## Weak value amplified optical activity measurements |

Optics Express, Vol. 19, Issue 17, pp. 16508-16517 (2011)

http://dx.doi.org/10.1364/OE.19.016508

Acrobat PDF (1145 KB)

### 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 Δ*θ* ∼ 1 nanoradian, which corresponds to a circular birefringence of Δ*n* ∼ 1 × 10^{−9}, could be measured with a relative error of less than 1%.

© 2011 OSA

## 1. Introduction

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]

*μ*m between the two linear-polarization components induced by a birefringent crystal. Hosten and Kwiat [4

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]

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]

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]

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]

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]

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

^{6}in the beam separation compared with the actual separation (in the absence of weak value amplification).

## 2. Theory

### 2.1. Weak measurements of small beam deflections

*pre-selection*, (2) a

*weak interaction*giving rise to a (small) polarization-dependent beam deflection, and finally (3)

*post-selection*of the final (polarization) state, which is chosen to be nearly orthogonal to the initial state. We now consider the case where the weak interaction is due to optical activity. A coherent light beam with a Gaussian beam profile is linearly polarized and is taken to be incident upon an optically active wedge (see Fig. 1). The wedge’s optical activity can be natural, e.g. a chiral liquid in a triangular cuvette, or magnetic, e.g. due to a static magnetic field that is parallel to the light beam’s direction of propagation (Faraday effect). The linear polarization state of the light is equivalent to a superposition of two circular polarization states (left- and right-circular) and these experience different refractive indices in the wedge. It follows that the two circular components refract with different angles of refraction at the second interface such that the light beam splits into two [8

**97**, 173002 (2006). [CrossRef] [PubMed]

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]

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]

*z*′ direction and is taken to be linearly polarized along

*x*′ (Fig. 1 and Fig. 2). Using Jones calculus the complex electric-field vector after the pre-selection polarizer is: where

*w*is the radius of the Gaussian beam and

**k**

_{0}||

**ẑ**′ the wavevector with |

**k**

_{0}| = 2

*πn*

_{0}/

*λ*. After refraction at the optically active prism surface the beam now propagates along

*z*(Fig. 2). The weak interaction at the interface causes an angular deflection of the left- (−) and right- (+) circularly polarized beam components. Their angular beam separation is Δ

*θ*=

*θ*

_{−}–

*θ*

_{+}(see Eq. (11)). For neat (undiluted) chiral liquids |Δ

*θ*| is already ≲ 10

^{−6}rad and the beam separation even after ∼ m is therefore always small (∼

*μ*m) compared to

*w*(∼ mm). A distance

*z*

_{0}after the interface the two Gaussian beam components are separated by 2 |Δ

*x*(

*z*

_{0})| with Δ

*x*(

*z*

_{0}) =

*z*

_{0}tan (Δ

*θ*/2), so that one can write:

*n*

_{0}(here air). The vectors |−〉 and |+〉 are respectively the normalized Jones vectors for the left- and right-circularly polarized components. Because the angular beam shift is so

*weak*, the circular polarization components spatially overlap except for the outermost wings of the Gaussian beam. It follows that the center of the resulting beam is still linearly polarized, and only the edges contain circular polarization components. Post-selection is achieved with an analyzer set to an angle

*β*placed after the interface and in front of the detector. If

_{pol}*β*is orthogonal to the initial polarization the center is extinguished and only the circular components at the wings of the Gaussian pass the analyzer. A weak value amplification takes place as the beam separation dramatically increases from a few ∼ nm or less to the width of the light beam, typically ∼ mm. This amplification is maximal for an orthogonal post-selection state [3

_{pol}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]

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]

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]

*θ*|. The bigger |Δ

*θ*| the more intensity passes the analyzer. This can be seen from the field vector and the expression for the intensity after the analyzer:

*c*

_{0}and

*ɛ*

_{0}are the speed of light and the permittivity of the vacuum, respectively. The difference between the two wavevectors

**k**

_{±}corresponding to the separated circularly polarized modes is given by: Figure 3 shows plots of the intensities

*I*

_{2}(

*x, y,z*

_{0}) and

*I*

_{3}(

*x, y,z*

_{0}) calculated for two different beam displacements Δ

*x*(

*z*

_{0}) and an orthogonal post-selection polarizer (i.e.

*β*= 90°).

_{pol}*I*

_{2}is a single Gaussian (Figs. 3a and 3c), while

*I*

_{3}shows two peaks separated by a distance comparable to the beam diameter (Figs. 3b and 3d). Here the peak intensity is proportional to the displacement of the beams. As can be seen from Fig. 3 and Eq. (3), the peak intensity after orthogonal post-selection is not sensitive to the sign of Δ

*x*(

*z*

_{0}), i.e. the left- and right-circular components can not be distinguished when

*β*= 90° as it is no longer clear which of the two polarization components deflects with Δ

_{pol}*x*(

*z*

_{0}) and which one deflects with −Δ

*x*(

*z*

_{0}). The sign information of the optical activity is consequently lost. Therefore it becomes necessary to combine weak value amplification with an appropriate polarization modulation scheme to retrieve the sign.

*α*(

*x*) also varies across the beam diameter. Eq. (3) has to be modified accordingly: Any polarization scheme that simply modulates the angle of the pre- or post-selection polarizers will adjust the amplification factor of the separation, but will also be sensitive to the optical rotation across the beam waist and will therefore primarily measure optical rotation (polarimetry) and not the angular deflection (beam separation). Therefore care must be taken when designing an appropriate polarization modulation scheme, if these two phenomena are to be distinguished.

### 2.2. Polarization modulated weak measurements

*x*′ (Eq. (1)), but then passes the FR in combination with the QWP. The FR rotates the plane of polarization by a small angle

*β*, but since the quarter-waveplate is oriented with its fast-axis parallel to the pre-selection polarizer, the beam now contains a circular polarization component and becomes elliptical: where the explicit x and y dependence has been suppressed for clarity. At the exiting surface of the optically active wedge the circular polarization components are split (see above), and after passing the post-selecting analyzer, set at

_{FR}*β*, the resultant intensity distribution at the detector becomes

_{pol}*ω*so that the intensity

_{FR}*J*

_{0},

*J*

_{1}, and

*J*

_{2}are the Bessel functions of the zeroth, first, and second order, respectively. It is seen that the second term is modulated at the fundamental,

*ω*, whereas the third term depends on the harmonic 2

_{FR}*ω*. Lock-in detection allows the second term and hence the beam separation to be discriminated from the third term, which is a function of the optical rotation

_{FR}*α*(

*x*), via Γ(

*x*). Assuming that a position sensitive split detector is used for detection with

*A*and

*B*of the photodetector, then the measured time-dependent power modulated at

*ω*on each side of the detector can be calculated from: The difference of the two signals is where

_{FR}*θ*| and the sign of the signal is proportional to the

*sgn*(Δ

*θ*) (Fig. 5b). It follows that with this setup it is now possible to detect both the magnitude and the sign of the angular beam deflection Δ

*θ*, whilst benefitting from the amplification offered by a weak value amplification scheme.

## 3. Measurements

### 3.1. Experimental setup

*λ*= 633 nm and an output power of

*P*

_{0}∼ 7 mW is taken as a light source. The (magneto-optical) Faraday effect in a glass prism serves as a model system for optical activity. The prism is made from SF11-glass (n=1.77 and Verdet constant

*V*= 14 rad/Tm at

*λ*= 633 nm) and placed inside a longitudinal magnetic field induced by an electromagnet. The magnetic field strength is detected with a Gaussmeter (MAGSYS HGM09, resolution 0.1 Gauss, accuracy ±0.5%). The amplitude and the phase of Δ

*P*(

*t*) are measured with a lock-in amplifier (LIA, Stanford Research Systems SR830).

### 3.2. Splitting of circular polarization beam components via the Faraday effect

**97**, 173002 (2006). [CrossRef] [PubMed]

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]

*θ*is the angle of incidence,

_{i}*θ*is the average of the two angles of refraction, Δ

*n*is the circular birefringence, and

*n*

_{0}is the refractive index of the surrounding medium. The angular divergence Δ

*θ*is directly proportional to the magnetic optical activity, i.e. the magnetic field strength

*B*and the Verdet constant

*V*of the glass prism.

### 3.3. Angular measurement

*B*across the glass prism, whilst detecting Δ

*P*(

*t*). Figure 6 shows the results for two measurements with the analyzer placed respectively at

*z*

_{0}= 0.65 m and

*z*

_{0}= 1.2 m after the prism. Each data point corresponds to an average over 100 measurements taken in 200 ms time intervals. It can be seen that as the distance between the interface and the post-selection polarizer increases, so does the intensity after the post-selection analyzer, in accordance with theory (dashed lines in Fig. 6). This is expected, as Δ

*P*is proportional to Δ

*x*(

*z*

_{0}) =

*z*

_{0}tan (Δ

*θ*/2) (see Eq. (10)), and therefore scales linearly with distance. The distance

*z*

_{0}increases 1.846 fold in Fig. 6 and this is in good agreement with the experimentally observed increase in the slope of 1.87 ± 0.08. This clearly demonstrates that the setup is sensitive to the angular divergence Δ

*θ*, as opposed to a lateral beam shift or some effect due to optical rotation.

### 3.4. Sign-recovery and sensitivity

*z*

_{0}= 1.2 m after the prism and measured Δ

*P*(

*t*) for opposite signs of the longitudinal magnetic field (optical activity). The results are depicted in Fig. 7a. Each data point is the average of 100 measurements taken in 200 ms time intervals. Both data sets match the theoretically predicted lines for Δ

*P*(

*t*) (Eq. (10)). It follows that the sign (i.e. the direction) of the angular splitting, and hence the handedness are faithfully recovered in this weak value amplification scheme.

*n*= 3 × 10

^{−9}. This in turn corresponds to a change of Δ

*θ*= 3 nrad in the relative angle of refraction, which could be resolved with a relative error of

*s*≤ 1%.

_{rel}## 4. Conclusions

^{6}in the separation of the beam components compared with the actual separation (in the absence of weak value amplification). The modulation scheme is chosen so that it measures angular deflections and not optical rotation. We demonstrate that the scheme can be used to determine the absolute sign of the optical activity and to detect beam separations of ∼ nanoradians with a relative error of

*s*≤ 1%, which correspond to a circular birefringence of Δ

_{rel}*n*∼ 1 × 10

^{−9}.

*α*=

*π*Δ

*n l*/

*λ*of about 0.03° in a polarimeter that uses a 10cm sample cell. However, unlike the polarimeter, which requires volumes of at least 800

*μ*l, the refraction method requires a volume of less than 4

*μ*l. An interesting prospect is to extend the present scheme by combining it with interferometric weak-value detection, where the resolution of ∼ 1 picorad in angular separation has been reported [5

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]

*μ*l volumes, and would surpass the sensitivity of any commercial laboratory polarimeter.

## Acknowledgments

## References and links

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. |

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 |

3. | N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. |

4. | O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science |

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. |

6. | L. D. Barron, |

7. | A. Fresnel, |

8. | A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. |

9. | A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. |

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 |

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 |

12. | I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. |

13. | A. Ghosh, W. Hill, and P. Fischer, “Observation of the Faraday effect via beam deflection in a longitudinal magnetic field,” Phys. Rev. A |

14. | A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,”Opt. Lett. |

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 |

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. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.5710) Instrumentation, measurement, and metrology : Refraction

(260.1440) Physical optics : Birefringence

(160.1585) Materials : Chiral media

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 11, 2011

Revised Manuscript: August 4, 2011

Manuscript Accepted: August 5, 2011

Published: August 11, 2011

**Citation**

Marcel Pfeifer and Peer Fischer, "Weak value amplified optical activity measurements," Opt. Express **19**, 16508-16517 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16508

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### References

- 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]
- 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]
- 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]
- O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science 319, 787–790 (2008). [CrossRef] [PubMed]
- 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]
- 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.
- A. Ghosh and P. Fischer, “Chiral molecules split light: reflection and refraction in a chiral liquid,” Phys. Rev. Lett. 97, 173002 (2006). [CrossRef] [PubMed]
- A. Ghosh, F. M. Fazal, and P. Fischer, “Circular differential double diffraction in chiral media,” Opt. Lett. 32, 1836–1838 (2007). [CrossRef] [PubMed]
- 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]
- 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]
- I. J. Lalov and E. M. Georgieva, “Multibeam interference, total internal reflection and optical activity,” J. Mod. Opt. 44, 265–278 (1997). [CrossRef]
- 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. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,”Opt. Lett. 33, 1437–1439 (2008). [CrossRef] [PubMed]
- 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]
- 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]

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