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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 24 — Dec. 1, 2003
  • pp: 3310–3319
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Talbot experiment re-examined: study of the chromatic regime and application to spectrometry

Nicolas Guérineau, Emmanuel Di Mambro, Jérôme Primot, and Frédéric Alves  »View Author Affiliations


Optics Express, Vol. 11, Issue 24, pp. 3310-3319 (2003)
http://dx.doi.org/10.1364/OE.11.003310


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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

In 1836, H.F. Talbot illuminated a Fraunhofer grating with a very small white light source. Behind the grating, he observed colourful bands resembling the thin slits of the grating [1

1. H. F. Talbot, “Facts relating to optical science. No IV,” Philos. Mag. 9, 401–407 (1836).

]. Since this discovery, the Talbot self-imaging effect is used in many areas of physics research [2

2. 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.

]. To our point of view, this simple effect can lead to compact and robust setups, in particular in the field of optical metrology of infrared staring arrays [3

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]

]. Recently [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]

],we have reconstituted the original Talbot experiment and observed the formation of achromatic and propagation-invariant thin lines above a certain distance Z given by:

Z=2d2Δλ,
(1)

where d is the period of the grating and Δλ the spectral bandwidth of the camera.

This regime of self-imaging that we call panchromatic Talbot effect offers a simple means to project high-resolution lines (or spots) in polychromatic light. In this paper, we propose to explore the chromatic and propagation-dependent regime that appears between the grating and the panchromatic zone. In this regime, colourful bands appear at fractional Talbot distances. Our goal is to exploit this chromatic dispersion to make measurements of detectors spectral response. A first Talbot spectrometer has been proposed by Lohmann [5

5. W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in Proceedings of the conference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58–61.

]. The principle is to use a cosine grating (or an approximation). In monochromatic light of wavelength λ, this grating generates a cosinusoidal intensity along the propagation axis of period zT , where zT is the Talbot distance:

zT=2d2λ.
(2)

In polychromatic light, the intensity profile along the propagation axis results of an incoherent superposition of cosinusoidal profiles of different period, leading to an interferogram whose envelope encodes the Fourier transform of the spectral distribution light B(λ). This original principle of Fourier spectrometer can lead to elegant solutions [6

6. H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).

,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]

] but requires a quasi cosine-grating to be efficient, whereas the projection of thin lines is preferable in our application. To our knowledge, Lokshin is the first to propose a Talbot spectrometer based on a thin-slit Fraunhofer grating [8

8. G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

,9

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]

]. The main principle is illustrated in Fig. 1. A slit source in the far-field of the grating and a moving slit detector are used. The exit slit is moved longitudinally in the neighbourhood of a self-image plane and delivers directly the spectral density of light B(λ), like a classic spectrograph.

Fig. 1. Talbot-Lokshin spectrometer. Basic set-up.

In Section 2, a simple model to describe the field diffracted by a Fraunhofer grating in polychromatic light is presented. Using this model, we will study in Section 3 the performances of the Lokshin spectrometer in terms of resolution and free spectral range. Then a new configuration based on the use of a detector array moving along the 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

In this section, a simple model is developed to describe the field diffracted by a binary-amplitude mask in polychromatic light. For this, we consider a periodic object of 1D periodicity and of infinite aperture defined by a transmittance t(x), which may be represented by a Fourier series:

t(x)=p=+Cpexp(2iπpxd).
(3)

In the case of a mask made of slits of width a, spaced at a distance d, the Fourier coefficients Cp are given by:

Cp=adsinc(pad),
(4)

where sinc(x) is defined by [sin(πx)]/(πx).

When illuminated by a plane monochromatic wave of wavelength λ and amplitude u i at normal incidence, this object generates a scalar field uλ(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]

]:

uλ(x,z)=uipmaxpmaxCpexp(2iπpxd)exp(2iπzλ1(λpd)2),
(5)

where pmax is the number of wavelengths λ in period d.

Using Edgar approach, we can write the intensity pattern Iλ (x,z)=|uλ (x,z)|2 as a Fourier series,

Iλ(x,z)=ui22pmax2pmaxDp,λ(z)exp(2iπpxd),
(6)

where the pth coefficient is given by:

Dp,λ(z)=qCp+qCqexp[2iπzλ(1λ2(p+q)2d21λ2q2d2)]
(7)

Equation (7) highlights the difference of two propagation phase terms Φp+qq that are currently approximated by a parabolic relation, so that:

Φp+qΦq=2iπz(p2+2pq)zT,
(8)

where zT is the Talbot distance, given by Eq.(2). This ordinary (Rayleigh-Fresnel) approximation is true when λ/a is small.

Substituting Eq.(8) into Eq.(7) and putting m=p+2q and zp =zT /p yields:

Dpodd(even),λ(z)=modd(even)Cmp2Cm+p2exp(2iπmzzp).
(9)

Equation (9) can be read as a Fourier series where the mth coefficient is non-null only if m and p have the same parity and shows that the pth coefficient Dp is a periodic function along the propagation axis of period zp (if p is odd) and zp /2 (if p is even). Substituted into Eq. (6), this relation offers a simple means to compute the monochromatic intensity distribution Iλ at any point (x,z) behind the grating.

Now we consider a polychromatic light source defined by a spectral density profile B(λ). Then, the polychromatic intensity distribution can be written as an incoherent sum of contributions Iλ weighted by the spectral density B(λ):

I(x,z)=λB(λ)Iλ(x,z)dλ.
(10)

Substituting Eq. (6) into Eq. (10) yields:

I(x,z)=pDp(z)exp(2iπpxd),
(11)

with

Dp(z)=B(λ)Dλ,p(z)dλ.
(12)

Substituting Eq. (9) into Eq. (12) yields

Dpodd(even)(z)=modd(even)Cmp2Cm+p2B˜(mpz2d2),
(13)

where (f) is the Fourier transform of B(λ) defined by

B˜(f)=λB(λ)exp(2iπλf)dλ.
(14)

Thus, Eqs. (11) and (13) offers a simple mean to compute the polychromatic intensity distribution behind the grating, apart from the Fourier coefficients Cp of the transmittance and the Fourier transform (f) of the spectral density of the polychromatic source.

As an illustration, we consider a grating of period 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 Δλ=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 zT at λ 0 is 40 mm and distance Z at which the panchromatic regime is reached is 100 mm, using Eq. (1). The curves Dp 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 Dp (z) are periodic functions of period zT/p for odd p and zT /(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 Dp 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]

].

Fig. 2: Simulation of the field diffracted by a binary-amplitude grating in polychromatic light (c) of known spectral density (a), using the Fourier coefficients of the transverse intensity profile (b) (dotted line, Fourier coefficients in monochromatic light).

3. Application to spectrometry

3.1 Lokshin spectrometer

Lokshin has proposed a spectrometer that exploits this property. The principle is to measure the intensity distribution along the propagation axis, using a moving slit detector. Thus, this detector delivers a signal S(z) given by:

S(z)=I(0,z)=pDp(z)
(15)

In monochromatic illumination, the signal S(z), that we can call the apparatus function of the Lokshin spectrometer, is periodic of period zT . 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

11. 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).

]:

δz=a2λ.
(16)

Working in the vicinity of the first Talbot image, one can potentially resolve two wavelengths λ and λλ with a theoretical resolving power λλ of:

λδλ=2d2a2.
(17)

For example, with 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.

Nevertheless, this spectrometer is not efficient for the measurement of broad spectral density profiles. Indeed, the monochromatic signal 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]

]. In polychromatic light, all these peaks are enlarged and are folding on each other. The presence of these peaks reduces drastically the free spectral range of this spectrometer. This free spectral range corresponds to the distance Δz=zT /(2M) 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 Δλ around a wavelength λ is given by:

Δλ=λa2d.
(18)

Numerically, we find Δλ=25 nm with a=10 µm, d=100 µm and λ=0.5 µm.

As an illustration we have plotted in dotted lines an ideal apparatus function in Fig 3(a). This ideal function is in fact a Dirac function δ(z-zT ). 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.

Fig. 3: Simulation of the signal delivered by the Talbot-Lokshin spectrometer in monochromatic (a) or polychromatic (b) light.

3.2 Recommended configuration

At each z location, an image is grabbed that is treated in order to extract the first Fourier coefficient D1 (z) of the projected intensity profile. In monochromatic illumination, the apparatus function D1 (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 D1,λ(z):

D1,λ(z)=mdmexp(2iπmzzT),
(19)

with dm =0 for m even, and

dm=Cm12Cm+12=(ad)2sinc(m1)a2dsinc(m+1)a2dCm2
(20)

for m odd.

Fig. 4. Recommended spectrometer. Basic set-up.

The triangular profile of D 1,λ(z) is due to Eq. (20) where the expression of dm can be approximated by Cm2, where Cm 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 zT and an elementary cell is made of two opposite triangles (spaced at a distance zT /2) of width δz=2ad/λ at 50%. This width at 50% of the apparatus function gives the spectral resolution δλ=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 zT /2. For this working distance, the resolving power is:

λδλ=d2a.
(20)

This function is plotted in Fig. 5(a) for 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 zT /2, zT , 3zT /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 D1(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 zT) spreads on the range [2d 2 σ min, 2d 2 σ max]. As a consequence, if no folding is tolerated, the following condition has to be respected: σ minσ max/2, i.e.:

λminλmax2.
(21)

Thus, for an analysis around a wavelength λ0, the free spectral range is [3λ 0/4, 3λ 0/2], i.e. Δλ=3λ 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. δλ=100 nm at λ=0.5 µm.

As an illustration, we have computed the curve 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

Bext(d2z)=D1(z)M1,
(22)

where 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 zT , 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 Bcorr extracted from the curve D 1(z) is then given by

Bcorr(d2z)=D1(z)M1Bext(2d2z)×M2M1,
(23)

where 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).

Fig. 5: Simulation of the signal delivered by the recommended spectrometer in monochromatic (a) or polychromatic (b) light and extraction of the spectral density (c).

4. Experimental study

For this experimental study, the Fraunhofer grating is a quartz window with a chromium mask of 300 slits 14 µm wide and spaced 140 µm apart. The experimental set-up is the following: a slit 100 µm wide is illuminated by a quartz-tungsten-halogen lamp and is located at an approximate distance of 1000 mm from the periodic object. Images are grabbed with a colour CCD camera mounted with a microscope objective. Each colour image consists of red, green, blue (RGB) images. We have recorded 1000 images at different distances from the mask, corresponding to a propagation-distance range of 100 mm at a translation step of 0.1 mm. A movie of this experiment is presented in Fig. 6.

Fig. 6. (2.2 MB) Movie of the reconstituted Talbot experiment.
Fig. 7: Experimental study in white light: (a) response of a line of CCD detectors versus propagation distance and extraction of the R-,G-, B- spectral responses (c) from the D1(z) curves.

From each image, the response of a line of 450 pixels is extracted. Figure 7(a) shows the evolution of this response (vertical axis) versus the propagation distance (horizontal axis). In the vicinity of the mask, we observe the white lobes of energy diffracted by the slits. Between 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 zT /2. At twice these distances, in the neighbourhood of the first Talbot distance, the colourful bands appear again.

Using our technique, we have extracted the first Fourier coefficient D1 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 zT /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

In this study, the Talbot effect in polychromatic light has been studied. For this, we have developed an analytical model based on Edgar approach but extended to a polychromatic illumination of known spectral density. This approach describes the evolution of the Fourier coefficients of the transverse intensity profile versus the propagation distance. In monochromatic illumination, these curves are made of narrow peaks spaced at a regular distance, corresponding to a fractional Talbot distance. In polychromatic light, these peaks are longitudinally dispersed leading to an enlarged peak which encodes directly the spectral density of the illuminating beam, like for a classic grating spectrograph. Making this observation, we have proposed a new concept of Talbot spectrometer based on the use of a linear array of detectors moving along the propagation axis. For every 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.

An extension of this technique could be considered by looking at higher harmonics Dp of the transverse intensity data. For example, the third order D3 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.

An experimental study has been performed using a colour CCD camera. This study has offered a nice illustration of the original Talbot effect in white light. In addition, it has permitted to measure the spectral responses (red, green, blue) of the camera using our technique. This approach is being implemented in a Talbot test bench, in order to make spectral and spatial characterisations of infrared focal plane arrays. In this case, all the Fourier-components will be computed in order to extract both the spectral response from D1 (z) and the modulation transfer function of the pixels from the spectrum of harmonics Dp 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. 9, 401–407 (1836).

2.

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.

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]

5.

W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in Proceedings of the conference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58–61.

6.

H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).

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]

8.

G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

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]

10.

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

11.

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).

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]

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

  1. H. F. Talbot, �??Facts relating to optical science. No IV,�?? Philos. Mag. 9, 401-407 (1836).
  2. 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.
  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]
  5. 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.
  6. H. Klages, �??Fourier spectrometry based on grating resonances,�?? J. Phys. Colloque 2, C2-40 (1967).
  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]
  8. G. R. Lokshin, V. E. Belonuchkin and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47
  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]
  10. R. F. Edgar, �??The Fresnel diffraction images of periodic structures,�?? Opt. Acta 16, 281-287 (1969). [CrossRef]
  11. 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).
  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]

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