Abstract

Two analysis procedures to determine the refractive-index profile using Interphako interference microscopy are compared for accuracy: (I) expressing the profile as a five-nomial equation (proposed previously by Ohtsuka and Shimizu); (II) considering ray bending due to the index gradient and using Abel inversion (proposed by Iga et al.). Computer simulations indicate that the errors of both procedures are comparable except for index mismatching between the specimen periphery (np) and immersion oil (n2) where Analysis (I) is more accurate than Analysis (II). The partial splitting method is more practicable than total splitting for a rodlike specimen. In addition, the improved partial splitting procedure facilitates reduction of error. The index profiles of the two selected light-focusing plastic rods (LFR) were calculated with these two procedures in various conditions. The correlation between resulting profiles and measuring conditions agreed with the simulations. It is concluded that Analysis (I) is preferable to Analysis (II) when npn2 and that the improved partial splitting procedure is a practicable method for determining the index profile of a LFR.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Ohtsuka, Y. Hatanaka, Appl. Phys. Lett. 29, 735 (1976).
    [CrossRef]
  2. Y. Ohtsuka, I. Nakamoto, Appl. Phys. Lett. 29, 559 (1976).
    [CrossRef]
  3. Y. Ohtsuka, Y. Shimizu, Appl. Opt. 16, 1050 (1977).
    [PubMed]
  4. Registered trade name of Carl Zeiss, Jena, East Germany.
  5. Y. Kokubun, K. Iga, Trans. IECE Jpn. E60, 702 (1977).
  6. M. Ikeda, M. Tateda, H. Yoshikiyo, Appl. Opt. 14, 814 (1975).
    [CrossRef] [PubMed]
  7. J. A. Arnaud, R. M. Derosier, Bell Syst. Tech. J. 55, 1489 (1976).
  8. H. M. Presby, I. P. Kaminow, Appl. Opt. 15, 3029 (1976).
    [CrossRef] [PubMed]
  9. T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
    [CrossRef] [PubMed]
  10. P. L. Chu, Electron. Lett. 13, 736 (1977).
    [CrossRef]
  11. E. Brinkmeyer, Appl. Opt. 16, 2802 (1977).
    [CrossRef] [PubMed]
  12. D. Marcuse, Appl. Opt. 18, 9 (1979).
    [CrossRef] [PubMed]
  13. P. L. Chu, T. Whitbread, Appl. Opt. 18, 1117 (1979).
    [CrossRef] [PubMed]
  14. L. S. Watkins, Appl. Opt. 18, 2214 (1979).
    [CrossRef] [PubMed]
  15. G. D. Kahl, D. C. Mylin, J. Opt. Soc. Am. 55, 364 (1965).
    [CrossRef]
  16. A. M. Hunter, P. W. Schreiber, Appl. Opt. 14, 634 (1975).
    [CrossRef]
  17. Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
    [CrossRef]
  18. Y. Ohtsuka, T. Sugano, Y. Koike, Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. 40, 382 (1979).

1979 (4)

1977 (4)

Y. Ohtsuka, Y. Shimizu, Appl. Opt. 16, 1050 (1977).
[PubMed]

P. L. Chu, Electron. Lett. 13, 736 (1977).
[CrossRef]

Y. Kokubun, K. Iga, Trans. IECE Jpn. E60, 702 (1977).

E. Brinkmeyer, Appl. Opt. 16, 2802 (1977).
[CrossRef] [PubMed]

1976 (5)

J. A. Arnaud, R. M. Derosier, Bell Syst. Tech. J. 55, 1489 (1976).

Y. Ohtsuka, Y. Hatanaka, Appl. Phys. Lett. 29, 735 (1976).
[CrossRef]

Y. Ohtsuka, I. Nakamoto, Appl. Phys. Lett. 29, 559 (1976).
[CrossRef]

T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
[CrossRef] [PubMed]

H. M. Presby, I. P. Kaminow, Appl. Opt. 15, 3029 (1976).
[CrossRef] [PubMed]

1975 (2)

1974 (1)

Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
[CrossRef]

1965 (1)

Arnaud, J. A.

J. A. Arnaud, R. M. Derosier, Bell Syst. Tech. J. 55, 1489 (1976).

Brinkmeyer, E.

Chu, P. L.

Derosier, R. M.

J. A. Arnaud, R. M. Derosier, Bell Syst. Tech. J. 55, 1489 (1976).

Hatanaka, Y.

Y. Ohtsuka, Y. Hatanaka, Appl. Phys. Lett. 29, 735 (1976).
[CrossRef]

Hotate, K.

Hunter, A. M.

Iga, K.

Y. Kokubun, K. Iga, Trans. IECE Jpn. E60, 702 (1977).

Ikeda, M.

Kahl, G. D.

Kaminow, I. P.

Koike, Y.

Y. Ohtsuka, T. Sugano, Y. Koike, Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. 40, 382 (1979).

Kokubun, Y.

Y. Kokubun, K. Iga, Trans. IECE Jpn. E60, 702 (1977).

Marcuse, D.

Mylin, D. C.

Nakamoto, I.

Y. Ohtsuka, I. Nakamoto, Appl. Phys. Lett. 29, 559 (1976).
[CrossRef]

Ohtsuka, Y.

Y. Ohtsuka, T. Sugano, Y. Koike, Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. 40, 382 (1979).

Y. Ohtsuka, Y. Shimizu, Appl. Opt. 16, 1050 (1977).
[PubMed]

Y. Ohtsuka, Y. Hatanaka, Appl. Phys. Lett. 29, 735 (1976).
[CrossRef]

Y. Ohtsuka, I. Nakamoto, Appl. Phys. Lett. 29, 559 (1976).
[CrossRef]

Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
[CrossRef]

Okoshi, T.

Presby, H. M.

Schreiber, P. W.

Senga, T.

Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
[CrossRef]

Shimizu, Y.

Sugano, T.

Y. Ohtsuka, T. Sugano, Y. Koike, Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. 40, 382 (1979).

Tateda, M.

Watkins, L. S.

Whitbread, T.

Yasuda, H.

Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
[CrossRef]

Yoshikiyo, H.

Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. (1)

Y. Ohtsuka, T. Sugano, Y. Koike, Am. Chem. Soc. Div. Org. Coat. Plast. Chem. Pap. 40, 382 (1979).

Appl. Opt. (9)

Appl. Phys. Lett. (3)

Y. Ohtsuka, Y. Hatanaka, Appl. Phys. Lett. 29, 735 (1976).
[CrossRef]

Y. Ohtsuka, I. Nakamoto, Appl. Phys. Lett. 29, 559 (1976).
[CrossRef]

Y. Ohtsuka, T. Senga, H. Yasuda, Appl. Phys. Lett. 25, 659 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

J. A. Arnaud, R. M. Derosier, Bell Syst. Tech. J. 55, 1489 (1976).

Electron. Lett. (1)

P. L. Chu, Electron. Lett. 13, 736 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Trans. IECE Jpn. (1)

Y. Kokubun, K. Iga, Trans. IECE Jpn. E60, 702 (1977).

Other (1)

Registered trade name of Carl Zeiss, Jena, East Germany.

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 (16)

Fig. 1
Fig. 1

Primary elements for the shearing method of Interphako interference microscopy: Lt, light source; Sl, slit; CL, condenser lens; St, stage; G, glass cell; S, sample; OL, object lens; PrI, beam splitting prism; RW, rotary wedge; PrII, beam combining prism; EP, eyepiece; Δyp, shearing distance.

Fig. 2
Fig. 2

Schematic representation of the interference pattern observed with Interphako interference microscopy.

Fig. 3
Fig. 3

Schematic representation of the ray trajectory traversing the LFR.

Fig. 4
Fig. 4

Fringe shapes in total splitting calculated by Eq. (11) when n0 = 1.495, np = 1.480, rp = 0.700 mm, and λ = 550 nm. mnpn2: A, 0.002; B, 0.0; C, −0.002; D, −0.004.

Fig. 5
Fig. 5

Fringe shapes in partial splitting calculated with the same conditions as that of total splitting in Fig. 4 and ΔYp = 0.02. m: A, 0.002; B, 0.0; C, −0.002.

Fig. 6
Fig. 6

Effect of mnpn2 on the index discrepancies for Analysis (I) resulting from the fringes in Fig. 4 (total splitting), m: A, 0.002; B, 0.0; C, −0.002; D, −0.004.

Fig. 7
Fig. 7

Effect of m on the index discrepancies for Analysis (I) resulting from the fringes in Fig. 5 (partial splitting). m: A, 0.002; B, 0.0; C, −0.002.

Fig. 8
Fig. 8

Index discrepancies for Analysis (II) calculated by using total splitting when DN = 100 and mnpn2 = 0: A, calculated with Eq. (12) without estimation of either (E1) or (E2A) in the Appendix; B, calculated with Eq. (13) instead of Eq. (12); C, calculated with estimation (E2A) of dR(yp)/dyp.

Fig. 9
Fig. 9

Index discrepancies for Analysis (II) calculated by using total splitting under DN = 20 and m = 0, where curves A, B, and C correspond to curves A, B, and C in Fig. 8, respectively.

Fig. 10
Fig. 10

Effect of m on index discrepancies for Analysis (II) resulting from the fringes in Fig. 4 (total splitting). m: A, 0.002; B, 0.0; C, −0.002.

Fig. 11
Fig. 11

Effects of shearing distance ΔYp and improved partial splitting procedure on index discrepancies for Analysis (II) where DN = 20 and m = 0: A, ΔYp = 0.10; B, ΔYp = 0.08; C, ΔYp = 0.01; A1 the data of one point located in the shearing zone added to that of ΔR where ΔYp = 0.10; A2, plus the data of two points located in the shearing zone where ΔYp = 0.10.

Fig. 12
Fig. 12

Index distribution resulting from the simulation for a specimen whose index is expressed by Eq. (8) up to r/rp = 0.7 and constant in the outer region where DN = 20 and ΔYp = 0.02: solid line, Analysis (I); ○, Analysis (II); dashed line, theoretical index distribution.

Fig. 13
Fig. 13

Effect of index mismatching of np with n2 on the index distribution for Analysis (II) in the experimental measurement of LFR(a) when ΔYp = 0.031. n2: ▵, 1.4848; ○,1.4818;▫, 1.4793.

Fig. 14
Fig. 14

Effect of index mismatching of np with n2 on the index distribution for Analysis (I) in the same condition as that of Fig. 13. n2: - · -, 1.4848; —, 1.4818; - - -, 1.4793.

Fig. 15
Fig. 15

Comparison of index distributions for Analyses (I) and (II) in experimental measurements of respective LFRs where npn2 and ΔYp is 0.031 for LFR(a) and 0.034 for LFR(b): solid line, Analysis (I); ○, Analysis II; straight line A, quadratic distribution of Eq. (9) evaluated from the reduction rate of the lens characteristics of the LFR.

Fig. 16
Fig. 16

Effect of the improved partial splitting procedure on the index distribution in the experimental measurement of LFR(a): ▵, adding the data of one point in the shearing zone to that of ΔR; ○, normal partial splitting procedure.

Equations (19)

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

Δ R ( y p ) = R ( y p + 1 2 Δ y p ) R ( y p 1 2 Δ y p ) ,
λ D R ( y · sec ψ ) = P 0 P 1 n ( r ) ds 2 n s ( r p 2 y 2 ) 1 / 2 n 2 y · tan ψ ,
P 0 P 1 n ( r ) ds = 2 n p [ r p 2 ( ν y ) 2 ] 1 / 2 2 n 2 y n p r p [ d ln n ( u ) du ] × u 2 · du [ u 2 ( n 2 y ) 2 ] 1 / 2 ,
ψ = ψ 0 + 2 [ sin 1 ( y / r p ) sin 1 ( ν y / r p ) ] ,
ψ 0 = 2 n 2 y n 2 y n p r p [ d ln n ( u ) du ] × du [ u 2 ( n 2 y ) 2 ] 1 / 2 ,
n ( r ) = n 0 ( 1 a r 2 + b r 4 + c r 6 + d r 8 ) ,
n ( u ) = n 2 · exp { 1 π u / n 2 r p [ λ D · dR ( y ) dy ] dy [ ( n 2 y ) 2 u 2 ] 1 / 2 } ,
n ( u ) = n 0 [ ( n 0 n p ) / ( r p n p ) 2 ] u 2 ,
n ( r ) = n 0 ( 1 1 2 A r 2 ) ,
d R / d y p [ dR d ( y · sec ψ ) ]
ψ = 2 ν Y [ Δ m ( ν Y ) 2 ] 1 / 2 ln ( { [ Δ m ( ν Y ) 2 ] 1 / 2 + [ 1 ( ν Y ) 2 ] 1 / 2 } 2 / ( Δ m 1 ) ) + 2 [ sin 1 ( Y ) sin 1 ( ν Y ) ] ,
R / D = r p λ ( n p { Δ m ν Y ψ 0 2 [ 1 ( ν Y ) 2 ] 1 / 2 } n 2 [ Y · tan ψ + 2 ( 1 Y 2 ) 1 / 2 ] ) ,
d ( R / D ) dY = r p n 2 λ tan ψ { 2 Y · tan ψ [ 1 Δ m ( ν Y ) 2 { ν 3 Y 2 + ν Δ m [ 1 ( ν Y ) 2 ] 1 / 2 Δ m ψ 0 2 Y } 1 ( 1 Y 2 ) 1 / 2 ] 1 } ,
T d ( R / D ) d Y p = r p n 2 λ · sin ψ ,
n ( r ) = n 0 ( 1 2.09 × 10 2 r 2 + 7.64 × 10 4 r 4 + 3.24 × 10 4 r 6 3.87 × 10 4 r 8 ) .
tan ψ = ψ + 1 3 ψ 3 + 2 15 ψ 5 + . . . , sec ψ = 1 + 1 2 ψ 2 + 5 24 ψ 4 + . . . .
dR ( y p ) d y p = Δ R ( y p ) Δ y p ,
A p = l · y p + m ,
u i / n 2 r p dR ( y ) dy · dy [ ( n 2 y ) 2 u 2 ] 1 / 2 = 1 n 2 j = 0 n i [ l ( y 2 y p , i 2 ) 1 / 2 + m ln [ y + ( y 2 y p , i 2 ) 1 / 2 ] ] y p , i + j y p , i + j + 1 ,

Metrics