Abstract

A reflection-type film–substrate retarder is an optical device that changes the relative phase but not the relative amplitude of light upon reflection from a film–substrate system. While there are several such device designs based on the common negative film–substrate system, very little has been done with the other two categories of systems, zero and positive. The system category is determined by the relationship between the refractive indices of the ambient N0, film N1, and substrate N2. If N1<N0N2, the system is negative; if N1=N0N2, the system is zero; and if N1>N0N2, the system is positive. The design procedure and characteristics of zero-system reflection retarders are discussed. The polarization and ellipsometric properties of the positive system preclude the existence of a reflection retarder. First, a brief characterization of the zero and positive systems by means of constant-angle-of-incidence contours and constant-thickness contours of the ellipsometric function is presented and discussed. Then an algorithm outlining the design procedures is presented, and the characteristics of the obtained designs are optimized, analyzed, and discussed. The exact retarder is valid for a single wavelength at a set angle of incidence. The design tolerance to changes in the design parameters is analyzed and discussed. In general, N1N1N2 is the condition to be satisfied to realize reflection-type retarders with film–substrate systems.

© 2005 Optical Society of America

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References

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  1. S. Kawabata, M. Suzuki, “MgF2-Ag tunable reflection retarder,” Appl. Opt. 19, 484–485 (1980).
    [Crossref] [PubMed]
  2. R. M.A. Azzam, A. R.M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film–substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975).
    [Crossref]
  3. A. R.M. Zaghloul, R. M.A. Azzam, N. M. Bashara, “Design of film–substrate single-reflection retarders,” J. Opt. Soc. Am. 65, 1043–1049 (1975).
    [Crossref]
  4. R. M.A. Azzam, M. EmdadurRahman Khan, “Single-reflection film–substrate half-wave retarders with nearly stationary reflection properties over a wide range of incidence angles,” J. Opt. Soc. Am. 73, 160–166 (1983).
    [Crossref]
  5. M. S.A. Yousef, A. R.M. Zaghloul, “Ellipsometric function of a film–substrate system: characterization and detailed study,” J. Opt. Soc. Am. A 6, 355–366 (1989).
    [Crossref]
  6. A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and representation of category-dependent film-thickness behavior of the ellipsometric function of film–substrate system,” submitted to Appl. Opt.
  7. A. R.M. Zaghloul, M. S.A. Yousef, “Ellipsometric function of a film–substrate system: detailed analysis and closed-form inversion,” J. Opt. Soc. Am. A 16, 2029–2044 (1999).
    [Crossref]
  8. A PPD was recognized in Ref. [9], without derivation through the CAICs and CTCs of the system.
  9. R. M.A. Azzam, “Simultaneous reflection and refraction of light without change of polarization by a single-layer-coated dielectric surface,” Opt. Lett. 10, 107–109 (1985).
    [Crossref] [PubMed]
  10. B. Van Zeghbroeck, Principles of Semiconductor Devices (Van Zeghbroeck, 2004) ece-www.colorado.edu/∼bart/book.

1999 (1)

1989 (1)

1985 (1)

1983 (1)

1980 (1)

1975 (2)

Azzam, R. M.A.

Bashara, N. M.

Kawabata, S.

Khan, M. EmdadurRahman

Suzuki, M.

Van Zeghbroeck, B.

B. Van Zeghbroeck, Principles of Semiconductor Devices (Van Zeghbroeck, 2004) ece-www.colorado.edu/∼bart/book.

Yousef, M. S.A.

Zaghloul, A. R.M.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

Opt. Lett. (1)

Other (3)

B. Van Zeghbroeck, Principles of Semiconductor Devices (Van Zeghbroeck, 2004) ece-www.colorado.edu/∼bart/book.

A PPD was recognized in Ref. [9], without derivation through the CAICs and CTCs of the system.

A. R.M. Zaghloul, M. S.A. Yousef, “Unified analysis and representation of category-dependent film-thickness behavior of the ellipsometric function of film–substrate system,” submitted to Appl. Opt.

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Figures (15)

Fig. 1
Fig. 1

Film–substrate system. An electromagnetic wave is reflected from the system at an angle of incidence ϕ 0 . The ambient material has a refractive index of N 0 , the film N 1 , and the substrate N 2 . The thickness of the film is d.

Fig. 2
Fig. 2

CAICs in the complex ρ plane for the 1.46 2.1316 zero system at 632.8 nm. At normal incidence (grazing incidence) ϕ 0 = 0 ( 90 ° ) , and the corresponding CAIC collapses to the point ρ = 1 ( ρ = + 1 ) .

Fig. 3
Fig. 3

CTCs of the 1.46 2.1316 zero system at λ = 632.8 nm . The film thickness d of each contour is labeled in nanometers.

Fig. 4
Fig. 4

CTC of the 1.46 2.1316 zero system at λ = 632.8 nm for a film thickness of 120 nm.

Fig. 5
Fig. 5

CAICs in the complex ρ-plane for the 1.38 1.5 positive system at λ = 632.8 nm . At normal incidence ( ϕ 0 = 0 ) the CAIC collapses to the point ρ = 1 . As ϕ 0 increases, the contours shift to the right and collapse to ρ = + 1 at grazing incidence ( ϕ 0 = 90 ° ) . Arrows indicate the direction of rotation as the film thickness d increases.

Fig. 6
Fig. 6

Positive system, 1.38 1.5 at 632.8 nm; CAICs in the ρ-plane. At any point of intersection of two contours, there is a single value of ρ that corresponds to two values of ϕ 0 .

Fig. 7
Fig. 7

CTCs in the complex ρ-plane for the 1.38 1.5 positive system at λ = 632.8 nm . The film thickness d of each contour is labeled in nanometers.

Fig. 8
Fig. 8

Exact-retarding thickness as a function of angle of incidence for the 1.46 2.1316 zero system at λ = 632.8 nm . The maximum and minimum of each curve sets the boundaries for the PTBs (Fig. 9). The center contour corresponds to d = D ϕ .

Fig. 9
Fig. 9

PTBs for the 1.46 2.1326 zero system at λ = 632.8 nm , for m = 0 (108.4–148.7) and 1 (325.0–446.1); see Fig. 8. The corresponding forbidden-thickness gaps are 0–108.4 and 148.7–325.0, respectively.

Fig. 10
Fig. 10

ψ and Δ as functions of ϕ 0 for the sample design using an IMP for the 1.46 2.1316 zero system at λ = 632.8 nm , d = 120 nm . At ϕ 0 = 38.85 ° , ψ = 45 ° , and Δ = 0 .

Fig. 11
Fig. 11

Plotted p and s reflectance for the sample design using an IMP. N 0 = 1 , N 1 = 1.46 , N 2 = 2.1316 , λ = 632.8 nm , and d = 120 nm . At the exact-retarder angle ϕ er = 38.85 ° , the p and s reflectance are equal and extremely small.

Fig. 12
Fig. 12

Plot of tan ψ as changed with the angle of incidence ϕ 0 for three different design angles for the 1.46 2.1316 zero system. For higher design angles there is a larger tolerance for the angle of incidence. For ϕ er of 10°, 30°, and 50°, 10 ° < δ ϕ 0 < 6.25 ° , 3.85 ° < δ ϕ 0 < 4.18 ° , and 5.3 ° < δ ϕ 0 < 9 ° , respectively, meet the criteria for the retarder design.

Fig. 13
Fig. 13

Thickness sensitivity for three different retarder designs for the 1.46 2.1316 zero system. Higher design angles have better tolerance, with the exception of very low angles of incidence as shown above. For ϕ 0 er = 10 ° any film thickness is a retarder design. The tolerance for ϕ 0 er = 30 ° (50°) is 1.9 nm δ d 2 nm ( 4.48 nm δ d 4.5 nm ) .

Fig. 14
Fig. 14

Film index sensitivity for three different designs for the 1.46 2.1316 zero system. Curves end at δ N 1 = 0.46 , which corresponds to N 1 = 1 , a bare substrate. Lower design angles are more sensitive to changes in N 1 . At ϕ er of 10°, 30°, and 50°, 7.8 × 10 5 < δ N 1 < 8.3 × 10 5 , 8.4 × 10 4 < δ N 1 < 8.9 × 10 4 , and 0.0034 < δ N 1 < 0.0036 , respectively.

Fig. 15
Fig. 15

Plot of tan ψ versus N 2 for ϕ 0 er of 10°, 30°, and 50°, for N 1 = 1.46 . The three designs all cross tan ψ = 1 at N 2 = 2.1316 , zero system. At ϕ er of 10°, 30°, and 50°, 2.4 × 10 4 < δ N 2 < 2.3 × 10 4 , 0.0026 < δ N 2 < 0.0025 , and 0.01 < δ N 1 < 0.0099 , respectively.

Tables (2)

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Table 1 Sample Designs for Several Retarding Angles for the 1.46 2.1316 Zero System at λ = 632.8 nm a

Tables Icon

Table 2 Exact-Retarder Designed by the IMP Method at ϕ 0 er = 75 ° , with Performance Parameters and the Range of Improved Design by the EROM for a Film Thickness of 113.4 nm a

Equations (19)

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ρ = R p R s ,
R v = r 01 v + r 12 v exp ( j 2 β ) 1 + r 01 v r 12 v exp ( j 2 β ) , v = p , s .
β = 2 π d λ [ N 1 2 N 0 2 sin 2 ( ϕ 0 ) ] 1 2 ,
ρ = r 01 p + r 12 p exp ( j 2 β ) 1 + r 01 p r 12 p exp ( j 2 β ) 1 + r 01 s r 12 s exp ( j 2 β ) r 01 s + r 12 s exp ( j 2 β ) .
ρ = tan ψ exp ( j Δ ) .
R p = a + b X 1 + a b X ,
R s = c + d X 1 + c d X ,
ρ = R p R s = ( a + b X ) ( 1 + c d X ) ( 1 + a b X ) ( c + d X ) ,
X = exp ( j 2 β ) ,
( a , b , c , d ) = ( r 01 p , r 12 p , r 01 s , r 12 s ) .
ρ = A + B X + C X 2 D + E X + F X 2 ,
A = a , B = b + a c d , C = b c d ,
D = c , E = d + a b c , F = a b d .
D ϕ = λ 2 ( N 1 2 N 0 2 sin 2 ϕ 0 ) 1 2 .
X = exp [ j 4 π ( d λ ) ( N 1 2 N 0 2 sin 2 ϕ 0 ) 1 2 ] .
X = exp [ j 2 π ( d D ϕ ) ] .
ρ = exp ( j Δ ) .
d er = D ϕ 2 + m D ϕ , m = 0 , 1 , 2 , 3 ,
ϕ 0 er = sin 1 [ ( N 1 N 0 ) 2 ( 4 d m λ N 0 ) 2 ] 1 2 .

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