Abstract

We present compensating methods that address inherent errors in quantitative phase reporting for low-coherence interferometric techniques. A brief theoretical treatment of the problem and experimental validation using spectral domain phase microscopy demonstrate mitigation of the degrading effects of phase leakage on accurate measurement of optical path length in the vicinity of closely spaced reflectors. This result has direct implications for phase-sensitive interferometry techniques, such as Doppler imaging, as well as amplitude-based quantitative reporting. Corrected phase retrieval is demonstrated for conversion of interferometric phase to optical path length in cell surface deflections of beating cardiomyocytes.

© 2007 Optical Society of America

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References

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2006

2005

2004

1982

Akkin, T.

Badizadegan, K.

Cense, B.

Choma, M. A.

Chu, M. C.

Cobb, M. J.

Creazzo, T. L.

Dasari, R. R.

de Boer, J. F.

Deflores, L. P.

Ellerbee, A. K.

Fang-Yen, C.

Feld, M. S.

Ina, H.

Iwai, H.

Izatt, J. A.

Joo, C.

Li, X.

MacDonald, D. J.

Park, B. H.

Popescu, G.

Ren, H.

Sarunic, M. V.

Seiji, K.

Seung, H. S.

Sun, T.

Takeda, M.

Vaughan, J. C.

Weinberg, S.

Yang, C.

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Figures (4)

Fig. 1
Fig. 1

SDPM setup used to demonstrate phase leakage. SLD, superluminescent diode; Spec, spectrometer; CS, coverslip, whose back surface served as the reference reflector; M, mirror mounted on a PZT actuator.

Fig. 2
Fig. 2

Phase leakage affects phase, as predicted by Eq. (2). (a) The measured phase versus system coherence length for two reflectors with Gaussian point spead functions is plotted as points fitted to theoretical lines for three values of k 0 ( β = 0.4 , y = 0.15 μ m , n = 1.5 ). Deviations can be explained by the power conservation shown in (b), which causes the β value of points at extreme ends to trail off for all coherence lengths (Lc) plotted. Inset: β versus coherence length for k 0 = 1195 mm 1 . The glass index is assumed to be 1.5.

Fig. 3
Fig. 3

Low-coherence phase leakage demonstrated and corrected. Sinusoidal modulation applied to a mirror positioned at various distances from the coverslip surface (C). (a) Overlapped representative A scans, with * denoting mirror position; (b) associated phase profiles. Note the diminished phase modulation amplitude and distorted modulation frequency for the mirror position nearest C ( 248 μ m ) , which is corrupted. (c) Corrective methods indicated in parentheses restore the corrupted phase to its proper magnitude (compare with black solid curve, the magnitude of an uncorrupted profile) and modulation frequency and vary slightly in their correction efficiency. The efficiency for methods 1 and 2 can be improved with signal strength matching through multiplicative constants. Artifacts of method 3 (identified with *) are attributed to poor SNRs for these deviation values, not phase unwrapping errors.

Fig. 4
Fig. 4

Method 3 applied to a biological sample, showing correction of phase-corrupted data. (a) Representative A scans showing a coverslip and a cardiomyocyte cell surface. The latter is the dominant reflection after correction. (b) Phase trace from the surface of the spontaneously beating cardiomyocyte, located 11.16 μ m from the coverslip reflection. Corrected data recover beating amplitudes.

Equations (5)

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i ( k ) m = 1 M ρ e s ( k ) δ k Δ t R r R m cos ( 2 n m k x m ) .
θ ( x ) = tan 1 ( m = 1 M P m ( x ) sin [ 2 n m k 0 ( x x m ) ] m = 1 M P m ( x ) cos [ 2 n m k 0 ( x x m ) ] ) .
θ ( x m + δ x ) = tan 1 { β exp ( δ x 2 σ 2 ) sin ( 2 k 0 δ x ) + exp [ ( y + δ x ) 2 σ 2 ] sin [ 2 k 0 ( y + δ x ) ] β exp ( δ x 2 σ 2 ) cos ( 2 k 0 δ x ) + exp [ ( y + δ x ) 2 σ 2 ] cos [ 2 k 0 ( y + δ x ) ] } ,
θ ( x m + δ x ) x = 2 k 0 2 y β sin ( 2 k 0 y ) σ 2 { exp [ ( y 2 + 2 y δ x ) σ 2 ] + β 2 exp [ ( y 2 + 2 y δ x ) σ 2 ] + 2 β cos ( 2 k 0 y ) } ,
θ actual ( x ) = { P m ( x ) exp [ j n m k 0 ( x x m ( t 1 ) ) ] P m ( x ) exp [ j n m k 0 ( x x m ( t 0 ) ) ] } = 2 j P m ( x ) sin ( n m k 0 Δ x 2 ) exp [ j n m k 0 ( x x m ( t 0 ) ) ] exp ( j n m k 0 Δ x 2 ) .

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