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The formation of quasi-nondiffracting focused beams with second-harmonic generation of
flower Laguerre–Gaussian modes
View the table of contents for this issue, or go to the journal homepage for more 2013 Laser Phys. 23 115405
(http://iopscience.iop.org/1555-6611/23/11/115405)
Laser Phys. 23 (2013) 115405 (6pp) doi:10.1088/1054-660X/23/11/115405
The formation of quasi-nondiffracting
focused beams with second-harmonic
generation of flower Laguerre–Gaussian
modes
Y C Lin, K F Huang and Y F Chen
Department of Electrophysics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan
E-mail:[email protected]
Received 21 August 2013, in final form 13 September 2013 Accepted for publication 16 September 2013
Published 7 October 2013
Online atstacks.iop.org/LP/23/115405
Abstract
We theoretically demonstrate that the second-harmonic generation of a flower
Laguerre–Gaussian mode can be exploited to form a quasi-nondiffracting focused beam. It is verified that the product of donut Laguerre–Gaussian modes with opposite topological charges leads to the formation of the quasi-nondiffracting focused beam. The divergence of the quasi-nondiffracting focused beam is numerically confirmed to be
√
2 l + 1 times less than that of the Gaussian TEM00mode with the same beam waist parameter, where l is the
azimuthal index of the flower Laguerre–Gaussian mode. In an experiment, we employ a diode-pumped solid-state laser resonator with intracavity second-harmonic generation to realize the formation of the quasi-nondiffracting focused beams.
(Some figures may appear in colour only in the online journal)
1. Introduction
Focused beams have been extensively studied and applied in diverse fields. Remarkable examples are in optical tweezers [1, 2], lithography [3], optical data storage [4], optical interconnects [5], biophotonics [6], and optical coherence tomography [7]. Gaussian TEM00 modes have
been the prevalent ones since they can be emitted by most laser cavities. In recent decades, the zeroth-order Bessel beam has become an alternative method for the Gaussian TEM00mode in related topics on beam focusing [5–9]. Unlike
the conventional Gaussian modes with the beam divergence of the order of the Rayleigh range, Bessel beams have gained attention because of their inherent characteristic of nondiffraction or non-spreading [10–12]. This fascinating feature has led to progressive developments such as in the alignment of rod-like samples, stacking and manipulation of multiple spheres, and guiding of spheres along the extended line focus of a Bessel beam [9]. The finding of such a
diffraction-free solution to the Bessel beam [13] has not only stimulated a great number of theoretical and experimental studies but has also paved the way to the discovery of other intriguing nondiffracting focused beams, including the Airy beam [14,15].
In addition to the Bessel beam, the Laguerre–Gaussian (LG) modes 9p,±l are also solutions to the Helmholtz
equation in cylindrical coordinates, where p is the radial index, ±l is the azimuthal index, and the sign ± signifies the opposite topological charges. The spatial distributions of high-order 9p,0 modes are similar to those of Bessel
beams which possess a sharp central peak with inherent nondiffracting properties [16, 17]. In contrast, the flower LG mode [18–24], consisting of 90,l and 90,−l modes,
possesses a null at the beam center. The flower LG modes have been experimentally explored in a large-aperture CO2 laser [18], optically pumped [21] and electrically
pumped [22] vertical-cavity surface emitting semiconductor lasers (VCSELs), and a solid-state laser cavity with a
c
Laser Phys. 23 (2013) 115405 Y C Lin et al
nonlinear medium [19, 23,24]. It has been shown that the attractive features of the flower LG modes lead to rich studies in pattern formation [19,23,24] and recent research on the generation of two-dimensional optical vortex arrays [20].
In this work we first show that the second-harmonic generation (SHG) of a flower LG mode can give rise to the formation of a nondiffracting focused beam. We verify that the SHG of the flower LG mode can yield a cross term 90,l90,−l. With the re-projection scheme, the
cross term 90,l90,−l can be analytically expressed to be a
coherent superposition of a set of9p,0 modes with 0 ≤ p ≤
l. The derived expression clearly reveals that the distinct Gouy phases between the different 9p,0 modes lead to the
formation of a bright central maximum. The bright central beam displays a nondiffracting feature with the divergence numerically calculated to be
√
2l + 1 times less than that of the Gaussian TEM00 mode with the same beam waist
parameter. Finally, we use a diode-pumped solid-state laser cavity with intracavity SHG to experimentally confirm the formation of the quasi-nondiffracting focused beam. Here, the term ‘quasi-nondiffracting’ is used to distinguish a nearly nondiffracting beam that actually diffracts during propagation from an ideal diffraction-free one.
2. Theoretical analysis
Under the paraxial approximation, the traveling LG mode 9p,±lwith the wavenumber k and the beam waist wois given
by [25] 9p,±l(r, φ, z; k, wo) = (−1)p Np,l w(z) √ 2r w(z) !l Llp 2r2 w(z)2
× exp(±i l φ) exp − r 2 w(z)2 × exp(iγ k z) × exp[i(2p + l + 1) θG(z)], (1) where w(z) = wo p
1 +(z/zR)2 is the spot size at z, wo is
the beam radius at z = 0, zR =k w2o/2 is the Rayleigh
range,γ = 1 + r2/[2(z2+z2R)], θG(z) = tan−1(z/zR) is the
Gouy phase, Np,l=p2p!/(1 + δ0,l)π(l + p)! is the constant
for normalization, where δ0,l = 1 for l = 0, and Llp(·) is the associated Laguerre polynomial of integer indices l and p in the azimuthal and radial directions, respectively. The sign ± in the representation 9p,±l indicates opposite
topological charges. The standing wave LG mode is the coherent superposition of a pair of traveling wave LG modes with equal but opposite topological charges. In general, the basis for the standing wave LG modes is expressed as
9c p,l= 9p,+l+9p,−l /√2 (2a) and 9s p,l= 9p,+l−9p,−l/ √ 2i, (2b)
where the subscripts c and s are employed to express wavefunctions with azimuthal distributions of cos(lφ) and
sin(lφ), respectively. Unlike the traveling wave LG mode, the standing wave LG mode of l 6= 0 possesses no net orbital angular momentum. For p = 0, the traveling wave LG mode 9p,±lexhibits a simple donut structure, whereas the standing
wave LG mode9pc,lor9ps,ldisplays a flower structure. Next we consider that the flower LG mode is used as the fundamental wave for the second-harmonic process. Under the non-depletion approximation, the amplitude of the frequency-doubled field is proportional to the square of the fundamental field [26]. Furthermore, the second-harmonic process is generally implemented with focusing of the fundamental wave into the nonlinear crystal; namely, the generation of the second-harmonic wave is near the region of the beam waist, z = 0. Therefore, the second-harmonic wave for the fundamental flower LG mode with the wavenumber k1
and the beam waist w1oat z = 0 is given by
8(r, φ, 0) = η[9c
0,l(r, φ, 0; k1, w1o)]2, (3)
whereη is a constant related to the conversion efficiency in the second-harmonic process. Here, we consider the case of the cosine LG mode for convenience. Substituting equation (2a) into equation (3), the second-harmonic wave8(r, φ, 0) at the beam waist can be expressed as
8(r, φ, 0) = η 2{[90,+l(r, φ, 0; k1, w1o)] 2 + [90,−l(r, φ, 0; k1, w1o)]2 +290,+l(r, φ, 0; k1, w1o) ×90,−l(r, φ, 0; k1, w1o)}. (4) With the given distribution 8(r, φ, 0) evaluated at z = 0, we can then derive the z-dependent expression of the second-harmonic wave 8(r, φ, z) in terms of the Fresnel diffraction integral [25], 8(r, φ, z) = k2 i 2π zexp i2πz k2 Z ∞ 0 Z 2π 0 8(r, φ, 0) × exp ik2r 2 2z rdφ dr. (5)
To analytically manifest the z-dependence of the second-harmonic wave 8(r, φ, z) with a closed form, the wavefunction in equation (4) needs to be expanded with the LG modes and the parameters need to be changed from k1
and w1o for the fundamental wave to k2 and w2o for the
second-harmonic wave, where k2 =2k1 and w22o =w21o/2.
With equation (1), it can be verified that 9 0,±l(r, φ, 0; k1, w1o)2= √ (2l)! √ 2π 2ll!w 2o ×90,± 2l(r, φ, 0; k2, w2o) . (6) Equation (6) indicates that the SHG of the donut LG mode 90,±l directly leads the orbital angular momentum per photon to double from l ¯h to 2l ¯h. The corresponding experimental realization was popularized in previous work by Dholakia et al [27]. On the other hand, the cross term 90,+l(r, φ, 0; k1, w1o)90,−l(r, φ, 0; k1, w1o) in equation (4) 2
Figure 1. Theoretical simulations for the fundamental flower LG modes 90c,l(r, φ, 0; k1, w1o) 2
at the beam waist, z = 0 (first column), and their second-harmonic counterparts |8(r, φ, z)|2from z = 0 to 10z
R. First row: l = 5, second row: l = 12, third row: l = 20.
can be expanded with the expression [28]
xl=l!
l
X
n=0
Cnl(−1)nLn(x) , (7)
where Cln=l!/[n!(l − n)!] is the binomial coefficient. With equation (7), the cross term can be expressed as
90,+l(r, φ, 0; k1, w1o) 90,−l(r, φ, 0; k1, w1o) =√ 1 π 2lw 2o l X n=0 Cnl9n,0(r, φ, 0; k2, w2o) . (8)
It is clear that the multiplication of the LG modes with opposite topological charges is derived to be the coherent superposition of a set of LG modes 9n,0(r, φ, 0; k2, w2o),
as shown in equation (8). Although all the LG modes 9n,0(r, φ, 0; k2, w2o) in equation (8) possess the
maxi-mum intensity at the beam center, r = 0, it is worth noting that the interference leads the superposed wave Pl
n=0Cnl9n,0(r, φ, 0; k2, w2o) to display a null at r = 0.
Substituting equations (6) and (8) into equation (4), we can obtain 8(r, φ, 0) = η 2l√π w 2o " √(2l)! 2(l!) 9 c 0,2l(r, φ, 0; k2, w2o) + l X n=0 Cln9n,0(r, φ, 0; k2, w2o) # . (9)
The well-known propagation characteristics of the LG modes 9p,±l(r, φ, z; k2, wo) in equation (1) and9pc,l(r, φ, z; k2, wo)
in equation (2a) can be used to predict the amplitude function for8(r, φ, z) at any cross section z via the Fresnel diffraction integral. Substituting the expansion in equation (9) for 8(r, φ, 0) in equation (5), the z-dependence of the
second-harmonic wave can thus be given by 8(r, φ, z) = η 2l√π w 2o " √(2l)! 2(l!) 9 c 0,2l(r, φ, z; k2, w2o) + l X n=0 Cln9n,0(r, φ, z; k2, w2o) # . (10)
From equation (1), it can be understood that the variations of the Gouy phases in the propagation are not the same for the LG modes9n,0(r, φ, z; k2, w2o) with different orders
n. As a consequence, even though the superposed wave Pl
n=0Cln9n,0(r, φ, z; k2, w2o) at the beam waist, z = 0,
displays a null at the beam center, r = 0, the influence of the Gouy phase leads to an intriguing phenomenon that the second-harmonic wave 8(r, φ, z) in the propagation starts to display a central maximum. To be brief, the cross term 90,+l90,−l shown in equation (8) results in the superposed
term Pl
n=0Cln9n,0(r, φ, z; k2, w2o) which forms a focused
beam in the propagation.
In figure1, we theoretically present the intensity profiles of the fundamental flower LG modes |90c,l(r, φ, 0; k1, w1o)|2
at the beam waist, z = 0, and their second-harmonic counterparts |8(r, φ, z)|2 at different longitudinal positions from z = 0 to 10zR. It can be seen that the second-harmonic
LG modes |8(r, φ, 0)|2 display the same flower structures as the fundamental LG modes |90c,l(r, φ, 0; k1, w1o)|2 with
a null at the beam center. From z = 0 to 10zR, we
observe the formation of bright central maxima for the second-harmonic fields |8(r, φ, z)|2 due to the interference of the 9n,0(r, φ, z; k2, w2o) modes with different Gouy
phases. For a higher-order |8(r, φ, z)|2 field such as l = 20 in the third row, a concentric-ringed structure is developed around the bright central maximum owing to the interference of some higher-order 9n,0(r, φ, z; k2, w2o)
Laser Phys. 23 (2013) 115405 Y C Lin et al
Figure 2. The intensity distribution of the cross sections for the frequency-doubled beams |8(r, φ, z)|2(solid line) and the Gaussian TEM00modes |90,0(r, φ, z; k2, w2o)|2(dashed line) at z = 0, 5zR, and 10zRwith (a)–(c) l = 5, (d)–(f) l = 12, (g)–(i) l = 20.
modes. Moreover, the influence of the superposed state Pl
n=0Cnl9n,0(r, φ, z; k2, w2o) on the propagation-invariant
term 90c,2l(r, φ, z; k2, w2o) in equation (10) is shown in
figure 1, where the flower structures evolve with their azimuthal indices transforming from l to 2l along the longitudinal direction.
Figure 2 illustrates the intensity distribution of the cross sections for the second-harmonic waves |8(r, φ, z)|2 and the Gaussian TEM00 mode |90,0(r, φ, z; k2, w2o)|2 at
z = 0zR, z = 5zR and z = 10zR to clearly reveal the
quasi-nondiffracting properties of the central lobes. It can be seen that the divergence of the central peaks is directly decided by the orders l of the flower LG modes 9c
0,l(r, φ, 0; k1, w1o). The divergence of the central peak
for the second-harmonic wave |8(r, φ, z)|2 is qualitatively estimated to be
√
2l + 1 times less than that of the Gaussian TEM00 mode |90,0(r, φ, z; k2, w2o)|2 with the same beam
waist parameter. As the order l of the flower LG mode 9c
0,l(r, φ, 0; k1, w1o) gets higher, the divergence of the bright
center for the second-harmonic wave |8(r, φ, z)|2 becomes smaller. The central parts of the high-order |8(r, φ, z)|2 modes conserve their profiles through several Rayleigh ranges without obvious divergence, showing the quasi-nondiffracting properties. Previous works [16, 17] have shown that the central lobes of high-order 9n,0(r, φ, z; k, wo) modes
are characterized by their nondiffracting features. In this investigation we validate that the bright center arising from the
coherent superposition of the group of 9n,0(r, φ, z; k2, w2o)
modes can exhibit the feature of a quasi-nondiffracting beam. The nondiffracting focused phenomena of optical waves have great importance for applications such as optical trapping and optical manufacturing. Therefore, it might be useful to generate such second-harmonic focused beams characterized by their inherent quasi-nondiffracting features through the SHG. In the following, we exploit a diode-pumped solid-state laser with intracavity SHG to experimentally confirm the formation of the quasi-nondiffracting focused beam.
3. Experimental realization
The schematic diagram for the experimental setup of a diode-pumped Nd:YVO4 laser with a KTP crystal as an
intracavity SHG medium is illustrated in figure 3. The experimental configuration was designed for SHG at 532 nm from a fundamental wavelength of 1064 nm. The laser medium was an a-cut 2.0 at.% Nd:YVO4 crystal with a
length of 2 mm and a cross section of 10 × 10 mm2. The SHG medium was a 10-mm-long KTP crystal with a cutting angle (θ = 90◦andϕ = 23.5◦) to satisfy the type II SHG phase matching condition. Both sides of the Nd:YVO4
and KTP crystals were coated for antireflection at 1064 nm (R < 0.2%). In addition, they were wrapped with indium foil and mounted in a water-cooled copper block. The front mirror was a 250 mm radius-of-curvature concave mirror with
Figure 3. The experimental setup of the diode-pumped solid-state laser with intracavity SHG.
Figure 4. The observed transverse intensity profiles of the fundamental flower LG modes |9c
0,l(r, φ, 0; k1, w1o)|2at z = 0 (first column) and their second-harmonic counterparts |8(r, φ, z)|2from z = 0 to 10z
R. First row: l = 5, second row: l = 12, third row: l = 20.
antireflection (R< 0.2%) coating at 808 nm on the entrance surface (R< 0.2%) and high-reflection coating at 1064 nm (R> 99.8%) and 532 nm (R > 99%) on the other surface. The output coupler was a flat mirror with partial-reflection coating at 1064 nm (R = 98%) and high-transmission coating at 532 nm (T > 85%). The pump source was an 808 nm fiber-coupled laser diode with a pump core of 100 µm in radius, a numerical aperture of 0.16, and a maximum output power of 1 W. A focusing lens with a focal length of 25 mm and 85% coupling efficiency was used to reimage the pump beam into the laser medium.
To generate the high-order flower LG modes of different orders, we employed a donut-shaped pump profile and defocused a standard fiber-coupled diode [23]. At the focal plane, the profile for a multi-mode fiber-coupled diode laser beam passing through a focusing lens is like a top-hat distribution; however, away from the focal plane it is like a donut-shaped distribution. Based on this property, we defocused a standard fiber-coupled diode to obtain a good
overlap with a high-order flower LG0,l mode and generate
it purely. Flower LG modes of varying orders correspond to differing sizes of the pump profiles. The pump spot sizes were controlled to be 50–200µm. The overall cavity length was nearly 80 mm. According to the experimental scheme, the beam radius of the fundamental TEM00 mode can be
calculated to be around 141µm. The difference of the beam radius between adjacent flower LG modes is of the order of 40µm. With a pump power of 800 mW, the average output power is found to be 80 mW for the second-harmonic TEM00
mode. The conversion efficiency for the second-harmonic TEM00 mode from the diode laser incident power to the
SHG output is about 12%. However, for the second-harmonic standing wave LG mode of l = 12, the average output power is measured to be 0.84 mW with a pump power of 745 mW. To scale up the output power, the laser diode can be replaced with a high-power one. In this case, a less Nd-doped laser medium should be employed to reduce the severe thermal issues owing to the concentration quenching. Moreover, filters
Laser Phys. 23 (2013) 115405 Y C Lin et al
placed after the laser cavity allow either the fundamental or the second-harmonic wave to be chosen and projected onto a screen. The projected patterns were observed through a CCD camera. Figure 4 displays the experimental results for the formation of the quasi-nondiffracting focused beams corresponding to figure 1. The experimental tomographic transverse patterns show good agreement with the theoretical simulations in figure 1. This investigation confirms that a nondiffracting focused beam can be experimentally realized through the intracavity SHG of flower LG modes.
4. Conclusions
In summary, we have theoretically demonstrated that the generation of quasi-nondiffracting focused beams can be realized through the nonlinear SHG of flower LG modes. The formation of the quasi-nondiffracting focused beam has been analytically verified to result from the cross term90,l90,−l. We have further confirmed that the divergence of the nondiffracting focused beam is approximately
√
2l + 1 times less than that of the Gaussian TEM00 mode with the same
beam waist parameter. Finally, we have employed a diode-pumped solid-state laser cavity to experimentally generate a quasi-nondiffracting focused beam via the intracavity SHG of flower LG modes.
Acknowledgment
The authors acknowledge the National Science Council of Taiwan (NSCT) for their financial support of this research under contract NSC-100-2628-M-009-001-MY3.
References
[1] Ashkin A, Dziedzic J M, Bjorkholm J E and Chu S 1986 Opt. Lett.11 288
[2] Ashkin A, Dziedzic J M and Yamane T 1987 Nature330 769
[3] Takashi I and Shinji O 2000 Nature406 1027
[4] Strickler J H and Webb W W 2002 Opt. Lett.16 1780
[5] Yu C, Wang M R, Varela A J and Chen B 2000 Opt. Commun.
177 369
[6] Tsampoula X, Garc´es-Ch´avez V, Comrie M, Stevenson D J, Agate B, Brown C T A, Gunn-Moore F and Dholakia K 2007 Appl. Phys. Lett.91 053902
[7] Ding Z, Ren H, Zhao Y, Nelson J S and Chen Z 2002 Opt. Lett.27 243
[8] Garc´es-Ch´avez V, McGloin D, Melville H, Sibbett W and Dholakia K 2002 Nature419 145
[9] Arlt J, Garces-Chavez V, Sibbett W and Dholakia K 2001 Opt. Commun.197 239
[10] Durnin J, Miceli J J and Eberly J H 1987 Phys. Rev. Lett.
58 1499
[11] McGloin D and Dholakia K 2005 Contemp. Phys.46 15
[12] Bouchal Z, Wagner J and Chlup M 1998 Opt. Commun.
151 207
[13] Durnin J 1987 J. Opt. Soc. Am. A4 651
[14] Siviloglou G A, Broky J, Dogariu A and Christodoulides D N 2007 Phys. Rev. Lett.99 213901
[15] Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N and Chen Z 2011 Opt. Lett.36 2883
[16] Flood C J, Giuliani G and van Driel H M 1990 Opt. Lett.
15 215
[17] Senatsky Y, Bisson J-F, Li J, Shirakawa A,
Thirugnanasambandam M and Ueda K-i 2012 Opt. Rev.
19 201
[18] Green C, Mindlin G B, D’Angelo E J, Solari H G and Tredicce J R 1990 Phys. Rev. Lett.65 3124
[19] Grynberg G, Maitre A and Petrossian A 1994 Phys. Rev. Lett.
72 2378
[20] Lin Y C, Lu T H, Huang K F and Chen Y F 2011 Opt. Express
19 10293
[21] Pereira S F, Willemsen M B, van Exter M P and Woerdman J P 1998 Appl. Phys. Lett.73 2239
[22] Deng Q, Deng H and Deppe D G 1997 Opt. Lett.22 463
[23] Chen Y F and Lan Y P 2001 Phys. Rev. A63 063807
[24] Ito A, Kozawa Y and Sato S 2010 J. Opt. Soc. Am. A27 2072
[25] Siegman A E 1986 Lasers (Mill Valley, CA: University Science Books)
[26] He G S and Liu S H 1990 Physics of Nonlinear Optics (Singapore: World Scientific)
[27] Dholakia K, Simpson N B and Padgett M J 1996 Phys. Rev. A
54 R3742
[28] Lebedev N N and Silverman R A 1965 Special Functions and their Applications(Englewood Cliffs, NJ: Prentice-Hall)
