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 OSA

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 [1416] 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 E1(t) and E2(t), where t is the time, the mutual coherence function is defined as Γ12(τ)=E1*(t)E2(t+τ), 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 Γjj(0)=Ej*(t)Ej(t)=Ij, and the complex degree of mutual coherence is subsequently introduced via the relation

γ12(τ)=Γ12(τ)I1I2.
This quantity is bounded such that 0 ≤ |γ12(τ)| ≤ 1 for all wavefields. The spatial coherence at the two points then is quantitatively represented by |γ12(0)|, regardless of whether the optical radiation is quasi-monochromatic or polychromatic [1, 2]. For strictly monochromatic fields |γ12(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.

 

Fig. 1 Measuring the spatial coherence. (a) Use of the conventional Young’s interferometer setup to measure the visibility, and thus the spatial coherence of polychromatic light (here an RGB source is considered), leads to problems. The spherical waves emerging from the pinholes interfere creating intensity fringes. However, the periods of the fringes scale as a function of the wavelength and the visibility will obviously be low as the fringes of various frequencies mix. (b) By placing an achromatic Fourier-transform system (AFT) between the pinhole screen and the detector plane, the emerging spherical waves are transformed into plane waves that arrive to the detector at a frequency-dependent angle θ(ω). Such an achromatization of the diffraction pattern results in scale-invariant intensity fringes and the visibility of the fringe pattern can be utilized to measure the spatial coherence.

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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 E1 and E2 at the two pinholes consist of distributions of random spectral components V1(ω) and V2(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 V1 and V2 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

V1(x,ω)=αV1(ω)exp[ikx(ω)x]
V2(x,ω)=αV2(ω)exp[ikx(ω)x],
where α is a proportionality factor, kx(ω) = (ω/c)sinθ(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) = 〈|V1(x, ω) + V2(x, ω)|2〉, where the angle brackets now denote ensemble average. This quantity may readily be shown to take on the form
S(x,ω)=S(1)(ω)+S(2)(ω)+2S(1)(ω)S(2)(ω)|μ12(ω)|cos[2kx(ω)xα12(ω)],
where S(j) (ω) = |α|2〈|Vj(ω)|2〉 = |α|2Sj(ω), j = (1, 2), denotes the spectral density at the screen when only pinhole j is open, and
μ12(ω)=|μ12(ω)|exp[iα12(ω)]=V1*(ω)V2(ω)S1(ω)S2(ω)
is the complex degree of spectral coherence [1, 2] between the light fields at the pinholes. It is clear from Eq. (3) that the spectral scaling is suppressed if
2kx(ω)=2ωcsinθ(ω)=2πd,
where 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 sinθ(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

I(x)=0S(x,ω)dω.
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(x)=I(1)+I(2)+2I(1)I(2)0s(1)(ω)s(2)(ω)|μ12(ω)|cos[2πx/dα12(ω)]dω,
where 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 α12(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 |μ12(ω)|, whereas the phase term α12(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 γ12(τ) and the spectral degree of coherence μ12(ω), at a pair of points, may be cast in the form

γ12(τ)=0s1(ω)s2(ω)μ12(ω)exp(iωτ)dω,
where sj(ω) = Sj(ω)/Ij is the normalized spectral density at position j, j = (1, 2). Taking these points as the pinhole locations and using the fact that s(j) (ω) = sj(ω), Eq. (8) enables the intensity pattern on the observation screen to admit the form
I(x)=I(1)+I(2)+2I(1)I(2)|γ12(0)|cos[2πx/dβ12(0)],
where β12(τ) = arg[γ12(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 Imax and Imin, respectively, we find from Eq. (9) that the visibility of the interference fringes is
V=ImaxIminImax+Imin=2I1I2I1+I2|γ12(0)|,
where we have used the fact that I(j) = |α|2Ij, j = (1, 2). If the optical intensities I1 and I2 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 γ12(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.

 

Fig. 2 Measurement setup. The beams from two diode-pumped solid-state lasers with wavelengths of 473 nm and 532 nm and a HeNe laser with the wavelength of 633 nm are combined into a single polychromatic (RGB) beam using mirrors and 50:50 non-polarizing beam splitters. The resulting beam field is directed onto the modified Young’s interferometer, where the pinhole size is 30 μm × 30 μm and their separation is 150 μm. The distance between the diffractive lens and Young’s pinholes is 60.2 mm whereas the distance from the diffractive lens to the detector is 159.8 mm.

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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 μm 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.

 

Fig. 3 The measurement setup in detail. An achromatic doublet (AC) with focal length of f1 = 150 mm is described by radii of curvature r1 = 91.62 mm, r2 = −66.68 mm, and r3 = −197.7 mm with thicknesses t1 = 5.7 mm in crown material N-BK7 and t2 = 2.2 mm in flint material SF5. The doublet is placed directly in front of Young’s double pinholes (DP) with slit size of 30 μm and separation of a = 150 μm. Distance d1 = 60.2 mm separates the achromatic lens and the diffractive lens (DL) with focal length of f2 = 89.940 mm at design wavelength 571 nm. The detector (CCD) is located at a distance of d2 = 159.8 mm from the diffractive lens. The system contained within the colored area is represented by system matrix M which is calculated using ray optics.

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Fig. 4 Profile of the diffractive lens. Only the first few periods are presented since the physical width w of the lens is 6 mm. The structure provides the diffraction efficiency as shown in Fig.5.

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Fig. 5 The spectral efficiency curve of the diffractive lens, the spectral response curve of the CCD detector, and the total efficiency curve. It is seen that the total efficiency, a product of the efficiencies of the CCD and the diffractive lens at each wavelength, is sufficiently constant within the spectral regime of 473 nm – 633 nm.

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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

Λ(λ)=2aλ|B(λ)|,
where 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.

 

Fig. 6 Wavelength dependence of the transverse scale factor Λ(λ) of the sinusoidal pattern emerging from the pinholes with achromatization done by AFT. Point sources are separated by distance a = 150 μm.

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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 |μ12(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., |γ12(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.

 

Fig. 7 (a) Experimental results for the interference pattern of polychromatic light measured with a CCD line detector. The modified Young’s interferometer is illuminated with three laser sources (RGB) with the spatial degree of coherence |μ12(0)| ≈ 1. The achromatic Fresnel-transform system adjusts the two-pinhole intensity fringe periods to be the same for each wavelength. Very high visibility is obtained and the degree of spatial coherence for the polychromatic radiation in this case will be |γ12(0)| ≈ 0.93. (b) Corresponding theoretical curves for the interference pattern of polychromatic light with a simulated system. Blue: 473 nm, Green: 532 nm, Red: 633 nm, and Black: superposition.

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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 SiO2 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 |μ12(ω)| = 1 at each wavelength in the RGB system, in view of Eq. (8) we obtain

γ12(0)=ls(ωl)exp[iα12(ωl)],
where l runs over all three frequencies. Setting α12(ω1) = α0, we may calculate that α12(ω2) = α0 − 0.6702 and α12(ω3) = α0 − 1.5139, where the angular frequencies ω1, ω2, and ω3 correspond the wavelengths λ1 = 473 nm, λ2 = 532 nm, and λ3 = 633 nm. Since now α12(ωl) 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 |γ12(0)| ≈ 0.82 and |γ12(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.

 

Fig. 8 (a) Experimental results for phase-shifted visibility. Incoming polychromatic (RGB) light experiences a wavelength-dependent phase delay between the two Young’s pinholes. This leads to a lateral displacement of the interference pattern that is different in magnitude for each wavelength. As a result, the total visibility and thus the degree of spatial coherence is reduced to |γ12(0)| ≈ 0.61. (b) Corresponding theoretical curves for phase-shifted visibility. The degree of coherence decreases to |γ12(0)| ≈ 0.81. Blue: 473 nm, Green: 532 nm, Red: 633 nm, and Black: superposition.

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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).

References and links

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

2. M. Born and E. Wolf, Principles of Optics, 7th exp. edition (Cambridge University Press, 1999).

3. E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt. 50, 251–273 (2007). [CrossRef]  

4. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938). [CrossRef]  

5. E. Wolf, “A macroscopic theory of diffraction and interference of light from finite sources — II. Fields with spectral range of arbitrary width,” Proc. R. Soc. London A 230, 246–265 (1955). [CrossRef]  

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

7. R. J. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).

8. J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37, 151–153 (2012). [CrossRef]   [PubMed]  

9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley-Interscience, New York, 2007).

10. R. R. Alfano, ed., The Supercontinuum Laser Source: Fundamentals with Updated References, 2nd ed. (Springer, 2006).

11. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B 28, 2301–2309 (2011). [CrossRef]  

12. C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt. 46, 1763–1774 (1999).

13. A. T. Friberg and E. Wolf, “Relationship between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett. 20, 623–625 (1995). [CrossRef]   [PubMed]  

14. E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt. 37, 6164–6173 (1998). [CrossRef]  

15. D. Faklis and G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988). [CrossRef]   [PubMed]  

16. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981). [CrossRef]   [PubMed]  

17. J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm. 136, 297–305 (1997). [CrossRef]  

18. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]  

19. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006). [CrossRef]   [PubMed]  

20. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006). [CrossRef]   [PubMed]  

References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  2. M. Born and E. Wolf, Principles of Optics, 7th exp. edition (Cambridge University Press, 1999).
  3. E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt.50, 251–273 (2007).
    [CrossRef]
  4. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica5, 785–795 (1938).
    [CrossRef]
  5. E. Wolf, “A macroscopic theory of diffraction and interference of light from finite sources — II. Fields with spectral range of arbitrary width,” Proc. R. Soc. London A230, 246–265 (1955).
    [CrossRef]
  6. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130, 2529–2539 (1963).
    [CrossRef]
  7. R. J. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).
  8. J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett.37, 151–153 (2012).
    [CrossRef] [PubMed]
  9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley-Interscience, New York, 2007).
  10. R. R. Alfano, ed., The Supercontinuum Laser Source: Fundamentals with Updated References, 2nd ed. (Springer, 2006).
  11. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B28, 2301–2309 (2011).
    [CrossRef]
  12. C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).
  13. A. T. Friberg and E. Wolf, “Relationship between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett.20, 623–625 (1995).
    [CrossRef] [PubMed]
  14. E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt.37, 6164–6173 (1998).
    [CrossRef]
  15. D. Faklis and G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett.13, 4–6 (1988).
    [CrossRef] [PubMed]
  16. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt.20, 2017–2025 (1981).
    [CrossRef] [PubMed]
  17. J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
    [CrossRef]
  18. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am.60, 1168–1177 (1970).
    [CrossRef]
  19. T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett.31, 2208–2210 (2006).
    [CrossRef] [PubMed]
  20. T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett.31, 2669–2671 (2006).
    [CrossRef] [PubMed]

2012

2011

2007

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt.50, 251–273 (2007).
[CrossRef]

2006

1999

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

1998

1997

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

1995

1988

1981

1970

1963

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

1955

E. Wolf, “A macroscopic theory of diffraction and interference of light from finite sources — II. Fields with spectral range of arbitrary width,” Proc. R. Soc. London A230, 246–265 (1955).
[CrossRef]

1938

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica5, 785–795 (1938).
[CrossRef]

Andrés, P.

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt.37, 6164–6173 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th exp. edition (Cambridge University Press, 1999).

Climent, V.

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt.37, 6164–6173 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

Collins, S. A.

Danner, M.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

Drexler, W.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

Faklis, D.

Fercher, A. F.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

Fernández-Alonso, M.

Friberg, A. T.

Genty, G.

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130, 2529–2539 (1963).
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R. J. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).

Hitzenberger, C. K.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

Lancis, J.

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt.37, 6164–6173 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Morris, G. M.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley-Interscience, New York, 2007).

Setälä, T.

Surakka, M.

Tajahuerce, E.

E. Tajahuerce, V. Climent, J. Lancis, M. Fernández-Alonso, and P. Andrés, “Achromatic Fourier transforming properties of a separated diffractive lens doublet: theory and experiment,” Appl. Opt.37, 6164–6173 (1998).
[CrossRef]

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley-Interscience, New York, 2007).

Tepichín, E.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
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E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt.50, 251–273 (2007).
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A. T. Friberg and E. Wolf, “Relationship between the complex degrees of coherence in the space-time and in the space-frequency domains,” Opt. Lett.20, 623–625 (1995).
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E. Wolf, “A macroscopic theory of diffraction and interference of light from finite sources — II. Fields with spectral range of arbitrary width,” Proc. R. Soc. London A230, 246–265 (1955).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th exp. edition (Cambridge University Press, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica5, 785–795 (1938).
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Appl. Opt.

J. Mod. Opt.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt.46, 1763–1774 (1999).

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Comm.

J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm.136, 297–305 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev.

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[CrossRef]

Physica

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica5, 785–795 (1938).
[CrossRef]

Proc. R. Soc. London A

E. Wolf, “A macroscopic theory of diffraction and interference of light from finite sources — II. Fields with spectral range of arbitrary width,” Proc. R. Soc. London A230, 246–265 (1955).
[CrossRef]

Prog. Opt.

E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt.50, 251–273 (2007).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

M. Born and E. Wolf, Principles of Optics, 7th exp. edition (Cambridge University Press, 1999).

R. J. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd edition (Wiley-Interscience, New York, 2007).

R. R. Alfano, ed., The Supercontinuum Laser Source: Fundamentals with Updated References, 2nd ed. (Springer, 2006).

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Figures (8)

Fig. 1
Fig. 1

Measuring the spatial coherence. (a) Use of the conventional Young’s interferometer setup to measure the visibility, and thus the spatial coherence of polychromatic light (here an RGB source is considered), leads to problems. The spherical waves emerging from the pinholes interfere creating intensity fringes. However, the periods of the fringes scale as a function of the wavelength and the visibility will obviously be low as the fringes of various frequencies mix. (b) By placing an achromatic Fourier-transform system (AFT) between the pinhole screen and the detector plane, the emerging spherical waves are transformed into plane waves that arrive to the detector at a frequency-dependent angle θ(ω). Such an achromatization of the diffraction pattern results in scale-invariant intensity fringes and the visibility of the fringe pattern can be utilized to measure the spatial coherence.

Fig. 2
Fig. 2

Measurement setup. The beams from two diode-pumped solid-state lasers with wavelengths of 473 nm and 532 nm and a HeNe laser with the wavelength of 633 nm are combined into a single polychromatic (RGB) beam using mirrors and 50:50 non-polarizing beam splitters. The resulting beam field is directed onto the modified Young’s interferometer, where the pinhole size is 30 μm × 30 μm and their separation is 150 μm. The distance between the diffractive lens and Young’s pinholes is 60.2 mm whereas the distance from the diffractive lens to the detector is 159.8 mm.

Fig. 3
Fig. 3

The measurement setup in detail. An achromatic doublet (AC) with focal length of f1 = 150 mm is described by radii of curvature r1 = 91.62 mm, r2 = −66.68 mm, and r3 = −197.7 mm with thicknesses t1 = 5.7 mm in crown material N-BK7 and t2 = 2.2 mm in flint material SF5. The doublet is placed directly in front of Young’s double pinholes (DP) with slit size of 30 μm and separation of a = 150 μm. Distance d1 = 60.2 mm separates the achromatic lens and the diffractive lens (DL) with focal length of f2 = 89.940 mm at design wavelength 571 nm. The detector (CCD) is located at a distance of d2 = 159.8 mm from the diffractive lens. The system contained within the colored area is represented by system matrix M which is calculated using ray optics.

Fig. 4
Fig. 4

Profile of the diffractive lens. Only the first few periods are presented since the physical width w of the lens is 6 mm. The structure provides the diffraction efficiency as shown in Fig.5.

Fig. 5
Fig. 5

The spectral efficiency curve of the diffractive lens, the spectral response curve of the CCD detector, and the total efficiency curve. It is seen that the total efficiency, a product of the efficiencies of the CCD and the diffractive lens at each wavelength, is sufficiently constant within the spectral regime of 473 nm – 633 nm.

Fig. 6
Fig. 6

Wavelength dependence of the transverse scale factor Λ(λ) of the sinusoidal pattern emerging from the pinholes with achromatization done by AFT. Point sources are separated by distance a = 150 μm.

Fig. 7
Fig. 7

(a) Experimental results for the interference pattern of polychromatic light measured with a CCD line detector. The modified Young’s interferometer is illuminated with three laser sources (RGB) with the spatial degree of coherence |μ12(0)| ≈ 1. The achromatic Fresnel-transform system adjusts the two-pinhole intensity fringe periods to be the same for each wavelength. Very high visibility is obtained and the degree of spatial coherence for the polychromatic radiation in this case will be |γ12(0)| ≈ 0.93. (b) Corresponding theoretical curves for the interference pattern of polychromatic light with a simulated system. Blue: 473 nm, Green: 532 nm, Red: 633 nm, and Black: superposition.

Fig. 8
Fig. 8

(a) Experimental results for phase-shifted visibility. Incoming polychromatic (RGB) light experiences a wavelength-dependent phase delay between the two Young’s pinholes. This leads to a lateral displacement of the interference pattern that is different in magnitude for each wavelength. As a result, the total visibility and thus the degree of spatial coherence is reduced to |γ12(0)| ≈ 0.61. (b) Corresponding theoretical curves for phase-shifted visibility. The degree of coherence decreases to |γ12(0)| ≈ 0.81. Blue: 473 nm, Green: 532 nm, Red: 633 nm, and Black: superposition.

Equations (13)

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γ 12 ( τ ) = Γ 12 ( τ ) I 1 I 2 .
V 1 ( x , ω ) = α V 1 ( ω ) exp [ i k x ( ω ) x ]
V 2 ( x , ω ) = α V 2 ( ω ) exp [ i k x ( ω ) x ] ,
S ( x , ω ) = S ( 1 ) ( ω ) + S ( 2 ) ( ω ) + 2 S ( 1 ) ( ω ) S ( 2 ) ( ω ) | μ 12 ( ω ) | cos [ 2 k x ( ω ) x α 12 ( ω ) ] ,
μ 12 ( ω ) = | μ 12 ( ω ) | exp [ i α 12 ( ω ) ] = V 1 * ( ω ) V 2 ( ω ) S 1 ( ω ) S 2 ( ω )
2 k x ( ω ) = 2 ω c sin θ ( ω ) = 2 π d ,
I ( x ) = 0 S ( x , ω ) d ω .
I ( x ) = I ( 1 ) + I ( 2 ) + 2 I ( 1 ) I ( 2 ) 0 s ( 1 ) ( ω ) s ( 2 ) ( ω ) | μ 12 ( ω ) | cos [ 2 π x / d α 12 ( ω ) ] d ω ,
γ 12 ( τ ) = 0 s 1 ( ω ) s 2 ( ω ) μ 12 ( ω ) exp ( i ω τ ) d ω ,
I ( x ) = I ( 1 ) + I ( 2 ) + 2 I ( 1 ) I ( 2 ) | γ 12 ( 0 ) | cos [ 2 π x / d β 12 ( 0 ) ] ,
V = I max I min I max + I min = 2 I 1 I 2 I 1 + I 2 | γ 12 ( 0 ) | ,
Λ ( λ ) = 2 a λ | B ( λ ) | ,
γ 12 ( 0 ) = l s ( ω l ) exp [ i α 12 ( ω l ) ] ,

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