## Measuring the thickness of soap bubbles with phase-shift interferometry |

Optics Express, Vol. 21, Issue 17, pp. 19657-19667 (2013)

http://dx.doi.org/10.1364/OE.21.019657

Acrobat PDF (1114 KB)

### Abstract

A model for the optical path difference introduced by a soap bubble in transmission is described. The model is then used with interferometric data to solve for the fringe order, and to define a procedure to extract the global film thickness in presence of turbulence flows occurring during the drainage process due to gravity. Experimental results on soap bubbles examined in single-pass phase-shift interferometry are presented.

© 2013 OSA

## 1. Introduction

5. O. Bélorgey and J. J. Benattar, “Structural properties of soap black films investigated by X-ray reflectivity,” Phys. Rev. Lett. **66**(3), 313–316 (1991). [CrossRef] [PubMed]

8. S. Lionti-Addad and J. M. Di Meglio, “Stabilization of aqueous foam by hydrosoluble polymers. 1. Sodium dodecyl sulphate-poly(ethylene oxide) system,” Langmuir **8**(1), 324–327 (1992). [CrossRef]

9. A. S. Dimitrov, P. A. Kralchevsky, A. D. Nikolov, and D. T. Wasan, “Contact angles of thin liquid films: interferometric determination,” Colloids Surf. **47**, 299–321 (1990). [CrossRef]

10. A. D. Nikolov, P. A. Kralchevsky, and I. B. Ivanov, “Film and line tension effects on the attachment of particles to an interface,” J. Colloid Interface Sci. **112**(1), 122–131 (1986). [CrossRef]

11. G. Pretzler, H. Jäger, and T. Neger, “High-accuracy differential interferometry for the investigation of phase objects,” Meas. Sci. Technol. **4**(6), 649–658 (1993). [CrossRef]

13. A. González-Cano and E. Bernabáu, “Automatic interference method for measuring transparent film thickness,” Appl. Opt. **32**(13), 2292–2294 (1993). [CrossRef] [PubMed]

14. V. Greco, C. Lemmi, S. Ledesma, G. Molesini, G. P. Puccioni, and F. Quercioli, “Measuring soap black films by phase shifting interferometry,” Meas. Sci. Technol. **5**(8), 900–903 (1994). [CrossRef]

15. V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. **66**(7), 3729–3734 (1995). [CrossRef]

16. V. Greco and G. Molesini, “Monitoring the thickness of soap films by polarization homodyne interferometry,” Meas. Sci. Technol. **7**(1), 96–101 (1996). [CrossRef]

17. O. Greffier, Y. Amarouchene, and H. Kellay, “Thickness fluctuations in turbulent soap films,” Phys. Rev. Lett. **88**(19), 194101 (2002). [CrossRef] [PubMed]

20. A. Kariyasaki, Y. Yamasaki, M. Kagawa, T. Nagashima, A. Ousaka, and S. Morooka, “Measurement of liquid film thickness by a fringe method,” Heat Transf. Eng. **30**(1-2), 28–36 (2009). [CrossRef]

21. G. Ropars, D. Chauvat, A. Le Floch, M. N. O’Sullivan-Hale, and R. W. Boyd, “Dynamics of gravity-induced gradients in soap films,” Appl. Phys. Lett. **88**(23), 234104 (2006). [CrossRef]

22. S. N. Tan, Y. Yang, and R. G. Horn, “Thinning of a vertical free-draining aqueous film incorporating colloidal particles,” Langmuir **26**(1), 63–73 (2010). [CrossRef] [PubMed]

23. L. Liggieri, F. Ravera, and A. Passerone, “Dynamic interfacial tension measurements by a capillary pressure method,” J. Colloid Interface Sci. **169**(1), 226–237 (1995). [CrossRef]

24. W. Lv, H. Zhou, C. Lou, and J. Zhu, “Spatial and temporal film thickness measurement of a soap bubble based on large lateral shearing displacement interferometry,” Appl. Opt. **51**(36), 8863–8872 (2012). [CrossRef] [PubMed]

25. F. Seychelles, Y. Amarouchene, M. Bessafi, and H. Kellay, “Thermal convection and emergence of isolated vortices in soap bubbles,” Phys. Rev. Lett. **100**(14), 144501 (2008). [CrossRef] [PubMed]

*OPD*) introduced by a soap bubble, and we describe a procedure of data processing that accordingly provides a value representing the film thickness of the bubble on the whole. The method we use also avoids a major inconvenience of standard interferometry approaches when dealing with interference patterns made of two unconnected regions, as it occurs with soap bubbles on a reference background. In fact in that case the integer number of wave lengths making up the gap between the bubble and the background is unknown, so that the customary determination of the

*OPD*would be affected by intrinsic ambiguity.

## 2. Geometrical *OPD* model of an ideal soap bubble

*OPD*map that is expected when such a bubble is passed by a light beam. Such an expression will be used in Section 4 to compute the bubble thickness from

*OPD*data.

*R*the radius of curvature of the outer sphere of the bubble,

*d*the film thickness and

*n*the refractive index of water. Locally the film can be assimilated to a plane parallel plate made of a water layer. In normal section, the general case of a ray impinging on such plate at oblique incidence is illustrated in Fig. 1(a). The quantity of interest is the

*OPD*defined as the difference between the optical path of a ray when the bubble is not there and when it is in place. Indicating with

*OPD*is expressed bywhere

*d*is the film thickness.

*(z,y)*is schematically shown in Fig. 1(b). The height

*y*is normalized by the radius R yielding the normalized ray height

*q*also represents the normalized radius of the

*OPD*map. The incidence angle is

*OPD*through the entrance and exit films of the bubble is thereforewithin other words, the radial

*OPD*profile is a standard function

*d*of the bubble film. Using for example the value

*d*= 100 nm,

*q*ranging from 0.0 to 0.8, and

*n*= 1.333, the resulting plot for

*OPD*(

*q*) is shown in Fig. 2. As a general notation, it is seen that the

*OPD*is always negative, meaning that the insertion of a bubble in the measuring arm of an interferometer produces an optical delay.

*OPD*so far discussed does not take into consideration grazing incidence phenomena and guided light propagation occurring when

*q*approaches unity; with soap bubbles in air, we conservatively limit our study to

## 3. Pressure and temperature effects of the inner air on the *OPD* induced by a bubble

*OPD*is to be taken into account. Edlén’s formula [26

26. B. Edlén, “The refractive index of air,” Metrologia **2**(2), 71–80 (1966). [CrossRef]

^{−2}Nm

^{−1},

^{5}Nm

^{−2},

^{−4}and

^{−3}m, it is found that the contribution due to pressure increase is

^{−7}.

^{−7}, and overallBeing it negative, the net effect is a decrease of the

*OPD*already computed with the geometric model of the bubble. The absolute value of the

*OPD*pertaining to a sphere of inner air has a maximum along a diameter; for the case in point we have

*OPD*at

*q*= 0 that is measured in experiments with the thinnest soap bubbles, and can be safely neglected for bubbles not reaching the condition of black film. It is also considered that reaching the condition of black film generally requires several minutes. During this time, thermal exchanges through the soap film between the inner and outer air are occurring, tending to reduce and eventually annul the temperature difference. As a consequence,

## 4. Thickness computation

*OPD*(

*q*= 0), expressed by

*d*at the bubble centre. Expected values for

*d*range from thousands or several hundreds of nm when the bubble is first formed, to a few tens of nm at the end of the thinning process [16

16. V. Greco and G. Molesini, “Monitoring the thickness of soap films by polarization homodyne interferometry,” Meas. Sci. Technol. **7**(1), 96–101 (1996). [CrossRef]

*OPD*regions are obtained, one of which corresponds to the bubble and the other to the surrounding area; the latter defines the reference optical path for the determination of the

*OPD*(

*q*= 0). The intrinsic inconvenience is that such two regions are not connected to each other, so that situations of ambiguity due to integer multiples of the wavelength λ cannot be excluded. For example, only considering a measured

*OPD*of −66 nm at the center of the interference pattern as in Fig. 2, it is not possible to distinguish whether the true

*OPD*is −66 nm, −66 nm ± λ, −66 nm ± 2λ or more, since all of them are within the expected

*OPD*range. In practice, limiting the analysis to the bubble center with respect to the outside region, only the fractional part of the

*OPD*(in λ units) can be reliably retrieved. In addition, only referring to the

*OPD*at the bubble center, such a fractional part would also be severely affected by the disturbance due to turbulence.

*f*scales times the geometrical thickness

*d*of the bubble film to yield the actual

*OPD*. As a consequence, fitting the

*OPD*map to the mathematical representation of

*f*would directly produce a coefficient equal to

*d*. However, as given in Eq. (3), such mathematical representation is somewhat involved for use with analysis programs. By numerical fitting with least squares techniques it is easily seen that a convenient polynomial expression for

*f*(

*q*,

*n*) isFor the case in point, again with

*n*= 1.333, limiting the range of

*q*to

*N*= 5 (so selecting even powers of the normalized radius

*q*up to

*q*

^{10}), best fit is obtained with the parameters listed in Table 1.

^{−5}. In case it is required, one might also express Eq. (3) in terms of radial Zernike polynomials, either repeating the fitting procedure or using appropriate conversion formulas [28].

*a*

_{1}, …

*a*

_{5}accounting for the standard function as in Table 1. The set of fitting functions we use is thenwith azimuthal terms up to the fifth angular order. As noted, the function

*c*

_{i}, one for each fitting function of the set in Eq. (11); addressing in particular

*c*

_{4}, Eq. (2) directly providesThe pertaining uncertainty is given by the fourth diagonal element of the autocovariance matrix associated to the fitting. Such an uncertainty value shall then be combined with the uncertainty contributions from the data acquisition process (mainly, the uncertainty on the bubble diameter). As to the least squares fitting itself, we use standard routines available from general mathematics packages, just specifying the set of functions

*OPD*, and by the maximum value of the normalized radius

*q*that is considered, the compliance with the source thickness data is within the uncertainty given by the amount of Gaussian noise that was added. For example, simulating the

*OPD*induced by a soap bubble with a film thickness of 100 nm and a Gaussian noise of 1%, the recovered thickness after processing is 100 ± 1 nm. The uncertainty in this case is very small because the data were generated by computer, and no uncertainty contribution from the bubble diameter was considered. Dealing with real data, where the diameter of the bubble is also to be determined, more contributions to the uncertainty balance as mentioned above shall be added.

## 5. Experimental results

*OPD*map shown in Fig. 6. As anticipated, however, ambiguity effects due to the missing connection are showing up. In fact the bubble induces an optical delay, so that the central hemisphere should appear shifted downwards, below the level of the background. The interferometric data are therefore re-processed by singling out a central portion of normalized radius

*q*= 0.8, and submitting it to the fitting with the set of functions given in Eq. (11). The resulting coefficient of the fourth function is then taken as the bubble thickness, according to Eq. (12). In our case such a thickness is 313 ± 14 nm; the stated uncertainty is estimated numerically, repeating the analysis with a bubble diameter at the extremes of its error range (147 and 149 pixels), and quadratically combining the half-difference of the resulting diameters with the uncertainty from the autocovariance matrix of the fitting mentioned above.

*OPD*map given in Fig. 6. The

*OPD*profile selected with a line through the central part of such a map is shown in Fig. 7. Using inspectors provided by the software of the interferometer, it is seen that the bubble occupies a lateral distance of 144 pixels. Taking such a distance as the bubble diameter, a reduction by a factor 0.8 leads to 115 pixels. Still working with inspectors, it is seen that the peak-to-valley of the central part of the profile for a width of 115 pixels is 88 nm. In Fig. 2 it is shown that the

*OPD*difference between

*q*= 0.0 and

*q*= 0.8 for

*d*= 100 nm is approximately 27 nm. The thickness estimate of our soap bubble is then (88/27)·100 nm = 326 nm. Although it is obtained using a single profile of the bubble instead of the entire bubble cap within

*q*= 0.8, and also with some rough approximations, such a value is compatible with the result of 313 ± 14 nm computed with the general procedure described in Section 4.

## 6. Concluding remarks

## Acknowledgments

## References and links

1. | C. V. Boys, |

2. | K. J. Mysels, K. Shinoda, and S. Frankel, |

3. | J. N. Israelashvili, |

4. | I. B. Ivanov, ed., |

5. | O. Bélorgey and J. J. Benattar, “Structural properties of soap black films investigated by X-ray reflectivity,” Phys. Rev. Lett. |

6. | J. J. Benattar, J. Daillant, O. Bélorgey, and L. Bosio, “Langmuir monolayers and Newton black films: two-dimensional systems investigated by X-ray reflectivity,” Physica A |

7. | S. Cohen-Addad, J. M. Di Meglio, and R. Ober, “Épaisseur d’un film noir de savon contenant un polymère hydrosoluble,” C. R. Acad. Sci, Paris Série II |

8. | S. Lionti-Addad and J. M. Di Meglio, “Stabilization of aqueous foam by hydrosoluble polymers. 1. Sodium dodecyl sulphate-poly(ethylene oxide) system,” Langmuir |

9. | A. S. Dimitrov, P. A. Kralchevsky, A. D. Nikolov, and D. T. Wasan, “Contact angles of thin liquid films: interferometric determination,” Colloids Surf. |

10. | A. D. Nikolov, P. A. Kralchevsky, and I. B. Ivanov, “Film and line tension effects on the attachment of particles to an interface,” J. Colloid Interface Sci. |

11. | G. Pretzler, H. Jäger, and T. Neger, “High-accuracy differential interferometry for the investigation of phase objects,” Meas. Sci. Technol. |

12. | T. Mishima and K. C. Kao, “Determination of 2-D thickness distributions of low absorbing thin films by new laser interferometry,” Appl. Opt. |

13. | A. González-Cano and E. Bernabáu, “Automatic interference method for measuring transparent film thickness,” Appl. Opt. |

14. | V. Greco, C. Lemmi, S. Ledesma, G. Molesini, G. P. Puccioni, and F. Quercioli, “Measuring soap black films by phase shifting interferometry,” Meas. Sci. Technol. |

15. | V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum. |

16. | V. Greco and G. Molesini, “Monitoring the thickness of soap films by polarization homodyne interferometry,” Meas. Sci. Technol. |

17. | O. Greffier, Y. Amarouchene, and H. Kellay, “Thickness fluctuations in turbulent soap films,” Phys. Rev. Lett. |

18. | M. Tebaldi, L. Angel, N. Bolognini, and M. Trivi, “Speckle interferometric technique to assess soap films,” Opt. Commun. |

19. | X. Wang and H. Qiu, “Fringe probing of gas-liquid interfacial film in a microcapillary tube,” Appl. Opt. |

20. | A. Kariyasaki, Y. Yamasaki, M. Kagawa, T. Nagashima, A. Ousaka, and S. Morooka, “Measurement of liquid film thickness by a fringe method,” Heat Transf. Eng. |

21. | G. Ropars, D. Chauvat, A. Le Floch, M. N. O’Sullivan-Hale, and R. W. Boyd, “Dynamics of gravity-induced gradients in soap films,” Appl. Phys. Lett. |

22. | S. N. Tan, Y. Yang, and R. G. Horn, “Thinning of a vertical free-draining aqueous film incorporating colloidal particles,” Langmuir |

23. | L. Liggieri, F. Ravera, and A. Passerone, “Dynamic interfacial tension measurements by a capillary pressure method,” J. Colloid Interface Sci. |

24. | W. Lv, H. Zhou, C. Lou, and J. Zhu, “Spatial and temporal film thickness measurement of a soap bubble based on large lateral shearing displacement interferometry,” Appl. Opt. |

25. | F. Seychelles, Y. Amarouchene, M. Bessafi, and H. Kellay, “Thermal convection and emergence of isolated vortices in soap bubbles,” Phys. Rev. Lett. |

26. | B. Edlén, “The refractive index of air,” Metrologia |

27. | C. Isenberg, |

28. | V. Greco, G. Molesini, F. Quercioli, and A. Novi, “Interferometric testing of weak aspheric surfaces versus design specifications,” Optik (Stuttg.) |

29. | D. Malacara, |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 25, 2013

Revised Manuscript: May 26, 2013

Manuscript Accepted: June 5, 2013

Published: August 14, 2013

**Citation**

Maurizio Vannoni, Andrea Sordini, Riccardo Gabrieli, Mauro Melozzi, and Giuseppe Molesini, "Measuring the thickness of soap bubbles with phase-shift interferometry," Opt. Express **21**, 19657-19667 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19657

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### References

- C. V. Boys, Soap Bubbles and the Forces that Mould Them (Dover, 1958).
- K. J. Mysels, K. Shinoda, and S. Frankel, Soap Films, Studies of Their Thinning (Pergamon, 1959).
- J. N. Israelashvili, Intermolecular and Surface Forces (Academic, 1985).
- I. B. Ivanov, ed., Thin Liquid Films, Fundamentals and Applications (Dekker, 1988).
- O. Bélorgey and J. J. Benattar, “Structural properties of soap black films investigated by X-ray reflectivity,” Phys. Rev. Lett.66(3), 313–316 (1991). [CrossRef] [PubMed]
- J. J. Benattar, J. Daillant, O. Bélorgey, and L. Bosio, “Langmuir monolayers and Newton black films: two-dimensional systems investigated by X-ray reflectivity,” Physica A172(1-2), 225–241 (1991). [CrossRef]
- S. Cohen-Addad, J. M. Di Meglio, and R. Ober, “Épaisseur d’un film noir de savon contenant un polymère hydrosoluble,” C. R. Acad. Sci, Paris Série II315, 39–44 (1992).
- S. Lionti-Addad and J. M. Di Meglio, “Stabilization of aqueous foam by hydrosoluble polymers. 1. Sodium dodecyl sulphate-poly(ethylene oxide) system,” Langmuir8(1), 324–327 (1992). [CrossRef]
- A. S. Dimitrov, P. A. Kralchevsky, A. D. Nikolov, and D. T. Wasan, “Contact angles of thin liquid films: interferometric determination,” Colloids Surf.47, 299–321 (1990). [CrossRef]
- A. D. Nikolov, P. A. Kralchevsky, and I. B. Ivanov, “Film and line tension effects on the attachment of particles to an interface,” J. Colloid Interface Sci.112(1), 122–131 (1986). [CrossRef]
- G. Pretzler, H. Jäger, and T. Neger, “High-accuracy differential interferometry for the investigation of phase objects,” Meas. Sci. Technol.4(6), 649–658 (1993). [CrossRef]
- T. Mishima and K. C. Kao, “Determination of 2-D thickness distributions of low absorbing thin films by new laser interferometry,” Appl. Opt.21(16), 2894–2896 (1982). [CrossRef] [PubMed]
- A. González-Cano and E. Bernabáu, “Automatic interference method for measuring transparent film thickness,” Appl. Opt.32(13), 2292–2294 (1993). [CrossRef] [PubMed]
- V. Greco, C. Lemmi, S. Ledesma, G. Molesini, G. P. Puccioni, and F. Quercioli, “Measuring soap black films by phase shifting interferometry,” Meas. Sci. Technol.5(8), 900–903 (1994). [CrossRef]
- V. Greco, G. Molesini, and F. Quercioli, “Accurate polarization interferometer,” Rev. Sci. Instrum.66(7), 3729–3734 (1995). [CrossRef]
- V. Greco and G. Molesini, “Monitoring the thickness of soap films by polarization homodyne interferometry,” Meas. Sci. Technol.7(1), 96–101 (1996). [CrossRef]
- O. Greffier, Y. Amarouchene, and H. Kellay, “Thickness fluctuations in turbulent soap films,” Phys. Rev. Lett.88(19), 194101 (2002). [CrossRef] [PubMed]
- M. Tebaldi, L. Angel, N. Bolognini, and M. Trivi, “Speckle interferometric technique to assess soap films,” Opt. Commun.229(1-6), 29–37 (2004). [CrossRef]
- X. Wang and H. Qiu, “Fringe probing of gas-liquid interfacial film in a microcapillary tube,” Appl. Opt.44(22), 4648–4653 (2005). [CrossRef] [PubMed]
- A. Kariyasaki, Y. Yamasaki, M. Kagawa, T. Nagashima, A. Ousaka, and S. Morooka, “Measurement of liquid film thickness by a fringe method,” Heat Transf. Eng.30(1-2), 28–36 (2009). [CrossRef]
- G. Ropars, D. Chauvat, A. Le Floch, M. N. O’Sullivan-Hale, and R. W. Boyd, “Dynamics of gravity-induced gradients in soap films,” Appl. Phys. Lett.88(23), 234104 (2006). [CrossRef]
- S. N. Tan, Y. Yang, and R. G. Horn, “Thinning of a vertical free-draining aqueous film incorporating colloidal particles,” Langmuir26(1), 63–73 (2010). [CrossRef] [PubMed]
- L. Liggieri, F. Ravera, and A. Passerone, “Dynamic interfacial tension measurements by a capillary pressure method,” J. Colloid Interface Sci.169(1), 226–237 (1995). [CrossRef]
- W. Lv, H. Zhou, C. Lou, and J. Zhu, “Spatial and temporal film thickness measurement of a soap bubble based on large lateral shearing displacement interferometry,” Appl. Opt.51(36), 8863–8872 (2012). [CrossRef] [PubMed]
- F. Seychelles, Y. Amarouchene, M. Bessafi, and H. Kellay, “Thermal convection and emergence of isolated vortices in soap bubbles,” Phys. Rev. Lett.100(14), 144501 (2008). [CrossRef] [PubMed]
- B. Edlén, “The refractive index of air,” Metrologia2(2), 71–80 (1966). [CrossRef]
- C. Isenberg, The Science of Soap Films and Soap Bubbles (Dover, 1992, pp. 14).
- V. Greco, G. Molesini, F. Quercioli, and A. Novi, “Interferometric testing of weak aspheric surfaces versus design specifications,” Optik (Stuttg.)87, 159–162 (1991).
- D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).

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