## Abstract

Partial spatial coherence is a fundamental concept in optical systems. Theoretically, the normalized mutual coherence function gives a quantitative measure for partial spatial coherence regardless of the spectral nature of the radiation. For narrowband light the degree of spatial coherence can be measured in terms of the fringe modulation in the classic Young’s two-pinhole interferometer. Though not commonly appreciated, with polychromatic radiation this is not the case owing to the wavelength dependence of diffraction. In this work we show that with a modified two-beam interferometer containing an achromatic Fresnel transformer the degree of spatial coherence is again related to the visibility of intensity fringes in Young’s experiment for any polychromatic light. This result, which is demonstrated both theoretically and experimentally, thus restores the usefulness of the two-pinhole interferometer in the measurement of the spatial coherence of light beams of arbitrary spectral widths.

© 2013 Optical Society of America

## 1. Introduction

The concept of optical coherence is at the root of much of the science and applications of light [1]. In scalar description, the cornerstone of optical coherence theory is the mutual coherence function [2]. Under certain conditions this quantity can be measured with a Young’s two-pinhole interference setup that historically was instrumental in bringing into evidence the wave nature of light and later has contributed greatly especially to the development of statistical optics and quantum physics [3]. The importance of Young’s interferometer in the measurement of the coherence of light was first realized by Zernike who, in 1938, introduced the ‘degree of coherence’ in terms of the visibility of the intensity fringes produced in this interference arrangement [4]. However, Zernike considered only equal-time coherence of the light at the pinholes. A full description of classical optical coherence theory including time delays was subsequently introduced by Wolf [5] and the corresponding quantum coherence theory was formulated by Glauber [6, 7].

The mutual coherence function provides a quantitative measure for the space–time coherence of light regardless of the spectral width of the radiation. However, the intimate relation between the phase correlations of the field at the pinholes and the visibility of the intensity fringes in Young’s experiment [8] relies on the illumination being quasi-monochromatic, i.e., the frequency bandwidth has to be narrow and the path differences sufficiently short (see [1], Sec. 4.3.1). This limitation, which often is not explicitly stated, thus severely restricts the usefulness of Young’s interferometer in coherence measurements as it excludes all broadband radiation. Young’s two-pinhole interferometer is diffractive in nature and while the interference pattern is periodic at each frequency, the period scales linearly as a function of the wavelength. Therefore polychromatic illumination leads to a mix of colored fringes on the observation screen and breaks down the connection between the complex degree of coherence and the two-beam interference fringes in the space–time domain.

Nonetheless, several types of broadband fields that are effectively stationary in time, such as thermal light [1], white-light LED and superluminescent diode (SLD) emission [9], and quasi-stationary supercontinuum radiation [10,11], are increasingly employed in various optical applications, making the characterization of their spatial coherence important in practice [12]. It is possible to measure the degree of spatial coherence of broadband radiation indirectly, for instance by using a spectrally resolved Young’s interferometer. In such an instrument, a spectrometer with cylindrical optics would disperse the fringes in the direction perpendicular to the pinholes. By observing the visibility across lines parallel with the pinholes, corresponding to different frequencies, values for the spectral degree of coherence in the space–frequency domain can be acquired. Combined with the knowledge of the spectra at the pinholes, the relation by Friberg and Wolf [13] may then be applied to numerically calculate the time–domain complex degree of coherence.

Diffraction is a universal dispersive property of light that can be compensated for. In this paper, we put forward and demonstrate an achromatic Young’s two-pinhole interferometer that enables the measurement of the degree of spatial coherence directly from the intensity fringes even when the radiation is polychromatic. The approach is based on the achromatization of the fringe pattern such that the period of the intensity fringes is the same for all wavelengths. This is done in practice by using an achromatic Fresnel-transform system that consists of a diffractive lens and an achromatic doublet set in a specific geometry, a system resembling a previously introduced Fourier-achromat [14–16] or Fresnel-achromat [17] component. Our configuration thus performs the Friberg–Wolf integration optically, and allows one to measure the spatial coherence of stationary light fields of arbitrary spectral widths as in the conventional quasi-monochromatic case. Pulsed and electromagnetic fields are briefly addressed in the conclusions.

## 2. Measurement principle and theory

#### 2.1. Polychromatic spatial coherence

Denoting the randomly fluctuating, statistically stationary, polychromatic scalar field at two spatial points by *E*_{1}(*t*) and *E*_{2}(*t*), where *t* is the time, the mutual coherence function is defined as
${\mathrm{\Gamma}}_{12}\left(\tau \right)=\u3008{E}_{1}^{*}\left(t\right){E}_{2}\left(t+\tau \right)\u3009$, where the asterisk denotes complex conjugate and the brackets stand for time (or ensemble) average [2]. The mean optical intensity at point *j*, with *j* = (1, 2), can be taken to be
${\mathrm{\Gamma}}_{jj}\left(0\right)=\u3008{E}_{j}^{*}\left(t\right){E}_{j}\left(t\right)\u3009={I}_{j}$, and the complex degree of mutual coherence is subsequently introduced via the relation

*γ*

_{12}(

*τ*)| ≤ 1 for all wavefields. The spatial coherence at the two points then is quantitatively represented by |

*γ*

_{12}(0)|, regardless of whether the optical radiation is quasi-monochromatic or polychromatic [1, 2]. For strictly monochromatic fields |

*γ*

_{12}(0)| = 1.

Let us now examine the behavior of the interference fringes when polychromatic illumination is incident onto the traditional Young’s two-pinhole interferometer. The situation is illustrated in Fig. 1(a). Waves from the two pinholes interfere on the observation screen to form periodic intensity fringes at each frequency component of the radiation. The period of the interference fringes depends on the pinhole separation and increases linearly with the wavelength. It is thus obvious that with polychromatic light the fringe pattern will be colored and spatially smeared out. The visibility changes along the lateral position on the observation screen, even when spatially fully coherent light sources are used, making the measurement of spatial coherence impossible.

#### 2.2. Modified Young’s two-pinhole interferometer

If we assume that an achromatic Fourier- or Fresnel-transform (AFT) system is placed between the two screens in Young’s interferometer, the situation is changed dramatically, as is sketched in Fig. 1(b). The polychromatic fields *E*_{1} and *E*_{2} at the two pinholes consist of distributions of random spectral components *V*_{1}(*ω*) and *V*_{2}(*ω*), where *ω* is the angular frequency. The action of the AFT system is formally such that it converts the spherical waves originating from the pinholes with amplitudes proportional to *V*_{1} and *V*_{2} into plane waves which emerge from the system at specified angles *θ* that depend on *ω*. Hence the transverse profiles of these waves at the observation plane can be expressed as

*α*is a proportionality factor,

*k*(

_{x}*ω*) = (

*ω*

*/c*)sin

*θ*(

*ω*), with

*c*being the vacuum speed of light, and

*x*is the transverse position in the detection plane. The spectral density [1] on the observation screen then is

*S*(

*x*,

*ω*) = 〈|

*V*

_{1}(

*x*,

*ω*) +

*V*

_{2}(

*x*,

*ω*)|

^{2}〉, where the angle brackets now denote ensemble average. This quantity may readily be shown to take on the form

*S*

^{(j)}(

*ω*) = |

*α*|

^{2}〈|

*V*(

_{j}*ω*)|

^{2}〉 = |

*α*|

^{2}

*S*(

_{j}*ω*),

*j*= (1, 2), denotes the spectral density at the screen when only pinhole

*j*is open, and

*d*is the ensuing constant fringe period. Hence the modified Young’s interferometer produces, under polychromatic illumination, intensity fringes with a given period

*d*, if the AFT system is designed so that the exit angles of the plane waves at different frequencies satisfy the condition sin

*θ*(

*ω*) =

*πc*/

*ωd*.

#### 2.3. Polychromatic fringe visibility

We next assume, for simplicity, that the spectral response of the detector at the observation screen is constant within the frequency band involved. In such a case, the detected optical intensity may be expressed simply as

Integrating both sides of Eq. (3) with respect to*ω*, and using the condition of Eq. (5), reveals that the intensity profile on the observation plane is of the form

*I*

^{(j)},

*j*= (1, 2), denote the optical intensities when only pinhole

*j*is open and

*s*

^{(j)}(

*ω*) =

*S*

^{(j)}(

*ω*)/

*I*

^{(j)}are the corresponding normalized spectral densities. Since the period of the cosine term in Eq. (7) remains unchanged when

*ω*varies, and assuming that

*α*

_{12}(

*ω*) is effectively constant as a function of

*ω*, the interference pattern

*I*(

*x*) will have clear minima and maxima, as is illustrated in Fig. 1(b). The extreme values depend on the spectral coherence |

*μ*

_{12}(

*ω*)|, whereas the phase term

*α*

_{12}(

*ω*) is related to the transverse shift of the spectral interference pattern [1, 2].

Using the theorem by Friberg and Wolf [13], the relationship between the time-domain complex degree of coherence *γ*_{12}(*τ*) and the spectral degree of coherence *μ*_{12}(*ω*), at a pair of points, may be cast in the form

*s*(

_{j}*ω*) =

*S*(

_{j}*ω*)/

*I*is the normalized spectral density at position

_{j}*j*,

*j*= (1, 2). Taking these points as the pinhole locations and using the fact that

*s*

^{(j)}(

*ω*) =

*s*(

_{j}*ω*), Eq. (8) enables the intensity pattern on the observation screen to admit the form

*β*

_{12}(

*τ*) = arg[

*γ*

_{12}(

*τ*)]. This expression resembles closely the classical formula for the diffraction pattern in a Young’s two-pinhole experiment [1, 4]. Denoting the maximum and minimum intensities in close vicinity of a given position by

*I*

_{max}and

*I*

_{min}, respectively, we find from Eq. (9) that the visibility of the interference fringes is

*I*

^{(j)}= |

*α*|

^{2}

*I*,

_{j}*j*= (1, 2). If the optical intensities

*I*

_{1}and

*I*

_{2}at the pinholes are the same, the fringe visibility is equal to the modulus of the degree of spatial coherence. We also see from Eq. (9) that the phase of the degree of spatial coherence is associated with the transverse position of the fringes. To conclude, the relation between the visibility

*V*and the degree of spatial coherence

*γ*

_{12}(0) for polychromatic light of any spectral width in the modified Young’s interferometer is exactly of the same form as for quasi-monochromatic light in the conventional two-pinhole interferometer in the neighborhood of

*x*= 0.

## 3. Experimental setup

We have confirmed the theoretical predictions by experiments. Our measurement setup, illustrated in Fig. 2, consists of three laser sources, a Young’s two-pinhole interferometer, an achromatic Fresnel-transform system, and a CCD detector. Radiations from the lasers which separately can be regarded as quasi-monochromatic are combined into one polychromatic beam and directed onto the pinholes thus creating an interference pattern on the observation screen at each laser frequency component. The periods of the intensity fringes, which otherwise would be different at each color, are corrected by the achromatic Fresnel transformer that is designed and constructed to produce frequency-independent two-dimensional Fresnel transforms. The patterns in the observation plane are then detected by the CCD array.

The achromatic Fresnel-transform system, shown in detail in Fig. 3, consists of an achromatic doublet and a diffractive lens. The achromatic lens has a 150 mm focal length and is placed directly behind the pinholes. The diffractive lens is 6 mm in diameter and was constructed to have a focal length of 89.94 mm at the design wavelength of 571 nm. The diffractive element profile, the central part of which is illustrated in Fig. 4, was optimized for high efficiency and also designed to accommodate the frequency response of the detector, a CCD line camera with a 7 *μ*m pixel width. The overall spectral sensitivity is shown in Fig. 5 along with the diffractive lens efficiency and CCD sensitivity. The element was fabricated by electron beam lithography and reactive ion etching.

To further examine the performance of the achromatic Fresnel-transform system we need to study the wavelength dependence of the transverse scale factor. Assuming that two point sources, in this case the pinholes, are separated by a distance *a*, the scale of the sinusoidal output pattern is given by

*B*(

*λ*) is an element of the wavelength-dependent system matrix

*M*(see Fig. 3). The variation of Λ(

*λ*) in our system is shown in Fig. 6. While this is not the best possible performance (ideally the factor would remain unchanged as a function of the wavelength), it is sufficient in the scope of our work. Better results could be obtained with more complex AFT systems. It is worthwhile to note that the visibility is better in a Fresnel system where any zero-order stray light will be spread out, while in a Fourier system it will be focused in the observation plane.

## 4. Experimental results

We first studied the fundamental issue with the conventional (unmodified) Young’s two-pinhole interferometer, namely the inequality of the intensity fringe periods at different wavelengths. With three separate lasers forming a red-green-blue (RGB) source, the light is effectively spatially fully coherent at each frequency, i.e., we have |*μ*_{12}(*ω*)| = 1 at all *ω* at the pinholes. The laser beams are naturally mutually uncorrelated, so their intensities simply add up. Now, without the achromatization, the superposition of the diffraction patterns of the three laser beams of different colors, illustrated schematically in Fig. 1(a) (experimental result not shown), demonstrates how the frequency scaling of the interference fringe period affects the measurement. Visibility is low and, even at best, only the most central fringes could be used to obtain some information about spatial coherence.

We next proceed to the results obtained with the modified Young’s interferometer described above and compare them to the theoretical predictions. The theoretical analysis was performed by standard matrix optics with the help of Collins’ formula [18]. Now, if the laser beams are well aligned, we should get a nearly 100 percent visibility with our achromatic setup. The experimentally obtained value for the visibility *V*, measured using the first few fringe minima and maxima, is approximately 93 percent. Since the intensities at the two pinholes are essentially equal, the degree of spatial coherence has by Eq. (10) the same value with the visibility, i.e., |*γ*_{12}(0)| ≈ 0.93. As can be observed from Fig. 7, the experimental results [Fig. 7(a)] correspond quite well to the theoretical curves [Fig. 7(b)], especially recalling the possible error sources such as those in the fabrication of the diffractive lens, in laser and detector positioning, and possible scattered background radiation. It should also be noted that more accurate results could be obtained with involved and better optimized AFT systems.

To gain further insight into polychromatic spatial coherence, we subsequently placed a phase-shifting element before the double pinhole such that light passing pinhole 1 experiences a greater phase shift than light traversing pinhole 2. We used a simple SiO_{2} plate covering both pinholes, but a sufficiently large indentation (dent) of 930 nm depth was fabricated on the plate by reactive ion etching and positioned in front of the second pinhole. Such an arrangement creates a wavelength-dependent phase delay between the light fields at the two openings. Because we still have the same normalized spectra *s*(*ω*) at the pinholes and |*μ*_{12}(*ω*)| = 1 at each wavelength in the RGB system, in view of Eq. (8) we obtain

*l*runs over all three frequencies. Setting

*α*

_{12}(

*ω*

_{1}) =

*α*

_{0}, we may calculate that

*α*

_{12}(

*ω*

_{2}) =

*α*

_{0}− 0.6702 and

*α*

_{12}(

*ω*

_{3}) =

*α*

_{0}− 1.5139, where the angular frequencies

*ω*

_{1},

*ω*

_{2}, and

*ω*

_{3}correspond the wavelengths

*λ*

_{1}= 473 nm,

*λ*

_{2}= 532 nm, and

*λ*

_{3}= 633 nm. Since now

*α*

_{12}(

*ω*) is different for each frequency, the transverse locations of the spectral interference patterns differ and hence the resulting visibility is less than unity. Figure 8 shows this effect both experimentally [Fig. 8(a)] and theoretically [Fig. 8(b)]. In this particular example, we have |

_{l}*γ*

_{12}(0)| ≈ 0.82 and |

*γ*

_{12}(0)| ≈ 0.61 for the theoretical and experimental approaches, respectively. The deviation between the values comes from various sources like fabrication errors and difficulties in positioning the gap in the laser beam but, nevertheless, the experiment shows a clear reduction of the degree of spatial coherence for the polychromatic radiation as predicted by the theory.

## 5. Conclusions

The partial spatial coherence of light sources and radiation fields is a fundamental characteristic in most optical systems. Theoretically the degree of spatial coherence is given for stationary scalar beams of arbitrary spectral widths by the normalized version of the mutual coherence function, but in the measurement practical difficulties arise with polychromatic light due to the dispersive nature of diffraction and interference. However, the spectral variation of diffraction can be balanced by suitable chromatic elements of physical optics. The main contribution of our work is that we identify the problems associated with the spatial coherence measurement of polychromatic radiation in a Young’s double-slit instrument and rectify them by means of an achromatic Fresnel-transform system. We emphasize that the principle of the modified Young’s two-slit interferometer is universal, whereas the fidelity of the results then depends on the actual practical design and implementation of the elements. The modification thereby restores the usefulness of the classic Young’s two-pinhole interferometer for the quantitative measurement of the spatial coherence of arbitrary broadband light beams. Our experimental results agree well with the theoretical predictions.

Besides stationary polychromatic light, short pulses such as those from Q-switched or mode-locked lasers or fs-scale supercontinuum radiation also exhibit broad spectral distributions. For a train of fluctuating pulses the degree of spatial coherence can be measured in a similar manner where the ensemble averaging then takes over the sequence of individual pulses. For electromagnetic (vectorial) partially coherent and partially polarized light the situation is considerably more involved. In two-beam interference with random vector fields not only the intensity but also the polarization state shows modulation on the observation screen and both modulation contrasts must be fully accounted for to obtain the degree of spatial coherence [19, 20]. In principle, however, these modulations can be measured using a polarization-insensitive Fresnel transformer in Young’s two-pinhole interferometer.

## Acknowledgments

This work was partially funded by the Academy of Finland (projects 118951, 128331, and 135027).

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