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Sum frequency generation with two orbital angular momentum carrying laser beams Yan Li,1,2,† Zhi-Yuan Zhou,1,2,† Dong-Sheng Ding,1,2 and Bao-Sen Shi1,2,* 1 2
Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China *Corresponding author:
[email protected] Received August 15, 2014; revised January 16, 2015; accepted January 16, 2015; posted January 16, 2015 (Doc. ID 221070); published February 12, 2015 The frequency sum of two laser beams carrying orbital angular momentum (OAM) in quasi-phase matching crystals was reported. The situations in which one beam carried OAM, the other was in Gaussian mode, and both beams carried OAM were studied in detail. The sum of the two beams’ OAM was demonstrated in the conversion process, which verified OAM conservation in the sum frequency generation process. We give an analytical expression for the frequency conversion of two OAM-carrying beams; the experimental results are well matched with the theoretical simulations. The methods shown here provide an effective way for OAM light generation. Our study is very promising in constructing hybrid OAM-based optical communication networks and all-optical spatial mode switching. © 2015 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (050.4865) Optical vortices. http://dx.doi.org/10.1364/JOSAB.32.000407
1. INTRODUCTION A light with orbital angular momentum (OAM) has stimulated great research interest since it was first introduced in 1992 by Allen et al. [1]. It has been used in many fields, such as optical manipulation and trapping [2–5], high precision optical measurements [6,7], high capacity optical communications [8], and quantum information processing [9–14]. Frequency conversion is a basic technique to expand the frequency range of a fundamental light. It is widely used in optical communications [15] and up-conversion detection of infrared light [16,17]. Frequency conversion of OAM-carried lights using nonlinear crystals (atomic vapor) in second (and third, or even higher) order nonlinear optical processes has been studied widely [18–30]. Usually, there are two kinds of phase matching methods for frequency conversion with nonlinear crystals: one is birefringence phase matching (BPM), and the other is quasi-phase matching (QPM). A BPM crystal used to realize second harmonic generation (SHG) of an OAMcarrying light was reported in ref. [19]. Recently, we reported SHG of an OAM-carrying light in a QPM crystal [18]. QPM crystals allow accessing stronger elements of the nonlinear tensor to achieve a greater efficiency and no walk-off effect in comparison with BPM crystals. Thus, QPM crystals are promising in realizing high efficiency and high quality frequency conversion of an OAM-carried light [31]. For SHG of an OAM-carried light, only even values of OAM are available in SHG light, while all the OAM values are available by using sum frequency generation (SFG) as we can imprint OAM on both pump beams. In this article, SFG of two OAM-carrying beams in QPM crystals was studied. The situations when one beam carried OAM, the other beam was in Gaussian mode, and both beams carried OAM were studied in detail. The SFG light carried 0740-3224/15/030407-05$15.00/0
OAM that is the sum of the two pump beams’ OAM, which is consistent with OAM conservation in the conversion process. All the experimental results can be well explained with the analytical expression we calculated. The up-conversion of an OAM-carried light in the telecom band is preferable for building OAM-based up-conversion optical communication networks. For the situation when both pump beams carried OAM, the change of OAM in one pump beam affects that of the SFG generated beam. Such an effect could be used to realize all-optical mode switching, which will be applicable in mode de-multiplexing for OAM-based high capacity optical communications. The article is organized as follows: we first introduce the experimental setup; after the theoretical calculation, we show the experimental results of the two situations in detail; finally, we come to the conclusion, where we discuss further perspectives of SFG of the OAM-carried lights.
2. EXPERIMENTAL SETUP The experimental setup is depicted in Fig. 1. The wavelengths of the two pump beams are 795 nm (from a Ti:sapphire laser, Coherent, MBR110) and 1550 nm (from a diode laser, Toptica, prodesign), respectively. Both beams are imprinted with OAM using vortex phase plates (VPPs, from RPC photonics), and focused using lenses L1 and L2 separately before being combined using a dichromatic mirror (DM). The periodically poled KTP (PPKTP, supplied by Raicol Crystals) crystal used here has a dimension of 1 mm × 2 mm × 10 mm, and has a poling period of 9.375 μm to get QPM for SFG of the 795 and 1550 nm beams to 525.5 nm; both end faces are anti-reflective coated for these wavelengths. The temperature of the crystal is controlled with a homemade semi-conductor Peltier temperature cooler with a temperature stability of 2 mK. The pump beams behind the crystal are filtered out using filters (FESH1000 and © 2015 Optical Society of America
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M2
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M3
QWP
L2 1550nm M4 VPP1
L1
DM
PPKTP
F HWP
PBS M5
BS
CCD
Fig. 1. Experimental setup. VPP1, VPP2: vortex phase plate; L1, L2: lenses; DM, dichromatic mirror; PPKTP, periodically poled KTP crystal; F, filters; HWP (QWP), half-wave plate (quarter-wave plate); M1-M4, mirror; PBS (BS), polarization beam splitter (beam splitter); CCD, charge coupled device camera.
FBH520-40, from Thorlabs). The polarization of the SFG light beam is rotated by a half-wave plate with its optical axes placed at 22.5 deg with respect to the vertical direction. Then the SFG light entered an interferometer to determine the OAM value of the input light. The interferometer has the same structure as our previous work [18]. The function of the interferometer is to convert an input OAM light of state jli into an output state of jli eiβ j − li; then the output state has a flower-like spatial shape with 2l petals. The spatial shape of the interference pattern is monitored and acquired using a charge coupled device camera (CCD).
3. THEORETICAL CALCULATION Before showing our experimental results, we will give some theoretical analysis. OAM light can be described by the Laguerre–Gaussian (LG) mode; the normalized wave function of an LG mode in the cylindrical coordinates is given by [18]
LGlp r; φ; x
s p jlj 2p! 1 2r 2 2r Ljlj p πjlj p! wx wx wx2 r2 r2 exp ik × exp − 2 2Rx wx × exp−i2p jlj 1ςx exp−ilφ;
n E SFG r 0 ;φ0 ;0 ∝ LGm 0 r 0 ;φ0 ;0LG0 r 0 ;φ0 ;0 s p jmjjnj 2r 0 2 1 jmj1 π jmj!jnj! w1 wjnj1 2 1 1 exp−imnφ0 ; (3) ×exp −r 20 w21 w22
where w1 and w2 are the beam waists of 780 and 1550 nm pump beams, respectively. Within the frame of paraxial approximation, the propagation of the generated SFG light through a stigmatic ABCD optical system can be studied with the help of the following Collins integral: [32] Er; φ; x
Z 2π Z ∞ i exp−ikx E SHG r 0 ; φ0 ; 0 λB 0 0
ik × exp − × Ar 20 − 2rr 0 cosφ − φ0 Dr 2 2B × r 0 dr 0 dφ0 :
(4)
The ABCD transfer matrix for free space of distant x reads as
A C
B D
1 x : 0 1
(5)
(1)
where l and p are the azimuthal index and radial index, respectively; wx is the beam radius at position x; x is the axial distant from the beam waist; Ljlj p is the generalized Laguerre polynomials; and k 2π∕λ is the wave number. Rx is the radius of curvature of the beam’s wavefronts; φ represents the azimuthal angle; and ςx is called the Gouy phase, which is an extra spatial contribution to the phase. To investigate the SFG conversion processes in QPM crystals, the coupling wave functions are used to describe the interaction of these waves [18]: 8 dE1z −iΔkx > < dx iK 1 E 2z E 3z e dE2z −iΔkx ; dx iK 2 E 1z E 3z e > : dE3z iK E E eiΔkx 3 1z 2z dx
frequency, nj is the refractive index, c is the speed of light, and d33 is the nonlinear coefficient of KTP. Δk k3z − k1z − k2z − Gm is the phase mismatch, where k represents the wave vector. We assume the phase matching condition is fulfilled in the conversion process (Δk 0), which means the momentum mismatching is fully compensated by the reciprocal vector Gm of the QPM crystals. In the tight focusing approximation, the main contribution to the SFG beam is from the beam spot. Assuming the two pump beams at 795 and 1550 nm are in modes LGm 0 and LGn0 , respectively, where m and n are the corresponding OAM values carried by the two pump beams, then the SFG light at the beam waist is determined by
2
where Ejz LGlp r; φ; xj 1; 2; 3 represent the amplitudes of the three waves involved and z stands for the polarization. The coefficient K j 2ωj d33 ∕πnj cj 1; 2; 3, ω is the angular
When combining Eqs. (1)–(5), the field at distant x from the source point is in the form pjmjjnj 2 2jmj jnj − jm nj∕2! imn1 p E SFG r; α; z jmj1 jnj1 λx jmj!jnj! w1 w2 2 2 ik k r × exp − r 2 − ikx exp − 2x 4ξx2 jmnj kr Ljmnj × ξ−jmjjnjjmnj∕2−1 jmjjnj−jmnj∕2 2x 2 2 k r exp−im nφ; (6) × 4ξx2 where k and λ are the wave number and wavelength of the SFG beam, respectively, and ξ
1 ik ; w2s 2x
w1 w2 ; ws q w21 w22
(7)
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where ws represents the beam waist of the SFG beam. In the above derivations, we have used the following integral formulas: Z 0
2π
exp−inθ1 ikbr cosθ1 − θ2 dθ1 π 2π exp in − θ2 J n kbr; 2
Z
∞
0
(8)
exp−ax2 J v 2bxx2nv1 dx 2 2 n! v −n−v−1 b b b a exp − Lv : 2 a n a
(9)
In the far-field approximation (x ≫ kw2s ∕2), Eq. (6) reduces to 2 p jmnj imn1 r 2r E SFG r; α; z Γ exp−ikx exp − 2 w λx w 2 r exp−im nφ × Ljmnj jmjjnj−jmnj∕2 w2 ∝ LGmn jmjjnj−jmnj∕2 r; φ; x;
(10)
where Γ is a constant depending on m, n, w1 , and w2 . w xws ∕xR and xR kw2s ∕2 with w and xR being the spot radius of the SFG beam at x and the Rayleigh range, respectively. Though the final result in Eq. (10) is written in the form of a standard LG mode, there are two cases that one needs to notice. In case mn ≥ 0, the SFG mode is a standard LG mode as the Laguerre polynomials are constant; while for the case mn < 0, the SFG mode is not a standard LG mode. By comparing Eq. (10) with the standard LG mode Eq. (1), we can find that there is a factor of 2 in the Laguerre polynomials for the radial coordinate. This difference leads to the intensity of the SFG beam decreasing much more rapidly in the radial direction than standard LG mode. Upon obtaining the expression of the SFG light, the experimental results can be explained well; the comparison between the experimental results and the simulation results is shown in detail in the following text.
4. EXPERIMENTAL RESULTS The results when only the 1550 nm pump light carried OAM and the 795 nm pump beam was in Gaussian mode are shown in Fig. 2. The group of images in Fig. 2(a) shows the spatial shapes of the SFG beam at 525.5 nm when one arm of the interferometer is blocked; the group of images in Fig. 2(b) shows the interference pattern of the SFG light. The central image is the schematic layout of our VPP. For m 0, n ranges from 1 to 8 and the SFG beam is in mode LGn0 ; therefore the OAM carried by the 1550 nm infrared light is transferred to the SFG light at 525.5 nm. The beam size of the SFG light increases with the increasing of the OAM carried by 1550 nm pump beam. The numbers of petals in the right group of images are in agreement with the theoretical analysis. Figures 2(c) and 2(d) are the corresponding theoretical simulation results based on Eq. (10), respectively. We can see the good agreements between them. The experimental results when both pump beams carried OAM are shown in Fig. 3. We have investigated two cases.
Fig. 2. Experimental and simulation results when the 1550 nm pump beam carried OAM and the 795 nm pump beam is in Gaussian mode. (a) Images of the spatial shapes of the SFG light when one arm of the interferometer is blocked; (b) images of the interference patterns of the interferometer of the SFG light; (c) and (d) images of of the corresponding simulation results for (a) and (b), respectively; the central image is the schematic layout of the VPP.
In case one, two pump beams carried OAM with the same sign, and the 795 nm pump light carried OAM of 2, and the OAM of the 1550 nm varied from 1 to 8. The results are shown in Figs. 3(a) and 3(b). Figures 3(a) and 3(b) have the same meanings as those in Figs. 2(a) and 2(b). From the numbers of petals of interference patterns shown in Fig. 3(b), the SFG light carried OAM that equals the sum of the two pump beams. For m 2, n ranges from 1 to 8, and the SFG light has the form of LGmn . Figures 3(c) and 3(d) are the case 0 of m −2 and n ranging from 1 to 8; the SFG light has the n−2 form of LG−1 for n ≥ 2. For n 1; 2; 3, 1 for n 1 and LG2 we see an outer ring in Fig. 3(c). The intensity of the outer ring is very weak; we therefore cannot see all the ring structures clearly. To see the outer rings, we need to saturate the CCD camera or block the central part of the beam. As explained in the previous text, the nonstandard LG mode of the SFG beam has a much weaker outer ring intensity; it is difficult to observe it experimentally. The images in Fig. 3(d) show the interference patterns, and the petals in the pattern show that the OAM of the two pump beams sum to the OAM of the SFG light again. For the special situation when the two pump beams carried the same value of OAM but an opposite sign, there is no interference structure in the azimuthal direction (m −2 and n 2). In the experiment, the images are obtained by focusing the SFG beam into the CCD camera, which means the images obtained are in the near field region of the SFG beam; thus a donut shape is obtained, while the far field image should be a central spot with two concentric rings. Figures 3(e)–3(h) are the corresponding simulation results for Figs. 3(a)–3(d); we can see the excellent matching between them. In the simulation, the special case m −2, n 2, the exact solution Eq. (6) is used in the near field region.
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Fig. 3. Experimental and simulation results when both pump beams carried OAM. In (a),(b), the 795 nm pump beam carried OAM of 2; in (c),(d), the 795 nm pump beam carried OAM of -2. Images in (a) and (c) are the spatial shapes of the SFG beams; images in (b) and (d) are the corresponding interference patterns. Images in (e)–(h) are the simulation results for (a)–(d), respectively.
5. SUMMARY In summary, the frequency up-conversion and transferring of OAM based on SFG in QPM crystals have been studied thoroughly in this article. The sum of two pump beams’ OAM is verified in the SFG process. The situations when one pump beam carried OAM, the other was in Gaussian mode, and both beams carried OAM were experimentally investigated and well explained with our theoretical analysis. The present study paves the way toward high efficient and high quality engineering and processing of an OAM-carried light by applying the SFG process. The present study can be used for upconverted detecting of an infrared OAM-carried light which may be used in astrophysical observation, generating a light with OAM at a specific wavelength using an OAM-carried light at wavelengths that are easy to obtain, and realizing all-optical mode switching using the sum property of OAM in the SFG process. In the future, we will place the crystal in an optical cavity and realize high efficiency up-conversion and transfer OAM of a light.
ACKNOWLEDGMENTS This work was supported by the National Fundamental Research Program of China (Grant No. 2011CBA00200), the National Natural Science Foundation of China (Grant Nos. 11174271, 61275115, 61435011), and the Innovation Fund from the Chinese Academy of Science. †These two authors have contributed equally to this article.
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