm u þ v Hm u v Iðx; zÞ ¼ w0 w0 2pz 2 2 ! ! 2 2 2u v k k exp i uv exp i xv : exp 2 exp 2 z z w0 2w0 u¼
ð4Þ The second moment /x2S (i.e., beam width) is defined as [24] R 2 x Iðx; zÞ dx : ð5Þ /x2 S ¼ R Iðx; zÞ dx Upon substituting from Eq. (4) into Eq. (5), recalling the integral formulae Z 00 ð7Þ x2 expði2pxsÞ dx ¼ d ðsÞ=ð2pÞ2 ; Z Z
pffiffiffiffi expðx2 ÞHm ðx þ yÞHm ðx þ zÞ dx ¼ 2m pLm ð2yzÞ; 00
00
f ðxÞd ðxÞ dx ¼ f ð0Þ;
D¼
C¼
ð9Þ
where Lm denotes the mth-order Laguerre polynomial, d denotes the Dirac delta function and d00 is its second derivative, f is an arbitrary function and f00 is its second derivative; after very
ð12Þ
ði1 i2 Þ2 x2d ; w20
ð13Þ
ðN1=2Þ X
ðN1=2Þ X
expðD=2ÞLm ðDÞ
ð14Þ
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
with LðlÞ m being lth (l= 1, 2, 3, y) derivative of Lm. According to Ref. [24], the free space propagation law of the second moments /x2S and /xyS of an arbitrary field can be expressed as 2
/x2 S ¼ /x2 S0 þ /y S0 z2 ; 2
/xyS ¼ /y S0 z;
ð15Þ ð16Þ
2
where /y S represents the far-field divergence of the beam, and the angle brackets with the subscript 0 denote the second moments in the plane z =0. On comparing Eq. (15) with Eq. (10), we obtain /x2S0 =A and 2 /y S0 = B. Thus, the second moments /x2S and /xyS of the H–G array beam are given by /x2 S ¼ A þBz2 ;
ð17Þ
/xyS ¼ Bz:
ð18Þ
The effective radius of curvature of an arbitrary field is defined as [24] R¼
/x2 S : /xyS
ð19Þ
Upon substituting from Eqs. (17) and (18) into Eq. (19), yields R¼
ð8Þ
2 2 þ4Lð2Þ m ðDÞDgexpðD=2Þ=2w0 k C;
A þz: Bz
ð20Þ
Eq. (20) indicates that for the coherent combination, the effective radius of curvature of H–G array beams depends on N, m, xd, w0 and l. It is noted that, for m= 0 Eq. (20) reduces to the effective radius of curvature of Gaussian array beams for the coherent combination, and for N= 1 Eq. (20) reduces to the effective radius of curvature of a single H–G beam centered at the origin.
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Eq. (20) can be rewritten as zR z ; ð21Þ þ R ¼ zR z zR pffiffiffiffiffiffiffiffiffi where zR ¼ A=B is the Rayleigh range of the H–G array beams for the coherent combination. From Eq. (21) we conclude that the effective radius of curvature reaches its minimum Rmin = 2zR when the propagation distance zmin = zR. From Eq. (20) we obtain R1 ¼ lim R z: z-1
For N=1, Eq. (24) reduces to the effective radius of curvature of a single H–G beam centered at the origin, i.e., R¼
k2 w40 þz: 4z
ð28Þ
Eq. (28) is also the effective radius of curvature of a single Gaussian beam centered at the origin.
3. Numerical calculation results and analysis ð22Þ
Eq. (22) implies that for the coherent combination the wavefront of H–G array beams can be regarded as the spherical surface when the free space propagation distance is large enough, which is independent of the beam parameters (i.e., N, m, xd, w0 and l). The result is also valid for the incoherent beam combination.
2.2. Incoherent combination For the non-phase-locked case, the N individual H–G beams combine incoherently, i.e., the superposition of the intensity should be considered. Similarly, the second moment /x2S of H–G array beams for the incoherent combination can be obtained, which is given by ! 1 N 2 1 2 1 1 þ 2m 2 2 2 z : ð23Þ ð1 þ 2mÞw0 þ xd þ 2 /x S ¼ 4 3 k w20 From Eqs. (15), (16), (19) and (23), we obtain the effective radius of curvature of the H–G array beam for the incoherent combination, i.e., " # ðN 2 1Þx2d k2 w20 2 w0 þ þz: ð24Þ R¼ 4z 3ð1 þ2mÞ
Numerical calculation results are given in Figs. 2–6 to show changes of the effective radius of curvature R of H–G array beams with different values of the beam parameters (i.e., N, m, xd, w0 and l), where solid and dashed curves represent R for the coherent combination and the incoherent combination, respectively. From Figs. 2–6 it can be seen that for the two types of beam combination a minimum Rmin of the effective radius of curvature appears as the propagation distance z increases. For H–G array beams with the same beam parameters, R for the coherent combination is larger than that for the incoherent combination, and the position zmin where the effective radius of curvature reaches its minimum is further away from the source plane (z= 0) for the coherent combination than that for the incoherent combination. The effective radius of curvature R of H–G array beams with different values of the beam number N=1, 5 and 9 versus the propagation distance z is shown in Fig. 2. Fig. 2 indicates that R increases with increasing N for the two types of combination. In particular, R of a single H–G beam is always smaller than that of an H–G array beam. In addition, zmin increases with increasing N. For example, when N= 1, 5 and 9, we have zmin =3.0, 13.6 and 24.3 m for the coherent combination, and zmin = 3.0, 5.7 and 9.3 m for the incoherent combination. Fig. 3 gives the effective radius of curvature R of H–G array beams with different values of the beam order m= 0, 1 and 4 versus the propagation distance z. It can be
Eq. (24) can also be rewritten as R¼
z2R þz; z
ð25Þ
100
ð26Þ
80
zR ¼
" # ðN2 1Þx2d k2 w20 w20 þ 4 3ð1þ 2mÞ
is the Rayleigh range of the H–G array beams for the incoherent combination. From Eq. (25) it is clear that the effective radius of curvature R increases with increasing the Rayleigh range zR. It means that the laser beam with longer Rayleigh range will have larger effective radius of curvature. On the other hand, Eq. (26) shows that the H–G array beam with larger beam number N, beam separation distance xd, waist width w0, and smaller beam order m and wavelength l will have longer Rayleigh range. Therefore, we can conclude that R increases with increasing N, xd, w0, and with decreasing m and l. It implies that R of single H–G beams (i.e., N= 1) is always smaller than that of H–G array beams (i.e., N41), and R of Gaussian array beams (i.e., m= 0) is always larger than that of H–G array beams (i.e., m40). For m =0, Eq. (24) reduces to " # ðN2 1Þx2d k2 w20 w20 þ þz: ð27Þ R¼ 4z 3 Eq. (27) is the effective radius of curvature of Gaussian array beams for the incoherent combination.
R (m)
where
60 N=9 40 N=5 N=9
20
N=5 N =1
0 1
19
37
55 z (m)
73
91
Fig. 2. Effective radius of curvature R of H–G array beams with different values of the beam number N versus the propagation distance z. The calculation parameters are m = 1, w0 = xd = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
ARTICLE IN PRESS G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
1057
120
80
100 60
40
R (m)
R (m)
80
m= 0
w0 = 2.5mm w0 = 2mm
60
w0 = 2.5mm
m= 1 40
m= 4
20
m= 0 m= 1
w0 = 2mm
20
m= 4
w0 = 0.5mm
0 1
16
31
46
61
0
76
1
z (m)
16
31
46
61
76
91
z (m)
Fig. 3. Effective radius of curvature R of H–G array beams with different values of the beam order m versus the propagation distance z. The calculation parameters are N = 5, w0 = xd = 1 mm, l =1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
Fig. 5. Effective radius of curvature R of H–G array beams with different values of the waist width w0 versus the propagation distance z. The calculation parameters are m =1, N= 5, xd = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
120
100
100
80
80
60
= 0.6328um
R (m)
R (m)
60 xd = 5mm
40 40
= 1.06um
xd = 1mm
= 1.54um 20
xd = 0.5mm
20
= 0.6328um = 1.06um
xd = 1mm
= 1.54um
xd = 0.5mm
0 1
16
31
46 z (m)
61
76
91
0
1
16
31
46 z (m)
61
76
Fig. 4. Effective radius of curvature R of H–G array beams with different values of the beam separation distance xd versus the propagation distance z. The calculation parameters are m= 1, N= 5, w0 = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
Fig. 6. Effective radius of curvature R of H–G array beams with different values of the wavelength l versus the propagation distance z. The calculation parameters are m =1, N =5, w0 = xd =1 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
seen that for the two types of combination R decreases with increasing m. In particular, R of Gaussian array beams is always larger than that of H–G array beams if other beam parameters are the same for the two types of array beam. Furthermore, zmin
decreases with increasing m. For instance, when m= 0, 1 and 4, we have zmin = 17.6, 13.6 and 9.7 m for the coherent combination, and zmin =8.9, 5.7 and 4.1 m for the incoherent combination. The effective radius of curvature R of H–G array beams with different
ARTICLE IN PRESS 1058
G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
values of the beam separation distance xd = 0.5, 1 and 5 mm versus the propagation distance z is plotted in Fig. 4. Fig. 4 shows that R and zmin increase with increasing xd for the two types of combination. The effective radius of curvature R of H–G array beam with different values of the waist width w0 =0.5, 2 and 2.5 mm versus the propagation distance z is shown in Fig. 5, which indicates that R and zmin increase with increasing w0 for the two types of combination. In particular, from Figs. 4 and 5 it can also been seen that the curves of R are the same for the two types of combination when xd is large enough (e.g., xd =5 mm in Fig. 4) or w0 is small enough (e.g., w0 = 0.5 mm in Fig. 5). The physical reason is that each beam expands independently and there is no superposition between them when xd is large enough or w0 is small enough. For such cases the coherent beam combination can be regarded as the incoherent beam combination. The effective radius of curvature R of H–G array beams with different values of the wavelength l =0.6328, 1.06 and 1.54 mm versus the propagation distance z is depicted in Fig. 6. Fig. 6 shows that R and zmin decrease with increasing l for the two types of combination. The R for the coherent combination when l = 1.54 mm is almost the same as that for the incoherent combination when l = 0.6328 mm.
4. Conclusions The easy analytical expressions for the effective radius R of curvature of H–G array beams in free space have been derived, where both coherent and incoherent combinations have been considered. The analytical formulae of R of Gaussian array beams and single H–G beams have been treated as special cases of those of H–G array beams. It has been shown that for the two types of combination there exists a minimum Rmin of the effective radius of curvature as the propagation distance z increases. For the coherent combination R is larger than that for the incoherent combination, and the position zmin where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. However, the R is almost the same for the two types of combination when xd is large enough or w0 is small enough. For the two types of combination, R and zmin increase with increasing N, xd, w0, and decreasing m and l. In particular, the R of single H–G beams is always smaller than that of H–G array beams; R of Gaussian array beams is always larger than that of H–G array beams if other beam parameters are the same for the two types of array beam.
Acknowledgments The authors are very thankful to the two anonymous reviewers for valuable comments. This work was supported by the National Natural Science Foundation of China under Grant 60778048. References [1] Siegman A E. New developments in laser resonators. Proc SPIE 1990;1224:2–14. [2] Pu J, Zhang H. Intensity distribution of Gaussian beams focused by a lens with spherical aberration. Opt Commun 1998;151:331–8. [3] Amarande S A. Beam propagation factor and the kurtosis parameter of flattened Gaussian beams. Opt Commun 1996;129:311–7. [4] Gbur G, Wolf E. Spreading of partially coherent beams in random media. J Opt Soc Am A 2002;19:1592–8. [5] Shirai T, Dogariu A, Wolf E. Directionality of Gaussian-Schell-model beams propagating in atmospheric turbulence. Opt Lett 2003;28:610–2. [6] Cai Y, He S. Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere. Opt Lett 2006;31:568–70. [7] Eyyuboglu H T, Baykal Y. Average intensity and spreading of cosh-Gaussian ˘ laser beams in the turbulent atmosphere. Appl Opt 2005;44:976–83. [8] Eyyuboglu HT, Baykal Y, Cai Y. Complex degree of coherence for partially ˘ coherent general beams in atmospheric turbulence. J Opt Soc Am A 2007;24:2891–900. [9] Dan Y, Zhang B. Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere. Opt Express 2008;16:15563–75. [10] Miguel AP, Javier A, Eusebio B. Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams. Appl Opt 1992;31:6389–402. [11] Ricklin JC, Davidson FM. Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication. J Opt Soc Am A 2002;19:1794–802. [12] Strohschein JD, Herb JJ, Clarence EC. Beam propagation constants for a radial laser array. Appl Opt 1998;37:1045–8. [13] Brunel M, Floch AL, Bretenaker F, Marty J, Molva E. Coherent addition of adjacent lasers by forked eigenstate operation. Appl Opt 1998;37:2402–6. [14] Lu¨ B, Ma H. Beam propagation properties of radial laser arrays. J Opt Soc Am A 2000;17:2005–9. [15] Deng D. Propagation properties of off-axis cosh Gaussian beam combinations through a first-order optical system. Phys Lett A 2004;333:485–94. [16] Cai Y, Chen Y, Eyyuboglu HT, Baykal Y. Propagation of laser array beams in a turbulent atmosphere. Appl Phys B 2007;88:467–75. [17] Du X, Zhao D. Propagation characteristics of stochastic electromagnetic array beams. Appl Phys B 2008;93:901–5. [18] Zhu Y, Zhao D, Du X. Propagation of stochastic Gaussian Schell model array beams in turbulent atmosphere. Opt Express 2008;16:18437–42. [19] Ji X, Zhang E, Lu¨ B. Superimposed partially coherent beams propagating through atmospheric turbulence. J Opt Soc Am B 2008;25:825–33. [20] Ji X, Li X. Directionality of Gaussian array beams propagating in atmospheric turbulence. J Opt Soc Am A 2009;26:236–43. [21] Ji X, Zhang T, Jia X. Beam propagation factor of partially coherent Hermite– Gaussian array beams. J Opt A: Pure Appl Opt 2009;11:105705. [22] Eyyuboglu HT, Baykal Y, Cai Y. Scintillations of laser array beams. Appl Phys B ˘ 2008;91:265–71. [23] Siegman AE. Lasers. California: University Science Books; 1986. [24] Weber H. Propagation of higher-order intensity moments in quadratic-index media. Opt Quant Electron 1992;24:1027–49. [25] Mandel L, Wolf E. Optical coherence and quantum optics. Cambridge: Cambridge University Press; 1995.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Effective radius of curvature of Hermite–Gaussian array beams Guangming Ji a,, Xiaoling Ji b a b
College of Information Management, Chengdu University of Technology, Chengdu 610059, China Department of Physics, Sichuan Normal University, Chengdu 610068, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 20 October 2009 Received in revised form 16 January 2010 Accepted 19 January 2010 Available online 6 February 2010
Analytical expressions for the effective radius of curvature, R, of Hermite–Gaussian (H–G) array beams propagating in free space for both coherent and incoherent combinations are derived. It is shown that for the two types of beam combination a minimum of the effective radius of curvature, Rmin, appears as the propagation distance z increases. For the coherent combination, R is larger than that for the incoherent combination. The position zmin where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. For the two types of beam combination, R and zmin increase with increasing beam number, increasing beam separation distance, increasing waist width, and decreasing beam order and wavelength. In particular, the R of single H–G beams is always smaller than that of H–G array beams; the R of Gaussian array beams is always larger than that of H–G array beams. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Effective radius of curvature Hermite–Gaussian (H–G) array beam Coherent and incoherent beam combinations
1. Introduction The propagation of laser beams is a topic that has been of considerable theoretical and practical interest. Much work has been carried out concerning the propagation properties of different laser beams, such as the intensity, beam width, far-field divergence angle, coherence length, kurtosis parameter and beam propagation factor (i.e., M2-factor) [1–9]. However, until now few papers have dealt with the radius of curvature of laser beams. In 1992, the radius of curvature of non-Gaussian and nonspherical beams was defined in Ref. [10] by using the spherical wave front that best fits the actual wave front. However, based on the definition in Ref. [10], it is difficult to obtain an easy analytical expression for the radius of curvature of non-Gaussian and nonspherical beams, since the derivatives of the field are involved in the definition (see Eq. (16) in Ref. [10]). In 2002, the radius of curvature of Gaussian Schell-model (GSM) beams in turbulence was evaluated from a mutual coherence expression [11]. However, the method used in Ref. [11] is only applicable to laser beams expressed by a single Gaussian exponential term. Beam combination is a useful method for scaling diode arrays, CO2 lasers and hydrogen fluoride/deuterium fluoride chemical lasers up to high-power level [12,13]. Beam combination has been used widely in high-power system, inertial confinement fusion and high-energy weapons. Some propagation properties of laser array beams have been studied [14–22]. In practice, there are certain scenarios to minimize excitation of nonlinearities within Corresponding author.
E-mail address: [email protected] (G. Ji). 0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.01.030
the crystal of a laser, or when the received spot needs to have a multiple spot pattern. In addition, the conversion efficiency, power out/power in, is greater for higher-order modes. Higherorder solutions of the paraxial wave equation can take the form of the either Hermite–Gaussian (H–G) functions in a rectangular coordinate system or Laguerre–Gaussian (L–G) functions in a cylindrical coordinate system. In real lasers, the Brewster windows and other tilted surfaces or distorted elements usually provide a small but inherent rectangular symmetry to the laser cavity [23]. Real lasers, therefore, overwhelmingly elect to oscillate in near Hermite–Gaussian (H–G) modes. In this paper, based on the definition of effective radius of curvature of an arbitrary field [24], the easy analytical expressions for the effective radius of curvature of H–G array beams propagating in free space are derived, where coherent and incoherent beam combinations are considered. The effective radius of curvature of H–G array beams are studied both analytically and numerically, and some interesting results are obtained.
2. Theoretical formulae As shown in Fig. 1, a one-dimensional array beam in linear symmetry consists of N individual off-axis H–G beams positioned at the plane z = 0, the waist width of the corresponding TEM00 mode is w0, and the beam separation distance is xd. For the sake of simplicity, let N= odd number. It is clear Fig. 1 reduces to a single H–G beam centered at the origin when N= 1.
ARTICLE IN PRESS G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
1055
y x 0 xd Fig. 1. A schematic diagram of the one-dimensional array beam.
tedious integral calculations we obtain
2.1. Coherent combination For the phase-locked case, the N individual H–G beams combine coherently. The cross-spectral density function of the H–G array beam at the plane z =0 is expressed as "pffiffiffi # ðN1=2Þ ðN1=2Þ X X 2ðx01 i1 xd Þ Hm Wðx01 ; x02 ; 0Þ ¼ w0 i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ "pffiffiffi # ( ) ðx01 i1 xd Þ2 þðx02 i2 xd Þ2 2ðx02 i2 xd Þ exp ; Hm w0 w20 ð1Þ
/x2 S ¼ A þBz2 ;
ð10Þ
where A¼
ðN1=2Þ X
ðN1=2Þ X
2 2 fw20 ½Lm ðDÞ2Lð1Þ m ðDÞ þði1 þ i2 Þ xd Lm ðDÞg
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
expðD=2Þ=8C;
B¼
ðN1=2Þ X
ðN1=2Þ X
ð11Þ
ð1Þ fLm ðDÞ2Lð1Þ m ðDÞ½Lm ðDÞ4Lm ðDÞ
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
where * denotes the complex conjugate, Hm is the mth-order Hermite polynomial. Based on the Huygens–Fresnel principle, the intensity of the H–G array beam represented by Eq. (1) propagating in free space reads as [25] ZZ k dx01 dx02 Wðx01 ; x02 ; 0Þ Iðx; zÞ ¼ 2pz ik exp ð2Þ ½ðx0 21 x0 22 Þ2xðx01 x02 Þ ; 2z where k is the wave number related to the wavelength l by k= 2p/l. To obtain the analytical results, the new variables of integration are introduced as x02 þ x0 ; v ¼ x02 x01 ; ð3Þ 2 Eq. (2) then becomes "pffiffiffi # "pffiffiffi # ZZ k 1 1 2 2 du dvHm u þ v Hm u v Iðx; zÞ ¼ w0 w0 2pz 2 2 ! ! 2 2 2u v k k exp i uv exp i xv : exp 2 exp 2 z z w0 2w0 u¼
ð4Þ The second moment /x2S (i.e., beam width) is defined as [24] R 2 x Iðx; zÞ dx : ð5Þ /x2 S ¼ R Iðx; zÞ dx Upon substituting from Eq. (4) into Eq. (5), recalling the integral formulae Z 00 ð7Þ x2 expði2pxsÞ dx ¼ d ðsÞ=ð2pÞ2 ; Z Z
pffiffiffiffi expðx2 ÞHm ðx þ yÞHm ðx þ zÞ dx ¼ 2m pLm ð2yzÞ; 00
00
f ðxÞd ðxÞ dx ¼ f ð0Þ;
D¼
C¼
ð9Þ
where Lm denotes the mth-order Laguerre polynomial, d denotes the Dirac delta function and d00 is its second derivative, f is an arbitrary function and f00 is its second derivative; after very
ð12Þ
ði1 i2 Þ2 x2d ; w20
ð13Þ
ðN1=2Þ X
ðN1=2Þ X
expðD=2ÞLm ðDÞ
ð14Þ
i1 ¼ ðN1=2Þ i2 ¼ ðN1=2Þ
with LðlÞ m being lth (l= 1, 2, 3, y) derivative of Lm. According to Ref. [24], the free space propagation law of the second moments /x2S and /xyS of an arbitrary field can be expressed as 2
/x2 S ¼ /x2 S0 þ /y S0 z2 ; 2
/xyS ¼ /y S0 z;
ð15Þ ð16Þ
2
where /y S represents the far-field divergence of the beam, and the angle brackets with the subscript 0 denote the second moments in the plane z =0. On comparing Eq. (15) with Eq. (10), we obtain /x2S0 =A and 2 /y S0 = B. Thus, the second moments /x2S and /xyS of the H–G array beam are given by /x2 S ¼ A þBz2 ;
ð17Þ
/xyS ¼ Bz:
ð18Þ
The effective radius of curvature of an arbitrary field is defined as [24] R¼
/x2 S : /xyS
ð19Þ
Upon substituting from Eqs. (17) and (18) into Eq. (19), yields R¼
ð8Þ
2 2 þ4Lð2Þ m ðDÞDgexpðD=2Þ=2w0 k C;
A þz: Bz
ð20Þ
Eq. (20) indicates that for the coherent combination, the effective radius of curvature of H–G array beams depends on N, m, xd, w0 and l. It is noted that, for m= 0 Eq. (20) reduces to the effective radius of curvature of Gaussian array beams for the coherent combination, and for N= 1 Eq. (20) reduces to the effective radius of curvature of a single H–G beam centered at the origin.
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Eq. (20) can be rewritten as zR z ; ð21Þ þ R ¼ zR z zR pffiffiffiffiffiffiffiffiffi where zR ¼ A=B is the Rayleigh range of the H–G array beams for the coherent combination. From Eq. (21) we conclude that the effective radius of curvature reaches its minimum Rmin = 2zR when the propagation distance zmin = zR. From Eq. (20) we obtain R1 ¼ lim R z: z-1
For N=1, Eq. (24) reduces to the effective radius of curvature of a single H–G beam centered at the origin, i.e., R¼
k2 w40 þz: 4z
ð28Þ
Eq. (28) is also the effective radius of curvature of a single Gaussian beam centered at the origin.
3. Numerical calculation results and analysis ð22Þ
Eq. (22) implies that for the coherent combination the wavefront of H–G array beams can be regarded as the spherical surface when the free space propagation distance is large enough, which is independent of the beam parameters (i.e., N, m, xd, w0 and l). The result is also valid for the incoherent beam combination.
2.2. Incoherent combination For the non-phase-locked case, the N individual H–G beams combine incoherently, i.e., the superposition of the intensity should be considered. Similarly, the second moment /x2S of H–G array beams for the incoherent combination can be obtained, which is given by ! 1 N 2 1 2 1 1 þ 2m 2 2 2 z : ð23Þ ð1 þ 2mÞw0 þ xd þ 2 /x S ¼ 4 3 k w20 From Eqs. (15), (16), (19) and (23), we obtain the effective radius of curvature of the H–G array beam for the incoherent combination, i.e., " # ðN 2 1Þx2d k2 w20 2 w0 þ þz: ð24Þ R¼ 4z 3ð1 þ2mÞ
Numerical calculation results are given in Figs. 2–6 to show changes of the effective radius of curvature R of H–G array beams with different values of the beam parameters (i.e., N, m, xd, w0 and l), where solid and dashed curves represent R for the coherent combination and the incoherent combination, respectively. From Figs. 2–6 it can be seen that for the two types of beam combination a minimum Rmin of the effective radius of curvature appears as the propagation distance z increases. For H–G array beams with the same beam parameters, R for the coherent combination is larger than that for the incoherent combination, and the position zmin where the effective radius of curvature reaches its minimum is further away from the source plane (z= 0) for the coherent combination than that for the incoherent combination. The effective radius of curvature R of H–G array beams with different values of the beam number N=1, 5 and 9 versus the propagation distance z is shown in Fig. 2. Fig. 2 indicates that R increases with increasing N for the two types of combination. In particular, R of a single H–G beam is always smaller than that of an H–G array beam. In addition, zmin increases with increasing N. For example, when N= 1, 5 and 9, we have zmin =3.0, 13.6 and 24.3 m for the coherent combination, and zmin = 3.0, 5.7 and 9.3 m for the incoherent combination. Fig. 3 gives the effective radius of curvature R of H–G array beams with different values of the beam order m= 0, 1 and 4 versus the propagation distance z. It can be
Eq. (24) can also be rewritten as R¼
z2R þz; z
ð25Þ
100
ð26Þ
80
zR ¼
" # ðN2 1Þx2d k2 w20 w20 þ 4 3ð1þ 2mÞ
is the Rayleigh range of the H–G array beams for the incoherent combination. From Eq. (25) it is clear that the effective radius of curvature R increases with increasing the Rayleigh range zR. It means that the laser beam with longer Rayleigh range will have larger effective radius of curvature. On the other hand, Eq. (26) shows that the H–G array beam with larger beam number N, beam separation distance xd, waist width w0, and smaller beam order m and wavelength l will have longer Rayleigh range. Therefore, we can conclude that R increases with increasing N, xd, w0, and with decreasing m and l. It implies that R of single H–G beams (i.e., N= 1) is always smaller than that of H–G array beams (i.e., N41), and R of Gaussian array beams (i.e., m= 0) is always larger than that of H–G array beams (i.e., m40). For m =0, Eq. (24) reduces to " # ðN2 1Þx2d k2 w20 w20 þ þz: ð27Þ R¼ 4z 3 Eq. (27) is the effective radius of curvature of Gaussian array beams for the incoherent combination.
R (m)
where
60 N=9 40 N=5 N=9
20
N=5 N =1
0 1
19
37
55 z (m)
73
91
Fig. 2. Effective radius of curvature R of H–G array beams with different values of the beam number N versus the propagation distance z. The calculation parameters are m = 1, w0 = xd = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
ARTICLE IN PRESS G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
1057
120
80
100 60
40
R (m)
R (m)
80
m= 0
w0 = 2.5mm w0 = 2mm
60
w0 = 2.5mm
m= 1 40
m= 4
20
m= 0 m= 1
w0 = 2mm
20
m= 4
w0 = 0.5mm
0 1
16
31
46
61
0
76
1
z (m)
16
31
46
61
76
91
z (m)
Fig. 3. Effective radius of curvature R of H–G array beams with different values of the beam order m versus the propagation distance z. The calculation parameters are N = 5, w0 = xd = 1 mm, l =1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
Fig. 5. Effective radius of curvature R of H–G array beams with different values of the waist width w0 versus the propagation distance z. The calculation parameters are m =1, N= 5, xd = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
120
100
100
80
80
60
= 0.6328um
R (m)
R (m)
60 xd = 5mm
40 40
= 1.06um
xd = 1mm
= 1.54um 20
xd = 0.5mm
20
= 0.6328um = 1.06um
xd = 1mm
= 1.54um
xd = 0.5mm
0 1
16
31
46 z (m)
61
76
91
0
1
16
31
46 z (m)
61
76
Fig. 4. Effective radius of curvature R of H–G array beams with different values of the beam separation distance xd versus the propagation distance z. The calculation parameters are m= 1, N= 5, w0 = 1 mm, l = 1.06 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
Fig. 6. Effective radius of curvature R of H–G array beams with different values of the wavelength l versus the propagation distance z. The calculation parameters are m =1, N =5, w0 = xd =1 mm. ‘‘—’’, coherent combination; ‘‘U U U’’, incoherent combination.
seen that for the two types of combination R decreases with increasing m. In particular, R of Gaussian array beams is always larger than that of H–G array beams if other beam parameters are the same for the two types of array beam. Furthermore, zmin
decreases with increasing m. For instance, when m= 0, 1 and 4, we have zmin = 17.6, 13.6 and 9.7 m for the coherent combination, and zmin =8.9, 5.7 and 4.1 m for the incoherent combination. The effective radius of curvature R of H–G array beams with different
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G. Ji, X. Ji / Optics & Laser Technology 42 (2010) 1054–1058
values of the beam separation distance xd = 0.5, 1 and 5 mm versus the propagation distance z is plotted in Fig. 4. Fig. 4 shows that R and zmin increase with increasing xd for the two types of combination. The effective radius of curvature R of H–G array beam with different values of the waist width w0 =0.5, 2 and 2.5 mm versus the propagation distance z is shown in Fig. 5, which indicates that R and zmin increase with increasing w0 for the two types of combination. In particular, from Figs. 4 and 5 it can also been seen that the curves of R are the same for the two types of combination when xd is large enough (e.g., xd =5 mm in Fig. 4) or w0 is small enough (e.g., w0 = 0.5 mm in Fig. 5). The physical reason is that each beam expands independently and there is no superposition between them when xd is large enough or w0 is small enough. For such cases the coherent beam combination can be regarded as the incoherent beam combination. The effective radius of curvature R of H–G array beams with different values of the wavelength l =0.6328, 1.06 and 1.54 mm versus the propagation distance z is depicted in Fig. 6. Fig. 6 shows that R and zmin decrease with increasing l for the two types of combination. The R for the coherent combination when l = 1.54 mm is almost the same as that for the incoherent combination when l = 0.6328 mm.
4. Conclusions The easy analytical expressions for the effective radius R of curvature of H–G array beams in free space have been derived, where both coherent and incoherent combinations have been considered. The analytical formulae of R of Gaussian array beams and single H–G beams have been treated as special cases of those of H–G array beams. It has been shown that for the two types of combination there exists a minimum Rmin of the effective radius of curvature as the propagation distance z increases. For the coherent combination R is larger than that for the incoherent combination, and the position zmin where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. However, the R is almost the same for the two types of combination when xd is large enough or w0 is small enough. For the two types of combination, R and zmin increase with increasing N, xd, w0, and decreasing m and l. In particular, the R of single H–G beams is always smaller than that of H–G array beams; R of Gaussian array beams is always larger than that of H–G array beams if other beam parameters are the same for the two types of array beam.
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