Abstract

We present a new form of optical activity measurement based on a modified weak value amplification scheme. It has recently been shown experimentally that the left- and right-circular polarization components refract with slightly different angles of refraction at a chiral interface causing a linearly polarized light beam to split into two. By introducing a polarization modulation that does not give rise to a change in the optical rotation it is possible to differentiate between the two circular polarization components even after post-selection with a linear polarizer. We show that such a modified weak value amplification measurement permits the sign of the splitting and thus the handedness of the optically active medium to be determined. Angular beam separations of Δθ ∼ 1 nanoradian, which corresponds to a circular birefringence of Δn ∼ 1 × 10−9, could be measured with a relative error of less than 1%.

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  1. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
    [CrossRef] [PubMed]
  2. I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
    [CrossRef]
  3. N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
    [CrossRef]
  4. O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [CrossRef] [PubMed]
  5. P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
    [CrossRef] [PubMed]
  6. L. D. Barron, Molecular Light Scattering and Optical Activity , 2nd ed. (Cambridge University Press, 2004).
    [CrossRef]
  7. A. Fresnel, Œuvres complètes d’Augustin Fresnel , H. d. Sénarmont, E. Verdet, and L. Fresnel, eds. (Imprimerie impériale, Paris, 1866), Vol. 1.
  8. A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006).
    [CrossRef] [PubMed]
  9. A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. 32, 1836–1838 (2007).
    [CrossRef] [PubMed]
  10. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  11. M. P. Silverman and J. Badoz, “Interferometric enhancement of chiral asymmetries: ellipsometry with an optically active Fabry-Perot interferometer,” J. Opt. Soc. Am. A 11, 1894–1917 (1994).
    [CrossRef]
  12. I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997).
    [CrossRef]
  13. A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
    [CrossRef]
  14. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,”Opt. Lett. 33, 1437–1439 (2008).
    [CrossRef] [PubMed]
  15. J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
    [CrossRef]
  16. A. J. Leggett, “Comment on ‘How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100’,” Phys. Rev. Lett. 62, 2325 (1989).
    [CrossRef] [PubMed]

2010 (1)

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

2009 (1)

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

2008 (2)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,”Opt. Lett. 33, 1437–1439 (2008).
[CrossRef] [PubMed]

2007 (2)

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. 32, 1836–1838 (2007).
[CrossRef] [PubMed]

2006 (1)

A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006).
[CrossRef] [PubMed]

2004 (1)

L. D. Barron, Molecular Light Scattering and Optical Activity , 2nd ed. (Cambridge University Press, 2004).
[CrossRef]

1997 (1)

I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997).
[CrossRef]

1994 (1)

1990 (1)

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
[CrossRef]

1989 (2)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

A. J. Leggett, “Comment on ‘How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100’,” Phys. Rev. Lett. 62, 2325 (1989).
[CrossRef] [PubMed]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef] [PubMed]

1986 (1)

1866 (1)

A. Fresnel, Œuvres complètes d’Augustin Fresnel , H. d. Sénarmont, E. Verdet, and L. Fresnel, eds. (Imprimerie impériale, Paris, 1866), Vol. 1.

Aharonov, Y.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef] [PubMed]

Aiello, A.

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef] [PubMed]

Badoz, J.

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity , 2nd ed. (Cambridge University Press, 2004).
[CrossRef]

Dixon, P. B.

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Duck, I. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Fazal, F. M.

Fischer, P.

A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. 32, 1836–1838 (2007).
[CrossRef] [PubMed]

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006).
[CrossRef] [PubMed]

Fresnel, A.

A. Fresnel, Œuvres complètes d’Augustin Fresnel , H. d. Sénarmont, E. Verdet, and L. Fresnel, eds. (Imprimerie impériale, Paris, 1866), Vol. 1.

Georgieva, E. M.

I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997).
[CrossRef]

Ghosh, A.

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. 32, 1836–1838 (2007).
[CrossRef] [PubMed]

A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006).
[CrossRef] [PubMed]

Hill, W.

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

Horwell, J. C.

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

Hulet, R. G.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
[CrossRef]

Jordan, A. N.

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

Lalov, I. J.

I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997).
[CrossRef]

Leggett, A. J.

A. J. Leggett, “Comment on ‘How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100’,” Phys. Rev. Lett. 62, 2325 (1989).
[CrossRef] [PubMed]

Ritchie, N. W. M.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
[CrossRef]

Silverman, M. P.

Starling, D. J.

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

Stevenson, P. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Story, J. G.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
[CrossRef]

Sudarshan, E. C. G.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Vaidman, L.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef] [PubMed]

Vudyasetu, P. K.

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

Woerdman, J. P.

J. Mod. Opt. (1)

I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Phys. Rev. A (2)

A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A 76, 055402 (2007).
[CrossRef]

J. C. Horwell, D. J. Starling, P. B. Dixon, P. K. Vudyasetu, and A. N. Jordan, “Interferometric weak value deflections: quantum and classical treatments,” Phys. Rev. A 81, 033813 (2010).
[CrossRef]

Phys. Rev. D (1)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[CrossRef]

Phys. Rev. Lett. (5)

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1990).
[CrossRef]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Horwell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[CrossRef] [PubMed]

A. J. Leggett, “Comment on ‘How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100’,” Phys. Rev. Lett. 62, 2325 (1989).
[CrossRef] [PubMed]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351–1354 (1988).
[CrossRef] [PubMed]

A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006).
[CrossRef] [PubMed]

Science (1)

O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[CrossRef] [PubMed]

Other (2)

L. D. Barron, Molecular Light Scattering and Optical Activity , 2nd ed. (Cambridge University Press, 2004).
[CrossRef]

A. Fresnel, Œuvres complètes d’Augustin Fresnel , H. d. Sénarmont, E. Verdet, and L. Fresnel, eds. (Imprimerie impériale, Paris, 1866), Vol. 1.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a weak measurement. It is based on three steps: (1) preparation of the system in a defined (polarization-) state (pre-selection), (2) weak interaction (perturbation) with the system giving rise to different angles of refraction for the two orthogonal circular polarization states (CP, and C P ¯ ), and finally (3) post-selection of the final (polarization-) state.

Fig. 2
Fig. 2

Refraction geometry at a chiral-achiral interface for a positive (a) and negative (b) circular birefringence Δn.

Fig. 3
Fig. 3

Calculated intensities for the sum (solid blue line, a) and c)) of two separated and orthogonally polarized Gaussians (dashed lines, a) and c)) before and after post-selection (red line, (b) and (d)) for different beam displacements Δx. The amplitude of the post-selected intensity is directly proportional to |Δx| but not to the sign of Δx.

Fig. 4
Fig. 4

Setup used for polarization modulated weak measurements, consisting of a HeNe-laser, two Glan-Thompson-polarizers (P1 and P2), a Faraday-Rotator (FR), a quaterwave-plate (QWP), a glass prism, and a dual-anode photomultiplier tube (PMT). A lock-in amplifier (LIA) was used to detect the time varying difference signal ΔP.

Fig. 5
Fig. 5

(a) Intensity I 3 m o d ( x , y , z , t ) at y = 0 for beam displacements Δx = ±100 nm calculated for different fractions of the period T. (b) Resulting time-dependent power difference ΔP(t) calculated with Eq. (10) for Δx = ±100 nm. Both signals have the same amplitude but are phase-shifted by π.

Fig. 6
Fig. 6

Measurements of ΔP for two different distances z 0 between the prism surface and the post-selection analyzer P2 (see text for further details). The theoretically predicted power differences (Eq. (10)) are shown by the dashed lines.

Fig. 7
Fig. 7

(a) Measurement of ΔP for d = 1.2 m with B⃗|| (black) and B⃗|| − (red). Both data sets have the same magnitude but opposite sign. The measurements are in good agreement with theoretical predictions according Eq. (10) (dashed lines). (b) Zoom of a part of the measurement data in (a) marked by the box. The step size of ΔB = 10 Gauss corresponds to Δn = 3 × 10−9 and Δθ = 3 nrad and is well resolved. In both diagrams Δx corresponds to the separation of the circular polarization components before the post-selection analyzer, and ΔP is the intensity difference measured with the position sensitive detector after weak value amplification.

Equations (12)

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E 1 ( x , y , z ) = A 0 ( x , y ) e i k 0 z | x ^ = 0 exp [ x 2 + y 2 w 2 ] e i k 0 z ( 1 0 )
E 2 ( x , y , z 0 ) = 1 2 [ A e i k r | + A + e i k + r | + ] with A ± = A 0 ( x Δ x ( z 0 ) , y ) and k ± = 2 π n 0 λ [ ± sin ( Δ θ / 2 ) x ^ + cos ( Δ θ / 2 ) z ^ ] ,
E 3 ( x , y , z 0 ) = 1 2 [ A e i ( k r β p o l ) + A + e i ( k + r + β p o l ] ( cos β p o l | x ^ + sin β p o l | y ^ ) I 3 ( x , y , z 0 ) = 1 2 c 0 ɛ 0 | E 3 | 2 = c 0 ɛ 0 8 [ A + 2 + A 2 + 2 A + A cos ( Δ k r + 2 β p o l ) ]
Δ k = k + k = 2 k 0 sin [ Δ θ 2 ] x ^
I 3 ( x , y , z ) = c 0 ɛ 0 8 { A + 2 + A 2 + 2 A + A cos [ 2 ( k 0 sin ( Δ θ 2 ) x + β p o l α ( x ) ) ] }
E 1 m o d = A 0 2 e i k 0 r [ ( cos β F R + sin β F R ) | + ( cos β F R sin β F R ) | + ]
I 3 m o d = c 0 ɛ 0 8 [ ( A + 2 + A 2 ) + ( A + 2 A 2 ) sin 2 β F R + 2 A + A cos 2 Γ ( x ) cos 2 β F R ] with A ± = 0 exp [ ( x Δ x ) 2 + y 2 w 2 ] and Γ ( x ) = k 0 sin ( Δ θ 2 ) x α ( x ) + β p o l
β F R ( t ) = Δ β sin ω F R t ,
sin [ 2 Δ β sin ω F R t ] = 2 J 1 ( 2 Δ β ) sin ω F R t + cos [ 2 Δ β sin ω F R t ] = J 0 ( 2 Δ β ) + 2 J 2 ( 2 Δ β ) cos 2 ω F R t +
P A , ω F R ( t ) = d y 0 I 3 , ω F R m o d ( x , y , z 0 t ) d x and P B ω F R ( t ) = d y 0 I 3 , ω F R m o d ( x , y , z 0 , t ) d x
Δ P ω F R ( t ) = P A , ω F R ( t ) P B , ω F R ( t ) = P 0 J 1 ( 2 Δ β ) e r f [ 2 Δ x ( z 0 ) w ] sin [ ω F R t ]
Δ θ Δ n n 0 sin θ i cos θ = V B λ π n 0 sin θ i cos θ

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