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Dual-wavelength laser with topological charge

Haohai Yu, Miaomiao Xu, Yongguang Zhao, Yicheng Wang, Shuo Han, Huaijin Zhang, Zhengping Wang, and Jiyang Wang

Citation: AIP Advances 3, 092129 (2013); doi: 10.1063/1.4823583 View online: http://dx.doi.org/10.1063/1.4823583

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/3/9?ver=pdfcov Published by the AIP Publishing

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Dual-wavelength laser with topological charge

Haohai Yu,1 Miaomiao Xu,2 Yongguang Zhao,1 Yicheng Wang,1 Shuo Han,1 Huaijin Zhang,1,a Zhengping Wang,1 and Jiyang Wang1

1 State Key Laboratory of Crystal Materials and Institute of Crystal Materials,

Shandong University, Jinan 250100, China

2Taishan College, Shandong University, Jinan 250100, China

(Received 19 July 2013; accepted 13 September 2013; published online 24 September 2013)

We demonstrate the simultaneous oscillation of different photons with equal orbital angular momentum in solid-state lasers for the first time to our knowledge. Single tunable Hermite-Gaussian (HG0,n) (0 < n < 7) laser modes with dual wavelength were generated using an isotropic cavity. With a mode-converter, the corresponding Laguerre-Gaussian (LG0,n) laser modes were obtained. The oscillating laser modes have two types of photons at the wavelengths of 1077 and 1081 nm and equal orbital angular momentum of nh per photon. These results identify the possibility of simultaneous oscillation of different photons with equal and controllable orbital angular momentum. It can be proposed that this laser should have promising applications in many fields based on its compact structure, tunable orbital angular momentum, and simultaneous oscillation of different photons with equal orbital angular momentum. © 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4823583]

Orbital angular momentum of photons generates the optical vortex and phase singularity. Similar with the spin angular momentum, orbital angular momentum of photons is also quantized and can transfer torque.1,2 Its generation,3 collapse,4 propagation dynamics,5 generation of atomic rotational states,6 knots,7 entanglement,8 etc., have been investigated and attracted much interest, because it can discover the nature of the optical angular momentum. In the linear optical process, different kinds of angular momentum can indeed interact with each other when the light propagates in an optical inhomogeneous medium.9 Using Berry phase, the conservation can be observed with a q-plate9 and the zero spin angular momentum also contributes to the orbital angular momentum when the light passes through an optically active plate with topological charge.10 In the nonlinear optical process, the angular momentum is conserved in the process of second and high harmonic generation11,12 and whether it is conserved in the process of spontaneous parametric down-conversion depends on the symmetry of the nonlinear crystal used.13 However, up to now, the orbital angular momentum is related to a single wavelength. Rather few studies have addressed the possibility of simultaneous emission of different photons with the orbital angular momentum. It is motivating to investigate whether the simultaneous emission of different photons is possible. The lasers with different photons should be an efficient tool for study the interaction of photons and conservation of optical angular momentum13,14 and should have promising applications in the transferring torque because the interacted materials generally have different response to the photons with different wavelength or polarizations, and in the generation of terahertz radiation by difference-frequency generation.15,16

Up to now, spiral phase plate, hologram, q-plate and mode converter are the usual technology for the generation of optical angular momentum. For the mode converter, the only change is the

aCorresponding author: Huaijin Zhang, Address: State Key Laboratory of Crystal Materials and Institute of Crystal Materials, Shandong University, Jinan 250100, China, Fax: +86-531-88574135, Email: huaijinzhang@sdu.edu.cn

2158-3226/2013/3(9)/092129/5

3, 092129-1

) Author(s) 2013 k

FIG. 1. Room-temperature fluorescence spectrum of the Nd:Lu2Ö3 crystal.

Gouy phase during the converting process. If the wavelength difference of different photons is small, which indicates the dispersion of the cylindrical lenses negligible, the generated topological charge is only determined by the unconverted beam,17 which indicates that if the light possesses different photons localized in one state in the unconverted beam, the different photons in the converted beam should be possible to oscillate simultaneously. In this letter, we report the simultaneous oscillation of different photons at the wavelengths of 1077 and 1081 nm with equal and controllable orbital angular momentum.

Nd:Lu2O3 has a bixbyite structure and belongs to the cubic system, with the space group Ia3 and Nd concentration of about 1 at.%. The fluorescence spectrum at about 1 xm is shown in Fig. 1. It was discovered that its strong emission is located at about 1.08 xm with evident Stark splitting. Two peaks are from the same upper energy level (R1) but end at different levels (Y1 and Y2, respective) with emission cross-sections of 8.52 and 8.49 x 10-20 cm2, respectively.18 It is possible to get the simultaneous oscillation of different photons in a cavity with this material as the gain. The crystal used as the active element was 4 mm thick with an aperture of 2 x 2 mm2. The two 2 x 2 mm2 faces were polished and antireflection (AR) coated at 808 nm and 1.06-1.08 ¡m. The Nd:Lu2O3 sample was mounted in a Cu holder with circulating water at 15 °C.

The setup for the experiment is shown in Fig. 2. The pump source employed in the present experiment is a commercial fiber-coupled laser-diode (LD) with a central wavelength around 808 nm. The core size of the fiber is 100 ¡xm in radius with a numerical aperture of 0.22.The input plane mirror (M1) is high-transmission (HT) coated at the pump wavelength and highly reflective (HR) at 1081 nm. The output plane coupler (M2) has transmissions of 1.9% at 1077 nm and 2.1% at 1081 nm. The pump light is focused into the crystal by an imaging unit with a beam compression ratio of 1:1. The optimum physical length of the cavity was about 25 mm in laser operation. The laser pattern was recorded by a camera (EOS 5D Mark II, Canon) and the spectra were measured with an Ocean Optics Inc. model HR4000CG-UV-NIR optical spectrum analyzer.

In Cartesian coordinate, the pattern can be described by the Hermite-Gaussian (HGmn) modes and in Cylindrical coordinate, the generated modes are Laguerre-Gaussian (LGP,/). Here, m and n are the orders of the Hermite, and p and / are the orders of Laguerre polynomials. Since the components

FIG. 2. Experimental configuration for the generation of simultaneous oscillation of different photons with same orbital angular momentum.

of the setup such as the crystal, cavity and pump source can satisfy the Cartesian coordinate more easily,19 the resulted HGmn modes can be expressed as:17

EHG(x, ^ z) =

-)1/2 exp[-

-ik(x 2 +y2)z (x2+y2)

! m I ' —±-l 2(zR +z2) w2

exp[-i (n + m + 1) arctan(z/zR )] Hn (V2x/w) Hm (V2y /w)

kw2 = ( Z R + z2)/iR

Hn(x) is the Hermite polynomial of order n, k is the wave number, zR is the Rayleigh rang of the mode, and w is the beam waist. The HGmn modes can be decomposed as a superposition of Hermite-Gaussian modes with the Cartesian coordinate rotating by 45o. Introducing a i factor in front of each component of rotated Hermite-Gaussian modes, the HGmn mode would be converted to LGmn as:

eLG(r, t, z) = (-

)1/2 min(m, n)w exp[

x arctan(z/zR)]exp[-i(m - n)t](-1)min(m-n)(-—)

2(z2R + z2) w2

--2]exp[-i (m + n + 1)

Jm-nj j |m-n

jm-nj min(m,n^ w2

Here, Llp is the Laguerre polynomials. Tuning the Gouy phase arctan(z/zfl) to be n/2 with two 45o rotated cylindrical lenses, the ik factor can be introduced when the distance between the two lenses is V2 f and the Cartesian coordinate rotates by 45o, here, f is the focal lengths of the cylindrical lenses. Therefore, based on eq. (2), if the wavelength difference of different photons is small, which indicates the dispersion of the cylindrical lenses negligible, the conversion is only determined by the distance between the two lenses and, after conversion, the orbital angular momentum per photon is determined by the order of the HGmn modes and equal to (m - n)h.xl The maximum absolute topological charge appears when m or n equals 0, which requires HGmp or HG0n. The mode converter is shown in Fig. 2. Two cylindrical lenses shown in Fig. 2 have the focal length off = 30 mm, and the distance between them was precisely adjusted to be V2f

The order of the oscillating modes can be easily tuned by the astigmatism of the cavity.20 After getting the HG0n or HGm-0 mode, the symmetrical axis of the cylindrical lenses should be tuned to be 45o to the coordinates of HG0n or HG0m modes. The HG0n modes with 0 < n < 7 were achieved and those with 0 < n < 7 as shown in Fig. 3 by tuning the Fresnel number of the cavity. The HG00

FIG. 3. Transverse patterns of the laser beams before and after conversion.

1.0 0.80.60.40.20.0

' I ' I ' I ' I 1 I 1 1077.07

J 1^1081.12

1000 1020 1040 1060 1080 1100 1120

Wavelength (nm)

FIG. 4. Laser spectrum of simultaneously oscillating photons.

mode is round spot same with LG00. The output power decreased from 240 mW to about 20 mW with the increase of the order n from 0 to 7 due to the mode mismatching between the pump and oscillating modes. With the mode converter, LG0n modes with 0 < n < 7 were obtained, which are also shown in Fig. 3. From this figure, it can be found that the order of the LG0 n modes can be tuned, which determines the tunable orbital angular momentum from h to 7ft per photon.

With the spectral analyzer, both of the HG0 n and LG0nn modes have dual-wavelength located at 1077 and 1081 nm. The typical spectrum of the achieved lasers is shown in Fig. 4. It should be noted that the ratio between the different-wavelength components were almost stable, and oscillated simultaneously. Due to Nd:Lu2O3 belonging to the cubic crystal, both of the two components are unpolarized. In a word, this figure indentified the simultaneous oscillation of different photons with equal orbital angular momentum nh per photon with 0 < n < 7. It should be noted that the generated HG0n modes were stable which determined the generated LG0nn mode unchanged with time. The stable LG0nn modes also indicated that the different kind of photons contained in the laser beam did not interact with each other in space. The results identify the possibility of simultaneous oscillation of different photons with same and controllable orbital angular momentum, and it can also be proposed that more different photons (greater than two) can be generated with same orbital angular momentum if their wavelengths are close due to the dispersion of the focal length of the mode converter.

In conclusion, two kinds of photons with the wavelength of 1077 and 1081 nm, respectively, and same orbital angular momentum per photon are achieved in an isotropic laser system. Their angular momentum is tunable from 0 to 7h. This laser identifies the possibility of simultaneous oscillation of different photons with same and controllable orbital angular momentum, and should find promising applications in the study of angular momentum conversion by interacting with materials by linear or nonlinear optical processes, transferring torque, rotating particles, etc., based on its compact structure, tunable orbital angular momentum, and simultaneous oscillation of different photons.

This work is supported by the National Natural Science Foundation of China (Nos. 51025210, 51102156, 50721002, 60978027), Grant for State Key Program of China (No. 2010CB630702), the Program of Introducing Talents of Discipline to Universities in China (111 program).

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