Concept of annular vector beam generation at terahertz wavelengths via a nonlinear parametric process Kyosuke Saito,1,* Tadao Tanabe,2 and Yutaka Oyama1 1
Department of Materials Science, Graduate School of Engineering, Tohoku University, Aoba-yama 6-6-11-1021, Sendai 980-8579, Japan 2
Institute of Multidisciplinary Research for Advanced Materials, Tohoku University 2-1-1, Katahira, Aoba-ku, Sendai 980-8577, Japan *Corresponding author: k‑[email protected] Received 17 November 2014; revised 2 March 2015; accepted 3 March 2015; posted 3 March 2015 (Doc. ID 226982); published 27 March 2015
In this paper, we describe our theoretical investigation and calculations for a terahertz (THz)-wave profile generated by difference frequency mixing (DFM) of focused, cylindrically symmetric, and polarized optical vector beams. Using vector diffraction theory, the second-order nonlinear polarization was estimated from the electric field components of the optical pump beams penetrating uniaxial, birefringent nonlinear optics (NLO) crystals, GaSe and CdSe. The approximate beam patterns of the THz waves were simulated using DFM formulation. The intensity patterns of the THz waves for GaSe and CdSe showed sixfold symmetry and cylindrical symmetry, respectively, based on the nonlinear susceptibility tensor of the crystals. As the phase-matching angle θPM was constant with respect to the c axis of the NLO crystals, an annular vector beam with a narrow width was expected. © 2015 Optical Society of America OCIS codes: (000.4430) Numerical approximation and analysis; (170.3010) Image reconstruction techniques; (190.4223) Nonlinear wave mixing; (260.3090) Infrared, far; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.54.002769
1. Introduction
Optical vector beams have attracted a great deal of attention in the field of optics due to their donutshaped beam profile, spiral wavefront, and quantized orbital angular momentum characterized by a topological charge [1–4]. These unique properties have been applied in many research fields, such as optical manipulation, nanofabrication, and high-speed optical communications [5–12]. An optical vector beam also enables super-resolution microscopy beyond the diffraction limit, based on fluorescence depletion phenomena, such as stimulated emission depletion [13,14]. Techniques using mode converters, computergenerated holograms, and spiral phase plates have been used to generate optical vector beams [15–19]. An effective means of realizing ultimately high 1559-128X/15/102769-07$15.00/0 © 2015 Optical Society of America
resolution in laser microscopy is to use an annular vector beam with radial polarization [20,21]. This beam provides the ultimate small-focused spot size, expressed as 0.36λ∕NA, where NA is the numerical aperture of the focusing optics. Sato reported annular vector-beam generation through second-harmonic generation of 800 nm light, using a β-barium borate crystal with uniaxial birefringent properties; they successfully attained an annulus width that was less than 1∕40th of its radius [22]. Over the past several decades, the terahertz (THz) frequency regime has attracted a great deal of attention due to its application potential in imaging and nondestructive sensing realized by recent developments in THz technology. Imaging technologies using THz waves ranging in wavelength from 30 to 300 μm have been applied to molecular spectroscopy, biomedicine, security, and nondestructive inspection [23–27]. The THz frequency spectrum is referred to 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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as the molecular fingerprint region due to its sensitivity to the specific vibrational frequencies of molecules and polymers [28]; however, the spatial resolution of THz waves with a Gaussian-shaped beam profile is limited to the submillimeter order due to diffraction. In contrast, a THz vector beam with a donut-shaped profile has the potential to overcome the diffraction limit to achieve spatial imaging resolution on the order of micrometers for THz frequencies. To date, THz vector beams have been generated using segmented nonlinear optical (NLO) crystals, spiral phase plates, and antenna array structures [29–31]. However, the segmented NLO crystal generates a quasi-helical beam due to its finite segmentation, and THz beam converters are limited by their operation wavelength, which is dependent on their design. In this paper, we propose a vector-beam THz wave generation technique based on difference frequency mixing (DFM). In the DFM scheme, a uniaxial birefringent NLO crystal, such as c-cut GaSe and CdSe crystals, was used. The phase-matching angle was constant with respect to the c axis of the NLO crystals, enabling annular vector-beam generation with a narrow width. 2. Theoretical Description of THz Vector-Beam Generation via Difference Frequency Mixing
Figure 1(a) shows a schematic drawing of THz vectorbeam generation via DFM. The linearly polarized optical pump and signal beams were converted into azimuthally and radially polarized beams, using a segmented half-wave plate or liquid crystal-based polarization converter. These beams were focused by a high numerical aperture (NA) lens. A c-cut NLO crystal was placed at a defocused position to prevent diffraction of the generated THz wave. Here, selection of the uniaxial birefringent NLO crystal was especially important for vector-beam generation to ensure cylindrical symmetry for the difference frequency wave produced. Potential uniaxial NLO crystals for THz wave generation include GaSe, CdSe, and GaN. GaSe has a 62 m point-group symmetry, while CdSe and GaN have 6 mm point-group symmetry. The DFM output characteristics for GaSe and CdSe were calculated, as described below. THz waves with radial polarization (an extraordinary wave) should generate oee-type phase matching (PM) using the GaSe crystal [Fig. 1(b)]. On the other hand, a CdSe crystal can be used to generate azimuthally polarized THz waves (an ordinary wave), under oeo-type PM. We start with the formulation for the DFM process (ωTHz ωp − ωs ) based on optical vector-beam pumping. Youngworth and Brown developed an expression for the electric field components of the focused cylindrical vector beams [32]; this description was extended to the case including a dielectric interface by Biss and Brown [33]. The electric field cylindrical components of the transmitted azimuthally polarized beam through 2770
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Fig. 1. (a) Schematic drawing of the proposed optical setup for terahertz (THz) annular vector-beam generation from a nonlinear optics (NLO) crystal. The two optical pump and signal beams were azimuthally and radially polarized, respectively. These beams were focused using a high-numerical aperture (NA) lens, with propagation direction matched to the phase-matching (PM) angle θPM . (b) Intensity pattern and instantaneous electric field (arrow) of the generated THz wave by mixing of the vector pump and signal beams from GaSe and CdSe crystals.
the dielectric interface of an NLO crystal can be expressed as follows: 2 p 3 Er Zθ 6 p 7 max 6 Eϕ 7 A tp cos θ1∕2 sin θl0 θ expikp z0 cos θ 4 5 0
Ep z
" × exp
p inp 2 k
2
# s 2 n1 1 − p sin θ z − z0 n2 3
0 7 6 p × 4 J 1 k r sin θ 5dθ;
(1)
0 where θ represents the tangential angle with respect to the z axis, θmax sin−1 NA∕n1 is the maximum angle related to the NA of the lens, n1 and np 2 denote the refractive indices of air and the NLO medium for the pump beam, respectively, A is a constant, k is the wave number of the pump wave, z0 is the position of the NLO crystal interface, l0 θ is the relative amplitude of the electric field in front of the
pupil, and J m is the Bessel function of the first kind of order m. The description for a focused radially polarized beam is given by 2
Es r
3
6 Es 7 4 ϕ 5A Es z
Z
θmax 0
0 0 PNL
1
PTHz x B THz C @ Py A PTHz z
ts cos θ1∕2 sin θl0 θexpiks z0 cos θ
s 2 n1 s s × exp in2 k 1 − s sin θ z − z0 n2 3 2 2 r 1 − ns1 sin θ J 1 ks r sin θ n2 7 6 7 6 ×6 7dθ; 0 5 4 i ns1 sin θJ 0 ks r sin θ
(2)
n2
where tp and ts are the amplitude transmission coefficients for the parallel and perpendicular polarization states with respect to the incident plane. The relative amplitude l0 θ of the incident pump and signal beam is assumed to possess a doughnutshaped Laguerre–Gaussian intensity profile, expressed by 2 sin θ β20 sin θ 2β0 sin2 θ 1 L ; (3) exp − 2 l0 θ sin2 θmax sin θmax 0 sin2 θmax β20
where β0 is the ratio of the pupil radius to the input beam radius, and L10 indicates the generalized Laguerre polynomial of the radial and azimuthal indices 1 and 0, respectively. Figure 2 shows an example of the intensity distribution as a function of the angle θ, l0 θ. The angle corresponding to the intensity maximum is set to the external PM angle θext-PM in the DFM process, using a GaSe or CdSe crystal. DFM in the NLO crystal is dependent on the second-order nonlinear susceptibility and the electric field components. In the NLO medium, the secondorder nonlinear polarization components are given by
B C s B C Ep y Ey B C p s B C E E B C z z d2 B p s C; p s B Ey Ez Ez Ey C B C B p s s C @ Ex Ez Ep z Ex A s p s Ep x Ey Ey Ez
(4)
where Ep and Es are the electric field components of the pump and signal wave, respectively, and x, y, and z indicate the axes in the Cartesian coordinate system. d2 denotes the second-rank tensor of the second-order nonlinear susceptibility. As mentioned above, the THz wave was generated using GaSe and CdSe, under oee-and oeo-type PM configurations, respectively. GaSe is classified in the point-group symmetry 62 m. The corresponding nonlinear optical tensor dGaSe is described by [34] 0
dGaSe
0 @ −d22 0
0 d22 0
0 0 0 0 0 0
0 0 0
1 −d22 0 A: 0
(5)
For the CdSe crystal, the point-group symmetry is 6 mm, and the nonlinear optical tensor dCdSe is given by [34] 0
dCdSe
0 @ 0 d31
0 0 d31
0 0 d33
0 d31 0
d31 0 0
1 0 0 A: 0
(6)
The amplitude of the THz wave (ETHz ) is determined from the scalar wave equation, under a slowly varying amplitude approximation, as follows: ∇2 ETHz − kTHz2 ETHz
4πωTHz2 ∂2 NL P ; c2 ∂t2
(7)
where PNL is the induced nonlinear polarization at ωTHz ∕c the DF of ωTHz ωp − ωs, kTHz nTHz o THz is the wave vector, no is the refractive index of the medium for the ordinary DFM wave, and c is the speed of light in a vacuum. We assume the following factorized form for ETHz : ETHz 2π
Fig. 2. Example of the intensity profile of the incident vector beam as a function of the angle θ, l0 θ.
1
s Ep x Ex
ωTHz2 NL sinΔkL∕2 ; P ΔkL∕2 ikTHz c2
(8)
where L is the NLO crystal length, and Δk denotes the phase-mismatching factor Δk kp − ks − kTHz. For a uniaxial NLO crystal, the wave vector of the extraordinary wave is expressed by ke ne ω∕c, where ne θ is the refractive index of the extraordinary wave, which is dependent on the angle θ with respect to the c axis. To estimate the type-oee PM condition for the GaSe crystal, the refractive index of GaSe for the extraordinary wave at signal 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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THz wavelength λs , ns θ, is e θ, and THz wave, ne represented by
v u 1 tan2 θ 2 ; sorTHz no;GaSe λ tanθ 1 sorTHz
u sorTHz u nsorTHz e;GaSe λ;θ no;GaSe λs t
ne;GaSe λ
(9) where the wavelength-dependent refractive indices for the ordinary and extraordinary waves are given by [35] 0.405 0.0186 0.0061 λ2 λ4 λ6 2 3.1436λ ; 2 λ − 2194 0.3879 0.2288 0.1223 or THz 2 ns e;GaSe λ 5.76 λ2 λ4 λ6 1.855λ2 2 : (10) λ − 2194 2 np o;GaSe λ 7.37
Thus, the PM condition for type-oee PM in a GaSe crystal is expressed as s Δk kp − ks − kTHz ωp np o;GaSe λ − ωp ne;GaSe λ; θ
− ωp nTHz e;GaSe λ; θ∕c 0;
(11)
is the refractive index of the ordinary where np o wave at pump wavelength, and c is the speed of light. In the case of CdSe, the refractive index for the extra ordinary wave at signal wavelength λs , to estimate the oeo-type PM condition, is given by ns e;CdSe λ; θ
v u 1 tan2 θ 2 ; s no;CdSe λ tanθ 1 s
u u ns o;CdSe λt
Fig. 3. External PM angle for GaSe and CdSe crystals as a function of the difference frequency between pump and signal waves. THz where np represent the refractive indices o and no of the ordinary wave at the pump and produced THz wave wavelengths, respectively. Here, we assumed that the pump wavelength, generated by a Nd:YAG laser, was fixed at 1.0642 μm. Figure 3 shows the calculated external PM angle θext PM for GaSe and CdSe crystals as a function of the difference frequency between the pump and signal frequencies. The PM conditions were satisfied below 6 THz for GaSe and 3 THz for CdSe. In the following calculation, we focus on the vectorbeam output characteristics for 2 THz waves for both GaSe and CdSe crystals.
3. Calculated Results of the Vector-Beam THz Wave Output Characteristics through DFM
In this calculation, we assumed that the pump, signal, and THz wavelengths [λp , λs , and λTHz , respectively] were 1.0642 μm, 1.0718 μm, and 150 μm, respectively. The pump and signal beams were focused onto each NLO crystal, with the incident angle fixed by z0, β0 , and θmax ; specifically, 500 μm, 1.8, and 0.25 rad, respectively, for GaSe and 100 μm, 2.7, and 0.8 rad, respectively, for CdSe. To generate 2 THz waves, the corresponding external PM angles for GaSe and CdSe were 12.95° and 30.66°, respectively. The induced second-order nonlinear polarization in the NLO crystals, GaSe and CdSe, is expressed by
ne;CdSe λ
(12) where the wavelength-dependent refractive indices of CdSe for the ordinary and extraordinary wave are given as follows [36]: 1.768λ2 3.12λ2 2 ; 4.2243 2 λ − 0.227 λ − 3380 1.8875λ2 3.646λ2 2 : (13) λ 4.2009 ns e;CdSe λ2 − 0.2171 λ2 − 3629
or THz 2 np o;CdSe λ
The type-oeo PM condition for CdSe is given by s THz Δk ωp np ∕c 0; o − ωp ne θ − ωp no
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0
PNL x
1
0
s p s −d22 Ep x Ey Ey Ex
1
B NL C B p s p s C PNL GaSe @ Py A @ −d22 Ex Ex Ey Ey A; PNL z
0 (15)
and 0 B PNL CdSe @
s p s d31 Ep x Ez Ex Ez s p s d31 Ep y Ez Ey Ez
1 C A;
s p s p s d31 Ep x Ex Ey Ey d33 Ez Ez
(16) (14)
respectively. Here, PNL for CdSe is nearly zero, bez s cause the summation of the products of Ep x Ex
Fig. 4. Distribution diagrams of second-order nonlinear polarization induced at the near input face of (a) GaSe and (b) CdSe crystals. The polarization direction of the distributions is radial for GaSe and azimuthal for CdSe. s and Ep canceled out and Ep y Ey z 0. Thus, the longitudinal component of nonlinear polarization can be ignored for both NLO crystals. Both the nonlinear polarizations for GaSe and CdSe indicate azimuthal polarizations, corresponding to the origin of THz wave with ordinary wave in these NLO crystals. The distributions of the second-order nonlinear polarization induced by the pump and signal wave with radial and azimuthal polarization at the input face of GaSe and CdSe crystals are shown in Figs. 4(a) and 4(b), respectively. The input face of each NLO crystal was assumed to be placed at a position apart from the focal point of the incident pump and signal beams in order to avoid large divergence of the generated THz wave from a small spot size compared with its wavelength. For the GaSe crystal, the nonlinear polarization distribution was estimated to be a six-peak profile with sixfold symmetry. A donut-shaped pattern was obtained for the CdSe crystal. On the basis of the near-field pattern shown in Fig. 4, far-field THz wave patterns were generated from GaSe and CdSe crystals of crystal length of 0.5, 1.0, 2.5, and 5.0 mm, as shown in Figs. 5(a)–5(d) for GaSe and 5(e)–5(h) for CdSe. An annular pattern with sixfold symmetry was obtained for the GaSe crystal. The width of the annulus became narrower as the crystal length increased due to the longitudinal PM factor, sin cΔkL∕2. The GaSe crystal has a 62 m point-group symmetry. In this case, the effective second-order nonlinear susceptibility for type-oee PM is expressed by deff d22 cos θ sin3ϕ, where θ is the angle between the c axis and the propagation direction of the THz beam, and ϕ is the azimuthal angle around the c axis [37]. The dependence of sin3ϕ induces sixfold symmetry, as indicated in annular intensity patterns. The phase change by π at every null point of the annular profile can be considered as the superposition of two azimuthally polarized beams with a spiral phase shift of 3ϕ and −3ϕ (Fig. 4). In contrast, the CdSe crystal has 6 mm point-group symmetry and positive birefringence. The corresponding deff is expressed by d31 cos θ, which is
Fig. 5. Calculated far-field beam patterns of the THz wave from the end face of the NLO crystals with lengths of 0.5, 1.0, 2.5, and 5.0 mm for (a)–(d) GaSe and (e)–(h) CdSe; note that the total intensity profiles are shown, with each color scale normalized to 1.
independent of the azimuthal angle. Therefore, a perfect annular vector beam with azimuthal polarization can be obtained with use of a c-cut CdSe crystal [37]. The divergence angles of the annulus are 12.95° and 30.66° for GaSe and CdSe, respectively, estimated from the PM angle θPM and the refractive index of the THz wave for ordinary waves [35]. These large divergence angles can be collimated and focused by a high-NA lens, such as a hyperhemispherical lens made from high-resistivity silicon and a Cassegrain objective lens. Furthermore, the polarization conversion from azimuthal to radial polarization can be obtained easily using a quartz waveplate. The use of the annular vector beam in the THz frequency regime expands the THz application fields. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Tightly focused beams from a high-NA lens create a strong, localized longitudinal component of the electric field [38,39], which provides sensitive identification of highly oriented materials, such as carbon nanotubes and metallic and nonmetallic nanorods, as well as the direct excitation of surface plasmons without coupling optics (e.g., graphene nanofilm applications). The focused spot size of the vector beam was below the diffraction limit, much smaller than those of spatially homogeneous polarized Gaussian beams, thus making it suitable for confocal imaging with high spatial resolution. Combined with spectroscopic identification, THz nondestructive evaluation of a variety of composite materials can be realized. Furthermore, tightly focused spot sizes generate high electric fields for the THz wave, which induce NLO effects in the THz frequency regime. Thus, NLO in the THz regime is a promising application field. The intense THz waves interact with materials, such as gases and solid-state organic molecules, insulators, semiconductors, and superconductors, allowing characterization of their material properties through NLO phenomena [40–43]. High-power THz sources also produce frequency upconversion to optical frequencies via an optical parametric process [44], realized by bridging between the THz frequency region and optical telecom wavelengths, enabling highdensity THz communication applications. Therefore, the proposed scheme for THz vector-beam generation is expected to promote advances in other research fields, based on NLO phenomena, as well as linear optical phenomena. 4. Conclusions
We investigated direct THz vector-beam generation from NLO crystals, GaSe and CdSe, based on their uniaxial birefringent properties and effective nonlinear susceptibility. As a result, an annular beam with sixfold symmetry was generated with GaSe and a perfect profile was generated from CdSe using focused optical vector beams. THz waves with annular beam profiles can be focused beyond the diffraction limit. Therefore, super-resolution imaging and nonlinear spectroscopy can be realized for biological science, nondestructive evaluation, and security applications. References 1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974). 2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 3. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994). 4. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). 5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. RubinszteinDunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). 2774
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Department of Materials Science, Graduate School of Engineering, Tohoku University, Aoba-yama 6-6-11-1021, Sendai 980-8579, Japan 2
Institute of Multidisciplinary Research for Advanced Materials, Tohoku University 2-1-1, Katahira, Aoba-ku, Sendai 980-8577, Japan *Corresponding author: k‑[email protected] Received 17 November 2014; revised 2 March 2015; accepted 3 March 2015; posted 3 March 2015 (Doc. ID 226982); published 27 March 2015
In this paper, we describe our theoretical investigation and calculations for a terahertz (THz)-wave profile generated by difference frequency mixing (DFM) of focused, cylindrically symmetric, and polarized optical vector beams. Using vector diffraction theory, the second-order nonlinear polarization was estimated from the electric field components of the optical pump beams penetrating uniaxial, birefringent nonlinear optics (NLO) crystals, GaSe and CdSe. The approximate beam patterns of the THz waves were simulated using DFM formulation. The intensity patterns of the THz waves for GaSe and CdSe showed sixfold symmetry and cylindrical symmetry, respectively, based on the nonlinear susceptibility tensor of the crystals. As the phase-matching angle θPM was constant with respect to the c axis of the NLO crystals, an annular vector beam with a narrow width was expected. © 2015 Optical Society of America OCIS codes: (000.4430) Numerical approximation and analysis; (170.3010) Image reconstruction techniques; (190.4223) Nonlinear wave mixing; (260.3090) Infrared, far; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.54.002769
1. Introduction
Optical vector beams have attracted a great deal of attention in the field of optics due to their donutshaped beam profile, spiral wavefront, and quantized orbital angular momentum characterized by a topological charge [1–4]. These unique properties have been applied in many research fields, such as optical manipulation, nanofabrication, and high-speed optical communications [5–12]. An optical vector beam also enables super-resolution microscopy beyond the diffraction limit, based on fluorescence depletion phenomena, such as stimulated emission depletion [13,14]. Techniques using mode converters, computergenerated holograms, and spiral phase plates have been used to generate optical vector beams [15–19]. An effective means of realizing ultimately high 1559-128X/15/102769-07$15.00/0 © 2015 Optical Society of America
resolution in laser microscopy is to use an annular vector beam with radial polarization [20,21]. This beam provides the ultimate small-focused spot size, expressed as 0.36λ∕NA, where NA is the numerical aperture of the focusing optics. Sato reported annular vector-beam generation through second-harmonic generation of 800 nm light, using a β-barium borate crystal with uniaxial birefringent properties; they successfully attained an annulus width that was less than 1∕40th of its radius [22]. Over the past several decades, the terahertz (THz) frequency regime has attracted a great deal of attention due to its application potential in imaging and nondestructive sensing realized by recent developments in THz technology. Imaging technologies using THz waves ranging in wavelength from 30 to 300 μm have been applied to molecular spectroscopy, biomedicine, security, and nondestructive inspection [23–27]. The THz frequency spectrum is referred to 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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as the molecular fingerprint region due to its sensitivity to the specific vibrational frequencies of molecules and polymers [28]; however, the spatial resolution of THz waves with a Gaussian-shaped beam profile is limited to the submillimeter order due to diffraction. In contrast, a THz vector beam with a donut-shaped profile has the potential to overcome the diffraction limit to achieve spatial imaging resolution on the order of micrometers for THz frequencies. To date, THz vector beams have been generated using segmented nonlinear optical (NLO) crystals, spiral phase plates, and antenna array structures [29–31]. However, the segmented NLO crystal generates a quasi-helical beam due to its finite segmentation, and THz beam converters are limited by their operation wavelength, which is dependent on their design. In this paper, we propose a vector-beam THz wave generation technique based on difference frequency mixing (DFM). In the DFM scheme, a uniaxial birefringent NLO crystal, such as c-cut GaSe and CdSe crystals, was used. The phase-matching angle was constant with respect to the c axis of the NLO crystals, enabling annular vector-beam generation with a narrow width. 2. Theoretical Description of THz Vector-Beam Generation via Difference Frequency Mixing
Figure 1(a) shows a schematic drawing of THz vectorbeam generation via DFM. The linearly polarized optical pump and signal beams were converted into azimuthally and radially polarized beams, using a segmented half-wave plate or liquid crystal-based polarization converter. These beams were focused by a high numerical aperture (NA) lens. A c-cut NLO crystal was placed at a defocused position to prevent diffraction of the generated THz wave. Here, selection of the uniaxial birefringent NLO crystal was especially important for vector-beam generation to ensure cylindrical symmetry for the difference frequency wave produced. Potential uniaxial NLO crystals for THz wave generation include GaSe, CdSe, and GaN. GaSe has a 62 m point-group symmetry, while CdSe and GaN have 6 mm point-group symmetry. The DFM output characteristics for GaSe and CdSe were calculated, as described below. THz waves with radial polarization (an extraordinary wave) should generate oee-type phase matching (PM) using the GaSe crystal [Fig. 1(b)]. On the other hand, a CdSe crystal can be used to generate azimuthally polarized THz waves (an ordinary wave), under oeo-type PM. We start with the formulation for the DFM process (ωTHz ωp − ωs ) based on optical vector-beam pumping. Youngworth and Brown developed an expression for the electric field components of the focused cylindrical vector beams [32]; this description was extended to the case including a dielectric interface by Biss and Brown [33]. The electric field cylindrical components of the transmitted azimuthally polarized beam through 2770
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Fig. 1. (a) Schematic drawing of the proposed optical setup for terahertz (THz) annular vector-beam generation from a nonlinear optics (NLO) crystal. The two optical pump and signal beams were azimuthally and radially polarized, respectively. These beams were focused using a high-numerical aperture (NA) lens, with propagation direction matched to the phase-matching (PM) angle θPM . (b) Intensity pattern and instantaneous electric field (arrow) of the generated THz wave by mixing of the vector pump and signal beams from GaSe and CdSe crystals.
the dielectric interface of an NLO crystal can be expressed as follows: 2 p 3 Er Zθ 6 p 7 max 6 Eϕ 7 A tp cos θ1∕2 sin θl0 θ expikp z0 cos θ 4 5 0
Ep z
" × exp
p inp 2 k
2
# s 2 n1 1 − p sin θ z − z0 n2 3
0 7 6 p × 4 J 1 k r sin θ 5dθ;
(1)
0 where θ represents the tangential angle with respect to the z axis, θmax sin−1 NA∕n1 is the maximum angle related to the NA of the lens, n1 and np 2 denote the refractive indices of air and the NLO medium for the pump beam, respectively, A is a constant, k is the wave number of the pump wave, z0 is the position of the NLO crystal interface, l0 θ is the relative amplitude of the electric field in front of the
pupil, and J m is the Bessel function of the first kind of order m. The description for a focused radially polarized beam is given by 2
Es r
3
6 Es 7 4 ϕ 5A Es z
Z
θmax 0
0 0 PNL
1
PTHz x B THz C @ Py A PTHz z
ts cos θ1∕2 sin θl0 θexpiks z0 cos θ
s 2 n1 s s × exp in2 k 1 − s sin θ z − z0 n2 3 2 2 r 1 − ns1 sin θ J 1 ks r sin θ n2 7 6 7 6 ×6 7dθ; 0 5 4 i ns1 sin θJ 0 ks r sin θ
(2)
n2
where tp and ts are the amplitude transmission coefficients for the parallel and perpendicular polarization states with respect to the incident plane. The relative amplitude l0 θ of the incident pump and signal beam is assumed to possess a doughnutshaped Laguerre–Gaussian intensity profile, expressed by 2 sin θ β20 sin θ 2β0 sin2 θ 1 L ; (3) exp − 2 l0 θ sin2 θmax sin θmax 0 sin2 θmax β20
where β0 is the ratio of the pupil radius to the input beam radius, and L10 indicates the generalized Laguerre polynomial of the radial and azimuthal indices 1 and 0, respectively. Figure 2 shows an example of the intensity distribution as a function of the angle θ, l0 θ. The angle corresponding to the intensity maximum is set to the external PM angle θext-PM in the DFM process, using a GaSe or CdSe crystal. DFM in the NLO crystal is dependent on the second-order nonlinear susceptibility and the electric field components. In the NLO medium, the secondorder nonlinear polarization components are given by
B C s B C Ep y Ey B C p s B C E E B C z z d2 B p s C; p s B Ey Ez Ez Ey C B C B p s s C @ Ex Ez Ep z Ex A s p s Ep x Ey Ey Ez
(4)
where Ep and Es are the electric field components of the pump and signal wave, respectively, and x, y, and z indicate the axes in the Cartesian coordinate system. d2 denotes the second-rank tensor of the second-order nonlinear susceptibility. As mentioned above, the THz wave was generated using GaSe and CdSe, under oee-and oeo-type PM configurations, respectively. GaSe is classified in the point-group symmetry 62 m. The corresponding nonlinear optical tensor dGaSe is described by [34] 0
dGaSe
0 @ −d22 0
0 d22 0
0 0 0 0 0 0
0 0 0
1 −d22 0 A: 0
(5)
For the CdSe crystal, the point-group symmetry is 6 mm, and the nonlinear optical tensor dCdSe is given by [34] 0
dCdSe
0 @ 0 d31
0 0 d31
0 0 d33
0 d31 0
d31 0 0
1 0 0 A: 0
(6)
The amplitude of the THz wave (ETHz ) is determined from the scalar wave equation, under a slowly varying amplitude approximation, as follows: ∇2 ETHz − kTHz2 ETHz
4πωTHz2 ∂2 NL P ; c2 ∂t2
(7)
where PNL is the induced nonlinear polarization at ωTHz ∕c the DF of ωTHz ωp − ωs, kTHz nTHz o THz is the wave vector, no is the refractive index of the medium for the ordinary DFM wave, and c is the speed of light in a vacuum. We assume the following factorized form for ETHz : ETHz 2π
Fig. 2. Example of the intensity profile of the incident vector beam as a function of the angle θ, l0 θ.
1
s Ep x Ex
ωTHz2 NL sinΔkL∕2 ; P ΔkL∕2 ikTHz c2
(8)
where L is the NLO crystal length, and Δk denotes the phase-mismatching factor Δk kp − ks − kTHz. For a uniaxial NLO crystal, the wave vector of the extraordinary wave is expressed by ke ne ω∕c, where ne θ is the refractive index of the extraordinary wave, which is dependent on the angle θ with respect to the c axis. To estimate the type-oee PM condition for the GaSe crystal, the refractive index of GaSe for the extraordinary wave at signal 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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THz wavelength λs , ns θ, is e θ, and THz wave, ne represented by
v u 1 tan2 θ 2 ; sorTHz no;GaSe λ tanθ 1 sorTHz
u sorTHz u nsorTHz e;GaSe λ;θ no;GaSe λs t
ne;GaSe λ
(9) where the wavelength-dependent refractive indices for the ordinary and extraordinary waves are given by [35] 0.405 0.0186 0.0061 λ2 λ4 λ6 2 3.1436λ ; 2 λ − 2194 0.3879 0.2288 0.1223 or THz 2 ns e;GaSe λ 5.76 λ2 λ4 λ6 1.855λ2 2 : (10) λ − 2194 2 np o;GaSe λ 7.37
Thus, the PM condition for type-oee PM in a GaSe crystal is expressed as s Δk kp − ks − kTHz ωp np o;GaSe λ − ωp ne;GaSe λ; θ
− ωp nTHz e;GaSe λ; θ∕c 0;
(11)
is the refractive index of the ordinary where np o wave at pump wavelength, and c is the speed of light. In the case of CdSe, the refractive index for the extra ordinary wave at signal wavelength λs , to estimate the oeo-type PM condition, is given by ns e;CdSe λ; θ
v u 1 tan2 θ 2 ; s no;CdSe λ tanθ 1 s
u u ns o;CdSe λt
Fig. 3. External PM angle for GaSe and CdSe crystals as a function of the difference frequency between pump and signal waves. THz where np represent the refractive indices o and no of the ordinary wave at the pump and produced THz wave wavelengths, respectively. Here, we assumed that the pump wavelength, generated by a Nd:YAG laser, was fixed at 1.0642 μm. Figure 3 shows the calculated external PM angle θext PM for GaSe and CdSe crystals as a function of the difference frequency between the pump and signal frequencies. The PM conditions were satisfied below 6 THz for GaSe and 3 THz for CdSe. In the following calculation, we focus on the vectorbeam output characteristics for 2 THz waves for both GaSe and CdSe crystals.
3. Calculated Results of the Vector-Beam THz Wave Output Characteristics through DFM
In this calculation, we assumed that the pump, signal, and THz wavelengths [λp , λs , and λTHz , respectively] were 1.0642 μm, 1.0718 μm, and 150 μm, respectively. The pump and signal beams were focused onto each NLO crystal, with the incident angle fixed by z0, β0 , and θmax ; specifically, 500 μm, 1.8, and 0.25 rad, respectively, for GaSe and 100 μm, 2.7, and 0.8 rad, respectively, for CdSe. To generate 2 THz waves, the corresponding external PM angles for GaSe and CdSe were 12.95° and 30.66°, respectively. The induced second-order nonlinear polarization in the NLO crystals, GaSe and CdSe, is expressed by
ne;CdSe λ
(12) where the wavelength-dependent refractive indices of CdSe for the ordinary and extraordinary wave are given as follows [36]: 1.768λ2 3.12λ2 2 ; 4.2243 2 λ − 0.227 λ − 3380 1.8875λ2 3.646λ2 2 : (13) λ 4.2009 ns e;CdSe λ2 − 0.2171 λ2 − 3629
or THz 2 np o;CdSe λ
The type-oeo PM condition for CdSe is given by s THz Δk ωp np ∕c 0; o − ωp ne θ − ωp no
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0
PNL x
1
0
s p s −d22 Ep x Ey Ey Ex
1
B NL C B p s p s C PNL GaSe @ Py A @ −d22 Ex Ex Ey Ey A; PNL z
0 (15)
and 0 B PNL CdSe @
s p s d31 Ep x Ez Ex Ez s p s d31 Ep y Ez Ey Ez
1 C A;
s p s p s d31 Ep x Ex Ey Ey d33 Ez Ez
(16) (14)
respectively. Here, PNL for CdSe is nearly zero, bez s cause the summation of the products of Ep x Ex
Fig. 4. Distribution diagrams of second-order nonlinear polarization induced at the near input face of (a) GaSe and (b) CdSe crystals. The polarization direction of the distributions is radial for GaSe and azimuthal for CdSe. s and Ep canceled out and Ep y Ey z 0. Thus, the longitudinal component of nonlinear polarization can be ignored for both NLO crystals. Both the nonlinear polarizations for GaSe and CdSe indicate azimuthal polarizations, corresponding to the origin of THz wave with ordinary wave in these NLO crystals. The distributions of the second-order nonlinear polarization induced by the pump and signal wave with radial and azimuthal polarization at the input face of GaSe and CdSe crystals are shown in Figs. 4(a) and 4(b), respectively. The input face of each NLO crystal was assumed to be placed at a position apart from the focal point of the incident pump and signal beams in order to avoid large divergence of the generated THz wave from a small spot size compared with its wavelength. For the GaSe crystal, the nonlinear polarization distribution was estimated to be a six-peak profile with sixfold symmetry. A donut-shaped pattern was obtained for the CdSe crystal. On the basis of the near-field pattern shown in Fig. 4, far-field THz wave patterns were generated from GaSe and CdSe crystals of crystal length of 0.5, 1.0, 2.5, and 5.0 mm, as shown in Figs. 5(a)–5(d) for GaSe and 5(e)–5(h) for CdSe. An annular pattern with sixfold symmetry was obtained for the GaSe crystal. The width of the annulus became narrower as the crystal length increased due to the longitudinal PM factor, sin cΔkL∕2. The GaSe crystal has a 62 m point-group symmetry. In this case, the effective second-order nonlinear susceptibility for type-oee PM is expressed by deff d22 cos θ sin3ϕ, where θ is the angle between the c axis and the propagation direction of the THz beam, and ϕ is the azimuthal angle around the c axis [37]. The dependence of sin3ϕ induces sixfold symmetry, as indicated in annular intensity patterns. The phase change by π at every null point of the annular profile can be considered as the superposition of two azimuthally polarized beams with a spiral phase shift of 3ϕ and −3ϕ (Fig. 4). In contrast, the CdSe crystal has 6 mm point-group symmetry and positive birefringence. The corresponding deff is expressed by d31 cos θ, which is
Fig. 5. Calculated far-field beam patterns of the THz wave from the end face of the NLO crystals with lengths of 0.5, 1.0, 2.5, and 5.0 mm for (a)–(d) GaSe and (e)–(h) CdSe; note that the total intensity profiles are shown, with each color scale normalized to 1.
independent of the azimuthal angle. Therefore, a perfect annular vector beam with azimuthal polarization can be obtained with use of a c-cut CdSe crystal [37]. The divergence angles of the annulus are 12.95° and 30.66° for GaSe and CdSe, respectively, estimated from the PM angle θPM and the refractive index of the THz wave for ordinary waves [35]. These large divergence angles can be collimated and focused by a high-NA lens, such as a hyperhemispherical lens made from high-resistivity silicon and a Cassegrain objective lens. Furthermore, the polarization conversion from azimuthal to radial polarization can be obtained easily using a quartz waveplate. The use of the annular vector beam in the THz frequency regime expands the THz application fields. 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Tightly focused beams from a high-NA lens create a strong, localized longitudinal component of the electric field [38,39], which provides sensitive identification of highly oriented materials, such as carbon nanotubes and metallic and nonmetallic nanorods, as well as the direct excitation of surface plasmons without coupling optics (e.g., graphene nanofilm applications). The focused spot size of the vector beam was below the diffraction limit, much smaller than those of spatially homogeneous polarized Gaussian beams, thus making it suitable for confocal imaging with high spatial resolution. Combined with spectroscopic identification, THz nondestructive evaluation of a variety of composite materials can be realized. Furthermore, tightly focused spot sizes generate high electric fields for the THz wave, which induce NLO effects in the THz frequency regime. Thus, NLO in the THz regime is a promising application field. The intense THz waves interact with materials, such as gases and solid-state organic molecules, insulators, semiconductors, and superconductors, allowing characterization of their material properties through NLO phenomena [40–43]. High-power THz sources also produce frequency upconversion to optical frequencies via an optical parametric process [44], realized by bridging between the THz frequency region and optical telecom wavelengths, enabling highdensity THz communication applications. Therefore, the proposed scheme for THz vector-beam generation is expected to promote advances in other research fields, based on NLO phenomena, as well as linear optical phenomena. 4. Conclusions
We investigated direct THz vector-beam generation from NLO crystals, GaSe and CdSe, based on their uniaxial birefringent properties and effective nonlinear susceptibility. As a result, an annular beam with sixfold symmetry was generated with GaSe and a perfect profile was generated from CdSe using focused optical vector beams. THz waves with annular beam profiles can be focused beyond the diffraction limit. Therefore, super-resolution imaging and nonlinear spectroscopy can be realized for biological science, nondestructive evaluation, and security applications. References 1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. A 336, 165–190 (1974). 2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). 3. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994). 4. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). 5. H. He, M. E. J. Friese, N. R. Heckenberg, and H. RubinszteinDunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). 2774
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