Abstract

Phenomenological and mathematical analyses of oblique incidence in total-internal-reflection phase retarders and the procedure for constructing extremely achromatic quarter-wave retarders are presented. It is shown that the retardance can assume the same value twice at two definite spectral lines. As an example, a quarter-wave phase retarder is described that introduces a retardance of exactly 90° at two He–Ne laser lines of 632.8 and 1150 nm with a maximum deviation of 0.007° for other wavelengths between these two lines. This application is advantageous in conjunction with two-line and multiline lasers where the radiation is usually pumped in a state of circular polarization.

© 1997 Optical Society of America

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References

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  1. N. N. Nagib, “Oblique incidence in total internal reflection phase retarders. I. Theory and design considerations,” Opt. Pura Apl. (Spain) 27, 105–110 (1994).
  2. N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).
  3. N. N. Nagib, M. S. El-Bahrawy, “Phase retarders with variable angles of total internal reflection,” Appl. Opt. 33, 1218–1222 (1994).
    [CrossRef] [PubMed]
  4. N. N. Nagib, S. A. Khodier, “Optimization of a rhomb-type quarter-wave phase retarder,” Appl. Opt. 34, 2927–2930 (1995).
    [CrossRef] [PubMed]
  5. J. M. Bennett, “A critical evaluation of rhomb-type quarter-wave retarders,” Appl. Opt. 9, 2123–2129 (1970).
    [CrossRef] [PubMed]
  6. R. C. Weast, ed., Handbook of Chemistry and Physics (CRS Press, Boca Raton, Fla., 1981), p. E-380.

1995 (1)

1994 (3)

N. N. Nagib, “Oblique incidence in total internal reflection phase retarders. I. Theory and design considerations,” Opt. Pura Apl. (Spain) 27, 105–110 (1994).

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

N. N. Nagib, M. S. El-Bahrawy, “Phase retarders with variable angles of total internal reflection,” Appl. Opt. 33, 1218–1222 (1994).
[CrossRef] [PubMed]

1970 (1)

Bennett, J. M.

Demian, S. E.

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

El-Bahrawy, M. S.

N. N. Nagib, M. S. El-Bahrawy, “Phase retarders with variable angles of total internal reflection,” Appl. Opt. 33, 1218–1222 (1994).
[CrossRef] [PubMed]

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

Khodier, S. A.

N. N. Nagib, S. A. Khodier, “Optimization of a rhomb-type quarter-wave phase retarder,” Appl. Opt. 34, 2927–2930 (1995).
[CrossRef] [PubMed]

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

Nagib, N. N.

N. N. Nagib, S. A. Khodier, “Optimization of a rhomb-type quarter-wave phase retarder,” Appl. Opt. 34, 2927–2930 (1995).
[CrossRef] [PubMed]

N. N. Nagib, M. S. El-Bahrawy, “Phase retarders with variable angles of total internal reflection,” Appl. Opt. 33, 1218–1222 (1994).
[CrossRef] [PubMed]

N. N. Nagib, “Oblique incidence in total internal reflection phase retarders. I. Theory and design considerations,” Opt. Pura Apl. (Spain) 27, 105–110 (1994).

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

Weast, R. C.

R. C. Weast, ed., Handbook of Chemistry and Physics (CRS Press, Boca Raton, Fla., 1981), p. E-380.

Appl. Opt. (3)

Opt. Pura Apl. (Spain) (2)

N. N. Nagib, “Oblique incidence in total internal reflection phase retarders. I. Theory and design considerations,” Opt. Pura Apl. (Spain) 27, 105–110 (1994).

N. N. Nagib, M. S. El-Bahrawy, S. E. Demian, S. A. Khodier, “Oblique incidence in total internal reflection phase retarders. II. Application,” Opt. Pura Apl. (Spain) 27, 111–116 (1994).

Other (1)

R. C. Weast, ed., Handbook of Chemistry and Physics (CRS Press, Boca Raton, Fla., 1981), p. E-380.

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Figures (11)

Fig. 1
Fig. 1

Reflection angle θ (a) in a normal-incidence device and (b), (c) in oblique-incidence devices.

Fig. 2
Fig. 2

Retardance δ versus TIR angles θ for refractive-index values n of, a, 1.5; b, 1.6; c, 1.7; d, 1.8; e, 1.9; f, 2.0.

Fig. 3
Fig. 3

Schematic representation of the achromatization processes in oblique-incidence retarders. The arrows indicate the directions of increase for θ, δ, and n.

Fig. 4
Fig. 4

Variations of ∂δ/∂ n, (∂δ/∂θ)(dθ/dn), and dδ/dn for the device with n 0 = 1.65 and ε = 0.025°.

Fig. 5
Fig. 5

(a) achromatic phase retarder with n 0 = 1.65; (b) its retardance (2δ) versus refractive index n.

Fig. 6
Fig. 6

Two quarter-wave retarders for refractive-index values of (a) n = 1.60 and (b) n =1.70, drawn to scale. Dimensions are for a 10-mm aperture.

Fig. 7
Fig. 7

Variations of the device parameters θ, i, t, and α with refractive index n for a π/2 retardance.

Fig. 8
Fig. 8

Retardance versus refractive index for a retarder with an exact λ/4 phase shift at the He–Ne lines 632.8 and 1150 nm.

Fig. 9
Fig. 9

Variation of dθ/di with refractive index n.

Fig. 10
Fig. 10

Variations of dδ/dn with n for the retarders of Fig. 6. The curves are for external angles of incidence, 1, i; 2, i - 1; 3, i + 1.

Fig. 11
Fig. 11

Calibration curves for the retarders of Fig. 6 for external angles of incidence i, i - 1, and i + 1.

Equations (22)

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θ=α-t=α-sin-1sin i/n,
θ=α+t=α+sin-1sin i/n,
δ=2 tan-1cos θn2 sin2 θ-11/2/n sin2 θ, δ=f1n, θ, θ=f2n, n=f3λ,
θ=sin-1n2+1+n4-4n2C2-2n2+11/2/2n2C2+11/2,
dδ/dn=δ/n+δ/θdθ/dn.
sec2δ/2=1+tan2δ/2=n2 sin2 θ-cos2 θ/n2 sin4 θ,
δ/n=2 sin2 θ cos θ/n2 sin2 θ-cos2 θ×n2 sin2 θ-11/2,
δ/θ=2n sin θcos2 θ-n2 sin2 θ-1/×n2 sin2 θ-cos2 θn2 sin2 θ-11/2.
sinθ>2/n2+11/2,
dθ/dn=sin i/nn2-sin2 i1/2.
sin 2θ=2 sin in2+1sin2 θ-2/n2-sin2 i1/2.
sin2 i=n2 sin2 2θ/sin2 2θ+4n2+1sin2 θ-22.
dθ/dn=-sin i/nn2-sin2 i.
l/s=2 sin α tan θ0.
s=a/cos i,
l=2s sin α tan θ0,
A=ls sin α,
L=l/sin θ0.
θ1-θ2=t2-t1=Δt,
cos t1=n12-sin2 i1/2/n1,
sint1+Δt=sin i/n2.
dθ/di=-cos i/n cos t.

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