Abstract

Images of periodic phase objects are calculated for oblique illumination, the scanning method of Dekkers and de Lang, and differential interference contrast. The close relation between the scanning method and oblique illumination is shown. All the methods are capable of resolving grating periods larger than λ/2a, where a is the numerical aperture of both the objective and the condenser lenses. Differential interference gives higher contrast for object periods below λ/a; contrast is slightly higher above λ/a with oblique illumination.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
    [CrossRef]
  2. M. Francon, J. Opt. Soc. Am. 47, 528 (1957).
    [CrossRef]
  3. R. Hoffman and L. Gross, Appl. Opt. 14, 1169 (1975).
    [CrossRef] [PubMed]
  4. N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).
  5. W. T. Welford, J. Micros. 96, 105 (1972).
    [CrossRef]
  6. D. Kermisch, J. Opt. Soc. Am. 65, 887 (1975).
    [CrossRef]
  7. E. Zeitler and M. G. R. Thomson, Optik 3, 258 (1970).
  8. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 8.6.3.

1975 (2)

1974 (1)

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

1972 (1)

W. T. Welford, J. Micros. 96, 105 (1972).
[CrossRef]

1970 (1)

E. Zeitler and M. G. R. Thomson, Optik 3, 258 (1970).

1957 (1)

1953 (1)

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 8.6.3.

de Lang, H.

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

Dekkers, N. H.

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

Francon, M.

Gross, L.

Hoffman, R.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
[CrossRef]

Kermisch, D.

Thomson, M. G. R.

E. Zeitler and M. G. R. Thomson, Optik 3, 258 (1970).

Welford, W. T.

W. T. Welford, J. Micros. 96, 105 (1972).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 8.6.3.

Zeitler, E.

E. Zeitler and M. G. R. Thomson, Optik 3, 258 (1970).

Appl. Opt. (1)

J. Micros. (1)

W. T. Welford, J. Micros. 96, 105 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

Optik (2)

N. H. Dekkers and H. de Lang, Optik 41, 452 (1974).

E. Zeitler and M. G. R. Thomson, Optik 3, 258 (1970).

Proc. R. Soc. A (1)

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 8.6.3.

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 (9)

FIG. 1
FIG. 1

Schematic diagram of an optical scanning system using a coherent point source. The numerical apertures of the focusing and collector lenses are determined by the diameters of the aperture and the detector, respectively.

FIG. 2
FIG. 2

Schematic diagram of a conventional microscope using quasimonochromatic extended source illumination. The numerical apertures of the objective lens and the condenser are determined by the diameters of the aperture and the source, respectively.

FIG. 3
FIG. 3

Equivalent imaging systems. Reflective object consists of three parallel square-wave grating structures whose horizontal period is 5.7 μm. (a) and (b) are conventional micrographs with semicircular source aperture (a), and circular aperture (b). λ = 546 nm, a =0.95. (c) and (d) are detector waveforms by optically scanning one object grating. Semicircular detector aperture (c), circular detector aperture (d). λ = 442 nm, a =0.85. (a) and (c) give differential phase contrast by oblique illumination.

FIG. 4
FIG. 4

Schematic diagram of overlapping −1, 0, and +1 diffraction orders which occur in plane I of Fig. 1.

FIG. 5
FIG. 5

Calculated images of a square-wave phase grating (dashed) by oblique illumination. Normalized spatial frequency is μ = 0.1. The origin of the ordinate axis is for the Δϕ = π curve; the remaining curves are shifted successively upward for clarity.

FIG. 6
FIG. 6

Schematic diagram of ray paths in a differential interference contrast system used as a scanner. A polarizer and a modified Wollaston prism (not shown) are used in both plane III and plane I to first split and then recombine the wavefronts.

FIG. 7
FIG. 7

Calculated images of a square-wave phase grating (dashed) by differential interference contrast. Normalized spatial frequency is μ =0.1. The origin of the ordinate axis is for the Δϕ = π curve; the remaining curves are shifted successively upward for clarity.

FIG. 8
FIG. 8

Spatial frequency dependence of contrast with a weak sinusoidal phase object of peak phase excursion ϕ0. Oblique illumination (solid), differential interference contrast (dashed).

FIG. 9
FIG. 9

Uniform image level in differential phase contrast vs normalized slope of a constant phase gradient. (a) Oblique illumination, (b) split detector, (c) differential interference contrast.

Tables (2)

Tables Icon

TABLE I Differential phase-contrast methods.

Tables Icon

TABLE II Image level for a uniform phase wedge.

Equations (19)

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

I ( x , y ) α S U ( u , v ; x , y ) 2 d u d v .
I ( x , y ) = S U ( u , v ; x , y ) 2 d u d v .
t ( x 0 ) = n = - + b n exp ( i 2 π n x 0 T ) ,
b n = T - 1 0 T t ( x 0 ) exp ( - i 2 π n x 0 T ) d x 0 .
g ( u 0 , v 0 ) = circ [ ( u 0 2 + v 0 2 ) / h 2 ] exp [ i 2 π ( u 0 x + v 0 y ) / λ f ] ,
h = a f .
U ( u , v ; x , y ) = n = - + b n circ ( ( u - n λ h / a T ) 2 + v 2 h 2 ) exp ( i 2 π n x T ) .
O ( n μ ) = 2 π [ cos - 1 ( n μ ) - n μ ( 1 - n 2 μ 2 ) 1 / 2 ]             0 n μ 1 = 0 ,             n μ > 1 ,
μ = λ / 2 a T
I ± ( x ) = m = 1 N ( j = 1 m 1 2 ( 1 ± sgn [ j - 1 2 ( m + 1 ) ] ) × | k = 1 m b k - j exp [ i 2 π x ( k - j ) ] | 2 ) × { O [ ( m - 1 ) μ ] - 2 O ( m μ ) + O [ ( m + 1 ) μ ] } ,
b 0 = cos ϕ 0 , b n = [ 2 i sin ϕ 0 sin ( 1 2 π n ) ] / π n ,             n 0.
c n = 1 2 b n { exp ( i 2 π n δ / T ) + exp [ i ( β + γ - 2 π n δ / T ) ] } ,
t ( x 0 ) = exp [ i ϕ 0 cos ( 2 π x 0 / T ) ] ,
t ( x 0 ) = 1 + i ϕ 0 cos ( 2 π x 0 / T ) - ( 1 2 ϕ 0 2 ) cos 2 ( 2 π x 0 / T ) + .
I ± ( x ) = 1 2 ϕ 0 [ O ( μ ) - O ( 2 μ ) ] sin ( 2 π x / T ) - 1 4 ϕ 0 2 [ 1 - O ( μ ) ] .
C = 2 ϕ 0 [ O ( μ ) - O ( 2 μ ) ] .
I + ( x ) + I - ( x ) = cos 2 [ 1 2 ( β + γ ) ] + ϕ 0 sin ( β + γ ) sin ( 4 π a δ μ / λ ) O ( μ ) sin ( 2 π x / T ) - 1 4 ϕ 0 2 × { cos 2 [ 1 2 ( β + γ ) ] - [ 1 + cos ( 8 π a δ μ / λ ) cos ( β + γ ) ] O ( μ ) - 2 cos ( β + γ ) sin 2 ( 8 π a δ μ / λ ) O ( 2 μ ) cos ( 4 π x / T ) } ,
C = 2 ϕ 0 sin ( π μ ) O ( μ ) .
t ( x 0 ) = exp ( ± i 2 π x 0 / T ) ,