## Deterministic signal associated with a random field |

Optics Express, Vol. 21, Issue 18, pp. 20806-20820 (2013)

http://dx.doi.org/10.1364/OE.21.020806

Acrobat PDF (1235 KB)

### Abstract

Stochastic fields do not generally possess a Fourier transform. This makes the second-order statistics calculation very difficult, as it requires solving a fourth-order stochastic wave equation. This problem was alleviated by Wolf who introduced the coherent mode decomposition and, as a result, space-frequency statistics propagation of wide-sense stationary fields. In this paper we show that if, in addition to wide-sense stationarity, the fields are also wide-sense statistically homogeneous, then monochromatic plane waves can be used as an eigenfunction basis for the cross spectral density. Furthermore, the eigenvalue associated with a plane wave, **k**, *ω*). We show that the second-order statistics of these fields is fully described by the spatiotemporal power spectrum, a real, positive function. Thus, the second-order statistics can be efficiently propagated in the wavevector-frequency representation using a new framework of *deterministic signals* associated with random fields. Analogous to the complex analytic signal representation of a field, the deterministic signal is a mathematical construct meant to simplify calculations. Specifically, the deterministic signal associated with a random field is defined such that it has the identical autocorrelation as the actual random field. Calculations for propagating spatial and temporal correlations are simplified greatly because one only needs to solve a deterministic wave equation of second order. We illustrate the power of the wavevector-frequency representation with calculations of spatial coherence in the far zone of an incoherent source, as well as coherence effects induced by biological tissues.

© 2013 OSA

## 1. Introduction

*coherence theory*or

*statistical optics*[1, 2]. Besides its importance to basic science, coherence theory is crucial in predicting outcomes of many light experiments. For example, in

*quantitative phase imaging*(QPI), we often employ spatially and temporally broadband light to image phase shifts associated with the imaging field [3]. Such phase shifts are physically meaningful only when they are defined via averages, through field cross-correlations (see, e.g., [4

4. Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett. **96**(5), 051117 (2010). [CrossRef]

*interferometry*) the result of the statistical average performed by the detection process is strongly dependent on the coherence properties of the light. Importantly, half of the 2005 Nobel Prize in Physics was awarded to Roy Glauber “for his contribution to the quantum theory of optical coherence.” For a selection of Glauber’s seminal papers, see Ref [5

5. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963). [CrossRef]

*monochromatic plane wave*, we cannot find a function

*f*(

**r**,

*t*) that prescribes the field at each point in space and at each moment in time. Instead, we describe the source as emitting a random signal,

*s*(

**r**,

*t*), and describe its behavior via probability distributions.

*realizations*of a certain random variable is called

*ensemble averaging.*The importance of the ensemble averaging has been emphasized many times by both Wolf and Glauber [1, 5

5. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963). [CrossRef]

5. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963). [CrossRef]

*stochastic*wave equation (see Appendix), which is tedious as it involves a fourth order differential equation. In order to alleviate this problem, Wolf introduced the

*coherent mode decomposition (CMD)*theory [8

8. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. **38**(1), 3–6 (1981). [CrossRef]

*wide sense stationary*fields, a square integrable cross-spectral density,

*W*can be constructed from contributions of completely spatially coherent sources,where the convergence is uniform in the mean-square sense. In Eq. (1), the functions

*W*,where

*D*is the spatial domain of interest. Since

*W*is Hermitian and also a non-negative definite Hilbert-Schmidt kernel, the eigenvalues

*D*of the source. The main benefit of this expansion is that each eigenfunction (mode) satisfies the deterministic Helmholtz equation and, thus, propagating

*W*reduces to propagating each mode and adding up the results. Using CMD, Wolf developed a framework for studying partial coherence in the

*space-frequency*domain [9

9. E. Wolf, “New theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. **72**(3), 343–351 (1982). [CrossRef]

10. E. Wolf, “New theory of partial coherence in the space-frequency domain. 2. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A **3**, 76–85 (1986). [CrossRef]

*W*admits plane waves as eigenfunctions, i.e., we can write

*W*can be expressed as the Fourier transform of a real, positive function, which we refer to as the

*spatiotemporal power spectrum*. The eigenvalue associated with each plane wave is the spatiotemporal power spectrum evaluated at the spatiotemporal frequency, (

**k**,

*ω*). The second order statistics is recovered in full by replacing the stochastic field with a deterministic field of the same power spectrum. This

*deterministic field*can therefore be propagated via a second order (deterministic) wave equation, significantly simplifying the calculations (Section 3). This calculation gives the correct result when explaining coherence effects (see, e.g., optical coherence tomography [11

11. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**(5035), 1178–1181 (1991). [CrossRef] [PubMed]

12. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. **3**(2), 129–134 (2007). [CrossRef]

*wavevector-frequency*representation. Propagation in the wavevector-frequency space allows us to easily compute second order moments of the transverse wavevector and, thus, correlation areas. We illustrate the power of this formalism by re-deriving the classic result of the van Cittert-Zernike theorem and correlation-induced spectral changes in biological tissues.

## 2. Statistically homogeneous fields

*W*is both wide sense stationary and statistically homogeneous then we can choose plane waves as an orthonormal basis for

*W*. The proof is as follows. We expand

*W*in the form of Eq. (1) where the

*ψ*

_{m}(

**r**

_{1}, ω) and integrating

**r**

_{1}over

*D*and using orthonormality of the basis functions:Thus, plane waves are eigenfunctions of the Fredholm integral equation of

*W*with associated eigenvalues

*S*, evaluated at the temporal frequency

*W*is contained in

*S*.

**1)**that plane waves form a coherent mode decomposition of

*W*,

**2)**that plane waves are eigenfunctions of the Fredholm integral equation of

*W*, and

**3)**that the eigenvalues of

*W*can be computed with the spatiotemporal power spectrum

*S*. Note that there can be many other decompositions of

*W*besides plane waves.

## 3. Deterministic signal associated with a random field

*real*,

*positive*function. In this section, we introduce a new concept,

*deterministic signal associated with the random field*. This property of power spectrum carries the same second-order statistics as the original stochastic field.

**k**and

*ω*. Therefore, we can mathematically introduce a

*spectral amplitude*,

*modulus integrable*. However, the fact that

*modulus-squared integrable*in the

**k-**domain (the spatial power spectrum contains finite energy) does not necessarily ensure that

*deterministic signal*associated with the

*random field*can be defined as the inverse Fourier transform of

*V*

**is the spatial volume of interest and**

_{r}*V*

**is the 3D domain of the wavevector. With this definition of the deterministic signal, the fourth order stochastic wave equation (Eq. (36b)) can be reduced to the second order deterministic wave equationwhere**

_{k}*deterministic*wave equation, i.e.Comparing our original

*stochastic*wave equation (see Appendix) with Eq. (10), it is clear that the only difference is replacing the source field with its

*deterministic signal*, which in turn requires that we replace the stochastic propagating field with its deterministic counterpart.

*autocorrelation*Γ

*of*

_{U}*V*, or its

_{U}*spectrum*

*V*itself. By the method of constructing the deterministic signal

_{U}*V*associated with the random field

_{U}*U*, we ensure their respective autocorrelation functions are equal,In other words, the fictitious deterministic signal has identical second order statistics with the original field.

*spectral phase*. Any arbitrary phase (random or deterministic),

*ϕ*, used to construct a complex signal,

*S*(

*ω*), and a light pulse of the same spectrum have an identical respective deterministic signal, because their temporal correlations are the same. Not surprisingly, both short pulses and broadband CW light have been successfully used for low-coherence interferometry and coherence gating [13]. Spatially, a focused beam and a random field distribution of the same spatial power spectrum,

*S*(

**k**), have identical spatial correlations. For this reason, a speckle (random) field and a focused (deterministic) field of the same spatial spectrum have the same sectioning capabilities [14

14. D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett. **33**(16), 1819–1821 (2008). [CrossRef] [PubMed]

8. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. **38**(1), 3–6 (1981). [CrossRef]

## 4. Propagation of field coherence

### 4.1. Propagation of coherence from primary sources

*van Cittert-Zernike theorem*which establishes the spatial autocorrelation of the field radiated in the

*far-zone*by a completely incoherent source (Fig. 1).

*mutual intensity*, defined asIn Eq. (12), the angular brackets indicate ensemble averaging over a certain

*area*of interest (we are interested in the field distribution in a plane,

**r**

_{1,2}∈

**ℜ**). This function

^{2}*J*describes the spatial similarity (autocorrelation) of the field at a given instant,

*t*, and it has been used commonly in statistical optics (see, e.g., [2].). The theorem establishes a relationship between

*J*at the source plane and that of the field in the far zone. Such propagation of correlations has been described in detail by Mandel and Wolf [1]. Here, we derive the coherence area in the far-zone using the concept of the

*deterministic signal*associated with a random field, as follows.

*mutual intensity*,

*J*, is the

*spatiotemporal correlation function*introduced in the Appendix, evaluated at time delay

*τ*= 0,Using the

*central ordinate theorem*, the cross-correlation function evaluated at

*τ*= 0 is equivalent to the cross spectral density integrated over all frequencies,Therefore, we can obtain

*J*(

**ρ**) via the spatiotemporal power spectrum,

*S*(

**k**,

*ω*), followed by the Fourier transform with respect to

**k**and an integration over

*ω*.

*deterministic*wave equation, satisfied by the

*deterministic signal, V*associated with the random field,

_{U},*U*,where

*V*and

_{U}*V*are the

_{s}*deterministic signals*associated with the propagating field,

*U,*and a planar source field,

*s*, respectively. Therefore,

*z*, described in Eq. (16) by

*δ*(

*z*). By Fourier transforming Eq. (16), we readily obtain the solution in the

*z*and

*inward*) term,

*k*, we obtain the field

_{z}*z*, which is known as the

*plane wave decomposition*or Weyl’s formula (see, e.g., Section 3.2.4. in [1].),The modulus squared on both sides in Eq. (20), yields a

*z*-independent relation in terms of the respective power spectra,

*ω*, i.e.,

*far zone*of the source, which implies that

*θ*, because

*low-pass filter.*The farther the distance from the source, the smaller the solid angle Ω and, thus, the larger the coherence area.

### 4.2. Propagation of coherence from secondary sources

*secondary source*. Starting with the

*deterministic*Helmholtz equation, the secondary source term can be separated to the right hand side.where

**k**-domain. Considering only the positive spatial frequency component associated with

**k**-vector, (

*f*(

*z*), with a complex exponential yields a simple result, namely,

*first order Born scattering solution*for an arbitrary illumination field. Equation (29) allows us to calculate, at any plane

*z,*the power spectrum of the scattered field, which equals that of the respective deterministic signal,

*spatial light interference microscopy*(SLIM) to obtain

*quantitative phase images*of a tissue slice. SLIM provides the quantitative information about the optical path-length induced by the sample with 0.3 nm spatial sensitivity [15

15. Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express **19**(2), 1016–1026 (2011). [CrossRef] [PubMed]

*θ*, we simply make the substitutions

16. R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. **36**(21), 4209–4211 (2011). [CrossRef] [PubMed]

## 5. Summary and discussion

**1)**We first represent the coherence mode decomposition (CMD) in the wave-vector domain to prove that the plane waves can be used as an eigenfunction basis of the cross spectral density associated with statistically homogeneous fields.

**2)**We introduced the concept of a

*deterministic signal*associated with a random field and showed that it significantly simplifies calculations of second order correlations.

**3**) We described spatial and temporal coherence in terms of the second order statistics (variance) of the spatial and temporal power spectra. Thus, for an arbitrary stochastic field, we can define a temporal bandwidth and coherence time for each spatial frequency (wavevector

**k**) component and, vice versa, a spatial correlation for each temporal frequency

*ω*.

**4)**We reviewed the stochastic wave equation in the Appendix and, for wide-sense stationary and statistically homogeneous fields, we solved this equation in the (

**k**,

*ω*) domain. Essentially, fourth order differential equations in field correlations can be replaced by second order differential equations for deterministic signals, which are defined via a Fourier transform of the

*spectral amplitude.*These signals do not contain information about the

*spectral phase*associated with the field. For example, the deterministic signal representation cannot make the distinction between a focused beam and a speckle field distribution with the same spatial bandwidth, or a light pulse versus a continuous wave field of the same temporal bandwidth. Therefore, it is important to note that the deterministic signal solution should only be used to generate the power spectrum (or autocorrelation) of the propagating field. From this power spectrum, first order (mean frequency) and second order (variance) statistics can be calculated both spatially and temporally, i.e., one can study how coherence changes upon propagation.

**5)**In Section 4, we applied the deterministic signal associated with a random field to derive a well-known result of the van Cittert-Zernike’s theorem, e.g., the field emitted by a spatially incoherent source gains coherence upon propagation. First we established that the

*mutual intensity,*a quantity that is traditionally used for describing spatial coherence in a plane, is merely the frequency averaged cross-spectral density. This result allows us to easily calculate propagation of field correlations directly in the frequency (

**k**,

*ω*) domain.

**6)**If one is only interested to know the spatial and temporal variances, as measures of spatial and temporal coherence, we show that this second order statistics can be calculated straight from the wave equation in the frequency domain [e.g., Eq. (29)]. We illustrated this approach with correlations of fields propagating from primary and secondary sources.

## Appendix

## A1. Stochastic wave equation

*s*. We start with the scalar wave equation that has this random source as the driving term,The random source signal,

*s*, as introduced in Section A1, can be regarded as a realization of the fluctuating source field (

*U*is the complex analytic signal associated with the real propagating field). For generality, here we consider a 3D spatial field distribution,

*stochastic differential equation*. Notoriously, Langevin introduced such an equation (the

*Langevin equation*) to describe Brownian motion of particles [20]. The key difference with respect to the

*deterministic*wave equation is that the field

*s*in Eq. (32) does not have a prescribed form, i.e., we cannot express the source field via an analytic function. Instead, it is known only through average quantities, e.g. the autocorrelation function or, equivalently, the power spectrum. According to the focus of this paper, we assume the source field to be stationary and statistically homogeneous at least in the wide sense.

*autocorrelation of U*and not

*U*itself. In order to achieve this, we calculate the spatiotemporal autocorrelation of Eq. (32) on both sides (see, Section 4.4. in Mandel and Wolf [1])

**r**,

_{s}is the spatiotemporal autocorrelation function of

*s*. Since we assumed wide sense stationarity and statistical homogeneity, which gives Γ

_{s}dependence only on the differences

**ρ**and

*τ*, all the derivatives in Eq. (33) can be taken with respect to the shifts, i.e. (see pp. 194 in Ref [1].)After these simplifications, Eq. (33) can be re-written aswhere Γ

*is the spatiotemporal autocorrelation of*

_{U}*U*,

*fourth order*differential equation that relates the autocorrelation of the propagating field,

*U*, with that of the source,

*s*. From the Wiener-Khintchine theorem, we know that both Γ

*and Γ*

_{U}*have Fourier transforms, which are their respective power spectra,*

_{s}*S*and

_{U}*S*. Therefore, we can solve this differential equation by Fourier transforming it with respect to both

_{s}**ρ**and

*τ*, In Eq. (36a), we used the differentiation property of the Fourier transform,

*S*, with respect to the spectrum of the source,

_{U}*S*. Note that here the function

_{s}*transfer function*), which incorporates all the effects of free space propagation. Because the free space is isotropic, the transfer function is also isotropic, i.e., it depends only on the magnitude of the wavevector,

## A2. Coherence time and area

*plane*. The coherence time,

*τ*

_{c}, and coherence area,

*A*

_{c}, describe the spread (standard deviation) of the autocorrelation function,

*τ*and

**ρ**, respectively. Due to the

*uncertainty relation*,

*τ*

_{c}and

*A*

_{c}are inversely proportional to the bandwidths of their respective power spectra,where

*isotropic,*meaning that the spatial coherence at a plane is characterized by a scalar function,

*A*

_{c}. If this is not the case, i.e., when the field statistics depends on direction, the coherence area is no longer sufficient and the concept must be generalized to a tensor quantity, of the form,

**-**dependent coherence time is that each plane wave component of the field can have a specific temporal correlation and, thus,

*coherence time,*τ c ( k ⊥ ) = 1 / Δ ω ( k ⊥ ) . Conversely, each monochromatic component can have a particular spatial correlation and, thus,

*coherence area,*A c ( ω ) = 1 / Δ k 2 ( ω ) .

*averaged*with respect to these variables, such that they become constant, Equation (39a) yields a coherence time,

*τ*

_{c}as a function of

**k**or vice-versa; we implicitly assume averaging of the form in Eq. (39a) and (39b).

## Acknowledgments

## Reference and links

1. | L. Mandel and E. Wolf, |

2. | J. W. Goodman, |

3. | G. Popescu, |

4. | Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett. |

5. | R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. |

6. | M. Born and E. Wolf, |

7. | E. Wolf, |

8. | E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. |

9. | E. Wolf, “New theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. |

10. | E. Wolf, “New theory of partial coherence in the space-frequency domain. 2. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A |

11. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

12. | T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. |

13. | C. A. Puliafito, M. R. Hee, J. S. Schuman, and J. G. Fujimoto, |

14. | D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett. |

15. | Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express |

16. | R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett. |

17. | J. Shamir, “Optical Systems and Processes, Vol,” PM65 of the SPIE Press Monographs (SPIE, 1999). |

18. | P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris) |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(030.1670) Coherence and statistical optics : Coherent optical effects

(030.6600) Coherence and statistical optics : Statistical optics

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(290.5825) Scattering : Scattering theory

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: April 25, 2013

Revised Manuscript: July 31, 2013

Manuscript Accepted: August 1, 2013

Published: August 29, 2013

**Virtual Issues**

Vol. 8, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Taewoo Kim, Ruoyu Zhu, Tan H. Nguyen, Renjie Zhou, Chris Edwards, Lynford L. Goddard, and Gabriel Popescu, "Deterministic signal associated with a random field," Opt. Express **21**, 20806-20820 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-18-20806

Sort: Year | Journal | Reset

### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), pp. xxvi, 1166 p.
- J. W. Goodman, Statistical Optics, Wiley classics library ed., Wiley classics library (Wiley, 2000), pp. xvii, 550 p.
- G. Popescu, Quantitative Phase Imaging of Cells and Tissues, McGraw-Hill biophotonics (McGraw-Hill, 2011), p. 385.
- Z. Wang and G. Popescu, “Quantitative phase imaging with broadband fields,” Appl. Phys. Lett.96(5), 051117 (2010). [CrossRef]
- R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th expanded ed. (Cambridge University Press, 1999).
- E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007), pp. xiv, 222 p.
- E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38(1), 3–6 (1981). [CrossRef]
- E. Wolf, “New theory of partial coherence in the space-frequency domain. 1. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am.72(3), 343–351 (1982). [CrossRef]
- E. Wolf, “New theory of partial coherence in the space-frequency domain. 2. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A3, 76–85 (1986). [CrossRef]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254(5035), 1178–1181 (1991). [CrossRef] [PubMed]
- T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys.3(2), 129–134 (2007). [CrossRef]
- C. A. Puliafito, M. R. Hee, J. S. Schuman, and J. G. Fujimoto, Optical Coherence Tomography of Ocular Diseases (Slack, Inc., 1995).
- D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett.33(16), 1819–1821 (2008). [CrossRef] [PubMed]
- Z. Wang, L. J. Millet, M. Mir, H. Ding, S. Unarunotai, J. A. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express19(2), 1016–1026 (2011). [CrossRef] [PubMed]
- R. Zhu, S. Sridharan, K. Tangella, A. Balla, and G. Popescu, “Correlation-induced spectral changes in tissues,” Opt. Lett.36(21), 4209–4211 (2011). [CrossRef] [PubMed]
- J. Shamir, “Optical Systems and Processes, Vol,” PM65 of the SPIE Press Monographs (SPIE, 1999).
- P. Langevin, “On the theory of Brownian motion,” C. R. Acad. Sci. (Paris)146, 530 (1908).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.