Abstract

A temporal phase-unwrapping algorithm has been developed for the analysis of dynamic interference patterns generated with interference-contrast microscopy in micromachined picoliter vials. These vials are etched in silicon dioxide, have a typical depth of 6 µm, and are filled with a liquid sample. In this kind of microscopy, fringe patterns are observed at the air–liquid interface. These fringe patterns are caused by interference between the directly reflected part of an incident plane wave and the part of that wave that is reflected on the bottom of the vial. The optical path difference (OPD) between both parts is proportional to the distance to the reflecting bottom of the vial. Evaporation decreases the OPD at the meniscus level and causes alternating constructive and destructive interference of the incident light, resulting in an interferogram. Imaging of the space-varying OPD yields a fringe pattern in which the isophotes correspond to isoheight curves of the meniscus. When the bottom is flat, the interference pattern allows for monitoring of the meniscus as a function of time during evaporation. However, when there are objects on the bottom of the vial, the heights of these objects are observed as phase jumps in the fringes proportional to their heights. First, we present a classical electromagnetic description of interference-contrast microscopy. Second, a temporal phase-unwrapping algorithm is described that retrieves the meniscus profile from the interference pattern. Finally, this algorithm is applied to measure height differences of objects on the bottom in other micromachined vials with a precision of ∼5 nm.

© 2001 Optical Society of America

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References

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  1. K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).
  3. A. Lambacher, P. Fromherz, “Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer,” Appl. Phys. 63, 207–216 (1996).
    [CrossRef]
  4. S. Inoué, Video Microscopy (Plenum, New York, 1986).
    [CrossRef]
  5. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  6. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriquez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [CrossRef]
  7. J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
    [CrossRef]
  8. J. M. Huntley, H. Saldner, “Temporal phase unwrapping for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  9. A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Problem 3.66. in Signals and Systems (Prentice-Hall, New Jersey, 1997), p. 275.
  10. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
    [CrossRef]

2000 (1)

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

1999 (2)

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, R. Rodriquez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

1996 (1)

A. Lambacher, P. Fromherz, “Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer,” Appl. Phys. 63, 207–216 (1996).
[CrossRef]

1993 (1)

Bakajin, O.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Cuevas, F. J.

Deegan, R. D.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Dupont, T. F.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Fromherz, P.

A. Lambacher, P. Fromherz, “Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer,” Appl. Phys. 63, 207–216 (1996).
[CrossRef]

Greiner, A.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Hjelt, K. T.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Huber, G.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Huntley, J. M.

Inoué, S.

S. Inoué, Video Microscopy (Plenum, New York, 1986).
[CrossRef]

Jain, A. K.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Korvink, I. G.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Lambacher, A.

A. Lambacher, P. Fromherz, “Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer,” Appl. Phys. 63, 207–216 (1996).
[CrossRef]

Lubking, G. W.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Malacara, D.

Marroquin, J. L.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Nagel, S. R.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Problem 3.66. in Signals and Systems (Prentice-Hall, New Jersey, 1997), p. 275.

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Problem 3.66. in Signals and Systems (Prentice-Hall, New Jersey, 1997), p. 275.

Rodriquez-Vera, R.

Saldner, H.

Servin, M.

Strand, J.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Taxt, T.

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

van den Doel, L. R.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

van Vliet, L. J.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Vellekoop, M. J.

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Problem 3.66. in Signals and Systems (Prentice-Hall, New Jersey, 1997), p. 275.

Witten, T. A.

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

Appl. Opt. (2)

Appl. Phys. (1)

A. Lambacher, P. Fromherz, “Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer,” Appl. Phys. 63, 207–216 (1996).
[CrossRef]

IEEE Trans. Image Process. (1)

J. Strand, T. Taxt, A. K. Jain, “Two-dimensional phase unwrapping using a block least-squares method,” IEEE Trans. Image Process. 8, 375–386 (1999).
[CrossRef]

Phys. Rev. E Part B (1)

R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, T. A. Witten, “Contact line deposits in an evaporating drop,” Phys. Rev. E Part B 62, 756–765 (2000).
[CrossRef]

Other (5)

A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Problem 3.66. in Signals and Systems (Prentice-Hall, New Jersey, 1997), p. 275.

S. Inoué, Video Microscopy (Plenum, New York, 1986).
[CrossRef]

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

K. T. Hjelt, G. W. Lubking, M. J. Vellekoop, L. J. van Vliet, L. R. van den Doel, A. Greiner, I. G. Korvink, “Nanoliter droplet behavior in micromachined wells,” in Sensors Update, 1st ed., H. Baltes, W. Gopel, J. Hesse, eds. (Wiley-VCH, Weinheim, Germany, 2000), Vol. 8, pp. 39–72.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980).

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Figures (13)

Fig. 1
Fig. 1

Top, dynamic interference patterns as recorded in a 6.0-µm-deep vial. Bottom, one-dimensional height profiles of the meniscus along a diagonal from the corner to the center of the vial.

Fig. 2
Fig. 2

Time evolution of the fringes along a line through the center of the vial. In the center of the vial most modulations occur, whereas along the sidewalls of the vial no modulations occur.

Fig. 3
Fig. 3

Incident plane wave refracted at the air–liquid interface of the liquid sample in the vial. After refraction the direct part of this plane wave interferes with the part of this plane wave that is reflected on the bottom of the vial. Whether this interference is constructive or destructive depends on the height d with respect to the bottom of the vial.

Fig. 4
Fig. 4

Intensity of the electric field as a function of the height d above the reflecting bottom of the vial. One curve is based on evaluation of Eq. (11), whereas the other curve is experimentally acquired. The moment t = 0 s corresponds to the moment in the recording of the interference pattern where the meniscus is perfectly flat.

Fig. 5
Fig. 5

Temporal phase-unwrapping algorithm: (a) background-corrected signal I′[t] in the center point of the vial in the frame series 750–1500, (b) window function W[ t] and the product I sub [t] W[ t], (c) estimated phases, (d) the height profile, which follows by unwrapping of the estimated wrapped phases.

Fig. 6
Fig. 6

The frame, where the meniscus is perfectly flat, is determined by a minimum variation in intensity. This frame is used to trigger all relative height profiles when the number of counted 2π discontinuities is set to zero in this frame.

Fig. 7
Fig. 7

Bias of the phase estimation for a cosine function with a varying number of periods α in K w . The cosine function has unit amplitude and no noise. The bias is less than 0.01 rad. In the region below α = 1.58 the bias is at least 1 order of magnitude larger.

Fig. 8
Fig. 8

RMS error of the phase estimation as a function of the amplitude of the interferogram with a constant noise level. Even when the amplitude equals the standard deviation of the noise, the phase can still be estimated with a RMS error of 0.16. If the SNR is below 1, the RMS error increases rapidly.

Fig. 9
Fig. 9

Left graph shows a fraction of Fig. 11 (below), indicating the contact angle. The left image shows a one-dimensional fringe pattern generated at the meniscus interface as a function of the contact angle of the meniscus as indicated in the left graph: From the bottom upward the contact angle increases, which results in a steeper (straight) meniscus and therefore a spatially denser fringe pattern. We assume in this example that the fringe pattern is generated with monochromatic light such that the amplitude of the fringe pattern is constant for different heights. The right image shows the effects of imaging the left image onto a CCD camera. First, the left image is convolved with a uniform filter, the pixel transfer function. As a result of this convolution, the amplitude decreases with increasing frequency, and, at a certain frequency, the amplitude of the fringe pattern becomes negative. This phase jump of π radians is visualized in region 3. Second, the filtered fringe pattern is sampled with a one-dimensional impulse train, indicated by the dots. In region 1 the spatial sampling density SD is larger than the Nyquist frequency f N , defined by the contact angle: Spatial as well as temporal unwrapping is possible in this region. In region 2 the spatial sampling density satisfies f N /2 ≤SD ≤ f N : Aliasing occurs spatially; only temporal unwrapping is possible. In region 3 the spatial sampling density is below f N /2: No unwrapping is possible, owing to irrecoverable phase jumps.

Fig. 10
Fig. 10

Computed height profiles corresponding to the interferograms shown in Fig. 1 at t = 97 s (top) and t = 164 s (bottom). Because of symmetry only a quarter of the well is shown. Compare the isoheight curves in these figures with the isophote curves in the interferograms. For clarity, only half the number of isoheight curves is shown in the bottom figure.

Fig. 11
Fig. 11

Time evolution of a single line of the meniscus through the center of the vial. The time difference between two successive lines is 10 s. The angle αmax is the maximum angle between meniscus and bottom for spatial analysis. The glitches in the meniscus are introduced by spurious phase jumps in the unwrapping process.

Fig. 12
Fig. 12

Upper graph, remaining liquid volume as a function of time for square wells of various size. The depth of the wells is 6.1 µm. Bottom graph, evaporation rate as a function of the width of the wells.

Fig. 13
Fig. 13

Upper image, electrodes patterned on the bottom on both sides of the vial. Bottom figure, measured height difference between the bottom of the vial and the electrodes.

Tables (1)

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Table 1 Measurement Parametersa

Equations (23)

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OPD = 2 n liq   d   cos Θ in ,
Φ in = 4 π n liq   d   cos Θ in λ ,
E i = E cos γ 0 cos Θ 0 sin γ 0 cos γ 0 sin Θ 0 ,
E r = E cos γ 0 cos Θ 0 r 01 sin γ 0 r 01 cos γ 0 sin Θ 0 r 01 ,
E trt = E   exp i Φ in cos γ 0 cos Θ 0 t 01 r 12 t 10 sin γ 0 t 01 r 12 t 10 cos γ 0 sin Θ 0 t 01 r 12 t 10 ,
E = E cos γ 0 cos Θ 0 r 01 + t 01 r 12 t 10   exp i Φ in sin γ 0 r 01 + t 01 r 12 t 10   exp i Φ in cos γ 0 sin Θ 0 r 01 + t 01 r 12 t 10   exp i Φ in .
| E | 2 γ 0 = π r 01 2 + r 01 2 + t 01 2 r 12 2 t 10 2 + t 01 2 r 12 2 t 10 2 + 2 π r 01 t 01 r 12 t 10 + r 01 t 01 r 12 t 10 cos Φ in .
S λ = 1 + λ - λ c 2 2 σ λ 2 exp - λ - λ c 2 2 σ λ 2 ,
tan Θ 0 , max = NA n air 2 - NA 2 .
Θ in , max = arcsin n air n liq sin arctan 1 5 NA n air 2 - NA 2 .
| E d | 2 = Θ in = 0 Θ in , max λ = 0 tan Θ in S λ | E | 2 γ 0 d Θ in d λ ,
Φ d = 4 π n liq   d   cos Θ in eff λ eff ,
Δ d = λ eff 2 n liq   cos Θ in eff .
A d = Θ in = 0 Θ in , max F Θ in eff tan Θ in eff × σ k 4 n liq 2 d 2 σ k 2   cos 2 Θ in eff - 3 × exp - 2 3   n liq 2 d 2 σ k 2   cos 2 Θ in eff d Θ in ,
I m ,   n ;   t = A m ,   n ;   t cos Φ m ,   n ;   t + B m ,   n ;   t + N m ,   n ;   t ,
W t = 1 + 1 2 t - K w / 2 σ t 2 + 1 8 t - K w / 2 σ t 4 × exp - t - K w / 2 2 2 σ t 2 ,
f sub = 2 K w 2 N I t # ZC ,
StDev I i m ,   n - I m ,   n t = 1010 ,     , 1070 ,
σ N = StDev I t - I t   *   G t ;   σ = 2.0 ,
K w = 2 τ Δ d R sampling ,
v evap max 2 K w   Δ dR sampling .
SD 2   tan α Δ d .
Δ Φ = Δ Φ + π + k 2 π .

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