Abstract

The detailed theory of the anomalous Faraday effect in potassium is studied for the 4S1/2 → 4P1/2 transition including hyperfine effects. The theoretical results are compared with experimental measurements.

© 1996 Optical Society of America

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References

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  1. P. Yeh, “Dispersive magnetooptic filters,” Appl. Opt. 21, 2069–2075 (1982).
    [CrossRef] [PubMed]
  2. B. Yin and T. M. Shay, “Theoretical model for a Faraday anomalous dispersion optical filter,” Opt. Lett. 16, 1617–1619 (1991).
    [CrossRef] [PubMed]
  3. B. Yin and T. M. Shay, “Faraday anomalous dispersion optical filter for Cs 455 nm transmission,” IEEE Photonics Technol. Lett. 4, 488–490 (1992).
    [CrossRef]
  4. B. Yin and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Proc. SPIE 1653, 128–134 (1992).
    [CrossRef]
  5. B. Yinand and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Opt. Commun.30–32 (1992).
  6. J. Menders, K. Benson, S. H. Bloom, C. S. Liu, and E. Korevaar, “Ultranarrow line filtering using a Cs Faraday filter at 852 nm,” Opt. Lett. 16, 846–848 (1991).
    [CrossRef] [PubMed]
  7. J. Menders, P. Searcy, K. Roff, and E. Korevaar, “Blue cesium Faraday and Voight magneto-optic atomic line filters,” Opt. Lett. 17, 1388–1390 (1990).
    [CrossRef]
  8. H. Chen, C. Y. She, P. Searcy, and E. Korevaar, “Sodium-vapor dispersive Faraday filter,” Opt. Lett. 18, 1019–1021 (1993).
    [CrossRef] [PubMed]
  9. A. S. Davydov, Quantum Mechanics (Pergamon, London, 1965), pp. 316–321.
  10. H. Kopfermann, Nuclear Moments (Academic, New York, 1958), pp. 1–41.
  11. R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, 1981), pp. 94–97.
  12. I. I. Sobelman, Atomic Spectra and Radiative Transitions (Springer-Verlag, Berlin, 1992), pp. 205–211.
  13. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.
  14. A. N. Nesmeyanov, Vapor Pressure of the Chemical Elements (Elsevier, Amsterdam, 1963), pp. 137–144.
  15. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7.
  16. R. I. Billmers, S. K. Gayen, M. F. Squicciarini, V. M. Contarino, W. J. Scharpf, and D. M. Allocca, “Experimental demonstration of an excited-state Faraday filter operating at 532 nm,” Opt. Lett. 20, 106–108 (1995).
    [CrossRef] [PubMed]
  17. Z. Wu, M. Kitano, W. Happer, M. Hou, and J. Daniels, “Optical determination of alkali metal vapor number density using Faraday rotation,” Appl. Opt. 25, 4483–4492 (1986).
    [CrossRef] [PubMed]

1995 (1)

1993 (1)

1992 (3)

B. Yin and T. M. Shay, “Faraday anomalous dispersion optical filter for Cs 455 nm transmission,” IEEE Photonics Technol. Lett. 4, 488–490 (1992).
[CrossRef]

B. Yin and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Proc. SPIE 1653, 128–134 (1992).
[CrossRef]

B. Yinand and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Opt. Commun.30–32 (1992).

1991 (2)

1990 (1)

1986 (1)

1982 (1)

Allocca, D. M.

Benson, K.

Billmers, R. I.

Bloom, S. H.

Chen, H.

Contarino, V. M.

Cowan, R. D.

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, 1981), pp. 94–97.

Daniels, J.

Davydov, A. S.

A. S. Davydov, Quantum Mechanics (Pergamon, London, 1965), pp. 316–321.

Desai, P. D.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Gayen, S. K.

Gleiser, M.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Happer, W.

Hawkins, D. T.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Hou, M.

Hultgren, R.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Kelly, K. K.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Kitano, M.

Kopfermann, H.

H. Kopfermann, Nuclear Moments (Academic, New York, 1958), pp. 1–41.

Korevaar, E.

Liu, C. S.

Menders, J.

Nesmeyanov, A. N.

A. N. Nesmeyanov, Vapor Pressure of the Chemical Elements (Elsevier, Amsterdam, 1963), pp. 137–144.

Roff, K.

Scharpf, W. J.

Searcy, P.

Shay, T. M.

B. Yinand and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Opt. Commun.30–32 (1992).

B. Yin and T. M. Shay, “Faraday anomalous dispersion optical filter for Cs 455 nm transmission,” IEEE Photonics Technol. Lett. 4, 488–490 (1992).
[CrossRef]

B. Yin and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Proc. SPIE 1653, 128–134 (1992).
[CrossRef]

B. Yin and T. M. Shay, “Theoretical model for a Faraday anomalous dispersion optical filter,” Opt. Lett. 16, 1617–1619 (1991).
[CrossRef] [PubMed]

She, C. Y.

Sobelman, I. I.

I. I. Sobelman, Atomic Spectra and Radiative Transitions (Springer-Verlag, Berlin, 1992), pp. 205–211.

Squicciarini, M. F.

Wagman, D. D.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

Wu, Z.

Yeh, P.

Yin, B.

B. Yin and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Proc. SPIE 1653, 128–134 (1992).
[CrossRef]

B. Yin and T. M. Shay, “Faraday anomalous dispersion optical filter for Cs 455 nm transmission,” IEEE Photonics Technol. Lett. 4, 488–490 (1992).
[CrossRef]

B. Yin and T. M. Shay, “Theoretical model for a Faraday anomalous dispersion optical filter,” Opt. Lett. 16, 1617–1619 (1991).
[CrossRef] [PubMed]

Yinand, B.

B. Yinand and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Opt. Commun.30–32 (1992).

Appl. Opt. (2)

IEEE Photonics Technol. Lett. (1)

B. Yin and T. M. Shay, “Faraday anomalous dispersion optical filter for Cs 455 nm transmission,” IEEE Photonics Technol. Lett. 4, 488–490 (1992).
[CrossRef]

Opt. Commun. (1)

B. Yinand and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Opt. Commun.30–32 (1992).

Opt. Lett. (5)

Proc. SPIE (1)

B. Yin and T. M. Shay, “A potassium Faraday anomalous dispersion optical filter,” Proc. SPIE 1653, 128–134 (1992).
[CrossRef]

Other (7)

A. S. Davydov, Quantum Mechanics (Pergamon, London, 1965), pp. 316–321.

H. Kopfermann, Nuclear Moments (Academic, New York, 1958), pp. 1–41.

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, 1981), pp. 94–97.

I. I. Sobelman, Atomic Spectra and Radiative Transitions (Springer-Verlag, Berlin, 1992), pp. 205–211.

R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973), pp. 4–7.

A. N. Nesmeyanov, Vapor Pressure of the Chemical Elements (Elsevier, Amsterdam, 1963), pp. 137–144.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7.

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Figures (4)

Fig. 1
Fig. 1

Splitting of the energy levels as a function of magnetic-field strength (a) for the 4S1/2 potassium level and (b) the 4P1/2 excited state.

Fig. 2
Fig. 2

Experimental setup for Faraday rotation measurements: HG, second-harmonic generator; PDL1, pulsed dye laser; SM, stepper motor; A/D, analog-to-digital converter; ND filter, neutral-density filter.

Fig. 3
Fig. 3

Theoretical transmission coefficient for (a) T = 150 °C and B = 100 G and (b) T = 200 °C and B = 100 G.

Fig. 4
Fig. 4

Comparison between the theoretical (bold curve) and the experimental transmission coefficients for (a) T = 150 °C and B = 100 G and (b) T = 200 °C and B = 100 G. The theoretical curves take into account the 3-GHz experimental linewidth.

Equations (43)

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IJFm|H|IJFm=12δFFAK+B3K(K+1)-2I(I+1)2J(J+1)2I(2I-1)2J(2J-1)+μBgJBz(-1)I+J+m+1×J(J+1)(2J+1)(2F+1)(2F+1)×JF 1I JFF-m 10 Fm-μNgIBz(-1)I+J+m+1×I(I+1)(2I+1)(2F+1)(2F+1)×IF 1J IFF-m 10 Fm,
Fm|H|Fm=14δFFA[2F(F+1)-9]+μBgJBz×(-1)m+132(2F+1)(2F+1)1/2F 13/2 1/2FF-m 10 Fm,
Sq(γM,γM)=|IJγM|dq|IJγM|2,
Sq(γM,γM)=FmFmYγMFmIJFm|dq|IJFmYγMFm2,
IJFm|dq|IJFm=(-1)F-mF-m 1q Fm×(-1)I+J+F+I(2F+1)(2F+1)×JF I1 FJJdJ,
|JdJ|2=3(2J+1)0hc316π3ν03τ,
Sq1/2(γM,γM)=JdJFmFmYγMFm(2F+1)(2F+1)×1/2F 3/21 F1/2F-m 1q FmYγMFm,
DγM=e22hγM [γM|r|γM·E0]γM|r|γMν0+ΔνγMγM-ν-i/4πτ×exp[i(k·r-ωt)]+c.c.,
χγMq=e20hγM |γM|rq|γM|2ν0+ΔνγMγM-ν-i/4πτ,
e2|γM|rq|γM=|γM|dq|γM|=Sq(γM,γM),
χγMq=10hγM Sq(γM, γM)ν0+ΔνγMγM-ν-i/4πτ.
f(u)du=NγMM02πkT exp-M0u2kTdu,
χγMq=-cc 10hγM×Sq(γM, γM)ν0(1+u/c)+ΔνγMγM-ν-i/4πτ×NγMM02πkT exp-M0u2kTdu.
BγM=exp-hΔνγMkTγM exp-hΔνγMkT,
W(z)=iπ- exp(-t2)z-tdt,
χq=iN00hν0πM0c22kT×γMγMBγMSq(γMγM)×WM0c22kT Δν-ΔνγMγMν0+iM0c22kT 14πν0τ,
E(z)=E02[eˆ+ exp(ik+ z)+eˆ- exp(ik- z)]×exp(-iωt),
Ey(L)=12iE0[exp(ik+L)-exp(ik-L)]exp(-iωt).
T=12exp[-Im(k++k-)L]{cosh[Im(k++k-)L]-cos[Re(k+-k-)L]}.
k±=ωc1+χ±2πν0c1+12χ±
T=12exp-πν0LcIm(χ++χ-)×coshπν0LcIm(χ+-χ-)-cosπν0LcRe(χ+-χ-).
H=1m|H|1m2m|H|1m 1m|H|2m2m|H|2m.
H=1400 03A+bm,
H=14-5A-bm-b4-m2 -b4-m23A+bm,
E8=3A4+b2|γ8,+2=|2,+2,
E7=-A4+12G+|γ7,+1=1N+-b34|1,+1+A+b4+12G+|2,+1,
E6=-A4+12G0|γ6,+0=1N0-b2|1,+0+A+12G0|2,+0,
E5=-A4+12G-|γ5,-1=1N--b34|1,-1+A-b4+12G-|2,-1,
E4=3A4-b2|γ4,-2=|2,-2,
E3=-A4-12G-|γ3,-1=1N-A-b4+12G-|1,-1+b34|2,-1,
E2=-A4-12G0|γ2,+0=1N0A+12G0|1,+0+b2|2,+0,
E1=-A4-12G+|γ1,+1=1N+A+b4+12G+|1,+1+b34|2,+1,
G±=4A2±2Ab+b2,  G0=4A2+b2,
N±=3b216+A±b4+12G±2,
N0=b24+A+12G02.
S-11/2(γ7+1, γ8+2)S-11/2(γ1+1, γ8+2)=JdJYγ7+11+1Yγ1+11+1 Yγ7+12+1Yγ1+12+1×00 1/2(1/2)30Yγ8+22+2
S-11/2(γ6+0, γ7+1)S-11/2(γ2+0, γ7+1) S-11/2(γ6+0, γ1+1)S-11/2(γ2+0, γ1+1)=JdJYγ6+01+0Yγ2+01+0 Yγ6+02+0Yγ2+02+0(1/2)6(1/2)6 (-1/2)2(-1/2)2×Yγ7+11+1Yγ7+12+1 Yγ1+11+1Yγ1+12+1,
S-11/2(γ5-1, γ6+0)S-11/2(γ3-1, γ6+0) S-11/2(γ5-1, γ2+0)S-11/2(γ3-1, γ2+0)=JdJYγ5-11-1Yγ3-11-1 Yγ5-12-1Yγ3-12-1(-1/2)6(-1/2)2 (1/2)6(1/2)2×Yγ6+01+0Yγ6+02+0 Yγ2+01+0Yγ2+02+0,
S-11/2(γ4-2, γ5-1) S-11/2(γ4-2, γ3-1)=JdJ0 Yγ4-22-201/2 0(-1/2)3×Yγ5-11-1Yγ5-12-1 Yγ3-11-1Yγ3-12-1,
S+11/2(γ8+2, γ7+1) S+11/2(γ8+2, γ1+1)=JdJ0 Yγ8+22+201/2 0(1/2)3×Yγ7+11+1Yγ7+12+1 Yγ1+11+1Yγ1+12+1,
S+11/2(γ7+1, γ6+0)S+11/2(γ1+1, γ6+0) S+11/2(γ7+1, γ2+0)S+11/2(γ1+1, γ2+0)=JdJYγ7+11+1Yγ1+11+1 Yγ7+12+1Yγ1+12+1(1/2)6(-1/2)2 (1/2)6(-1/2)2×Yγ6+01+0Yγ6+02+0 Yγ2+01+0Yγ2+02+0,
S+11/2(γ6+0, γ5-1)S+11/2(γ2+0, γ5-1) S+11/2(γ6+0, γ3-1)S+11/2(γ2+0, γ3-1)=JdJYγ6+01+0Yγ2+01+0 Yγ6+02+0Yγ2+02+0(-1/2)6(1/2)6 (-1/2)2(1/2)2×Yγ5-11-1Yγ5-12-1 Yγ3-11-1Yγ3-12-1,
S+11/2(γ5-1, γ4-2)S+11/2(γ3-1, γ4-2)=JdJYγ5-11-1Yγ3-11-1 Yγ5-12-1Yγ3-12-1×00 1/2(-1/2)30Yγ4-22-2.

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