Abstract

An axial resolution created by a spatial incoherent source is investigated theoretically and experimentally for the Linnik-type interferometer. The axial resolution in interference microscopy depends on both the temporal coherence length of the source and the objective numerical aperture (NA). Here the problem is treated in a more general situation by considering the spatial and temporal coherence of the illumination source which may be important for deep coherence imaging application. The results show that the axial resolution is degraded at the depth much less by using the optimal spectral bandwidth of the incoherent source and high-NA objectives.

© 2008 Optical Society of America

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References

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2006 (2)

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376-380 (2006).
[CrossRef]

W. Y. Oh, B. E. Bouma, N. Iftimia, S. H. Yun, R. Yelin, and G. J. Tearney, “Ultrahigh-resolution full-field optical coherence microscopy using InGaAs camera,” Opt. Express 14, 726-735(2006).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

2000 (1)

1999 (2)

A. Dubois, A. C. Boccara, and M. Lebec, “Real-time reflectivity and topography imagery of depth-resolved microscopic surfaces,” Opt. Lett. 24, 309-311 (1999).
[CrossRef]

C. K. Hitzenberger, A. Baumgartner, W. Drexler, and A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J Biomed. Opt. 4, 144-151 (1999).
[CrossRef]

1998 (3)

1994 (1)

1990 (1)

Appl. Opt. (5)

J Biomed. Opt. (1)

C. K. Hitzenberger, A. Baumgartner, W. Drexler, and A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J Biomed. Opt. 4, 144-151 (1999).
[CrossRef]

Opt. Commun. (1)

Y. Watanabe, K. Yamada, and M. Sato, “In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera,” Opt. Commun. 261, 376-380 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Other (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic diagram of the Linnik-type interferometer illuminated by a spatial incoherent source S ( S 1 , S 2 ), virtual sources; MO, micro-objective; BS, cube beam-splitter; M, mirror; EA, entrance aperture; (b) diagram used for the calculation of the degree of longitudinal coherence (in points P 1 and P 2 ) produced by the virtual source S 1 , 2 .

Fig. 2
Fig. 2

Simulation of axial resolution degradation as a function of a source spectral bandwidth to dispersion mismatch in the two interferometer arms. A sample consisting of water is considered and a reference mirror imaged in air. Curves from one to six correspond to the different depths of z: 0, 0.25, 0.5, 1, 1.5, and 2 ( mm ) .

Fig. 3
Fig. 3

Numerical simulations of axial point spread functions are shown: | γ 12 ( τ ) | by a solid curve; | γ ( τ ) | by a dotted curve, and | μ 12 | by a dashed curve as functions of the distance Δ z . Curves shown in (a)–(d) are calculated at different source spectral bandwidths: 155, 110, 77, and 55 nm , correspondingly ( NA = 0.85 ). A sample consisting of water is considered and a reference mirror imaged in air.

Fig. 4
Fig. 4

Experimental setup to measure the degree of longitudinal coherence: D, rotating scattering plate; MO 1 , 2 , 3 , micro-objectives; BS, cube beam-splitter; M, mirror; L, lens; PZT, piezoelectric transducer; O, oscilloscope; FG, frequency generator.

Fig. 5
Fig. 5

Axial point spread function of the Linnik-type interferometer using spatial incoherent source with uniform intensity distribution: (a) experimental and (b) calculated using relation (3).

Fig. 6
Fig. 6

Axial point spread function of the Linnik-type interferometer using a spatial incoherent source with Gaussian intensity distribution: (a) experimental and (b) calculated using relation (8). In insert, the beam intensity profile is shown: experimental by a solid curve and Gaussian fit by a dash curve.

Tables (1)

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Table 1 Results of Numerical Simulations

Equations (16)

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V = V 0 | γ 12 ( τ ) | ,
μ 12 ( 2 Δ z ) = I ( x S , y S ) exp [ j π 2 Δ z λ 0 f 2 ( x S 2 + y S 2 ) ] d x S d y S I ( x S , y S ) d x S d y S ,
I ( ρ ) = { I 0 , ρ r 0 0 , ρ r 0 ,
μ ( 2 Δ z ) = | 0 2 π α d α 0 r 0 I 0 exp [ j π 2 Δ z λ 0 f 2 ρ 2 ] ρ d ρ 0 2 π α d α 0 r 0 I 0 ρ d ρ | = | sin ( π Δ z r 0 2 λ 0 f 2 ) π Δ z r 0 2 λ 0 f 2 | .
l coh = 2 Δ z FWHM 3.8 λ 0 f 2 π r 0 2 = 3.8 λ 0 π ( NA ) EF 2 15 λ 0 π ( NA ) 2 ,
I ( ρ ) = I 0 exp ( ρ 2 / w 0 2 ) ,
| μ ( 2 Δ z ) | = | 0 2 π α d α 0 r 0 I 0 exp ( ρ 2 w 0 2 ) exp [ j π 2 Δ z λ 0 f 2 ρ 2 ] ρ d ρ 0 2 π α d α 0 r 0 I 0 ρ d ρ | = | 0 r I 0 exp [ ( 1 w 0 2 + j π 2 Δ z λ 0 f 2 ) ρ 2 ] d ρ 2 I 0 w 0 2 [ 1 exp ( r 2 w 0 2 ) ] | = | ( 1 exp [ ( 1 w 0 2 + j π 2 Δ z λ 0 f 2 ) r 2 ] ) ( j π 2 Δ z w 0 2 λ 0 f 2 1 ) [ 1 exp ( r 2 w 0 2 ) ] | = { 1 4 sin 2 ( π 2 Δ z r 2 λ 0 f 2 ) / [ exp ( r 2 w 0 2 ) 1 ] } 0.5 [ 1 + ( π 2 Δ z w 0 2 λ 0 f 2 ) 2 ] 0.5 ,
| μ ( 2 Δ z ) | = 1 [ 1 + ( π 2 Δ z w 0 2 λ f 2 ) 2 ] 0.5 .
| μ ( 2 Δ z ) | = [ 1 2.3 sin 2 ( π 2 Δ z w 0 2 λ f 2 ) ] 0.5 [ 1 + ( π 2 Δ z w 0 2 λ f 2 ) 2 ] 0.5 .
l coh = 2 Δ z FWHM 1.7 λ 0 f 2 π w 0 2 .
| γ ( τ ) | = exp [ α ( Δ z ) 2 ] ,
Δ z T = 2 ln 2 n π ( λ 0 2 Δ λ ) .
Δ z eff = [ Δ z T 2 + ( 2 z Δ λ d n d λ ) 2 ] 0.5 .
( Δ λ ) opt = [ ln 2 λ 0 2 n π z ( d n / d λ ) ] 0.5 .
( Δ z eff ) min = ( 8 ln 2 z λ 0 2 n π d n d λ ) 0.5 .
V = V 0 | γ 12 ( τ ) | = V 0 exp [ α ( n Δ z ) 2 ] | sin ( π n Δ z r 0 2 λ 0 f 2 ) π n Δ z r 0 2 λ 0 f 2 | .

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