## Abstract

Parrent’s criterion for the complete coherence of an optical wave field is re-examined and a new proof is obtained using Fourier–Stieltjes integrals.

© 1966 Optical Society of America

Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 56,
- Issue 6,
- pp. 739-740
- (1966)
- •doi: 10.1364/JOSA.56.000739

Parrent’s criterion for the complete coherence of an optical wave field is re-examined and a new proof is obtained using Fourier–Stieltjes integrals.

© 1966 Optical Society of America

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- G. Parrent, Opt. Acta 6, 285 (1959).See also: M. Beran and G. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

[Crossref] - S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

[Crossref] - Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).

[Crossref] - The author prefers the use of real-valued functions, although the subsequent analysis can also be effectuated using complex-valued functions.

- The braces { } denote ensemble properties.

- M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.

- Actually there is a third component termed the singular continuous component; this component is continuous but is without a spectral density at every point. Such functions do not have any physical significance and are always omitted.

- H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).

- This theorem was first stated by Kampe de Feriet (in an almost completely unknown paper) for the case of an autocorrelation function; the extension to cross-correlation functions presented here is a straightforward extension of his proof. See: M. J. Kampe de Feriet, Ann. Soc. Sci. Bruxelles, Ser. 1 59, 145 (1939).

- E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.

Dr. Emil Wolf kindly informed the author that he has reached the same conclusion concerning Ref. 1. See: C. L. Metha, E. Wolf, and A. P. Balachandran, J. Math. Phys. 7, 133 (1966).

[Crossref]

G. Parrent, Opt. Acta 6, 285 (1959).See also: M. Beran and G. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

[Crossref]

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

[Crossref]

This theorem was first stated by Kampe de Feriet (in an almost completely unknown paper) for the case of an autocorrelation function; the extension to cross-correlation functions presented here is a straightforward extension of his proof. See: M. J. Kampe de Feriet, Ann. Soc. Sci. Bruxelles, Ser. 1 59, 145 (1939).

[Crossref]

M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.

H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

[Crossref]

[Crossref]

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

[Crossref]

G. Parrent, Opt. Acta 6, 285 (1959).See also: M. Beran and G. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

[Crossref]

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, Cambridge, England, 1935), 4th ed., p. 172.

[Crossref]

[Crossref]

G. Parrent, Opt. Acta 6, 285 (1959).See also: M. Beran and G. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), p. 47.

[Crossref]

S. F. Edwards and G. Parrent, Opt. Acta 6, 367 (1959).

[Crossref]

The author prefers the use of real-valued functions, although the subsequent analysis can also be effectuated using complex-valued functions.

The braces { } denote ensemble properties.

M. S. Bartlett, Stochastic Processes, Methods and Applications (Cambridge University Press, Cambridge, England, 1956), p. 189.

Actually there is a third component termed the singular continuous component; this component is continuous but is without a spectral density at every point. Such functions do not have any physical significance and are always omitted.

H. Bohr, Almost Periodic Functions (Chelsea Publishing Co., New York, 1951).

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$$E\{{V}_{j}(t){V}_{k}(t+\tau )\}={r}_{jk}(\tau ),$$

$${r}_{jk}(\tau )={\mathit{\int}}_{-\infty}^{\infty}{e}^{i2\pi \tau f}d{\mathrm{\Lambda}}_{jk}(f),$$

$${r}_{jk}(\tau )={r}_{kj}*(-\tau ),$$

$${\mathrm{\Lambda}}_{jk}(f)={A}_{kj}*(f).$$

$${\mathrm{\Lambda}}_{jk}(f)={{\mathrm{\Lambda}}_{jk}}^{c}(f)+{{\mathrm{\Lambda}}_{jk}}^{d}(f).$$

$${\mathrm{\Lambda}}_{jk}(f)={\mathit{\int}}_{0}^{f}{\varphi}_{jk}(f)df,$$

$$\begin{array}{ll}{\mathrm{\Lambda}}_{jk}(f)\hfill & ={\text{\u2211}}_{l}{A}_{jk}(l)H(f-{f}_{l})\hfill \\ \hfill & ={\text{\u2211}}_{l}{A}_{kj}*(l)H(f-{f}_{l}).\hfill \end{array}$$

$$H(f)=\{\begin{array}{ll}0\hfill & f<0\hfill \\ \frac{1}{2}\hfill & f=0\hfill \\ 1\hfill & f>0\hfill \end{array}$$

$$d{\mathrm{\Lambda}}_{jk}(f)={\varphi}_{jk}(f)df+{\text{\u2211}}_{l}{A}_{jk}(l)\delta (f-{f}_{l}).$$

$${r}_{jk}(\tau )={{r}_{jk}}^{c}(\tau )+{{r}_{jk}}^{d}(\tau ),$$

$${{r}_{jk}}^{c}(\tau )={\mathit{\int}}_{-\infty}^{\infty}{e}^{i2\pi \tau f}{\varphi}_{jk}(f)df,$$

$${{r}_{jk}}^{d}(\tau )={\text{\u2211}}_{l}{A}_{jk}(l){e}^{i2\pi \tau fl}.$$

$$\underset{T\to \infty}{lim}\frac{1}{T}{\mathit{\int}}_{0}^{T}{|{r}_{jk}(\tau )|}^{2}d\tau =0.$$

$${{\mathit{\int}}_{0}^{T}|{{r}_{jk}}^{d}(\tau )|}^{2}d\tau =\text{\u2211}_{l,m}{A}_{jk}(l){A}_{jk}*(m){\mathit{\int}}_{0}^{T}{e}^{i2\pi ({f}_{l}-{f}_{m})\tau}d\tau .$$

$$\underset{T\to \infty}{lim}\frac{1}{T}{{\mathit{\int}}_{0}^{T}|{{r}_{jk}}^{d}(\tau )|}^{2}d\tau =\text{\u2211}_{l}{|{A}_{jk}(l)|}^{2}.$$

$$\underset{\tau \to \infty}{lim}{r}_{jk}(\tau )\sim 0({\tau}^{-1})$$

$$\underset{T\to \infty}{lim}\frac{1}{T}{\mathit{\int}}_{0}^{T}{|{{r}_{jk}}^{c}(\tau )|}^{2}d\tau =0.$$

$${\mathit{\int}}_{0}^{T}{{r}_{jk}}^{c}(\tau ){{r}_{jk}}^{d}(\tau )d\tau =\text{\u2211}_{l}{A}_{jk}(l){\mathit{\int}}_{0}^{T}{e}^{i2\pi fl\tau}{{r}_{jk}}^{c}(\tau )d\tau .$$

$$\underset{T\to \infty}{lim}\frac{1}{T}{\mathit{\int}}_{0}^{T}{{r}_{jk}}^{c}(\tau ){{r}_{jk}}^{d}(\tau )d\tau =0.$$

$$|{\gamma}_{jk}(\tau )|=|{r}_{jk}(\tau )/{r}_{jk}(0)|\equiv 1$$

$$\underset{T\to \infty}{lim}\frac{1}{T}{\mathit{\int}}_{0}^{T}{|{r}_{jk}(\tau )|}^{2}d\tau ={|{r}_{jk}|}^{2};$$

$${r}_{jk}(\tau )={\mathit{\int}}_{-\infty}^{\infty}{e}^{i2\pi f\tau}{\varphi}_{jk}(f)df+{A}_{jk}(1){e}^{i2\pi {f}_{1}\tau},$$

$${r}_{jk}(0)={\mathit{\int}}_{-\infty}^{\infty}{\varphi}_{jk}(f)df+{A}_{jk}(1).$$

$${\varphi}_{jk}(f)\equiv 0.$$

$${r}_{jk}(\tau )={A}_{jk}{e}^{i2\pi f\tau}$$

$$V=A{e}^{i2\pi f\tau};$$

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