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.

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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,” 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).
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  16. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
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  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 (1)

2011 (1)

2007 (1)

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

2006 (2)

1999 (1)

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 (1)

1997 (1)

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 (1)

1988 (1)

1981 (1)

1970 (1)

1963 (1)

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

1955 (1)

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]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 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).
[Crossref]

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).
[Crossref]

Tervo, J.

Turunen, J.

Wolf, E.

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

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]

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]

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).

Zernike, F.

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

Appl. Opt. (2)

J. Mod. Opt. (1)

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. (1)

J. Opt. Soc. Am. B (1)

Opt. Comm. (1)

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. (5)

Phys. Rev. (1)

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

Physica (1)

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

Proc. R. Soc. London A (1)

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]

Prog. Opt. (1)

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

Other (5)

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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