## Talbot experiment re-examined: study of the chromatic regime and application to spectrometry

Optics Express, Vol. 11, Issue 24, pp. 3310-3319 (2003)

http://dx.doi.org/10.1364/OE.11.003310

Acrobat PDF (576 KB)

### Abstract

The original Talbot experiment in white light has been reconstituted, using an amplitude grating made of thin slits and a colour CCD camera and a model has been developed to describe the field diffracted by the grating illuminated in polychromatic light with a known spectral density. Above the historical interest of this study, the possibility of applying this effect to make spectral measurements is explored and a new concept of Talbot spectrometer is proposed.

© 2003 Optical Society of America

## 1. Introduction

3. N. Guérineau, J. Primot, M. Tauvy, and M. Caes, “Modulation transfer function measurement of an infrared focal plane array using the self-imaging property of a canted periodic target,” Appl. Opt. **38**, 631–637 (1999). [CrossRef]

4. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. **180**, 199–203 (2000). [CrossRef]

*Z*given by:

*d*is the period of the grating and Δ

*λ*the spectral bandwidth of the camera.

*λ*, this grating generates a cosinusoidal intensity along the propagation axis of period

*z*

_{T}, where

*z*

_{T}is the Talbot distance:

*B*(

*λ*). This original principle of Fourier spectrometer can lead to elegant solutions [6,7

7. H. L. Kung, A. Bhatnagar, and D. A. B. Miller, “Transform spectrometer based on measuring the periodicity of Talbot self-images,” Opt. Lett. **26**, 1645–1647 (2001). [CrossRef]

9. G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. **89**, 312–317 (2000). [CrossRef]

*B*(

*λ*), like a classic spectrograph.

*z*-axis will be proposed. This new configuration is particularly well-suited to our application. As an application, the Talbot experiment has been reconstituted using a colour camera. The experimental results are presented in Section 4.

## 2. Theoretical study

*t*(

*x*), which may be represented by a Fourier series:

*a*, spaced at a distance

*d*, the Fourier coefficients

*C*

_{p}are given by:

*x*) is defined by [sin(π

*x*)]/(π

*x*).

*u*

_{i}at normal incidence, this object generates a scalar field u

_{λ}(

*x,z*) that can be written as a sum of scalar fields produced by the Fourier components of

*t*(

*x*), using the approach of angular spectrum of plane waves [10

10. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta **16**, 281–287 (1969). [CrossRef]

*p*

_{max}is the number of wavelengths λ in period

*d*.

*I*

_{λ}(

*x,z*)=|

*u*

_{λ}(

*x,z*)|

^{2}as a Fourier series,

*p*th coefficient is given by:

_{p+q}-Φ

_{q}that are currently approximated by a parabolic relation, so that:

_{T}is the Talbot distance, given by Eq.(2). This ordinary (Rayleigh-Fresnel) approximation is true when

*λ*/

*a*is small.

*m*th coefficient is non-null only if

*m*and

*p*have the same parity and shows that the

*p*th coefficient

*D*

_{p}is a periodic function along the propagation axis of period

*z*

_{p}(if

*p*is odd) and

*z*

_{p}/2 (if

*p*is even). Substituted into Eq. (6), this relation offers a simple means to compute the monochromatic intensity distribution

*I*

_{λ}at any point (

*x,z*) behind the grating.

*B*(

*λ*). Then, the polychromatic intensity distribution can be written as an incoherent sum of contributions I

_{λ}weighted by the spectral density

*B*(

*λ*):

*B̃*(

*f*) is the Fourier transform of

*B*(

*λ*) defined by

*C*

_{p}of the transmittance and the Fourier transform

*B̃*(

*f*) of the spectral density of the polychromatic source.

*d*=100 µm and slits openness

*a*=10 µm illuminated by a plane wave coming from a polychromatic light of central wavelength

*λ*

_{0}=0.5 µm and bandwidth Δ

*λ*=0.2 µm. For the sake of simplicity, we have supposed that the spectral density profile follows a Gaussian curve, plotted in Fig. 2(a). For these values, the Talbot distance

*z*

_{T}at

*λ*

_{0}is 40 mm and distance

*Z*at which the panchromatic regime is reached is 100 mm, using Eq. (1). The curves

*D*

_{p}versus

*z*for the four first orders

*p*are reported in Fig. 2(b). In the monochromatic case (curves in dotted lines, calculated at

*λ*=0.5 µm), we verify that the

*D*

_{p}(

*z*) are periodic functions of period z

_{T}/p for odd

*p*and

*z*

_{T}/(2p) for even

*p*. At these fractional distances, narrow peaks appear. In polychromatic light, periodic curves of different period (proportional to 1/

*λ*) weighted by

*B*(

*λ*) are summed, leading to a succession of enlarged peaks, with an enlargement proportional to

*z*. Above a certain distance, these curves reach a constant value, corresponding to the panchromatic regime. Using the

*D*

_{p}curves, we have calculated the intensity distribution I(

*x,z*), depicted in grey levels in Fig. 2(c). On the left, we observe the lobes of energy issuing from the slits that interfere in the vicinity of the grating. Above a distance of few millimetres from the mask, longitudinal fringes appear, corresponding transversally to the formation of thin lines spaced at a distance

*d*/2. This simulation is confirmed by the experimental study of Ref. [4

4. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. **180**, 199–203 (2000). [CrossRef]

## 3. Application to spectrometry

### 3.1 Lokshin spectrometer

*S*(

*z*) given by:

*S*(

*z*), that we can call the apparatus function of the Lokshin spectrometer, is periodic of period

*z*

_{T}. This function obtained at wavelength

*λ*=0.5 µm is plotted in Fig 3(a). We can observe the formation of narrow peaks at the Talbot distances. The width δz of these peaks is the depth of field of the Talbot images given by [11]:

*λ*and

*λ*+δ

*λ*with a theoretical resolving power

*λ*/δ

*λ*of:

*d*=100 µm and

*a*=10 µm, the Lokshin spectrometer has potentially a resolving power of 200, that is, a resolution of 2.5 nm at a wavelength

*λ*=0.5 µm.

*S*(

*z*) has secondary peaks between the Talbot images, at the so-called fractional Talbot distances. At these distances, the so-called Fresnel images are made of thin lines of width

*a*, and spaced at a distance

*d*/

*M*where

*M*is an integer [12

12. J.J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. **55**, 373–381 (1965). [CrossRef]

*z*=

*z*

_{T}/(2

*M*) between the Talbot distance and the first Fresnel image, obtained for

*M*=

*d*/

*a*and assuming that

*M*is even. Thus the free spectral range Δ

*λ*around a wavelength

*λ*is given by:

*λ*=25 nm with

*a*=10 µm,

*d*=100 µm and

*λ*=0.5 µm.

*z*-

*z*

_{T}). As an example, we have computed the signal

*S*(

*z*) delivered by the Lokshin spectrometer when illuminated by a polychromatic source with a rather broad spectral bandwidth, depicted in Fig 2(a). We obtain the curve of Fig 3(b); this curve has to be compared to the ideal curve (in dotted lines) obtained with the ideal apparatus function.

### 3.2 Recommended configuration

*z*location, an image is grabbed that is treated in order to extract the first Fourier coefficient

*D*

_{1}(

*z*) of the projected intensity profile. In monochromatic illumination, the apparatus function

*D*

_{1}(

*z*) is plotted in dotted lines in Fig 2(b). Let us describe more precisely this function. Substituting

*p*=1 into Eq. (9) yields the expression of D

_{1,λ}(z):

*d*

_{m}=0 for

*m*even, and

*m*odd.

*D*

_{1,λ}(

*z*) is due to Eq. (20) where the expression of

*d*

_{m}can be approximated by

*C*

_{m}are the Fourier coefficients of the binary-amplitude transmittance of period

*d*and slits width

*a*. Thus, the apparatus function

*D*

_{1,λ}(

*z*) is periodic of period z

_{T}and an elementary cell is made of two opposite triangles (spaced at a distance

*z*

_{T}/2) of width δ

*z*=2

*ad*/

*λ*at 50%. This width at 50% of the apparatus function gives the spectral resolution δ

*λ*=

*λa*/

*d*of this spectrometer for the analysis of a wavelength

*λ*around the first Talbot distance. If we choose to work in the vicinity of a multiple

*k*of the Talbot distance, this resolution will be divided by a factor of

*k*. In practice, the distance between the detector and the target has to be as small as possible in order to reduce the effect of the finite width of the slit source [3]. For this reason, it is better to work in the vicinity of

*z*

_{T}/2. For this working distance, the resolving power is:

*λ*=0.5 µm,

*d*=100 µm and

*a*=10 µm. In comparison with the Lokshin apparatus function, the curve

*D*

_{1,λ}(

*z*) exhibits no secondary peak between distances

*z*

_{T}/2,

*z*

_{T}, 3

*z*

_{T}/2, etc. For this reason, the free spectral range is expected to be much broader. To estimate this free spectral range [

*σ*

_{min},

*σ*

_{max}], where

*σ*=1/

*λ*is the wavenumber, the thought process is similar to that used in the case of a grating monochromator when the grating diffracts several orders. For the spectral range [

*σ*

_{min},

*σ*

_{max}], a longitudinal dispersion of the first peak of D

_{1}(z) appears, covering the range [

*d*

^{2}

*σ*

_{min},

*d*

^{2}

*σ*

_{max}], whereas the second peak of

*D*

_{1}(

*z*) (around the Talbot distance z

_{T}) spreads on the range [2

*d*

^{2}

*σ*

_{min}, 2

*d*

^{2}

*σ*

_{max}]. As a consequence, if no folding is tolerated, the following condition has to be respected:

*σ*

_{min}≤

*σ*

_{max}/2, i.e.:

_{0}, the free spectral range is [3

*λ*

_{0}/4, 3

*λ*

_{0}/2], i.e. Δ

*λ*=3

*λ*

_{0}/4. Numerically, for an analysis around

*λ*=0.5 µm with

*d*=100 µm and

*a*=10 µm, the free spectral range is [0.375 µm, 0.750 µm] and the resolving power is 5, i.e. δ

*λ*=100 nm at

*λ*=0.5 µm.

*D*

_{1}(

*z*) using Eq. (13) delivered by this spectrometer when illuminated by a polychromatic source with a rather broad spectral bandwidth, depicted in Fig. 2(a). We obtain the curve of Fig. 5(b), this curve has to be compared to the ideal curve (in dotted lines) obtained with the Dirac apparatus function depicted in Fig. 5(a). Extracted from the computed curve

*D*

_{1}(z), the normalised spectral density is given by

*M*

_{1}corresponds to the value at the peak. This curve is plotted in Fig. 5(c). As expected, the spectral density extracted is affected by folding effects because the condition expressed in Eq. (22) is not respected: with

*λ*

_{max}=0.8 µm,

*λ*

_{min}should be greater than 0.4 µm. In order to recover the right spectral density in the range [0.2 µm, 0.4 µm], we propose to subtract the folded part belonging to the second peak of

*D*

_{1}(z) around

*z*

_{T}, as illustrated in Fig. 5(b). The folded part, corresponding to the left-hand side of the second peak, can easily be deduced from the measurement of the left-hand side of the first peak of

*D*

_{1}(

*z*) which is not affected by aliasing effects. The corrected spectral density B

_{corr}extracted from the curve

*D*

_{1}(

*z*) is then given by

*M*

_{2}is the value at the second peak of

*D*

_{1}(

*z*). This curve is also plotted in Fig. 5(c) and fits better the ideal spectral density

*B*(

*λ*).

## 4. Experimental study

*z*~35 mm and

*z*~50 mm, colourful bands appear in the red-green-blue-order, spaced at the grating period: we are in the neighbourhood of

*z*

_{T}/2. At twice these distances, in the neighbourhood of the first Talbot distance, the colourful bands appear again.

*D*

_{1}of the intensity profile measured at each distance

*z*and for each colour (RGB). The experimental curves corresponding to the absolute value of

*D*

_{1}for each colour are plotted in Fig. 7(b). In the neighbourhood of

*z*

_{T}/2, the three curves (respectively R, G and B) exhibit a bound centred respectively at

*d*

^{2}/

*λ*

_{R},

*d*

^{2}/

*λ*

_{G}and

*d*

^{2}/

*λ*

_{B}. Using Eq. (22), we have translated these curves into the three relative spectral responses of the camera, plotted in Fig. 7(c) in arbitrary units. We can notice that the blue-response is more important than the others, which explains the bluish aspect of the diagram of Fig. 7(a).

## 5. Conclusion

*z*-position of the detector, the transverse intensity profile is acquired and the first Fourier coefficient of this profile is extracted via a Fourier-transform. This operation of extraction is rather robust not only to noise and spatial filtering of the pixels but also to imperfect lateral positioning of the detector, due to taking the absolute value of that coefficient. This concept is particularly well-suited for spectral measurements on large spectral ranges with a rather low resolution.

*D*

_{p}of the transverse intensity data. For example, the third order

*D*

_{3}exhibits the same behaviour as the first order but on a propagation-distance range reduced by a scale factor of 3. Thus, this extension to higher orders requires the experimenter to reduce the working distances between the grating and the detector and to control these distances with a higher precision. In addition, these higher orders are more affected by pixels filtering effects, leading to a reduced signal-to-noise ratio.

*D*

_{1}(

*z*) and the modulation transfer function of the pixels from the spectrum of harmonics

*D*

_{p}at a known self-imaging distance. On a theoretical level, non-paraxial limitations are expected if we want to use a Fraunhofer grating with very thin slits (close to the wavelength) to make high resolution measurements. These non-paraxial effects on the apparatus function are being studied. In that case, the Fourier-transform spectrometer proposed by Lohmann is a priori recommended.

## References and links

1. | H. F. Talbot, “Facts relating to optical science. No IV,” Philos. Mag. |

2. | K. Patorski, “The self-imaging phenomenon and its applications,” in |

3. | N. Guérineau, J. Primot, M. Tauvy, and M. Caes, “Modulation transfer function measurement of an infrared focal plane array using the self-imaging property of a canted periodic target,” Appl. Opt. |

4. | N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. |

5. | W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in |

6. | H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque |

7. | H. L. Kung, A. Bhatnagar, and D. A. B. Miller, “Transform spectrometer based on measuring the periodicity of Talbot self-images,” Opt. Lett. |

8. | G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in |

9. | G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. |

10. | R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta |

11. | V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR) |

12. | J.J. Winthrop and C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. |

**OCIS Codes**

(110.6760) Imaging systems : Talbot and self-imaging effects

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 30, 2003

Revised Manuscript: November 20, 2003

Published: December 1, 2003

**Citation**

Nicolas Guérineau, Emmanuel Di Mambro, Jérôme Primot, and Frédéric Alves, "Talbot experiment re-examined: study of the chromatic regime and application to spectrometry," Opt. Express **11**, 3310-3319 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-24-3310

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### References

- H. F. Talbot, �??Facts relating to optical science. No IV,�?? Philos. Mag. 9, 401-407 (1836).
- K. Patorski, �??The self-imaging phenomenon and its applications,�?? in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol XXVII, pp 1-108.
- N. Guérineau, J. Primot, M. Tauvy and M. Caes, �??Modulation transfer function measurement of an infrared focal plane array using the self-imaging property of a canted periodic target,�?? Appl. Opt. 38, 631-637 (1999). [CrossRef]
- N. Guérineau, B. Harchaoui and J. Primot, �??Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,�?? Opt. Commun. 180, 199-203 (2000). [CrossRef]
- W. Lohmann, �??A new Fourier spectrometer consisting of a two-gratings-interferometer,�?? in Proceedings of theconference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58-61.
- H. Klages, �??Fourier spectrometry based on grating resonances,�?? J. Phys. Colloque 2, C2-40 (1967).
- H. L. Kung, A. Bhatnagar and D. A. B. Miller, �??Transform spectrometer based on measuring the periodicity of Talbot self-images,�?? Opt. Lett. 26, 1645-1647 (2001). [CrossRef]
- G. R. Lokshin, V. E. Belonuchkin and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47
- G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin and N. I. Eskin, �??On the spectra selectivity of Talbot and Lau effects,�?? Opt. Spectrosc. 89, 312-317 (2000). [CrossRef]
- R. F. Edgar, �??The Fresnel diffraction images of periodic structures,�?? Opt. Acta 16, 281-287 (1969). [CrossRef]
- V. V. Aristov, A. I. Erko and V. V. Martynov, �??Optics and spectrometry based on the Talbot effect,�?? Opt. Spectrosc. (USSR) 64, 376-380 (1988).
- J.J. Winthrop and C. R. Worthington, �??Theory of Fresnel images. I. Plane periodic objects in monochromatic light,�?? J. Opt. Soc. Am. 55, 373-381 (1965). [CrossRef]

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