Electrically controlled transfer of spin angular momentum of light in an

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OPTICS LETTERS / Vol. 31, No. 23 / December 1, 2006

Electrically controlled transfer of spin angular momentum of light in an optically active medium Lixiang Chen, Guoliang Zheng, Jie Xu, Bingzhi Zhang, and Weilong She State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou 510275, China Received August 18, 2006; revised September 19, 2006; accepted September 20, 2006; posted September 21, 2006 (Doc. ID 74213); published November 9, 2006 Spin is an intrinsic property of the photon. A method for using an externally applied dc electric field to manipulate the transfer of spin angular momentum of light in an optically active medium is presented. To discuss this, we first develop a wave coupling theory of the mutual action of natural optical activity and the linear electro-optic effect. Besides being used for analyzing the electrically controlled transfer of spin angular momentum of light, the theory can also be used to describe the propagation of light traveling along an arbitrary direction in any optically active medium with an external dc electric field along an arbitrary direction. © 2006 Optical Society of America OCIS codes: 190.0190, 160.2100, 270.0270.

Spin is an intrinsic property of the photon. Poynting1 first identified the mechanical property of momentum of light and associated angular momentum with circular polarization. Beth2 presented the mechanical detection and measurement of the angular momentum of light and verified that along the propagation direction of the beam each circularly polarized photon possesses a spin angular momentum of ប. Allen et al.3 identified the total angular momentum of a classical light field and discussed the separation of spin and orbital angular momentum components.4 Barnett5 further derived the classical relation between spin (orbital) angular momentum flux and energy flux carried by light. Recently, Piccirillo et al.6 reported an experiment in which an elliptically shaped laser beam was used to control the molecular alignment in liquid films by the transfer of orbital angular momentum of light. In this Letter we present a way of using an externally applied dc electric field to manipulate the transfer of spin angular momentum of light in an optically active medium. To discuss this, we first consider the linear electrooptic effect in a medium possessing optical activity. Generally, optical activity and the linear electro-optic effect have been investigated individually in the past, since from a traditional point of view it may be too complicated to consider the optical activity and linear electro-optic effect at the same time.7 Recently, Yin et al.8 developed an electromagnetic theory for their electro-optic Q switch made with an optically active crystal. But their theory, including the use of refractive index ellipsoid theory, is far from general yet. Refractive index ellipsoid theory has been extensively used to analyze the linear electro-optic effect.9 However, the standardization of the equation of the refractive index ellipsoid is not an easy task, even using the general Jacobi method.10 In 2001, She and Lee11 presented a wave coupling theory of the linear electro-optic effect that avoids the difficulty of standardizing the equation of the refractive index ellipsoid and makes it easier to analyze the general case of the effect. Based on their theory, theories of the linear electro-optic effect in an absorbent medium12,13 0146-9592/06/233474-3/$15.00

and in a quasi-phase-matched medium14 have been developed. We find that the theory can be further generalized to the case of a medium with optical activity. In this Letter we develop a wave coupling theory of mutual action of the natural optical activity and the linear electro-optic effect and use it to discuss manipulation of the spin angular momentum of light in crystalline quartz with an externally applied dc electric field. Spatial dispersion, which expresses the dependence of macroscopic properties of matter on the spatial inhomogeneity of the electromagnetic fields, is responsible for the phenomena of optical activity.15 The polarization relative to spatial dispersion can be expressed as16 P␮␻ = ⑀0␹␮␣␤ⵜ␤E␣␻ = i⑀0␹␮␣␤E␣␻k␤ .

共1兲

In general, there are two independent polarized components in a birefringent crystal for a monochromatic plane light wave with frequency ␻, i.e., E共␻兲 = E1共␻兲 + E2共␻兲 = E1共r兲exp共ik1 · r兲 + E2共r兲exp共ik2 · r兲,

共2兲

where E1共␻兲 and E2共␻兲 denote two cross components of the light field when k1 = k2, while they denote two independent components experiencing different refractive indices when k1 ⫽ k2. The second-order polarization resulting from the spatial dispersion is P1共2兲共␻兲 = i2⑀0␬1共2兲:E1共r兲k1 exp共ik1r兲 + i2⑀0␬2共2兲:E2共r兲k2 exp共ik2r兲,

共3兲

where ␬1共2兲 and ␬2共2兲 denote second-order susceptibility tensors of natural optical activity and generally ␬1共2兲 ⫽ ␬2共2兲. In addition, there exists another second-order polarization responsible for the linear electro-optic effect if the optically active material is subject to an external dc electric field E共0兲. The polarization is11 © 2006 Optical Society of America

December 1, 2006 / Vol. 31, No. 23 / OPTICS LETTERS

P2共2兲共␻兲 = 2⑀0␹共2兲共␻,0兲:E1共r兲E共0兲exp共ik1r兲

dE1共r兲

+ 2⑀0␹共2兲共␻,0兲:E2共r兲E共0兲exp共ik2r兲.

dr

共4兲

Then the total second-order polarization should be written as 共2兲

P 共␻兲 =

i2⑀0␬1共2兲:E1共r兲k1

dr

exp共ik1r兲

+ i2⑀0␬2共2兲:E2共r兲k2 exp共ik2r兲 + 2⑀0␹共2兲共␻,0兲:E1共r兲E共0兲exp共ik1r兲 + 2⑀0␹共2兲共␻,0兲:E2共r兲E共0兲exp共ik2r兲.

共5兲

Similarly to Ref. 11, starting from Maxwell’s equations, considering only the second-order nonlinearity, and neglecting high-order nonlinearity as well as linear absorption, we can derive, under the slowly varying amplitude approximation and no walk-off approximation, the wave coupling equations for the mutual action of natural optical activity and the linear electro-optic effect, as follows: dE1共r兲 dr

dE2共r兲 dr

=

冉

冉

1 n1

= −

冊

f0 − id1 E2共r兲exp共i⌬kr兲 − id2E1共r兲,

1 n2

冊

共6a兲

f0 − id3 E1共r兲exp共− i⌬kr兲

− id4E2共r兲,

dE2共r兲

共6b兲

where ⌬k = k2 − k1 and di 共i = 1 , 2 , 3 , 4兲 are the same as 共2兲 bkkˆl兲 those in Ref. 11 and f0 = −兺jkl共k02兲 · 共ajn2␬2jkl 共2兲 = 兺jkl共k02兲 · 共bjn1␬1jklakkˆl兲 (the second equation is the requirement of the law of conservation of energy), where aj, bk, and kˆl are components of three unit vecˆ parallel to E , E , and k, respectively. tors a, b, and k 1 2 It should be noted that in the derivation of Eqs. (6a) 共2兲 共2兲 = −␬kjl has been used, and (6b) the relation15 ␬jkl 共2兲 共2兲 akkˆl = 0 and 兺jklbj␬2jkl bkkˆl = 0. which leads to 兺jklaj␬1jkl For a crystal without natural optical activity 共f0 = 0兲, Eqs. (6a) and (6b) reduce to the equations describing the linear electro-optic effect discussed in Ref. 11. And, in the absence of the external dc electric field, namely, di = 0 共i = 1 , 2 , 3 , 4兲, Eqs. (6a) and (6b) become those describing the phenomena of pure natural gyrotropy. Finding the general analytic solutions of Eqs. (6a) and (6b) is without any difficulty. But now we prefer using the theory to discuss the transfer of spin angular momentum of light in crystalline quartz controlled by an applied dc electric field. For simplicity, assuming that the light propagates along the optical axis of this crystal with an applied dc field along y-axis direction, then we have kˆ = 共0 , 0 , 1兲, a = 共1 , 0 , 0兲, b = 共0 , 1 , 0兲, and c = 共0 , 1 , 0兲. In this case, n1 = n2 = no, ⌬k = 0, and ␬1共2兲 = ␬2共2兲 = ␬共2兲. According to the nonvanishing elements of the secondorder tensor of crystalline quartz16 (belonging to class 32), we obtain that d1 = d3 = k0n3o E0␥62 / 2 = d and d2 = d4 = 0. Subsequently, Eqs. (6a) and (6b) become

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= 共f − id兲E2共r兲,

共7a兲

= 共− f − id兲E1共r兲,

共7b兲

共2兲 . Assuming that the incident where f = f0 / no = −k02␬xyz light beam is linearly polarized with the initial condition E1共0兲 = Ein, E2共0兲 = 0, and f ⬎ 0, then we obtain the solutions as follows:

E1共r兲 = Ein cos共冑f2 + d2r兲,

共8a兲

E2共r兲 = − Ein sin共冑f2 + d2r兲exp共i␪兲,

共8b兲

where ␪ = arg共f + id兲. In the absence of the external electric field, ␪ = arg共f + id兲 = 0; Eqs. (8a) and (8b) become E1共r兲 = Ein cos共fr兲 and E2共r兲 = −Ein sin共fr兲, which are just the equations describing the phenomena of natural optical activity for dextrorotatory quartz, where the spin angular momentum remains zero throughout and f is just the known optical rotatory power. In the following, we focus our attention on the manipulation of spin angular momentum of light in crystalline quartz with an externally applied dc electric field. In general, the light field described by Eqs. (8a) and (8b) can be expressed as the superposition of two circularly polarized lightfields: one is left-handed and the other right-handed, i.e., E共r兲 =

冋 册 E1共r兲

E2共r兲

再 冋 册 冋 册冎

= Ein ␣L

1/冑2

− i/冑2

+ ␣R

1/冑2 i/冑2

, 共9兲

where

␣L =

1

冑2

关cos共冑f2 + d2r兲 − i sin共冑f2 + d2r兲exp共i␪兲兴, 共10a兲

␣R =

1

冑2

关cos共冑f2 + d2r兲 + i sin共冑f2 + d2r兲exp共i␪兲兴. 共10b兲

Suppose the input and the output surfaces of the crystalline quartz have perfect antireflection coatings. Then the numbers of left- and right-handed circularly polarized photons transmitted at the output surface per unit area per second are the respective average Poynting energy flows divided by the energy per photon, i.e., NL =

c⑀0兩␣LEin兩2 2ប␻

,

NR =

c⑀0兩␣REin兩2 2ប␻

.

共11兲

The angular momentums of left- and right-handed circularly polarized photons transmitted per unit area per second are −NLប and NRប, respectively. Then the total angular momentum transmitted at

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OPTICS LETTERS / Vol. 31, No. 23 / December 1, 2006

Fig. 1. (a) Left- or right-handed circularly polarized photon number normalized, and (b) the total spin angular momentum normalized of light controlled by an external dc electric field.

The authors acknowledge financial support from the National Natural Science Foundation of China (grant 10574167). Weilong She’s e-mail address is [email protected].

the output surface per unit area per second is M=

2 c⑀0Ein

2␻ −

共兩␣R兩2 − 兩␣L兩2兲 =

2 c⑀0Ein

2␻

medium can be controlled through the linear electrooptic effect. According to the law of conservation of angular momentum, we know that light should transfer angular momentum to the medium in return during this process and exert a torque on it. As is well known, spin and orbital angular momentum can be transferred from a light beam to particles held within optical tweezers, forming an optical spanner.17,18 We have shown in the present Letter that with the aid of the external dc electric field linearly polarized light can also form an optical spanner, which can exert a torque on the medium. In addition, the transfer of spin angular momentum can be continuously manipulated by tuning the dc electric field. In conclusion, we have shown theoretically that an externally applied dc electric field can control the transfer of spin angular momentum of light in an optically active medium and make linearly polarized light form an optical spanner.

References sin ␪ sin共2冑f2 + d2r兲.

共12兲

Without losing generality, we take a 1 cm long (along the z axis of the crystal) and 2 mm thick (along the y axis of the crystal) quartz crystal as the working material to demonstrate the application of our theory. Assuming that the wavelength of the incident light is 550 nm, then the corresponding optical rotatory power is f = 4.41568 cm−1, and the refractive indices for ordinary and extraordinary waves are no = 1.54 and ne = 1.55, respectively; the nonvanishing electro-optic coefficients of crystalline quartz are ␥41 = −␥52 = 0.2 and ␥62 = ␥21 = −␥11 = 0.93 (in 10−12 m / V), respectively.16 Using these parameters, we can get d = 0.097V cm−1, where V (kilovolts) denotes the applied dc voltage. Substituting all these parameters into Eqs. (11) and (12), we can calculate NL, NR, and M as shown in Fig. 1. From Fig. 1(a) we can see that the left- or righthanded circularly polarized photon number varies with the applied electric field, but the total number is conserved. From Fig. 1(b) we find that the total spin angular momentum of the photons is not conserved. It is zero at V = 0 and varies with the applied electric field. This reveals that there is a transfer of spin angular momentum from the medium to the light induced by the applied electric field; i.e., the spin angular momentum of the light in the optically active

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