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OPTICS LETTERS / Vol. 39, No. 15 / August 1, 2014
Incoherent averaging of phase singularities in speckle-shearing interferometry Klaus Mantel,1,* Vanusch Nercissian,1 and Norbert Lindlein2 1
2
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1, 91058 Erlangen, Germany Institute of Optics, Information, and Photonics, FAU Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany *Corresponding author: [email protected] Received May 14, 2014; revised June 24, 2014; accepted June 27, 2014; posted June 30, 2014 (Doc. ID 212135); published July 28, 2014 Interferometric speckle techniques are plagued by the omnipresence of phase singularities, impairing the phase unwrapping process. To reduce the number of phase singularities by physical means, an incoherent averaging of multiple p speckle fields may be applied. It turns out, however, that the results may strongly deviate from the expected N behavior. Using speckle-shearing interferometry as an example, we investigate the mechanism behind the reduction of phase singularities, both by calculations and by computer simulations. Key to an understanding of the reduction mechanism during incoherent averaging is the representation of the physical averaging process in terms of certain vector fields associated with each speckle field. © 2014 Optical Society of America OCIS codes: (030.6140) Speckle; (120.6165) Speckle interferometry, metrology. http://dx.doi.org/10.1364/OL.39.004510
Speckle-shearing interferometry is a well-established technique for the deformation measurement of rough objects [1–3], owing mainly to the common path property which makes the setup compact and robust against external disturbances like air turbulence or vibrations. As for most speckle techniques, however, phase singularities [4,5] appear in the measured phase, seriously impairing the phase unwrapping process (notable exceptions are correlation methods like [6–8]). Rather than developing sophisticated software algorithms [9–11] to calculate the unwrapped phase, nevertheless, it is promising to avoid the generation of phase singularities in the first place. This goal may be achieved by physical means, namely by an averaging of different, mutually incoherent speckle fields carrying the same systematic phase information [12]. In this work, the mechanism behind the incoherent averaging is investigated by a combination of analytic calculations and computer modeling. Although the following deals mainly with speckle-shearing interferometry, the results can be straightforwardly transferred to other speckle interferometry setups. From the theory of the speckle effect, it is known that phase singularities may occur at those points within a speckle field where the intensity vanishes. Furthermore, when N uncorrelated speckle fields are incoherently added, the contrast of the resultant pattern is reduced p by the factor N [12] while intensity values of zero are not the most likely any more [13]. It then stands to reason that an incoherent averaging of N speckle fields reduces the number p of phase singularities correspondingly by a factor of N. However, it should be noted that in speckle interferometry, phase singularities do not necessarily occur at locations with vanishing intensity. Since the speckle field is superposed with a reference wave—for speckle-shearing interferometry, a shifted copy of the field itself—phase singularities rather appear at locations which do not change their intensity value during phase shifting. In other words, it is the visibility of the interference pattern at these points that vanishes. Furthermore, as will be demonstrated in this Letter, speckle-shearing fields show a special structure 0146-9592/14/154510-04$15.00/0
different from conventional p speckle fields. Therefore, the reasoning behind the N reduction in the number of phase singularities is inadequate. To determine the reduction of phase singularities due to incoherent averaging in a quantitative way, the speckle-shearing interferometer is modeled by the computer. The model assumes, as Fig. 1 illustrates, a random phase in the plane of the specimen to account for its surface roughness. An additional systematic phase, which may be changed between two simulation runs, allows the simulation of deformations of the specimen. The light field is propagated through a lens by means of linear systems theory (see [14]); the ratio D∕f of the lens diameter D and its focal length f determines the speckle size. The shearing is incorporated by generating two laterally shifted copies up and un of the resulting speckle field u, which are then added to get the speckle-shearing field us on the detector: us up un . Phase shifting is simulated by adding π∕4, π∕2, 3π∕4, and π to the phase of up while simultaneously subtracting the same values from un to obtain, for example, the necessary phase differences for a four-phase algorithm (cf. Eq. 1). Off-axis light source points were included by adding linear terms to the random phase before propagation. Separately adding the corresponding intensities makes the light source points mutually incoherent.
Fig. 1. Model of the speckle-shearing interferometer used for the computer simulations. © 2014 Optical Society of America
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p
In contrast to a mere N reduction, a much more complex behavior becomes apparent when the actual reduction factor is plotted against speckle size. This was done in Fig. 2 for the incoherent superposition of nine speckle fields resulting from a single specimen with a fixed surface roughness (a possible systematic phase plays no role at this stage). It should be noted that the results do not depend on the statistical distribution used, and that the amplitude of the roughness was chosen large enough to obtain fully developed speckles. The reduction factor was calculated to be the ratio of the number of phase singularities in the averaged speckle-shearing field and the number originally present in one speckleshearing field before the averaging. The number of phase singularities in the respective speckle fields has been counted by a simple numerical routine: for each pixel, the routine takes the differences of phase values of neighboring pixels along a closed path around the central pixel under investigation. The difference values are each divided by 2π, rounded to the nearest whole number, i.e., 0 or 1, and then summed. If the resulting sum differs from zero, a phase singularity is present. Since the simulation data do not suffer from noise, the procedure detects the singularities reliably. Figure 2 shows that p the resulting reduction curve contradicts an alleged N behavior, which in the case described would mean a reduction factor of about 0.3, independent of speckle size. In contrast, for large speckles, a complete reduction of the phase singularities takes place, while for medium and small speckle sizes, almost no reduction occurs; the reduction factor amounts to only 0.8. Figure 3 illustrates the speckle sizes mentioned. To explain the counter-intuitive shape of the reduction curve, the mechanism behind the incoherent averaging of phase singularities has been investigated. Of course, the effect of the averaging process may be calculated as was described above, i.e., by summing the intensities for the various light source points combined with phase shifting. But such calculations leave the mechanism behind the form of the reduction curve obscure. As it turns out, however, it is possible to replace the phase-shifting process of the incoherent superposition of speckle fields by a different, but equivalent and better
Fig. 2. Phase-singularity reduction for a light source consisting of nine mutually incoherent points. Vertical, reduction of the number of singularities compared to a single point source. Horizontal: a measure of inverse speckle size, varying from large to small speckles (the circles indicate the x-values referred to in Fig. 3). Small image, reduction curve averaged over 64 runs.
Fig. 3. Different simulated speckle-shearing fields (b/winverted) of 1000 × 1000 pixels with a shear of 50 pixels, corresponding to the different areas visible in Fig. 2. Left: x 1, complete elimination of phase singularities. Middle left, x 71, no reduction. Middle right, x 91, reduction factor of 0.8. Right, x 100, no reduction.
visualizable process. Let uis uip uin be the complex amplitude of light source point i as it reaches the detector. Then the phase Φ for the incoherent addition of all light source points as obtained by phase shifting is, for example, given by Φ arctan
I 4 − I 2 ; I 1 − I 3
(1)
where I k denotes the overall intensity for the whole light source and phase shift k. Alternatively, to the amplitude uis we now associate the (artificial) complex amplitude vi : uip · uin , where the star denotes complex conjugation. Now, vi is conveniently interpreted as a vector field vi : Re vi ; Im vi , where Re and Im denote the real and imaginary parts. The incoherent superposition of the speckle fields uis can then be represented by the ordinary addition of the associated amplitudes: v
X vi ;
(2)
i
and, consequently, the corresponding vector fields vi . Indeed, it can be shown by expressing the I k in terms of the superposition of the (appropriately phase shifted) amplitudes uis whereby Φ is equal to the phase angle of v: Φ argv:
(3)
The incoherent superposition of different speckle fields uis can therefore be interpreted as the ordinary addition of the associated vector fields vi , where the phase obtained by phase shifting can be directly read off as the phase angle of the resulting sum vector field v. In this way, the process of incoherent addition may be much more easily visualized. Equation (3) is particularly true for one single light field uis ; phase shifting gives the phase difference Φp − Φn between the corresponding fields uip and uin , but this is precisely the phase angle of the associated vi . This result is helpful for an understanding of the special structure of speckle-shearing fields mentioned earlier. Figure 4 contrasts a regular speckle field with a speckle-shearing field derived from it, where the shear s was directed along the horizontal axis of the field. With a ratio of speckle size to shear distance of more than 3, the shear distance chosen was small compared to the speckle size so that the regular structure of the speckle-shearing field becomes more pronounced.
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Fig. 4. Phase field and phase statistic for a standard speckle field (left) at x 40 and the speckle-shearing field derived from it (right). In black, phase edges (lines), phase singularities (illustrated as circles), and shear s. Both phase (upper part) and phase statistic (lower part) are shown. The phase statistic for the standard speckle field deviates randomly from the uniform distribution because of the low number of speckle grains in the field, while the statistic for the speckle-shearing field is systematically peaked at a phase value of zero.
As a consequence of the shearing principle, each phase singularity is doubled and the resulting pair is joined by a phase edge. The distance between corresponding singularities is equal to the shear distance, so that the phase edges are significantly shorter than for the original speckle field. Figure 5 illustrates the creation of these phase edges via shearing from the irregular phase edges of the original speckle field using the vector field representation. The phase singularities are visible as a pair, formed by a source/sink (or, in general, a spiral) and a saddle. They are joined by a line of vectors pointing in the negative horizontal direction indicating a phase value of π along the line which constitutes the phase edge (and was chosen here to be vertical). Forming the associated vector field up · un , which was shown to be equivalent to the shearing process for a single light source, leads to the addition of the phase angles. This generates new phase edges now joining the shifted copies of the original singularities.
Fig. 5. Recombination of phase edges in the vector field representation. The basic complex amplitudes up and un , which differ by a horizontal shift and a complex conjugation, are multiplied to form the associated vector field up · un . The corresponding addition of phase angles explains the realignment of the phase edges due to shearing and the doubling of the initial singularities. The corresponding phase maps are also illustrated.
The regular structure of the phase fields has immediate consequences for the phase statistic in speckle-shearing interferometry. Especially for large speckle sizes compared to the shear distance, the phase statistic is notably different from the uniform distribution expected for a speckle field—it is peaked around zero. For smaller speckle sizes, the regular structure of the phase pattern visible in Fig. 4 gradually disappears as the phase edges approach each other and the speckle-shearing field resembles the standard field more or less, see Fig. 6. Yet for small speckle sizes, more pronounced deviations in the statistic may still exist. These distributions may even show a depletion at zero, not a peak, as Fig. 6 also demonstrates. Obviously, the three types of phase statistics as shown in Fig. 4, right, and Fig. 6 are the only types of phase statistics possible, only differing in the amount of their deviation from the uniform one. With these results, it is now possible to explain the reduction curve. The ratio of speckle size and shear distance determines the deviation of the phase statistic from the uniform one. This corresponds to a preferred direction in the corresponding vector fields. The random addition of such vector fields with a bias direction makes this preferred direction more and more pronounced until it dominates the whole vector field. Figure 7 illustrates this process for a large speckle size. Before the averaging takes place (top part), phase singularities (source/sink/ spiral, saddle) and edges joining them represented by vectors pointing along the negative horizontal direction are clearly visible. Since in this case the statistic centers around phase values of zero, the preferred direction in the vector field is the positive horizontal direction, as can also be seen from the figure. The addition of such vector fields will then make the locations with phase edges disappear (middle and bottom part). A similar process occurs for smaller speckle sizes, when it may happen that the phase statistic favors vectors pointing along the negative horizontal direction. The smaller the deviations from the uniform phase statistic, the more speckle fields have to be added to achieve the desired reduction, see Fig. 8. This overall behavior is characteristic for speckleshearing interferometry. When the phase statistic is close to uniform, no or almost no reduction takes place. When the statistic differs from the uniform one, partial or even complete reduction of the phase singularities occurs. The
Fig. 6. Phase field and phase statistic for medium speckle sizes. Left, x 69. Right, x 76. The phase statistic changed to uniform and, for even smaller speckle sizes, to concave.
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Fig. 7. Incoherent averaging represented by vector fields from the beginning (top) to the end of the averaging process (bottom). The vector field gradually changes direction according to the phase distribution of the individual speckle fields. The initial phase singularities and edges (sketched by dots) have disappeared, and the dominating direction is now along the positive horizontal axis.
Fig. 8. Number of remaining phase singularities versus number of averaging speckle fields for x 40 (left), x 69 (middle), and x 76 (right); n denotes the averaging step after which the number of phase singularities fell to zero for the first time.
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proceeds. Figure 9 shows the deformation phase— essentially the difference between two simulations with different systematic phases—after an averaging of 1200 light source points as compared to a point source. Since the averaging leads to a dominating direction in the sum vector field (see Fig. 7), the difference of two such vector fields will be negligible; only traces of the deformation fringes may still be seen. The intended unwrapping process is impossible. In conclusion, we investigated the mechanism behind the reduction of phase singularities during an incoherent averaging. When the phase statistic of the speckle fields differs from the uniform one, as is the case for speckleshearing interferometry, the reduction succeeds for a sufficiently high number of light source points. At the same time, however, it gradually destroys the deformation phase carrying the desired information. The reduction of phase singularities can then only succeed in the form of a compromise: The number of light source points has to be chosen to ensure sufficient reduction at the desired speckle size without notably changing the deformation phase. These results were shown for speckle-shearing interferometry, where deviations from the uniform statistic may be very pronounced; the results however hold in general. In particular, they imply that in a Michelson setup, where no shearing is present and therefore the phase statistic is uniformly distributed (as in conventional speckle fields [15]), no significant reduction of phase singularities can occur. In a forthcoming publication, a physical realization of incoherent averaging in speckle-shearing interferometry will be described. A periodic light source consisting of mutually incoherent points will be used with the distance between the light source points matched to the shear. Therefore, high contrast for the deformation fringes is maintained as the averaging of the phase singularities takes place. First measurements suggest a good agreement with the theoretical considerations presented in this Letter. The authors thank the Deutsche Forschungsgemeinschaft for supporting this work under the contract MA 5167/1-1 and LI 1612/4-1.
Fig. 9. Deformation phase for a shear in horizontal direction. Left, point source. The fringes show a high visibility. Right, 20 × 60 light source. The phase values are restricted to a much smaller interval, and the fringes can only barely be seen.
stronger the deviation from the uniform statistic, the greater the reduction. With a sufficient number of light source points, a complete elimination of the phase singularities in a single speckle field may therefore be obtained. However, at the same time, the systematic phase also present in the light fields is destroyed as the averaging
References 1. J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973). 2. Y. T. Hung, Opt. Eng. 21, 213391 (1982). 3. D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010). 4. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974). 5. J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007). 6. I. Yamaguchi, Opt. Acta 28, 1359 (1981). 7. M. Sjödahl, Appl. Opt. 34, 7998 (1995). 8. P. Šmíd, P. Horváth, and M. Hrabovský, Appl. Opt. 46, 3709 (2007). 9. S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007). 10. J. M. Huntley, Appl. Opt. 40, 3901 (2001). 11. D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998). 12. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976). 13. R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971). 14. S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006). 15. J. W. Goodman, Statistical Optics (Wiley, 1985).
OPTICS LETTERS / Vol. 39, No. 15 / August 1, 2014
Incoherent averaging of phase singularities in speckle-shearing interferometry Klaus Mantel,1,* Vanusch Nercissian,1 and Norbert Lindlein2 1
2
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1, 91058 Erlangen, Germany Institute of Optics, Information, and Photonics, FAU Erlangen-Nürnberg, Staudtstr. 7/B2, 91058 Erlangen, Germany *Corresponding author: [email protected] Received May 14, 2014; revised June 24, 2014; accepted June 27, 2014; posted June 30, 2014 (Doc. ID 212135); published July 28, 2014 Interferometric speckle techniques are plagued by the omnipresence of phase singularities, impairing the phase unwrapping process. To reduce the number of phase singularities by physical means, an incoherent averaging of multiple p speckle fields may be applied. It turns out, however, that the results may strongly deviate from the expected N behavior. Using speckle-shearing interferometry as an example, we investigate the mechanism behind the reduction of phase singularities, both by calculations and by computer simulations. Key to an understanding of the reduction mechanism during incoherent averaging is the representation of the physical averaging process in terms of certain vector fields associated with each speckle field. © 2014 Optical Society of America OCIS codes: (030.6140) Speckle; (120.6165) Speckle interferometry, metrology. http://dx.doi.org/10.1364/OL.39.004510
Speckle-shearing interferometry is a well-established technique for the deformation measurement of rough objects [1–3], owing mainly to the common path property which makes the setup compact and robust against external disturbances like air turbulence or vibrations. As for most speckle techniques, however, phase singularities [4,5] appear in the measured phase, seriously impairing the phase unwrapping process (notable exceptions are correlation methods like [6–8]). Rather than developing sophisticated software algorithms [9–11] to calculate the unwrapped phase, nevertheless, it is promising to avoid the generation of phase singularities in the first place. This goal may be achieved by physical means, namely by an averaging of different, mutually incoherent speckle fields carrying the same systematic phase information [12]. In this work, the mechanism behind the incoherent averaging is investigated by a combination of analytic calculations and computer modeling. Although the following deals mainly with speckle-shearing interferometry, the results can be straightforwardly transferred to other speckle interferometry setups. From the theory of the speckle effect, it is known that phase singularities may occur at those points within a speckle field where the intensity vanishes. Furthermore, when N uncorrelated speckle fields are incoherently added, the contrast of the resultant pattern is reduced p by the factor N [12] while intensity values of zero are not the most likely any more [13]. It then stands to reason that an incoherent averaging of N speckle fields reduces the number p of phase singularities correspondingly by a factor of N. However, it should be noted that in speckle interferometry, phase singularities do not necessarily occur at locations with vanishing intensity. Since the speckle field is superposed with a reference wave—for speckle-shearing interferometry, a shifted copy of the field itself—phase singularities rather appear at locations which do not change their intensity value during phase shifting. In other words, it is the visibility of the interference pattern at these points that vanishes. Furthermore, as will be demonstrated in this Letter, speckle-shearing fields show a special structure 0146-9592/14/154510-04$15.00/0
different from conventional p speckle fields. Therefore, the reasoning behind the N reduction in the number of phase singularities is inadequate. To determine the reduction of phase singularities due to incoherent averaging in a quantitative way, the speckle-shearing interferometer is modeled by the computer. The model assumes, as Fig. 1 illustrates, a random phase in the plane of the specimen to account for its surface roughness. An additional systematic phase, which may be changed between two simulation runs, allows the simulation of deformations of the specimen. The light field is propagated through a lens by means of linear systems theory (see [14]); the ratio D∕f of the lens diameter D and its focal length f determines the speckle size. The shearing is incorporated by generating two laterally shifted copies up and un of the resulting speckle field u, which are then added to get the speckle-shearing field us on the detector: us up un . Phase shifting is simulated by adding π∕4, π∕2, 3π∕4, and π to the phase of up while simultaneously subtracting the same values from un to obtain, for example, the necessary phase differences for a four-phase algorithm (cf. Eq. 1). Off-axis light source points were included by adding linear terms to the random phase before propagation. Separately adding the corresponding intensities makes the light source points mutually incoherent.
Fig. 1. Model of the speckle-shearing interferometer used for the computer simulations. © 2014 Optical Society of America
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p
In contrast to a mere N reduction, a much more complex behavior becomes apparent when the actual reduction factor is plotted against speckle size. This was done in Fig. 2 for the incoherent superposition of nine speckle fields resulting from a single specimen with a fixed surface roughness (a possible systematic phase plays no role at this stage). It should be noted that the results do not depend on the statistical distribution used, and that the amplitude of the roughness was chosen large enough to obtain fully developed speckles. The reduction factor was calculated to be the ratio of the number of phase singularities in the averaged speckle-shearing field and the number originally present in one speckleshearing field before the averaging. The number of phase singularities in the respective speckle fields has been counted by a simple numerical routine: for each pixel, the routine takes the differences of phase values of neighboring pixels along a closed path around the central pixel under investigation. The difference values are each divided by 2π, rounded to the nearest whole number, i.e., 0 or 1, and then summed. If the resulting sum differs from zero, a phase singularity is present. Since the simulation data do not suffer from noise, the procedure detects the singularities reliably. Figure 2 shows that p the resulting reduction curve contradicts an alleged N behavior, which in the case described would mean a reduction factor of about 0.3, independent of speckle size. In contrast, for large speckles, a complete reduction of the phase singularities takes place, while for medium and small speckle sizes, almost no reduction occurs; the reduction factor amounts to only 0.8. Figure 3 illustrates the speckle sizes mentioned. To explain the counter-intuitive shape of the reduction curve, the mechanism behind the incoherent averaging of phase singularities has been investigated. Of course, the effect of the averaging process may be calculated as was described above, i.e., by summing the intensities for the various light source points combined with phase shifting. But such calculations leave the mechanism behind the form of the reduction curve obscure. As it turns out, however, it is possible to replace the phase-shifting process of the incoherent superposition of speckle fields by a different, but equivalent and better
Fig. 2. Phase-singularity reduction for a light source consisting of nine mutually incoherent points. Vertical, reduction of the number of singularities compared to a single point source. Horizontal: a measure of inverse speckle size, varying from large to small speckles (the circles indicate the x-values referred to in Fig. 3). Small image, reduction curve averaged over 64 runs.
Fig. 3. Different simulated speckle-shearing fields (b/winverted) of 1000 × 1000 pixels with a shear of 50 pixels, corresponding to the different areas visible in Fig. 2. Left: x 1, complete elimination of phase singularities. Middle left, x 71, no reduction. Middle right, x 91, reduction factor of 0.8. Right, x 100, no reduction.
visualizable process. Let uis uip uin be the complex amplitude of light source point i as it reaches the detector. Then the phase Φ for the incoherent addition of all light source points as obtained by phase shifting is, for example, given by Φ arctan
I 4 − I 2 ; I 1 − I 3
(1)
where I k denotes the overall intensity for the whole light source and phase shift k. Alternatively, to the amplitude uis we now associate the (artificial) complex amplitude vi : uip · uin , where the star denotes complex conjugation. Now, vi is conveniently interpreted as a vector field vi : Re vi ; Im vi , where Re and Im denote the real and imaginary parts. The incoherent superposition of the speckle fields uis can then be represented by the ordinary addition of the associated amplitudes: v
X vi ;
(2)
i
and, consequently, the corresponding vector fields vi . Indeed, it can be shown by expressing the I k in terms of the superposition of the (appropriately phase shifted) amplitudes uis whereby Φ is equal to the phase angle of v: Φ argv:
(3)
The incoherent superposition of different speckle fields uis can therefore be interpreted as the ordinary addition of the associated vector fields vi , where the phase obtained by phase shifting can be directly read off as the phase angle of the resulting sum vector field v. In this way, the process of incoherent addition may be much more easily visualized. Equation (3) is particularly true for one single light field uis ; phase shifting gives the phase difference Φp − Φn between the corresponding fields uip and uin , but this is precisely the phase angle of the associated vi . This result is helpful for an understanding of the special structure of speckle-shearing fields mentioned earlier. Figure 4 contrasts a regular speckle field with a speckle-shearing field derived from it, where the shear s was directed along the horizontal axis of the field. With a ratio of speckle size to shear distance of more than 3, the shear distance chosen was small compared to the speckle size so that the regular structure of the speckle-shearing field becomes more pronounced.
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Fig. 4. Phase field and phase statistic for a standard speckle field (left) at x 40 and the speckle-shearing field derived from it (right). In black, phase edges (lines), phase singularities (illustrated as circles), and shear s. Both phase (upper part) and phase statistic (lower part) are shown. The phase statistic for the standard speckle field deviates randomly from the uniform distribution because of the low number of speckle grains in the field, while the statistic for the speckle-shearing field is systematically peaked at a phase value of zero.
As a consequence of the shearing principle, each phase singularity is doubled and the resulting pair is joined by a phase edge. The distance between corresponding singularities is equal to the shear distance, so that the phase edges are significantly shorter than for the original speckle field. Figure 5 illustrates the creation of these phase edges via shearing from the irregular phase edges of the original speckle field using the vector field representation. The phase singularities are visible as a pair, formed by a source/sink (or, in general, a spiral) and a saddle. They are joined by a line of vectors pointing in the negative horizontal direction indicating a phase value of π along the line which constitutes the phase edge (and was chosen here to be vertical). Forming the associated vector field up · un , which was shown to be equivalent to the shearing process for a single light source, leads to the addition of the phase angles. This generates new phase edges now joining the shifted copies of the original singularities.
Fig. 5. Recombination of phase edges in the vector field representation. The basic complex amplitudes up and un , which differ by a horizontal shift and a complex conjugation, are multiplied to form the associated vector field up · un . The corresponding addition of phase angles explains the realignment of the phase edges due to shearing and the doubling of the initial singularities. The corresponding phase maps are also illustrated.
The regular structure of the phase fields has immediate consequences for the phase statistic in speckle-shearing interferometry. Especially for large speckle sizes compared to the shear distance, the phase statistic is notably different from the uniform distribution expected for a speckle field—it is peaked around zero. For smaller speckle sizes, the regular structure of the phase pattern visible in Fig. 4 gradually disappears as the phase edges approach each other and the speckle-shearing field resembles the standard field more or less, see Fig. 6. Yet for small speckle sizes, more pronounced deviations in the statistic may still exist. These distributions may even show a depletion at zero, not a peak, as Fig. 6 also demonstrates. Obviously, the three types of phase statistics as shown in Fig. 4, right, and Fig. 6 are the only types of phase statistics possible, only differing in the amount of their deviation from the uniform one. With these results, it is now possible to explain the reduction curve. The ratio of speckle size and shear distance determines the deviation of the phase statistic from the uniform one. This corresponds to a preferred direction in the corresponding vector fields. The random addition of such vector fields with a bias direction makes this preferred direction more and more pronounced until it dominates the whole vector field. Figure 7 illustrates this process for a large speckle size. Before the averaging takes place (top part), phase singularities (source/sink/ spiral, saddle) and edges joining them represented by vectors pointing along the negative horizontal direction are clearly visible. Since in this case the statistic centers around phase values of zero, the preferred direction in the vector field is the positive horizontal direction, as can also be seen from the figure. The addition of such vector fields will then make the locations with phase edges disappear (middle and bottom part). A similar process occurs for smaller speckle sizes, when it may happen that the phase statistic favors vectors pointing along the negative horizontal direction. The smaller the deviations from the uniform phase statistic, the more speckle fields have to be added to achieve the desired reduction, see Fig. 8. This overall behavior is characteristic for speckleshearing interferometry. When the phase statistic is close to uniform, no or almost no reduction takes place. When the statistic differs from the uniform one, partial or even complete reduction of the phase singularities occurs. The
Fig. 6. Phase field and phase statistic for medium speckle sizes. Left, x 69. Right, x 76. The phase statistic changed to uniform and, for even smaller speckle sizes, to concave.
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Fig. 7. Incoherent averaging represented by vector fields from the beginning (top) to the end of the averaging process (bottom). The vector field gradually changes direction according to the phase distribution of the individual speckle fields. The initial phase singularities and edges (sketched by dots) have disappeared, and the dominating direction is now along the positive horizontal axis.
Fig. 8. Number of remaining phase singularities versus number of averaging speckle fields for x 40 (left), x 69 (middle), and x 76 (right); n denotes the averaging step after which the number of phase singularities fell to zero for the first time.
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proceeds. Figure 9 shows the deformation phase— essentially the difference between two simulations with different systematic phases—after an averaging of 1200 light source points as compared to a point source. Since the averaging leads to a dominating direction in the sum vector field (see Fig. 7), the difference of two such vector fields will be negligible; only traces of the deformation fringes may still be seen. The intended unwrapping process is impossible. In conclusion, we investigated the mechanism behind the reduction of phase singularities during an incoherent averaging. When the phase statistic of the speckle fields differs from the uniform one, as is the case for speckleshearing interferometry, the reduction succeeds for a sufficiently high number of light source points. At the same time, however, it gradually destroys the deformation phase carrying the desired information. The reduction of phase singularities can then only succeed in the form of a compromise: The number of light source points has to be chosen to ensure sufficient reduction at the desired speckle size without notably changing the deformation phase. These results were shown for speckle-shearing interferometry, where deviations from the uniform statistic may be very pronounced; the results however hold in general. In particular, they imply that in a Michelson setup, where no shearing is present and therefore the phase statistic is uniformly distributed (as in conventional speckle fields [15]), no significant reduction of phase singularities can occur. In a forthcoming publication, a physical realization of incoherent averaging in speckle-shearing interferometry will be described. A periodic light source consisting of mutually incoherent points will be used with the distance between the light source points matched to the shear. Therefore, high contrast for the deformation fringes is maintained as the averaging of the phase singularities takes place. First measurements suggest a good agreement with the theoretical considerations presented in this Letter. The authors thank the Deutsche Forschungsgemeinschaft for supporting this work under the contract MA 5167/1-1 and LI 1612/4-1.
Fig. 9. Deformation phase for a shear in horizontal direction. Left, point source. The fringes show a high visibility. Right, 20 × 60 light source. The phase values are restricted to a much smaller interval, and the fringes can only barely be seen.
stronger the deviation from the uniform statistic, the greater the reduction. With a sufficient number of light source points, a complete elimination of the phase singularities in a single speckle field may therefore be obtained. However, at the same time, the systematic phase also present in the light fields is destroyed as the averaging
References 1. J. Leendertz and J. Butters, J. Phys. E 6, 1107 (1973). 2. Y. T. Hung, Opt. Eng. 21, 213391 (1982). 3. D. Francis, R. P. Tatam, and R. M. Groves, Meas. Sci. Technol. 21, 102001 (2010). 4. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974). 5. J. Adachi and G.-O. Ishikawa, Nonlinearity 20, 1907 (2007). 6. I. Yamaguchi, Opt. Acta 28, 1359 (1981). 7. M. Sjödahl, Appl. Opt. 34, 7998 (1995). 8. P. Šmíd, P. Horváth, and M. Hrabovský, Appl. Opt. 46, 3709 (2007). 9. S. Liu and L. Yang, Opt. Eng. 46, 051012 (2007). 10. J. M. Huntley, Appl. Opt. 40, 3901 (2001). 11. D. C. Ghiglia and M. D. Pritt, Two-dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998). 12. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976). 13. R. Dändliker and F. M. Mottier, ZAMP 22, 369 (1971). 14. S. Equis and P. Jacquot, Proc. SPIE 6341, 634138 (2006). 15. J. W. Goodman, Statistical Optics (Wiley, 1985).
