Abstract

A general theory relating continuous monotonic refractive-index inhomogeneities to the interference pattern produced by a Mach–Zehnder interferometer is specialized to a one-dimensional refractive-index distribution using power-series expansions for the refractive index and ray traces. The coefficients of the series are related to derivatives of the observed interference pattern. Practical interferogram-evaluation equation result that apply for any selected object plane and incorporate necessary corrections. Errors of the evaluation equations are discussed. The effect of misalignment, or lateral extension, of the light source is demonstrated. The phenomenon of, and conditions for, apparent ray-crossing are discussed. The effects of test-section windows are shown to be negligible. Other effects arising in practical applications are mentioned.

© 1966 Optical Society of America

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Corrections

W. L. Howes and D. R. Buchele, "Erratum: Optical Interferometry of Inhomogeneous Gases," J. Opt. Soc. Am. 57, 971-971 (1967)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-57-7-971

References

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  1. R. B. Kennard, J. Res. Natl. Bur. Std. 8, 787 (1932).
    [Crossref]
  2. J. Winckler, Rev. Sci. Inst. 19, 307 (1948). (Contains references to early German work.)
    [Crossref]
  3. F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
    [Crossref]
  4. G. D. Kahl and E. H. Wedemeyer, Phys. Fluids 7, 596 (1964).
    [Crossref]
  5. W. L. Howes and D. R. Buchele, Natl. Advisory Coram. Aeron. TN2693 (1952).
  6. W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3340 (1955).
  7. W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3507 (1955).
  8. T. H. Stix, The Theory of Plasma Waves (McGraw-Hill Book Co., New York, 1962), pp. 9–12.
  9. R. A. Alpher and D. R. White, Phys. Fluids 2, 153, 162 (1959).
    [Crossref]
  10. F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
    [Crossref]
  11. This fault appears to be mentioned first by F. J. Weinberg, Optics of Flames (Butterworth, Washington, 1963), p. 211 and more recently by R. A. Alpher and D. R. White, “Optical Interferometry” in Plasma Diagnostic Techniques, R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, New York, 1965), Ch. 10, p. 431, although it was recognized much earlier. Improper configurations are depicted in F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill Book Co., New York, 1957), p. 257; M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves (John Wiley & Sons, New York, 1965), p. 369; R. L. Rowe, Instr. Soc. Am. Trans. 5, 44 (1966) and analyzed by G. D. Kahl and D. C. Mylin, J. Opt. Soc. Am. 55, 364 (1965). For a proper configuration see Fig. 1, or R. Ladenburg, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion (Princeton University Press, Princeton, N. J., 1954), Vol. IX, p. 54.
    [Crossref]
  12. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), pp. 24, 38.
  13. T. J. I’a. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Co., London, 1942), Ch. 12, p. 317 or K. Knopp, Theory and Application of Infinite Series (Hafner Publishing Co., New York), 2nd ed., Ch. 14, p. 518.
  14. This effect is in addition to that described by G. D. Kahl and F. D. Bennett, J. Appl. Phys. 23, 763 (1952).
    [Crossref]
  15. Alternative analyses are given in Refs. 5 and 6 and by G. D. Kahl and D. C. Mylin, Ref. 11.

1964 (2)

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

G. D. Kahl and E. H. Wedemeyer, Phys. Fluids 7, 596 (1964).
[Crossref]

1959 (1)

R. A. Alpher and D. R. White, Phys. Fluids 2, 153, 162 (1959).
[Crossref]

1955 (1)

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3340 (1955).

1952 (3)

F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
[Crossref]

This effect is in addition to that described by G. D. Kahl and F. D. Bennett, J. Appl. Phys. 23, 763 (1952).
[Crossref]

W. L. Howes and D. R. Buchele, Natl. Advisory Coram. Aeron. TN2693 (1952).

1948 (1)

J. Winckler, Rev. Sci. Inst. 19, 307 (1948). (Contains references to early German work.)
[Crossref]

1932 (1)

R. B. Kennard, J. Res. Natl. Bur. Std. 8, 787 (1932).
[Crossref]

Alpher, R. A.

R. A. Alpher and D. R. White, Phys. Fluids 2, 153, 162 (1959).
[Crossref]

Bennett, F. D.

F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
[Crossref]

This effect is in addition to that described by G. D. Kahl and F. D. Bennett, J. Appl. Phys. 23, 763 (1952).
[Crossref]

Bergdolt, V. E.

F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
[Crossref]

Bromwich, T. J. I’a.

T. J. I’a. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Co., London, 1942), Ch. 12, p. 317 or K. Knopp, Theory and Application of Infinite Series (Hafner Publishing Co., New York), 2nd ed., Ch. 14, p. 518.

Buchele, D. R.

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3340 (1955).

W. L. Howes and D. R. Buchele, Natl. Advisory Coram. Aeron. TN2693 (1952).

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3507 (1955).

Carter, W. C.

F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
[Crossref]

Howes, W. L.

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3340 (1955).

W. L. Howes and D. R. Buchele, Natl. Advisory Coram. Aeron. TN2693 (1952).

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3507 (1955).

Jahoda, F. C.

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

Kahl, G. D.

G. D. Kahl and E. H. Wedemeyer, Phys. Fluids 7, 596 (1964).
[Crossref]

This effect is in addition to that described by G. D. Kahl and F. D. Bennett, J. Appl. Phys. 23, 763 (1952).
[Crossref]

Kennard, R. B.

R. B. Kennard, J. Res. Natl. Bur. Std. 8, 787 (1932).
[Crossref]

Little, E. M.

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), pp. 24, 38.

Quinn, W. E.

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

Ribe, F. L.

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

Sawyer, G. A.

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

Stix, T. H.

T. H. Stix, The Theory of Plasma Waves (McGraw-Hill Book Co., New York, 1962), pp. 9–12.

Wedemeyer, E. H.

G. D. Kahl and E. H. Wedemeyer, Phys. Fluids 7, 596 (1964).
[Crossref]

Weinberg, F. J.

This fault appears to be mentioned first by F. J. Weinberg, Optics of Flames (Butterworth, Washington, 1963), p. 211 and more recently by R. A. Alpher and D. R. White, “Optical Interferometry” in Plasma Diagnostic Techniques, R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, New York, 1965), Ch. 10, p. 431, although it was recognized much earlier. Improper configurations are depicted in F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill Book Co., New York, 1957), p. 257; M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves (John Wiley & Sons, New York, 1965), p. 369; R. L. Rowe, Instr. Soc. Am. Trans. 5, 44 (1966) and analyzed by G. D. Kahl and D. C. Mylin, J. Opt. Soc. Am. 55, 364 (1965). For a proper configuration see Fig. 1, or R. Ladenburg, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion (Princeton University Press, Princeton, N. J., 1954), Vol. IX, p. 54.
[Crossref]

White, D. R.

R. A. Alpher and D. R. White, Phys. Fluids 2, 153, 162 (1959).
[Crossref]

Winckler, J.

J. Winckler, Rev. Sci. Inst. 19, 307 (1948). (Contains references to early German work.)
[Crossref]

J. Appl. Phys. (3)

F. C. Jahoda, E. M. Little, W. E. Quinn, F. L. Ribe, and G. A. Sawyer, J. Appl. Phys. 35, 2351 (1964).
[Crossref]

F. D. Bennett, W. C. Carter, and V. E. Bergdolt, J. Appl. Phys. 23, 453 (1952).
[Crossref]

This effect is in addition to that described by G. D. Kahl and F. D. Bennett, J. Appl. Phys. 23, 763 (1952).
[Crossref]

J. Res. Natl. Bur. Std. (1)

R. B. Kennard, J. Res. Natl. Bur. Std. 8, 787 (1932).
[Crossref]

Natl. Advisory Comm. Aeron. TN (1)

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3340 (1955).

Natl. Advisory Coram. Aeron. TN (1)

W. L. Howes and D. R. Buchele, Natl. Advisory Coram. Aeron. TN2693 (1952).

Phys. Fluids (2)

R. A. Alpher and D. R. White, Phys. Fluids 2, 153, 162 (1959).
[Crossref]

G. D. Kahl and E. H. Wedemeyer, Phys. Fluids 7, 596 (1964).
[Crossref]

Rev. Sci. Inst. (1)

J. Winckler, Rev. Sci. Inst. 19, 307 (1948). (Contains references to early German work.)
[Crossref]

Other (6)

W. L. Howes and D. R. Buchele, Natl. Advisory Comm. Aeron. TN3507 (1955).

T. H. Stix, The Theory of Plasma Waves (McGraw-Hill Book Co., New York, 1962), pp. 9–12.

Alternative analyses are given in Refs. 5 and 6 and by G. D. Kahl and D. C. Mylin, Ref. 11.

This fault appears to be mentioned first by F. J. Weinberg, Optics of Flames (Butterworth, Washington, 1963), p. 211 and more recently by R. A. Alpher and D. R. White, “Optical Interferometry” in Plasma Diagnostic Techniques, R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, New York, 1965), Ch. 10, p. 431, although it was recognized much earlier. Improper configurations are depicted in F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill Book Co., New York, 1957), p. 257; M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves (John Wiley & Sons, New York, 1965), p. 369; R. L. Rowe, Instr. Soc. Am. Trans. 5, 44 (1966) and analyzed by G. D. Kahl and D. C. Mylin, J. Opt. Soc. Am. 55, 364 (1965). For a proper configuration see Fig. 1, or R. Ladenburg, “Physical Measurements in Gas Dynamics and Combustion,” in High Speed Aerodynamics and Jet Propulsion (Princeton University Press, Princeton, N. J., 1954), Vol. IX, p. 54.
[Crossref]

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964), pp. 24, 38.

T. J. I’a. Bromwich, An Introduction to the Theory of Infinite Series (Macmillan and Co., London, 1942), Ch. 12, p. 317 or K. Knopp, Theory and Application of Infinite Series (Hafner Publishing Co., New York), 2nd ed., Ch. 14, p. 518.

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Figures (8)

Fig. 1
Fig. 1

Mach–Zehnder interferometer and characteristic rays.

Fig. 2
Fig. 2

Light-path detail within test section.

Fig. 3
Fig. 3

Relation of observed profile ND(y) to undistorted profile N0(y). (D < 0).

Fig. 4
Fig. 4

Graph for computing coefficient b2.

Fig. 5
Fig. 5

Examples of apparent ray crossing: upper—crossing for zL/2; lower—crossing for zL/2.

Fig. 6
Fig. 6

Light-path detail in y,z plane with windows and μ = μ(y).

Fig. 7
Fig. 7

Re-evaluation of exponential density profile. L = 4.57 cm, λ = 5.46 × 10−5 cm. Upper: One-term approximation. Recomputed points for K: 1 3 , 1 2, Δ—1, — – assumed profile. Center: two-term approximation. Lower: three-term approximation.

Fig. 8
Fig. 8

Re-evaluation of exponential density profile. L = 9.15 cm. λ = 5.46 × 10−5 cm. Upper: One-term approximation. Recomputed points for K: 1 3 , 1 2, Δ—1, — –assumed profile. Center: two-term approximation. Lower: three-term approximation.

Equations (102)

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μ = 1 + k ρ ,
ψ = c d t = μ d s = a stationary value ,
N = ( ψ 1 ψ 2 ) / λ ,
N = λ 1 { 0 L μ ( x , y , z ) [ 1 + ( d x / d z ) 2 + ( d y / d z ) 2 ] 1 2 d z μ r L sec φ r ( 0 z L ) μ a D sin φ ( z < 0 ) μ a K * L [ sec φ ( z > L ) sec φ r ( r > L ) ] } ,
K * = 1 ( Z * / L ) ,
K * = K tan φ ( z = L ) / tan φ ( z > L ) ,
K = 1 ( Z / L )
N = λ 1 [ μ L sec φ 0 μ a L sec φ r μ a D sin φ ( z < 0 ) ] ,
N = ( μ μ a ) L / λ
D = ± [ ( H h 2 ) + 2 h H ( 1 cos δ ) ] 1 2 ,
h = ± [ ( l l r ) 2 + 2 l l r ( 1 cos θ ) ] 1 2 ,
l K * L sec φ ( z > L ) ,
l r K * L sec φ r ( z > L ) ,
ξ x x 0
η y y 0
ζ z ,
μ ( η ) ν = 0 b ν η ν .
η σ = 0 c σ ζ σ
( ψ ) 2 = μ 2 ( η )
ψ = C 1 ξ + 0 η [ μ 2 ( η ) C 2 2 ] 1 2 d η + C 3 ζ
C 2 2 = C 1 2 + C 3 2 .
C 1 2 = μ 0 2 tan 2 γ 0 cos 2 φ 0
C 3 2 = μ 0 2 cos 2 φ 0 ,
sec 2 φ 0 = 1 + tan 2 β 0 + tan 2 γ 0 .
d η / d ζ = [ μ 2 ( η ) C 2 2 ] 1 2 / C 3 = tan β
d η / d ξ = [ μ 2 ( η ) C 2 2 ] 1 2 / C 1 = tan α
( d η / d ζ ) 2 + sec 2 γ 0 = [ μ ( η ) / μ 0 ] 2 sec 2 φ 0
( d η / d ξ ) 2 tan 2 γ 0 + sec 2 γ 0 = [ μ ( η ) / μ 0 ] 2 sec 2 φ 0 .
μ ( η ) = μ 0 + Δ μ ( η ) ,
μ 0 1
Δ μ ( η ) 1.
( d η / d ζ ) 2 = 2 Δ μ ( η ) sec 2 φ 0 + tan 2 β 0 ,
b ν ( 1 / ν ! ) ( d ν μ / d η ν ) η = 0 ( 1 / ν ! ) ( d ν μ / d y ν ) y 0
b 0 = μ 0 1
Δ μ ( η ) = ν = 1 b ν η ν 1.
c σ ( 1 / σ ! ) ( d σ η / d ζ σ ) ζ = 0 ( 1 / σ ! ) ( d σ y / d z σ ) z = 0 .
η = tan β 0 σ = 1 , 3 , 5 , c σ * ζ σ sec σ 1 φ 0 + σ = 2 , 4 , c σ * ζ σ sec σ φ 0
c 1 * = 1 c 2 * = b 1 / 2 c 3 * = b 2 / 3 c 4 * = b 1 b 2 / 12
η = σ = 2 , 4 , c σ * ζ σ .
ψ ( L ) = 0 L μ ( η ) [ 1 + ( d ξ / d ζ ) 2 + ( d η / d ζ ) 2 ] 1 2 d ζ ,
ψ ( L ) = 0 L μ ( η ) [ sec 2 γ 0 + ( d η / d ζ ) 2 ] 1 2 d ζ = ( sec φ 0 / μ 0 ) × 0 L μ 2 ( η ) d ζ = sec φ 0 ( μ 0 L + 2 0 L Δ μ ( η ) d ζ ) ,
N = λ 1 { ( μ 0 sec φ 0 μ sec φ r ) L + 2 sec φ 0 0 L Δ μ ( η ) d ζ μ α D sin φ ( z < 0 ) μ a K * L [ sec φ ( z > L ) sec φ r ( z < L ) ] } ,
N = λ 1 [ ( μ 0 μ ) L + 2 0 L Δ μ ( η ) d ζ K L Δ μ ( H ) ] ,
N = λ 1 [ ( μ 0 μ ) L + ν = 1 σ = 2 , 4 , 6 ( 2 σ + 1 K ) b ν c σ , ν L σ + 1 ] ,
σ = 1 c σ , ν ζ σ ( σ = 1 c σ * ζ σ ) ν ( ν = 1 , 2 , 3 , ) .
c σ , ν = ( c 1 0 c 2 c 1 2 c 3 2 c 1 c 2 c 4 c 2 2 + 2 c 1 c 3 ) .
c σ = { c σ * tan β 0 sec σ 1 φ 0 ( σ odd ) c σ * sec σ φ ( σ even ) ,
c σ = { 0 ( σ odd ) c σ * ( σ even ) .
c σ , ν = ( 0 0 b 1 / 2 0 b 1 b 2 / 12 b 1 2 / 4 ) .
D = H h ,
H = ± ( η L L tan β r ) ,
h = ± K * L [ tan β ( z > L ) tan β r ( z > L ) ] ,
D = ± [ η L K * L tan β ( z > L ) ] = ± [ η L K L tan β ( z = L ) ]
D = ± σ = 2 , 4 , 6 , ( 1 σ K ) c σ * L σ .
y D = y 0 + D
( Δ N 0 / Δ y 0 ) ( Δ y D / Δ N D ) = Δ y D / Δ y 0
Δ y D / Δ y 0 = 1 + ( Δ D / Δ y 0 ) ,
d N D / d y D = [ 1 + ( d D / d y 0 ) ] 1 ( d N 0 / d y 0 ) = D N 0 ,
d n N D / d y D n = D n N 0 .
d N D d y D = λ 1 [ d μ 0 d y 0 L + ν = 1 σ = 2 , 4 , 6 , ( 2 σ + 1 K ) d d y 0 ( b ν c σ , ν ) L σ + 1 ] × [ 1 ± σ = 2 , 4 , 6 , ( 1 σ K ) d c σ * d y 0 L σ ] 1 .
d n b ν / d y 0 n ( 1 / ν ! ) ( d n + ν μ 0 / d y n + ν ) [ ( n + ν ) ! / ν ! ] b n + ν
μ 0 μ = N λ / L
D = 0 ,
μ 0 μ = ( N λ / L ) 1 6 ( 2 3 K ) b 1 2 L 2
D = ± 1 2 ( 1 2 K ) b 1 L 2
y = y 0 = y D D
b 1 = | d N D d y D | λ L ,
lim n [ ν = 0 n b ν η ν μ ( η ) ] η n = ( η fixed ) , lim η 0 [ ν = 0 n b ν η ν μ ( η ) ] η n = 0 ( n fixed ) ,
| Δ ( μ 0 μ ) | | ( 1 / 15 ) ( 2 5 K ) b 1 , 3 2 b 2 L 4 1 6 ( 2 3 K ) b 1 , 2 2 L 2 Δ b 1 |
| Δ D | | ± ( 1 / 12 ) ( 1 4 K ) b 1 , 3 b 2 L 4 ± 1 2 ( 1 2 K ) b 1 , 2 L 2 Δ b 1 | ,
b 1 , 2 = λ L | d N D d y D |
b 1 , 3 = κ 1 κ 2 ( ± ) λ L | d N D d y D |
b 2 = κ 1 2 κ 2 ( ± ) λ 2 L d 2 N D d y D 2
Δ b 1 = ( κ 2 ( ± ) / κ 1 ) 1
κ 1 1 + ( 1 2 K ) b 2 L 2 + 1 6 ( 1 4 K ) b 2 2 L 4
κ 2 ( ± ) 1 ± 2 3 ( 2 3 K ) b 2 L 2 ± ( 1 / 15 ) ( 2 5 K ) b 2 2 L 4 .
| b 2 L 2 | | λ L 2 d 2 N D d y D 2 | > 1.
Δ N = [ μ ( sec φ 0 1 ) μ a ( sec φ r 1 ) ] ( L / λ ) ,
Δ N / N φ 0 2 / 2 .
r / f tan φ 0 ,
Δ N = λ 1 { [ μ ¯ 0 sec φ ¯ 0 μ 0 μ r ( sec φ ¯ r 1 ) ] L + 2 sec φ ¯ 0 0 L Δ μ ( η ¯ ) d ζ 2 0 L Δ μ ( η ) d ζ D ¯ sin φ ¯ ( z < 0 ) [ K * ( sec φ ¯ ( z > L ) sec φ ¯ r ( z > L ) ) K Δ μ ( H ) ] L } ,
η L = K L tan β ( z = L ) .
D ¯ = ± { η ¯ L L tan β ¯ r K * L [ tan β ¯ ( z > L ) tan β ¯ r ( z > L ) ] }
Δ N 1 2 ( L / λ ) β ¯ 0 2 [ N ( λ / L ) + ( 1 / 12 ) b 1 2 L 2 ]
μ 0 = μ ¯ 0 + 1 2 b 1 L tan β ¯ 0
y 0 + Δ y 0 + D ( y 0 + Δ y 0 ) y 0 + D ( y 0 ) ,
d D / d y 1 ,
Δ D = D ( y 0 ) = D ( y 0 + Δ y 0 )
Δ y = y 0 + Δ y 0 y 0 = Δ y 0
| σ = 2 , 4 , 6 , ( 1 σ K ) d c σ * d y L σ | 1.
| ( 1 2 K ) b 2 L 2 + 1 6 ( 1 4 K ) b 2 2 L 4 | 1
N = λ 1 { 0 L μ ( x , y , z ) [ 1 + ( d x / d z ) 2 + ( d y / d z ) 2 ] 1 2 d z μ r L sec φ r μ a D sin φ ( z < t ) μ a ( K * L + t ) [ sec φ ( z > L + t ) sec φ r ( z > L + t ) ] + μ w t [ sec φ ( L < z < L + t ) sec φ r ( L < z < L + t ) ] } ,
K * L = [ tan φ ( z > L + t ) ] 1 { K L tan φ ( z = L ) t [ tan φ ( z > L + t ) tan φ ( L < z L + t ) ] } .
N = λ 1 ( 0 L μ ( x , y , z ) [ 1 + ( d x / d z ) 2 + ( d y / d z ) 2 ] 1 2 d z μ r L sec φ r μ a D sin φ ( z < t ) μ a { K L [ tan φ ( z = L ) / tan φ ( z > L + t ) ] + t [ tan φ ( L < z < L + t ) / tan φ ( z > L + t ) ] } × [ sec φ ( z > L + t ) sec φ r ( z > L + t ) ] + μ w t [ sec φ ( L < z < L + t ) sec φ r ( L < z L + t ) ] ) .
l = ( K * L + t ) sec φ ( z > L + t )
l r = ( K * L + t ) sec φ r ( z > L + t )
H = ± { η L L tan β r + t [ tan β ( L < z < L + t ) tan β r ( L < z < l + t ) ] }
h = ± ( K * L + t ) [ tan β ( z > L + t ) tan β r ( z > L + t ) ] ,
D = ± ( η L L tan β r + t [ tan β ( L < z < L + t ) tan β r ( L < z < L + t ) ] { K L [ tan β ( z = L ) / tan β ( z > L + t ) ] + t ( tan β ( L < z < L + t ) / tan β ( z > L + t ) } [ tan β ( z > L + t ) tan β r ( z > L + t ) ] ) ,
μ = μ [ μ μ ( y = 0 ) ] e y / α
μ = 1.0000792 , μ ( y = 0 ) = 1.0000475 , and a = 0.01475 cm .
ρ * = ( μ 1 ) / ( μ 1 ) .