## Spatial coherence measurement of polychromatic light with modified Young’s interferometer |

Optics Express, Vol. 21, Issue 4, pp. 4061-4071 (2013)

http://dx.doi.org/10.1364/OE.21.004061

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### 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

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]

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

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]

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]

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]

## 2. Measurement principle and theory

### 2.1. Polychromatic spatial coherence

*E*

_{1}(

*t*) and

*E*

_{2}(

*t*), where

*t*is the time, the mutual coherence function is defined as

*j*, with

*j*= (1, 2), can be taken to be

*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.

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

*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 where

*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 where

*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 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 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 sin

*θ*(

*ω*) =

*πc*/

*ωd*.

### 2.3. Polychromatic fringe visibility

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]

*γ*

_{12}(

*τ*) and the spectral degree of coherence

*μ*

_{12}(

*ω*), at a pair of points, may be cast in the form where

*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 where

*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

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

*I*

_{max}and

*I*

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

*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

*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.

*a*, the scale of the sinusoidal output pattern is given by 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.

## 4. Experimental results

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

*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.

_{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 where

*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

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]

## Acknowledgments

## References and links

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

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

3. | E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt. |

4. | F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica |

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 |

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

7. | R. J. Glauber, |

8. | J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. |

9. | B. E. A. Saleh and M. C. Teich, |

10. | R. R. Alfano, ed., |

11. | G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B |

12. | C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of super-luminescent diodes,” J. Mod. Opt. |

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. |

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. |

15. | D. Faklis and G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. |

16. | G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. |

17. | J. Lancis, E. Tajahuerce, P. Andrés, V. Climent, and E. Tepichín, “Single-zone-plate achromatic Fresnel-transform setup: pattern tunability,” Opt. Comm. |

18. | S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

19. | T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. |

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. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.1940) Diffraction and gratings : Diffraction

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(260.3160) Physical optics : Interference

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: November 20, 2012

Revised Manuscript: January 11, 2013

Manuscript Accepted: January 11, 2013

Published: February 11, 2013

**Citation**

Kimmo Saastamoinen, Jani Tervo, Jari Turunen, Pasi Vahimaa, and Ari T. Friberg, "Spatial coherence measurement of polychromatic light with modified Young’s interferometer," Opt. Express **21**, 4061-4071 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4061

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### References

- 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).
- E. Wolf, “The influence of Young’s interference experiment on the development of statistical optics,” Prog. Opt.50, 251–273 (2007). [CrossRef]
- F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica5, 785–795 (1938). [CrossRef]
- 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]
- 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).
- J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett.37, 151–153 (2012). [CrossRef] [PubMed]
- 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).
- G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B28, 2301–2309 (2011). [CrossRef]
- 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).
- 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. 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]
- D. Faklis and G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett.13, 4–6 (1988). [CrossRef] [PubMed]
- G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt.20, 2017–2025 (1981). [CrossRef] [PubMed]
- 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]
- S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am.60, 1168–1177 (1970). [CrossRef]
- 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]
- 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]

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