Abstract

Using Jones’s formalism, we prove three optical reversibility theorems that relate the polarization ellipticity at the output of an optical system to the polarization of the retroreflected light at the input. We describe how these theorems can be used to measure the ellipticity of a polarization remotely and thus to control it remotely. As an example, we use this method to create a linear or a circular polarization after a total internal reflection inside a prism, and the impurity of polarization is found to be better than 10−3. Finally we describe the use of this remote control to create polarization configurations that are useful for laser cooling of atoms.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. C. Jones, “A new calculus for the treatment of optical systems I. Description and discussion of the calculus,”J. Opt. Soc. Am. 31, 488–493 (1941).
    [Crossref]
  2. See, for example, the papers published in S. Chu, C. Wieman eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B6, 2019–2278 (1989). In particular, the importance of the gradient of polarization is emphasized in the contributions of J. Dalibard, C. Cohen Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” pp. 2023–2045, and of P. J. Ungar, D. S. Weiss, E. Riis, S. Chu, “Optical molasses and multilevel atoms: theory,” pp. 2058–2071.
  3. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
    [Crossref] [PubMed]
  4. Note that if the elements in the system have polarization-independent losses, they can be represented by a unitary matrix multiplied by a real number between 0 and 1. The results that we demonstrate in this paper then are unchanged.
  5. For real mirrors there can be losses that are dependent on polarization, corresponding to different reflection coefficients for sand ppolarizations. For a typical metallic mirror the reflection coefficients in amplitude will differ at most by 5%; see, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 13.2, p. 616. For a dielectric mirror, in the middle of the wavelength range where the reflection coefficient is maximum, the ratio is even closer to 1, typically above 99.5%.
  6. R. J. C. Spreeuw, National Institute of Standards and Technology, Phys. A167, Gaithersburg, Md. 20899 (personal communication, 1992).
  7. R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,”J. Opt. Soc. Am. 31, 493–499 (1941).
    [Crossref]
  8. See, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 1.5.4, p. 49.
  9. All the extinction ratios d have been evaluated visually by means of calibrated neutral-density filters with discrete values, which explains why we quote values that are exactly the same for different measurements. The accuracies are related to the density change that gives a distinguishable result.

1988 (1)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

1941 (2)

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Born, M.

For real mirrors there can be losses that are dependent on polarization, corresponding to different reflection coefficients for sand ppolarizations. For a typical metallic mirror the reflection coefficients in amplitude will differ at most by 5%; see, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 13.2, p. 616. For a dielectric mirror, in the middle of the wavelength range where the reflection coefficient is maximum, the ratio is even closer to 1, typically above 99.5%.

See, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 1.5.4, p. 49.

Cohen-Tannoudji, C.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Jones, R. C.

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Spreeuw, R. J. C.

R. J. C. Spreeuw, National Institute of Standards and Technology, Phys. A167, Gaithersburg, Md. 20899 (personal communication, 1992).

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 1.5.4, p. 49.

For real mirrors there can be losses that are dependent on polarization, corresponding to different reflection coefficients for sand ppolarizations. For a typical metallic mirror the reflection coefficients in amplitude will differ at most by 5%; see, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 13.2, p. 616. For a dielectric mirror, in the middle of the wavelength range where the reflection coefficient is maximum, the ratio is even closer to 1, typically above 99.5%.

J. Opt. Soc. Am. (2)

Phys. Rev. Lett. (1)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
[Crossref] [PubMed]

Other (6)

Note that if the elements in the system have polarization-independent losses, they can be represented by a unitary matrix multiplied by a real number between 0 and 1. The results that we demonstrate in this paper then are unchanged.

For real mirrors there can be losses that are dependent on polarization, corresponding to different reflection coefficients for sand ppolarizations. For a typical metallic mirror the reflection coefficients in amplitude will differ at most by 5%; see, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 13.2, p. 616. For a dielectric mirror, in the middle of the wavelength range where the reflection coefficient is maximum, the ratio is even closer to 1, typically above 99.5%.

R. J. C. Spreeuw, National Institute of Standards and Technology, Phys. A167, Gaithersburg, Md. 20899 (personal communication, 1992).

See, for example, M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, London, 1959), Chap. 1.5.4, p. 49.

All the extinction ratios d have been evaluated visually by means of calibrated neutral-density filters with discrete values, which explains why we quote values that are exactly the same for different measurements. The accuracies are related to the density change that gives a distinguishable result.

See, for example, the papers published in S. Chu, C. Wieman eds., feature on laser cooling and trapping of atoms, J. Opt. Soc. Am. B6, 2019–2278 (1989). In particular, the importance of the gradient of polarization is emphasized in the contributions of J. Dalibard, C. Cohen Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” pp. 2023–2045, and of P. J. Ungar, D. S. Weiss, E. Riis, S. Chu, “Optical molasses and multilevel atoms: theory,” pp. 2058–2071.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Plane wave of electric field ɛ1 incident upon an optical system S, which may include any number of mirrors, birefringent elements, or natural polarization rotators (i.e., it possesses optical activity). It then is retroreflected at normal incidence by the mirror M. ɛ2, ɛ3, and ɛ4 represent the polarizations of the three plane waves propagating as shown by the arrows. By analyzing ɛ4, we obtain information on polarization ɛ2 (or ɛ3 = ɛ2).

Fig. 2
Fig. 2

Convention on the reference axes for a mirror used at an angle of incidence i. The x and y axes after reflection are deduced from the x and y axes before reflection by a rotation of (π − 2i) around a direction normal to the plane of incidence. Note that this convention applies for all the mirrors included in the optical system S except the retroreflecting mirror M that is used for remote control of the polarization.

Fig. 3
Fig. 3

Experimental setup for remote control of the polarization after total internal reflection. The incident plane wave from a collimated laser diode is linearly polarized in the plane of the figure, so that it is completely transmitted through the polarizing cube P. The plane wave is then incident upon a system S composed of two wave plates, B1 and B2, and a prism used at total internal reflection with an angle of incidence close to 60°. The light is retroreflected by M, and we analyze the retroreflected polarization, using the intensities S1 and S2 reflected off the cube P and the beam splitter BS. The polarizations ɛ1, ɛ2, ɛ3, and ɛ4 are defined as on Fig. 1. The rotations of the wave plates allow us to adjust 2 to be either linear or circular, using the remote measurement of S1 and S2. As a test of our method, we can measure directly the output polarization ɛ2 right after the prism.

Fig. 4
Fig. 4

Part of the experimental setup for the laser cooling experiment described in Ref. 3 (we represent only the part that concerns the control of polarization). A laser beam from an LNA laser (λ = 1.08 μm) is passed through a polarizing beam-splitter cube and two orientable wave plates and then is expanded by a telescope to a beam of diameter 40 mm. The beam is then reflected with an incidence close to 45° on a large dielectric mirror (yielding an important dephasing on polarization) and sent inside the vacuum chamber, orthogonally to an atomic beam (not shown in the figure) in region I. The beam then can be either directly reflected upon a mirror inside the vacuum system [Fig. 4(a)] or first passed through a quarter-wave plate L placed in front of the retroreflecting mirror [Fig. 4(b)]. The polarizations ɛi defined earlier are shown on both (a) and (b). We want to control the polarization in region I, where the laser beam interacts with the atoms.

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

ɛ = [ E x E y ] .
ɛ · ( ɛ ) * = 0 ,
ɛ lin = ( ɛ lin ) * .
ɛ circ ( ɛ circ ) * .
ɛ = M ɛ .
S ( ω ) = [ cos ω - sin ω sin ω cos  ω ] .
S ( - ω ) = S T ( ω ) ,
G = [ exp ( i γ ) 0 0 exp ( - i γ ) ] ,
G T = G .
G = S ( ω ) G S ( - ω ) .
M rev = M T .
M = M n M n - 1 M i M 1 ,
M rev = M 1 T M i T M n T ,
ɛ 2 = M ɛ 1 ,
ɛ 4 = M T ɛ 3 ,
ɛ 4 = M T M ɛ 1 .
M - 1 = ( M T ) * .
ɛ 4 = ɛ 1 * .
( M - 1 ) * M ɛ 1 = ɛ 1 * .
M ɛ 1 = M * ɛ 1 *
ɛ 2 = ɛ 2 * ,
ɛ 2 ɛ 2 * .
M - 1 ɛ 2 M - 1 ɛ 3 * .
ɛ 1 ɛ 4 * .
ɛ 1 = [ 1 0 ] .
M = [ a b c d ] ,             M T = [ a c b d ] .
ɛ 2 = [ a c ] ,             ɛ 4 = [ a 2 + c 2 a b + c d ] .
M = S ( ω 2 ) G S ( ω 1 ) ,
M = S ( ω ) G .
ɛ 4 = G T S ( - ω ) S ( ω ) G ɛ 1 = G 2 ɛ 1 .
d = β 2 / ( α 2 + β 2 ) ,
ɛ = α ɛ + β ɛ

Metrics