Efficient beam converter for the generation of high-power femtosecond ...

2660

OPTICS LETTERS / Vol. 35, No. 15 / August 1, 2010

Efficient beam converter for the generation of high-power femtosecond vortices Vladlen G. Shvedov,1,2,3,* Cyril Hnatovsky,1 Wieslaw Krolikowski,1 and Andrei V. Rode1 1

2

Laser Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia

Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia 3

Department of Physics, Taurida National University, Simferopol 95007 Crimea, Ukraine *Corresponding author: [email protected] Received March 12, 2010; revised June 8, 2010; accepted June 22, 2010; posted July 9, 2010 (Doc. ID 125316); published July 30, 2010

We describe an optical beam converter for an efficient transformation of Gaussian femtosecond laser beams to single- or double-charge vortex beams. The device achieves a conversion efficiency of 75% for single- and 50% for double-charge vortex beams and can operate with high-energy broad bandwidth pulses. We also show that the topological charge of a femtosecond vortex beam can be determined by analyzing its intensity distribution in the focal area of a cylindrical lens. © 2010 Optical Society of America OCIS codes: 320.7080, 320.7160, 140.3300, 260.6042, 080.4865.

High-intensity ultrashort optical vortex pulses provide an opportunity to investigate the effects of the angular momentum on atomic or molecular systems and transient nonequilibrium states of matter [1,2]. Current methods of generating femtosecond optical vortices with spiral phase plates and holograms are inherently chromatic and therefore require the introduction of correcting elements to compensate topological charge dispersion caused by the broad spectral bandwidth associated with ultrashort light pulses [3,4]. We have recently demonstrated the formation of white-light cw vortices using polarization singularities in birefringent crystals [5]. In this Letter, we extend this idea to femtosecond pulses and propose a compact beam converter for the generation of single- and double-charge femtosecond laser vortex beams. The converter (i) yields high conversion efficiencies of 75% for single- and 50% for double-charge vortex beams; (ii) works with high-energy pulses, as it is nominally free of any light-absorbing elements, operates only with divergent laser beams, and uses standard high optical damage threshold optics (i.e., 0.01–0.1 J=cm2 for subpicosecond pulses); (iii) has no topological charge dispersion for double-charge and a weak dispersion for single-charge vortices; and (iv) can be adapted with minimum adjustment for a wide range of wavelength and beam power conditions. A similar idea using an axially symmetric polarizer has been recently implemented to generate supercontinuum double-charge optical vortices without topological charge dispersion [6]. This achromatic scheme, however, has a lower conversion efficiency of 25% and is limited to several μJ=cm2 of input fluence because of the relatively low damage threshold of the axial polarizer, which dissipates 50% of the incident circularly polarized beam. To synthesize femtosecond vortex beams, we exploit polarization singularities that can be created when a beam of light propagates through an anisotropic medium [7–10]. These singularities can be aligned along the same or very close directions within a broad spectral range of the incident light field [11,12] and can be converted into phase singularities by a polarization filter [9]. In particu0146-9592/10/152660-03$15.00/0

lar, it was shown in [8,9] that focusing a beam along the optical axis of a uniaxial crystal generates optical vortices. When a circularly polarized Gaussian beam E
in ¼ ⊥ ðE=ξÞ expð−r 2 =ðw20 ξÞÞc
enters a uniaxial crystal along its optical axis, a superposition of two vector states of the field is generated inside the crystal: (i) a nonvortex state of the same handedness and (ii) a double-charge optical vortex of the opposite handedness (Fig. 1) [10]: E
in ⇒ E⊥ ¼ 0:5ðc
ðGo þ Ge Þ − c∓ ððr 2 þ w20 ξo ÞGo ⊥ − ðr 2 þ w20 ξe ÞGe Þ expð
2iφÞr −2 Þ;

ð1Þ

where E⊥ denotes solutions of the paraxial wave equation in an anisotropic medium ð∇2⊥ þ 4πino λ−1 ∂z ÞE⊥ ¼ −γ∇⊥ ð∇⊥ E⊥ Þ for the transverse components of the electric field. Here, γ ¼ ðn2o − n2e Þ=n2e , where no and ne are the

Fig. 1. (Color online) Setup for the generation of doublecharge femtosecond vortex pulses: λ=4, achromatic quarterwave plate; L1, negative lens; CR, uniaxial crystal; L2, positive lens; PBS, polarization beam splitter. Polarization states after each optical element are indicated by arrows. Bottom right: CCD image (4:2 × 4:2 mm2 ) of a double-charge vortex recorded ~5 m after the converter. © 2010 Optical Society of America

August 1, 2010 / Vol. 35, No. 15 / OPTICS LETTERS

ordinary and extraordinary refractive indices, Go;e ¼ ðE=ξo;e Þ expð−r 2 =ðw20 ξo;e ÞÞ; E=const.; ξ ¼ 1 þ izλ=ðπw20 Þ; ξo ¼ 1 þ izλ=ðπno w20 Þ; ξe ¼ 1 þ izλno =ðπn2e w20 Þ; r, φ, and z are cylindrical coordinates; wo is the beam waist; λ is the wavelength in vacuum; and c
¼ ðer
ieφ Þ pffiffiffi expð
iφÞ= 2 is the unit polarization vector. As the conversion of the incident beam into a vortex along the optical axis of a uniaxial crystal is wavelength independent, it facilitates the generation of axially symmetric polychromatic femtosecond beams with zero intensity at their axes. One can see from Eq. (1) that the power-conversion efficiency of the incident beam into an optical vortex depends on both the crystal and the laser beam parameters: P ∓ ðzÞ ¼ ð1 − ð1 þ z2 =L2 Þ−1 ÞP
ð0Þ=2, where P ∓ ðzÞ is the power of an optical vortex after traveling a distance z along the axis of the crystal, P
ð0Þ is the power of the incident circularly polarized Gaussian beam, and w2 n2

L ¼ 2πλ n20−ne2 , and where (
) denotes the handedness of o e circular polarization in the incident and transformed beams. Hence, the conversion efficiency increases with the crystal length and the divergence of the beam. The maximum ratio of the powers in the double-charge vortex and the nonvortex beams is therefore 1:1, or 50% of the input power. Previously [9,11,12], the generation of polychromatic vortices was achieved by focusing the incident beam inside a uniaxial crystal or close to its surface. This approach is unsuitable for high-energy femtosecond pulses due to possible supercontinuum generation or optical breakdown inside the crystal. To avoid this problem, our design operates with only divergent beams, which still ensures efficient light conversion (Fig. 1). Here, the circularly polarized femtosecond beam is defocused with a negative lens L1 and, after propagating along the optical axis of a 10-mm-long Ca2 CO3 crystal, is collimated with a positive lens L2. The second quarter-wave plate together with a polarization beam splitter (PBS) separate the emerging double-charge vortex from a nonvortex beam. In such a setup, the laser power is not dissipated inside optical elements, which is important for the conversion of high-energy pulses. This scheme also allows one to synthesize femtosecond vortices with opposite topological charges simply by reversing the handedness of the input polarization. In our experiments, we used the output beam of a Coherent Mira 900 laser oscillator with a central wavelength at 800 nm. The pulse duration before the converter (based on measurements with a Femtochrome FR-103 autocorrelator) was 120 fs FWHM. After the converter, we observed a moderate temporal pulse broadening to ∼150 fs FWHM, which can be corrected, if necessary, by prechirping the input pulses. Figure 2 shows the experimental setup used to determine the topological charge of the generated femtosecond vortices. This simple noninterferometric technique was used earlier for cw beams [13]. It involves the astigmatic transformation of the vortex beam with a cylindrical lens. The resulting intensity distribution in the focal plane of a lens consists of tilted dark bands. Their number determines the absolute value of the topological charge, whereas the tilt α (see Figs. 2 and 4) gives its sign.

2661

Fig. 2. (Color online) Diagnostics of femtosecond vortex pulses after the converter in Fig. 1. Top, simulated behavior of a double-charge vortex at the focus of a positive cylindrical lens. Bottom, experimentally obtained images of femtosecond double-charge vortices 0:5 m after the converter (left) and their focal caustics (right). Horizontal size of all experimental images is 4:2 mm.

A similar approach has been applied to generate singlecharge femtosecond vortex pulses (Fig. 3). If the input Gaussian beam is linearly x polarized, then after the crystal, we again observe a superposition of two states of the field: (i) a nonvortex y-polarized beam and (ii) a multiple x-polarized vortex state. An expression for the vortex state is given as E x ¼ 0:5ðGo þ Ge − ððξo Go − ξe Ge Þw20 r −2 þ Go − Ge Þ cos 2φÞ [12]. Hence, the vortices lie in the four half-planes intersecting at the crystal’s axis. These halfplanes are oriented at an angle φ ¼ πð2m þ 1Þ=4 (m ¼ 0, 1, 2, 3) to the beam polarization plane. The odd and even values of m correspond to vortices with opposite topological charges. The positions of the vortices in the half-planes are given by the solutions of the equation Go þ Ge ¼ 0. A single vortex closest to the axis could be selected by tilting the crystal by an angle θ ≈ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ=ð2zðno − ne ÞÞ in one of these half-planes. This expression shows that θ is a function of the wavelength

Fig. 3. (Color online) Experimental setup for the generation of single-charge femtosecond vortex pulses (the notations are as in Fig. 1). o denotes the optical axis of the crystal. Bottom right, CCD image (4:2 × 4:2 mm2 ) of a single-charge vortex recorded ∼5 m after the converter.

2662

OPTICS LETTERS / Vol. 35, No. 15 / August 1, 2010

There was no deterioration of the generated vortices in the “high power” operation regime using ∼170 fs pulses of a Clark-MXR Ti:sapphire amplifier with a central wavelength at λ ¼ 780 nm. The single pulse fluence in this case was ∼5 × 10−3 J=cm2 . In summary, we have presented a simple and costeffective optical design for the generation of femtosecond optical vortices based on light propagation inside uniaxial birefringent crystal. The simplicity and adaptability, high conversion efficiency, achromaticity, and suitability for high power applications make this converter useful for a wide range of ultrafast intense-field experiments and laser micromachining applications. We thank the National Health and Medical Research Council of Australia and the Australian Research Council for financial support. Fig. 4. (Color online) Diagnostics of femtosecond vortex beam after the converter in Fig. 3. Top, simulated behavior of a single-charge vortex at the focus of a positive cylindrical lens. Bottom, experimental images of femtosecond singlecharge vortices 0:5 m after the converter (left) and their focal caustics for the opposite topological charges (right). Horizontal size of all experimental images is 4:2 mm.

λ and refractive indices no and ne , which nominally introduce topological charge dispersion for single-charge vortices. However, the dispersion is very weak and, even for ultrabroadband pulses, can be neglected in most applications. For example, for a Fourier-transform limited 10 fs (∼100 nm FWHM bandwidth) pulse with a central wavelength at 800 nm, the angular topological charge dispersion will be only ∼0:054°. Following [12], the maximum ratio of the powers in the single-charge vortex and the nonvortex beams is 3:1, giving a 75% conversion efficiency (Fig. 3). The inherent spatial asymmetry of single-charge vortices generated by this method can be partially corrected by focusing them with a lens [14]. To characterize single-charge vortices we again use the astigmatic transformation with a cylindrical lens (Fig. 4). The high 75% and 50% conversion efficiencies for single- and double-charge vortices, respectively, have been confirmed experimentally with the converter comprising −50 mm and þ100 mm lenses (i.e., L1 and L2 in Figs. 1 and 2). The input beam diameters giving the maximum conversion efficiency for single- and double-charge vortex beams were ∼1:5 mm and ∼3:0 mm, respectively.

References 1. J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, Opt. Express 18, 2144 (2010). 2. A. Picón, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, Opt. Express 18, 3660 (2010). 3. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, Opt. Express 13, 7599 (2005). 4. K. Bezuhanov, A. Dreischuh, G. Paulus, M. G. Schätzel, H. Walther, D. Neshev, W. Królikowski, and Y. Kivshar, J. Opt. Soc. Am. B 23, 26 (2006). 5. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, Opt. Express 14, 3724 (2006). 6. Y. Tokizane, K. Oka, and R. Morita, Opt. Express 17, 14517 (2009). 7. F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, Phys. Rev. Lett. 95, 253901 (2005). 8. A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 19, 792 (2002). 9. A. Volyar and T. Fadeyeva, Opt. Spectrosc. 94, 235 (2003). 10. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Progress in Optics, E. Wolf, ed. (Elsevier, 2009), Vol. 52. 11. E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. Kivshar, Opt. Lett. 34, 1021 (2009). 12. Ya. Izdebskaya, E. Brasselet, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Yu. Kivshar, Opt. Express 17, 18196 (2009). 13. V. Denisenko, V. Shvedov, A. Desyatnikov, D. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Yu. Kivshar, Opt. Express 17, 23374 (2009). 14. Ya. Izdebskaya, V. Shvedov, and A. Volyar, Opt. Lett. 30, 2530 (2005).