Quantifying phases in homogeneous twisted birefringent medium Dipti Banerjee Department of Physics, Vidyasagar College for Women 39, Sankar ghosh lane, Kolkata‐700006, West Bengal, INDIA Srutarshi Banerjee Department of Electrical Engineering Indian Institute of technology Kharagpur‐721302,West Bengal, INDIA 1 Introduction .Theory of twisted birefringent material (liquid crystal) was developed by Gennes and Chandrashekhar. .Three kinds of twist pitch exist‐short, comparable and very long with respect to optical wavelength. We consider very long pitch‐Maguin limit/ GOA. .The property of birefringence develops quantum phases. 1. Dynamical 2. Geometrical .Phases have origin in spin or orbital angular momentum of polarized photon. Last decades there are many important works in the field of angular momentum of photon. 2 Spinorial representation of photon We studied GP of polarized photon in connection with helicity in 1994. As light gets a fixed polarization, its helicity also fixes. The ± helicity implies respective right/left circular polarization. In analogy with spin system we may suggest the coordinate of polarized photon as i A Z=μ Xμ + iYμ → AA X' + θ θA ' , 2 0 1 2 3 AA ' 1 x⎡ − x x+ ix⎤ μ 2 X = ⎢ 2 3 0 1−⎥ −( 1 − ),Y |= | 0 2x⎣ − ix x+ ⎦ x From the twistor equation, we can define the helicity operator for massless spinor A A' θS= θ − πA πA=' − ε ε Photon with helicity 1 as massless spinor of helicity 1/2. The commutation relation of Linear momentum and conserved angular momentum becomes x k []p p= με i , i, j ijk r 3 J= r x − pμ − r −( − 2 − ) JL2 =2 −μ 2 0μ= , 1 ± / 2 , ± 1 ,... i δ i δ μ is eigenvalue of h δχ and m is the eigenvalue of h δθ The function Φ(Zμ) is holomorphic in anisotropic space and apart from r, and Ф the variable χ is needed to specify helicity. Thus the anisotropic sphere with variable (,Φ,χ ) is extended. 4 5 m,μ • Incorporating μ the spherical harmonics Yl • In view of this the polarized photon may be represented in two ways ⎛ θ (iφ− ) χ / 2⎞ ⎛ θ ⎞ ⎛ Y 2/1,2/1 ⎞ ⎜ sin e ⎟ ⎜ sin ⎟ ε = ⎜ 2/1 ⎟ = ⎜ 2 ⎟ = ⎜ 2 −⎟ − −)3( 1⎜ / 2 ,− 1⎟ / 2 Y θ (iφ+ ) χ / 2 θ iχ ⎝ 2/1 ⎠ ⎜ cos e ⎟ ⎜ cos e ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ( 1997DB , 2006 ) ⎛ θ (iφ+ ) χ / 2⎞ ⎛ θ iφ ⎞ ⎛ Y − 2/1,2/1 ⎞ ⎜ cos e ⎟ ⎜ cos e ⎟ ε = ⎜ 2/1 ⎟ = ⎜ 2 ⎟ = ⎜ 2 −⎟ − −)4( ⎜1− / 2 , − 1⎟ / 2 Y θ −(iφ − ) χ / 2 θ ⎝ 2/1 ⎠ ⎜ sin e ⎟ ⎜ sin ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ (DB 1994 ) 6 Method The birefringence and dichroism of a medium can be described by differential matrix N. We followed the methods of Jones and Berry to find M and N. dM dM d θ n⎡1 n 2⎤ N = M −1 = M −1 ;N =⎢ ⎥ dz d θ dz n⎣3 n 4⎦ In a laminated crystal eight constants are paired to specify the four matrix elements of N by Linear and circular birefringence and dichroism . •We study the pure birefringent ⎡ 0 −τ −i ρ⎤ ⎡ 0 n2 ⎤ material N = ⎢ ⎥ = ⎢ ⎥ τ⎣ +i ρ 0 ⎦ ⎣n3 0 ⎦ The evolution ray vector shows exchange of optical power dξ 1 dξ 2 = n2ξ 2, = n3ξ 1, dz dz 7 The evolution of ray is the spatial change of vector as it passes through the crystal d p = Ωx p, dz d α = Ωdz Here we consider long twist pitch (GOA approximation/Maguin limit) dθ = kdz The twisted matrix for uniform angle of twist dependent on crystal thickness, ' N = S(θ )N0 θ S(- ) For angle of twist independent of crystal thickness N′ = N0 −( kSπ / 2 ), 8 omecOut • eHer we paraM spherArized eOmety eb,Φ and SAM sphere ,χ. eTher is otw to one espondenceorrc eenbetw spin and alorbit angular tummomen space. The tionapolariz trixma in SAM space cos⎛− θ sine θ−iχ ⎞ M(,)θ χ=⎜ −⎟ −( 5 − ) ⎜ iχ ⎟ ⎝sinθe cos θ⎠ eHer M is edtructonsc on the espher of and Фby M=(|ψ ψ|‐ 1/2 ψ⎛ ψ − 1 ψ ψ ⎞ 1 + + + − M(,)θ φ = 2 ⎜ ⎟ ψ⎝ ψ− + + ψ + − ψ1⎠ where 1 1 /− 2 , 11 / /− 2 2 , − 1 / 2 −1 1 / 2 ,1 1 / / 2 2 ,− 1 / 2 Y1≈ψ− ψ +YY ≈1 / 2 .1 / 2 andY1 ≈ψ+ ψ −YY ≈1 / 2 .1 / 2 9 0 Y(≈1 ψ+ ψ 1 + − )or (ψ 1− ψ − ) 2/1,2/1 1− / 2 , − 1 / 2 − 12/1,2/1 / 2 ,− 1 / 2 YYYY≈ 2/1 . 2/1 − 2/1 . 2/1 cos⎛ θ sine θiφ ⎞ YY⎛ 0 1 ⎞ M = 1 ⎜ ⎟= 1 ⎜ 1 −1 ⎟ − −)6( 2⎜ −iφ ⎟ 2⎜ −1 0 ⎟ ⎝sinθe −cos θ⎠ YY⎝ 1 1 ⎠ Here the polarization matrix constructed from spinor in eq.(4) lie in the OAM space for l=1. Higher orbital angular momentum gives the polarization matrix in different OAM sphere. 10 The birefringent medium becomes ⎛ 0 − eiφ ⎞ θN(,) φ= ⎜ η −⎟ − )7( ⎜ −iφ ⎟ ⎝e 0 ⎠ possessing circular and linear birefringence by ηcosФ and –ηsinФ. The eigenvalues±iη represent the ⎛ ± ie iφ ⎞ internal birefringence and eigenvectors ⎜ ⎟ ε = ⎜ ⎟ ⎝ 1 ⎠ When the angle of twist has dependence on the crystal thickness by =kz, the twisted matrix N’ and twisted ray becomes will be iφ ⎛ 0 −ηeiφ + k⎞ ⎛ iecosθ+ sin θ ⎞ N'= ⎜ −⎟ − )8( ε ' = ⎜ ⎟ ⎜ −iφ ⎟ ⎜ iφ ⎟ ⎝ηe− k 0 ⎠ ⎝ − iesinθ+ cos θ⎠ 11 In birefringent medium (N) the dynamical phase γ varies uniformly with η.‐ γ= ε*N =2 ε i η− ( 9 − ) External twist for dd’ develops the phase γ’ that contain both dynamical and geometric part. '* ' ' '* ' ' 2 γ ε=' εn1 2 2 ε2 + [2 n ε3 1 sin = i η cos θ− 2k φ cos φ ] ' 2γ [π 2/ cos= i η− 2 φ k − cos − φ − ] − − ( 10 ) 12 Variation of total phase γ’ with ηand k 13 Variation of total phase γ’ with cosφ and k 14 The Geometric phase could be isolated by elimination (γ’‐γ)=Γ . [( 1Γ cos =iη 2− )θ cosφ − 2− ik 2φ ] 2 cos At normal incidence π/2, the GP will be 2 [Γ (cosπ 2/ =iη 2φ 1− )−k φ cos−− ] ( 11 ) At 0, the geometric phase becomes 2Γ [1 =i −η + cos k φ − ]− − ( 12 ) Graphical analysis shows that the presence of two birefringences introduces a spiral behavior in the geometric phase visualized in OAM space. 15 The passage of circularly polarized light also studied. The |L> and |R> states gives rise to the dynamical phases γ L =<L| | N L>= 2− η i cosφ γ R =<R| | N R>= 2η i cos φ For twisted birefringent crystal the total phase γ’ for |L> becomes 'γLL |=< ' N | L>= 2− ik 2η φ i cos '-Γ =γ 2ik γ =- - - -(13) Here the geometric phase depends only on the external birefringence k of the medium. 16 Variation of GP-Γ with η and k 17 Variation of GP-Γ with η and k 18 Variation of GP(0) with η and φ 19 • Future Interest • 1.To study the twist dependent on the thickness of the medium • in connection with STOC. • 2.Non‐uniform twist • 3. 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