Achromatic Polarization Retarder Realized with Slowly Varying Linear and Circular Birefringence

2716 OPTICS LETTERS / Vol. 36, No. 14 / July 15, 2011

Achromatic polarization retarder realized with slowly varying linear and circular birefringence

Andon A. Rangelov Department of Physics, Sofia University, James Bourchier 5 Boulevard, 1164 Sofia, Bulgaria ([email protected]‐sofia.bg) Received May 3, 2011; revised June 3, 2011; accepted June 17, 2011; posted June 17, 2011 (Doc. ID 146965); published July 14, 2011 Using the phenomena of linear and circular birefringence, we propose a device that can alter general elliptical polarization of a beam by a predetermined amount, thereby allowing conversion between linearly polarized light and circularly polarized light or changes to the handedness of the polarization. Based on an analogy with two-state adiabatic following of quantum optics, the proposed device is insensitive to the frequency of the light—it serves as an achromatic polarization retarder. © 2011 Optical Society of America OCIS codes: 260.5430, 260.1440, 260.1180, 260.2710.

Birefringence, the separation of a ray of light into two proven to be inefficient, whereas efficient polarization rays when it passes through some anisotropic materials converters are narrowband. such as quartz, was first described in 1669 by Bartholi- The equations for the transverse components Ex and nus, who observed it in calcite [1–5]. Some 30 years later, Ey of a monochromatic plane wave propagating along Huygens explained the phenomenon of birefringence by the z axis through an optically active uniaxial linear suggesting different refractive indices for ordinary and crystal with negligible absorption is [4,5,12] extraordinary rays. Another type of birefringence is the 2 2 optical activity; as light passes through material in which ∂ Ex ∂Ex ω þ 2ik − k2E ¼ − ðn2E þ iGE Þ; ð1aÞ the left and right circularly polarized components travel ∂z2 ∂z x c2 e x y with different phase velocities [1–5], the plane of linearly ∂2E ∂E 2 polarized light rotates, a phenomenon also known as cir- y y 2 ω 2 þ 2ik − k Ey ¼ ðiGEx − noEyÞ; ð1bÞ cular birefringence. ∂z2 ∂z c2 Natural optical activity was first observed in quartz by Arago in 1811. In 1845 Faraday discovered that a similar where k ¼ ωno=c is the wave vector, no and ne are the phenomenon could be induced by a static magnetic field: ordinary and extraordinary refractive indices, respec- when linearly polarized, light propagated in a dielectric tively, and the element G is responsible for the optical medium parallel to a strong static magnetic field the activity [5]. Let us neglect any abrupt change of dielectric plane of polarization rotated. This artificial optical activ- properties, such as occurs at interfaces (these are ity is now known as the Faraday effect [1–5]. responsible for reflection), and consider only slowly The theory underlying changes to optical polarization varying fields, for which as a beam of light passes through matter relies on two 2 components of the electric field, often parameterized ∂ Ex;y ∂Ex;y ≪ 2k : ð2Þ by the Jones vector [6], which is a simple analogy with ∂z2 ∂z the complex-valued probability amplitudes of the traditional two-state atom. As will be shown, such a treatment This condition requires that, over a distance comparable allows the design of a device that will convert any input to the optical wavelength, the fractional change of the polarization state into a prescribed polarization state, in- field amplitude should be much smaller than unity. dependent of the wavelength of the light—it will serve as The resulting slowly varying amplitude approximation an achromatic polarization retarder. The proposed tech- [16,17] leads to the equations nique is analogous to that of adiabatic following in quantum optics [7–11] and has the same advantages of ∂ E ω n2 − n2 −iG E efficiency and robustness. i x ¼ e o x : ð Þ E E 3 A numerical investigation of broadband conversion ∂z y 2cno iG 0 y from circularly polarized light into linearly polarized light for the wavelength range of 434 to 760 nm for crys- These are the equations upon which the present proposal talline quartz was made by Darsht et al. [12]. This result is is based. a special case of the general adiabatic technique de- The two complex-valued field amplitudes Ex and Ey are elements of a Jones vector J [6]. We symmetrize scribed here. 2 2 At the same time, there exist broadband polarization the diagonal terms ne − no and 0 in Eq. (3) by incorpor- devices proposed by different authors [13–15], but unfor- ating an identical phase in the amplitudes Ex and Ey. tunately they serve only as optical rotators. Indeed, a Upon making this choice, we have broadband polarization retarder device, known as a ∂ E 1 −Δ −iΩ E Fresnel rhomb [1,2,4], does exist, but the quality of polar- i x ¼ x ; ð4Þ ization transformation is limited by the dispersive proper- ∂z Ey 2 iΩΔ Ey ties of the material the rhomb is made of. Except the Fresnel rhomb, broadband polarization retarders have where 0146-9592/11/142716-03$15.00/0 © 2011 Optical Society of America July 15, 2011 / Vol. 36, No. 14 / OPTICS LETTERS 2717

2 2 ωG ωðno − neÞ For a pure adiabatic evolution, the evolution matrix Ω ¼ ; Δ ¼ : ð5Þ cno 2cno in adiabatic basis is diagonal and contains only phase factors: As a final step we map the coordinate dependence into z ¼ ct expðiηÞ 0 time dependence, . By so doing, Eq. (4) becomes UAðL; 0Þ¼ ; ð12Þ the traditional time-dependent Schrödinger equation 0 expð−iηÞ for a two-state atom in the rotating-wave approximation [7–11]. The two field amplitudes Ex and Ey are analogs of with the adiabatic phase the probability amplitudes for the ground state (horizon- Z L 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tal polarization) and the excited state (vertical polariza- η ¼ Ω2 þ Δ2dz: ð13Þ tion). The off-diagonal element Ω in Eq. (4) is known as 0 2 the Rabi frequency, while the element Δ corresponds to the atom-laser detuning [7,8]. The evolution matrix in the original basis (the Jones Let us assume that Ω and Δ are functions of the coor- matrix) is dinate z. Then we can write Eq. (4) in the so-called adiabatic basis [7–11] (for the two-state atom, this is the basis UðL; 0Þ¼RðLÞUAðL; 0ÞRð0Þ; ð14Þ of the instantaneous eigenstates of the Hamiltonian): or explicitly pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ EA − 1 Ω2 þ Δ2 ∂φ=∂z EA x 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x iη −iη i A ¼ A : ð6Þ U11ðL; 0Þ¼e cos φð0Þ cos φðLÞþe sin φð0Þ sin φðLÞ; ∂z Ey ∂φ=∂z 1 Ω2 þ Δ2 Ey 2 ð15aÞ The connection between the Jones vector JðzÞ¼ ðE ;E Þ JAðzÞ¼ x y in the original basis and the Jones vector iη −iη A A U12ðL; 0Þ¼ie sin φð0Þ cos φðLÞ − ie cos φð0Þ sin φðLÞ; ðEx ;Ey Þ in the adiabatic basis is given by ð15bÞ JðzÞ¼RðzÞJAðzÞ; ð7Þ

−iη iη where RðzÞ is the involutory matrix U21ðL; 0Þ¼ie sin φð0Þ cos φðLÞ − ie cos φð0Þ sin φðLÞ; ð15cÞ cosð2φÞ i sinð2φÞ RðzÞ¼ ; ð8Þ −i sinð2φÞ − cosð2φÞ −iη iη U22ðL; 0Þ¼e cos φð0Þ cos φðLÞþe sin φð0Þ sin φðLÞ: φ with angle defined by the equation ð15dÞ Ω tanð2φÞ¼ : ð9Þ Equation (15) gives the general adiabatic evolution sce- Δ nario for an arbitrary polarization. Achromatic conversion: When initially the light is For adiabatic evolution of the system there are no tran- Jð0Þ¼ EA EA jEAj linearly polarized in the horizontal direction, sitions between the amplitudes x and y . Hence, x ð1; 0Þ φð0Þ¼π=2 jEAj – , and if we ensure the initial condition , and y remain constant [7 11]. Mathematically, adia- then from Eq. (15) we have batic evolution means that, in Eq. (6), the nondiagonal terms can be neglected compared to the diagonal terms. −iη ExðLÞ¼e sin φðLÞ; ð16aÞ This occurs when [7–11] E ðLÞ¼ie−iη φðLÞ: ð Þ ∂φ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y cos 16b 2 2 ≪ Ω þ Δ : ð10Þ ∂z 2 Obviously from the last equation the global phase η is un- important and can be removed from final Jones vector z Thus, adiabatic evolutionpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi requires smooth dependence components; therefore, such polarization conversion will Ω Δ Ω2 þ Δ2 of , , and large . be frequency independent. For such achromatic conver- The adiabatic condition (10) is differential, and thus sion we canpffiffiffi endp upffiffiffi with left circularly polarized light, local condition, if it is integrated, will give a global adia- JðLÞ¼ð1= 2;i= 2Þ, if we set the final angle φðLÞ¼π=4. batic condition [9,10] This process is reversible: if we start with left circularly Z φð0Þ¼π=4 L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi polarized light with initial angle and we fin- Ω2 þ Δ2 dz ≫ 4π: ð11Þ ished with angle φðLÞ¼π=2, then we achieve reversal 0 of the direction of motion and we end up with a horizontal polarization. Analogously, if we set the final angle to For example,R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for adiabatic evolution it is sufficient to be φðLÞ¼−π=4, then wep endffiffiffi upp withffiffiffi right circularly po- L 2 2 have 0 Ω þ Δ dz ≳ 40π. Here L is the thickness of larized light, JðLÞ¼ð1= 2; −i= 2Þ, instead of left circu- the medium of propagation. larly polarized light. Again, the process is reversible. 2718 OPTICS LETTERS / Vol. 36, No. 14 / July 15, 2011

1.0 Table 1. Quartz Optical Properties

−1 −1 2 λ Ω ð Þ Δ ð Þ |E | 2 (nm) 0 m 0 m x |E | y 434 298 1321 0.8 589 156 973 λ=434nm 760 85 753

0.6 ΩðzÞ¼Ω0; ΔðzÞ¼Δ0 sin½πðz=L − 1=2Þ: ð18Þ

λ=589nm The proposed achromatic retarder can be realized in a 0.4 slab of multiple layers of optically active uniaxial linear crystals or could be demonstrated with a single-mode fi- Electric field amplitudes λ=760nm ber, which exhibits both stress-induced linear birefringence and circular birefringence (using the Faraday 0.2 effect in combination of a torsion of the fiber). In conclusion, we have used the analogy between the equation that describes the polarization state of light pro- 0 pagating through an optically active anisotropic medium 0 0.2 0.4 0.6 0.8 1.0 Distance z [m] and the time-dependent Schrödinger equation applied to 2 2 adiabatic change of a two-state atom to propose an effec- Fig. 1. (Color online) Electric field amplitudes jExj , jEyj as a function of the propagation distance z for three different wave- tive and achromatic polarization retarder. lengths: λ ¼ 434 nm (red dotted curve); λ ¼ 589 nm (blue solid This work has been supported by the European Com- λ ¼ 760 curve); nm (black dashed curve). Simulations are made mission project Ultrafast Control of Quantum Systems by with the smooth function choice from Eq. (18) and for a total Strong Laser Fields (FASTQUAST), the Bulgarian Na- thickness of the medium L ¼ 1 m. tional Science Fund (NSF) grants D002-90/08, DMU02- 19/09, and IRC-CoSiM, and the Sofia University grant Alternatively, we can start with horizontal polarized φð0Þ¼π=2 φðLÞ¼0 022/2011. The author is grateful to Nikolay Vitanov for light and, applying the conditions , , stimulating discussions and to Bruce Shore for critical we will end in vertical polarized light. reading of the manuscript. Finally, combining two or more then two achromatic conversions from the above processes, we can have References achromatic conversion from left circularly polarized light 1. M. Born and E. Wolf, Principles of Optics (Pergamon, to right circularly polarized light and vice versa, or from 1975). left (right) circularly polarized light to vertical polarized 2. M. A. Azzam and N. M. Bashara, Ellipsometry and light and vice versa. Polarized Light (North Holland, 1977). From the definition of the angle φ [Eq. (9)], we see that 3. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of the needed values of 0, π=4, and π=2 would be achieved Photoelasticity (Springer, 1979). when asymptotically the optical activity or the linear bi- 4. D. Goldstein and E. Collett, Polarized Light (CRC Press, refringence have negligible values: 2003). 5. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics φ → 0; φ → π=4; φ → π=2: ð17Þ (Wiley, 2007). Ω=Δ→0þ Ω=Δ→∞ Ω=Δ→0− 6. R. C. Jones, J. Opt. Soc. Am. 31, 500 (1941). The proposed adiabatic retarder has the useful property 7. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987). that the polarization conversion depends only on the in- φ 8. B. W. Shore, The Theory of Coherent Atomic Excitation itial and final values of angle . The retarder is frequency (Wiley, 1990). independent and it is robust against variations of the pro- 9. N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, pagation length, rotary power, etc., in contrast to the tra- Annu. Rev. Phys. Chem. 52, 763 (2001). ditional retarders. The only restriction that we have to 10. N. V. Vitanov, M. Fleischhauer, B. W. Shore, and follow during the polarization conversion is the adiabatic K. Bergmann, Adv. At. Mol. Opt. Phys. 46, 55 (2001). evolution condition [see Eq. (10)], thus slow change of 11. B. W. Shore, Acta Phys. Slovaca 58, 243 (2008). angle φ. 12. M. Ya. Darsht, B. Ya. Zel’dovich, and N. D. Kundikova, Russ. Phys. J. 40, 891 (1997). The achromatic transformation from horizontal to 63 vertical polarized light for quartz is demonstrated nu- 13. D. W. Berreman, J. Opt. Soc. Am. , 1374 (1973). merically in Fig. 1. Numerical values of the linear birefrin- 14. Z. Zhuang, Y. J. Kim, and J. S. Patel, Appl. Phys. Lett. 76, gence and the optical activity for the wavelength range of 3995 (2000). 434 to 760 nm, which are taken from Darsht et al. [12], are 15. P. Hariharan, Prog. Opt. 48, 149 (2005). given in Table 1. For the simulations of Fig. 1, we choose 16. R. W. Boyd, Nonlinear Optics (Academic, 1992). the following smooth functions: 17. H. K. Aben, Exp. Mech. 6, 13 (1966).