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The Lau effect (a diffraction experiment with incoherent illumination)
Volume 28, number 3
March 1979
OPTICS COMMUNICATIONS
THE LAU EFFECT (A DIFFRACTION
EXPERIMENT
WITH INCOHERENT
ILLUMINATION)
J. JAHNS and A.W. LOHMANN Physikalisches Institut, 8520 Erlangen, Fed. Rep. Germany
Received 26 December 1978
In 1948 Ernst Lau described a diffraction experiment, whereby the object was illuminated by an extended white light source. The object consisted of two coarse gratings, positioned behind each other at a distance of a few centimeters. Colored fringes appear at infinity. We present an explanation, based on diffraction theory. We also note similarities to Talbot’s effect called “self-imaging”, or sometimes “Fourier-imaging”. A specific application of the Talbot effect, the Talbot interferometer, is equivalent to the Lau effect in terms of the theory of partial coherence. Based on this equivalence we suggest two applications of the Lau effect, a simple interferometer and a Fourier spectrometer.
1. Introduction Talbot self-imaging is a well-known effect in coherent optics [I]. If a grating G(x) is illuminated with light from a monochromatic point source the interference patterns in specific planes behind this object will be similar to G(x). This happens in collimated or uncollimated light. Actually, the class of objects suitable for self-imaging contains not only gratings [2] . But gratings are the most important members of this class. With collimated monochromatic light the selfimages will appear at 2 d2/h and integer multiples thereof. For example, with a grating constant d = l/l0 mm and a wavelength h = l/2000 mm the Talbot distance 2 d2/h is 40 mm. Self-imaging has found applications in Fourier spectrometry [3] and in interferometry [4]. In the Talbot interferometer a second grating is placed into the selfimage plane of the first grating. No light will emerge from this interferometer, if the opaque bars of the second grating cover up the bright lines in the self-image of the first grating. The situation of complete darkness requires a careful adjustment, that can be disturbed easily by a phase object 0 between the two gratings of the Talbot interferometer (fig. 1). We mentioned the Talbot interferometer, because its setup is very similar as in the Lau experiment [S] which is the aim of our study. Instead of a monochromatic point source Lau
Gl
0
G2
F
I
Lz& l----l
CfA-f---i-rAti
Fig. 1. Talbot-Interferometer. Gl, G2: gratings; 0: phase object; I: Interference image of 0.
used an extended white source. Colorful fringes appear in plane F of fig. 1. The colors change when the distance z,, between the two gratings is varied.
2. Description
of the Lau effect
We now want to describe the setup of Lau’s experiment more thoroughly (fig. 2). To provide incoherent illumination, an extended light source is imaged onto the first grating, Cl. Gl can thus be considered as a self-luminous object. For a first understanding the following simplifying assumption can be made: the slit width of the two identical gratings should be very small compared to the slit separation d. Then a slit S of Gl acts as a line-source from which emerges a cylindrical wave (fig. 3). This wave will create secondary light 263
Volume 28, number 3 Gl
l-z,+
G2
F
L
L
f
-
at the slits P,, P, . . . of the second grating G2.
These secondary light sources will radiate with equal phases, if the path lengths (SPo), (SP1) . . . differ by integer multiples of h. This is possible indeed, if the grating separation is z. = d2/2h, a condition which follows from a simple Pythagoras consideration. For this specific wavelength, one slit of G 1 will produce interference fringes at infinity. Since the illumination is incoherent, the interference patterns generated by all the slits of Gl add in their intensities. For z. = d2/2h the maxima of all these fringe patterns coincide, so that in total one obtains interference fringes of high contrast. Fringe patterns can also be observed at infinity for a certain wavelength whenever zu = n d2/2X (n = 1,2,3,. . .). For white light illumination, the greatest fringe contrast will occur for the wavelength that satisfies the above equation. We will then observe a two color fringe pattern consisting of alternating fringes at wavelength h and at the complementary visual color. Until now, we dealt with Ronchi-rulings with very small slit width. In order to consider the case of more Gl
G2
9(x”)
h ( x’)
X
L
L
I
p&y0 h
Fig. 2. Lau’s experiment. The two gratings, Gl and G2, are illuminated by an extended light source. Observation takes place in the back focal plane F of the converging lens L (f: focal length of L). sources
March 1979
OPTICS COMMUNICATIONS
f --
Fig. 4. The setup as assumed in our analysis.
general objects, we now want to change from the geometrical to a wave-optical point of view. We consider two arbitrary transparencies h and R in the planes z = 0 and z=z o, respectively (see fig. 4). Our aim is to calculate the monochromatic intensity Z*(x) in the back focal plane of the lens L. To this end, we will first compute the complex amplitude uh (x,x1) in the observation plane due to a unit point source located at x’ = x, in the plane z = 0. The whole object h (x’) can be regarded as consisting of many point sources. Since the illumination is incoherent, the monochromatic intensity Ih is therefore obtained by summing up all the intensity contributions: ~~(~)=~1~(~,)121~,(~,~,)17
dx,.
(1)
The first step will be the calculation of uh (x,x1). We denote the unit point source at x’ = x1 in the plane z = 0 by a delta function. The complex amplitude immediately before the object I in the plane z = z. -0 is obtained by a Fresnel-transformation: u,(x”,x,) inz=z
a exp{m(x”
- x,)~/XZ,)
u -0.
(2)
After a multiplication with the complex amplitude transmission g(x)‘), the lens L performs a Fouriertransformation. This means: u,(x,xI)
a FOU’&(x”) exp[irr(x” - x,)~/XZ,]}
in the observation
plane,
(3)
(FOU {. . .} = Fourier-transform of {. . .}). The explicit calculation of the Fourier-transform yields:
u*(x,xr ) = I &“) Fig. 3. Geometrical explanation. The distances from one slit in Gl to several slits in G2 differ by integer multiples of h, if the gratings are separated by ze = d2/2k.
264
X exp {in[x” - (x1 + zofl =&x,
+,($-I
x).
x)] 2/ xZo} dx” (4)
OPTICS COMMUNICATIONS
Volume 28, number 3
March 1979
Here, we dropped a phase factor exp {ip(x,xl)} that vanishes anyway when computing the modulus square. The g(x) means the Fresnel-transform of g calculated at the point x. According to eq. (1) we get the final result by integrating over the contributions from all point sources inz=O ~~(X)=Jk(XJ21&xl
+zof-lx)12
dx,.
4
r
l--
,X Ida
-i
-ladi-
(5)
This integral just represents the correlation of Ih I2 with the modulus square of the Fresnel transform of g: z* = Ih I2 @I&
g(x)
Fig. 5. Complex amplitude transmission of a Ronchi-ruling with grating constant d and slit width ad.
(6)
(0 denotes the correlation operation). Next we will apply this result to the special case of two Ronchi-rulings as input functions (see fig. 5) which we describe by
g(x) = h(x) =
(5 rect (x*). m
z,= 31d2
From the Talbot effect is it known that the Fresneltransform of a Ronchi-ruling will again be a Ronchiruling whenever z,, = n d2/2h (n = 1,2,3,. . .), although with modified contrast, in general. The correlation of two grating-slits, i.e. of two rect (x/Ax)-functions results in a trian (x/Ax)-function. Therefore, we obtain a grating of trian-functions whenever z. = n d2/2h. We have to distinguish four cases:
C: C trian(x Cm) Ctrian(x
-zDi2)
+ “‘$-
2h
Fig. 6. Graphical representation 01= l/8 (see eq. (8)).
X
D
of I&(x) for n = 1,2,3,4, for
, n = 1,5,9,. . .
““1,
n = 2,6,10,.
..
I*(x) = f Ctrian(*)
,
c
trian(*)
with D = d f/zo.
, n=3,7,11,...
, n =4,8,12 ,... 09
Notice that the period D of the fringe patterns depends on the grating separation zo. This means that essentially the fringes become finer with increasing zo, with the special feature that the number of fringes per unit interval D is two if n is an odd number and one if n is even.
Fig. 7. Lau fringes for three different separations of two rulings (zo = 3.7 cm 4 2d2/2h; z. = 7.4 cm 2 4d2/2h; zo = 11.1 cm g 6d2/2A with d = 0.012 cm, A = 436 nm).
26.5
Volume
28, number
3
OPTICS COMMUNICATIONS
March
1979
This is visualized in fig. 6 and experimentally verified in fig. 7. It should be noted, however, that the amount of light falling on the observation plane is the same for any given value of n, or otherwise stated: AX s 0
I*(x)
dw =
const.
(9) --2
where Ax = d -f - (d2/2h)-l = 2 Xf/d is the period of the fringes for n = 1. A special case occurs for QI= l/2 i.e. when the opaque and transparent bars of the gratings have equal width. Then, IA(x) = const. whenever n is an odd number, as can be verified from eq. (8).
3. Comparison
g(x”) (b)
extended light source
I f
i
-20
I2 Ih(x’ + x) I2 dx’.
(10)
Like eq. (5) this represents a correlation: I, = lgl* @Ih 12.
(11)
In the case of the Lau effect, the same intensity is measured by a point detector as shown in fig. 8b. The agreement of eqs. (6) and (11) expresses mathematically the geometrical reciprocity between the experiments of Lau and Talbot. Obviously the extended light source and the point detector used for Lau’s experiment yield the same result as the extended detector and the point source used for the Talbot interferometer. It should be noted that the regions of Fresnel- and Fraunhoferpropagation of the light are interchanged, as well as the sequence of the two masks g and h. In this sense, we speak of Lau’s experiment and the Talbot-interferometer-experiment as reciprocal experiments. 266
h(x’+x)
of the Lau effect and the Talbot effect
We now want to compare Lau’s experiment to Talbot self-imaging. For both the Fresnel propagation of light is essential. Therefore in both cases the same specific distances z. = n d2/2X are significant. For a more specific comparison, we consider a modified setup of the Talbot effect as shown in fig. 8a. Gratingg is illuminated by a plane wave and produces self-images. Another transparency h may be used to detect these images (e.g. by means of MoirC). The total intensity IA(x) is measured by an extended photo-detector as a function of the lateral displacement x of the transparency h. The mathematical result for Zh(x) is easily derived to be: Q(x) =JI&‘)
o-
9(X”)
h (x’) Fig. 8. Comparison experiment (b).
4. Conclusions
of-the Talbot
interferometer
(a) and Lau’s
and possible extensions
As a consequence of the theoretical discussion in the previous sections, it follows that some applications of the Talbot effect should be transferable to the Lau experiment. As was said before, the Talbot effect is applicable to Fourier spectroscopy [3] and interferometry [4]. The setup of the Talbot-Fourier-spectrometer is shown in fig. 8a. Therefore a Lau-Fourier-spectrometer will look like fig. 8b. In addition, the Lau experiment can be modified to visualize phase structures. This will be explained in a subsequent paper. In both cases, i.e. Fourier spectroscopy and interferometry, the use of an extended light source as is present in the Lau experiment might be convenient since it avoids the need of a laser and requires less stable conditions. Yet another application of the Lau effect and the Talbot effect is suggested by the correlation equations (6) and (11): I& = Jhl2 @li12. Notice, neither g nor h had to be periodic. Hence, general images can be correlated in a setup as in fig. 4 (with the lens immediately behind g). Spectral width Ah
Volume 28, number 3
OPTICS COMMUNICATIONS
will have to be restricted since the Fresnel transformation is wavelength dependent. Such a correlator had been suggested previously [6 ] . Finally, we would like to thank H.O. Bartelt for many fruitful discussions and S.K. Case for his editorial assistance.
References
March 1979
[2] W.D. Montgomery, J. Opt. Sot. Am. 58 (1968) 1112. [3] A. Lohmann, Proc. ICO Conf. on Optical instruments, London (1961) p. 58; H. Klages, Journal de Physique 2 (1967) C2-40. [4] A.W. Lohmann and D.A. Silva, Optics Comm. 2 (1971) 413; 4 (1972) 326; S. Yokozeki and T. Suzuki, Appl. Opt. 10 (1971) 1575, 1690; D.A. Silva, Appl. Opt. 11 (1972) 2613. [5] E. Lau, Ann. Phys. 6 (1948) 417; Wiss. Ann. 1 (1952) 43. [6] A.W. Lohmann, IEEE Conf. Opt. Computing, Washington D.C. (1975) p. 142, esp. methods 7,8.
[l] F. Talbot, Phil. Mag. 9 (1936) 401; J.M. Cowley and A.F. Moodie, Proc. Phys. Sot. B70 (1957) 486,497,505; B76 (1960) 378; J. Winthrop and C. Worthington, J. Opt. Sot. Am. 55 (1965) 373.
267
March 1979
OPTICS COMMUNICATIONS
THE LAU EFFECT (A DIFFRACTION
EXPERIMENT
WITH INCOHERENT
ILLUMINATION)
J. JAHNS and A.W. LOHMANN Physikalisches Institut, 8520 Erlangen, Fed. Rep. Germany
Received 26 December 1978
In 1948 Ernst Lau described a diffraction experiment, whereby the object was illuminated by an extended white light source. The object consisted of two coarse gratings, positioned behind each other at a distance of a few centimeters. Colored fringes appear at infinity. We present an explanation, based on diffraction theory. We also note similarities to Talbot’s effect called “self-imaging”, or sometimes “Fourier-imaging”. A specific application of the Talbot effect, the Talbot interferometer, is equivalent to the Lau effect in terms of the theory of partial coherence. Based on this equivalence we suggest two applications of the Lau effect, a simple interferometer and a Fourier spectrometer.
1. Introduction Talbot self-imaging is a well-known effect in coherent optics [I]. If a grating G(x) is illuminated with light from a monochromatic point source the interference patterns in specific planes behind this object will be similar to G(x). This happens in collimated or uncollimated light. Actually, the class of objects suitable for self-imaging contains not only gratings [2] . But gratings are the most important members of this class. With collimated monochromatic light the selfimages will appear at 2 d2/h and integer multiples thereof. For example, with a grating constant d = l/l0 mm and a wavelength h = l/2000 mm the Talbot distance 2 d2/h is 40 mm. Self-imaging has found applications in Fourier spectrometry [3] and in interferometry [4]. In the Talbot interferometer a second grating is placed into the selfimage plane of the first grating. No light will emerge from this interferometer, if the opaque bars of the second grating cover up the bright lines in the self-image of the first grating. The situation of complete darkness requires a careful adjustment, that can be disturbed easily by a phase object 0 between the two gratings of the Talbot interferometer (fig. 1). We mentioned the Talbot interferometer, because its setup is very similar as in the Lau experiment [S] which is the aim of our study. Instead of a monochromatic point source Lau
Gl
0
G2
F
I
Lz& l----l
CfA-f---i-rAti
Fig. 1. Talbot-Interferometer. Gl, G2: gratings; 0: phase object; I: Interference image of 0.
used an extended white source. Colorful fringes appear in plane F of fig. 1. The colors change when the distance z,, between the two gratings is varied.
2. Description
of the Lau effect
We now want to describe the setup of Lau’s experiment more thoroughly (fig. 2). To provide incoherent illumination, an extended light source is imaged onto the first grating, Cl. Gl can thus be considered as a self-luminous object. For a first understanding the following simplifying assumption can be made: the slit width of the two identical gratings should be very small compared to the slit separation d. Then a slit S of Gl acts as a line-source from which emerges a cylindrical wave (fig. 3). This wave will create secondary light 263
Volume 28, number 3 Gl
l-z,+
G2
F
L
L
f
-
at the slits P,, P, . . . of the second grating G2.
These secondary light sources will radiate with equal phases, if the path lengths (SPo), (SP1) . . . differ by integer multiples of h. This is possible indeed, if the grating separation is z. = d2/2h, a condition which follows from a simple Pythagoras consideration. For this specific wavelength, one slit of G 1 will produce interference fringes at infinity. Since the illumination is incoherent, the interference patterns generated by all the slits of Gl add in their intensities. For z. = d2/2h the maxima of all these fringe patterns coincide, so that in total one obtains interference fringes of high contrast. Fringe patterns can also be observed at infinity for a certain wavelength whenever zu = n d2/2X (n = 1,2,3,. . .). For white light illumination, the greatest fringe contrast will occur for the wavelength that satisfies the above equation. We will then observe a two color fringe pattern consisting of alternating fringes at wavelength h and at the complementary visual color. Until now, we dealt with Ronchi-rulings with very small slit width. In order to consider the case of more Gl
G2
9(x”)
h ( x’)
X
L
L
I
p&y0 h
Fig. 2. Lau’s experiment. The two gratings, Gl and G2, are illuminated by an extended light source. Observation takes place in the back focal plane F of the converging lens L (f: focal length of L). sources
March 1979
OPTICS COMMUNICATIONS
f --
Fig. 4. The setup as assumed in our analysis.
general objects, we now want to change from the geometrical to a wave-optical point of view. We consider two arbitrary transparencies h and R in the planes z = 0 and z=z o, respectively (see fig. 4). Our aim is to calculate the monochromatic intensity Z*(x) in the back focal plane of the lens L. To this end, we will first compute the complex amplitude uh (x,x1) in the observation plane due to a unit point source located at x’ = x, in the plane z = 0. The whole object h (x’) can be regarded as consisting of many point sources. Since the illumination is incoherent, the monochromatic intensity Ih is therefore obtained by summing up all the intensity contributions: ~~(~)=~1~(~,)121~,(~,~,)17
dx,.
(1)
The first step will be the calculation of uh (x,x1). We denote the unit point source at x’ = x1 in the plane z = 0 by a delta function. The complex amplitude immediately before the object I in the plane z = z. -0 is obtained by a Fresnel-transformation: u,(x”,x,) inz=z
a exp{m(x”
- x,)~/XZ,)
u -0.
(2)
After a multiplication with the complex amplitude transmission g(x)‘), the lens L performs a Fouriertransformation. This means: u,(x,xI)
a FOU’&(x”) exp[irr(x” - x,)~/XZ,]}
in the observation
plane,
(3)
(FOU {. . .} = Fourier-transform of {. . .}). The explicit calculation of the Fourier-transform yields:
u*(x,xr ) = I &“) Fig. 3. Geometrical explanation. The distances from one slit in Gl to several slits in G2 differ by integer multiples of h, if the gratings are separated by ze = d2/2k.
264
X exp {in[x” - (x1 + zofl =&x,
+,($-I
x).
x)] 2/ xZo} dx” (4)
OPTICS COMMUNICATIONS
Volume 28, number 3
March 1979
Here, we dropped a phase factor exp {ip(x,xl)} that vanishes anyway when computing the modulus square. The g(x) means the Fresnel-transform of g calculated at the point x. According to eq. (1) we get the final result by integrating over the contributions from all point sources inz=O ~~(X)=Jk(XJ21&xl
+zof-lx)12
dx,.
4
r
l--
,X Ida
-i
-ladi-
(5)
This integral just represents the correlation of Ih I2 with the modulus square of the Fresnel transform of g: z* = Ih I2 @I&
g(x)
Fig. 5. Complex amplitude transmission of a Ronchi-ruling with grating constant d and slit width ad.
(6)
(0 denotes the correlation operation). Next we will apply this result to the special case of two Ronchi-rulings as input functions (see fig. 5) which we describe by
g(x) = h(x) =
(5 rect (x*). m
z,= 31d2
From the Talbot effect is it known that the Fresneltransform of a Ronchi-ruling will again be a Ronchiruling whenever z,, = n d2/2h (n = 1,2,3,. . .), although with modified contrast, in general. The correlation of two grating-slits, i.e. of two rect (x/Ax)-functions results in a trian (x/Ax)-function. Therefore, we obtain a grating of trian-functions whenever z. = n d2/2h. We have to distinguish four cases:
C: C trian(x Cm) Ctrian(x
-zDi2)
+ “‘$-
2h
Fig. 6. Graphical representation 01= l/8 (see eq. (8)).
X
D
of I&(x) for n = 1,2,3,4, for
, n = 1,5,9,. . .
““1,
n = 2,6,10,.
..
I*(x) = f Ctrian(*)
,
c
trian(*)
with D = d f/zo.
, n=3,7,11,...
, n =4,8,12 ,... 09
Notice that the period D of the fringe patterns depends on the grating separation zo. This means that essentially the fringes become finer with increasing zo, with the special feature that the number of fringes per unit interval D is two if n is an odd number and one if n is even.
Fig. 7. Lau fringes for three different separations of two rulings (zo = 3.7 cm 4 2d2/2h; z. = 7.4 cm 2 4d2/2h; zo = 11.1 cm g 6d2/2A with d = 0.012 cm, A = 436 nm).
26.5
Volume
28, number
3
OPTICS COMMUNICATIONS
March
1979
This is visualized in fig. 6 and experimentally verified in fig. 7. It should be noted, however, that the amount of light falling on the observation plane is the same for any given value of n, or otherwise stated: AX s 0
I*(x)
dw =
const.
(9) --2
where Ax = d -f - (d2/2h)-l = 2 Xf/d is the period of the fringes for n = 1. A special case occurs for QI= l/2 i.e. when the opaque and transparent bars of the gratings have equal width. Then, IA(x) = const. whenever n is an odd number, as can be verified from eq. (8).
3. Comparison
g(x”) (b)
extended light source
I f
i
-20
I2 Ih(x’ + x) I2 dx’.
(10)
Like eq. (5) this represents a correlation: I, = lgl* @Ih 12.
(11)
In the case of the Lau effect, the same intensity is measured by a point detector as shown in fig. 8b. The agreement of eqs. (6) and (11) expresses mathematically the geometrical reciprocity between the experiments of Lau and Talbot. Obviously the extended light source and the point detector used for Lau’s experiment yield the same result as the extended detector and the point source used for the Talbot interferometer. It should be noted that the regions of Fresnel- and Fraunhoferpropagation of the light are interchanged, as well as the sequence of the two masks g and h. In this sense, we speak of Lau’s experiment and the Talbot-interferometer-experiment as reciprocal experiments. 266
h(x’+x)
of the Lau effect and the Talbot effect
We now want to compare Lau’s experiment to Talbot self-imaging. For both the Fresnel propagation of light is essential. Therefore in both cases the same specific distances z. = n d2/2X are significant. For a more specific comparison, we consider a modified setup of the Talbot effect as shown in fig. 8a. Gratingg is illuminated by a plane wave and produces self-images. Another transparency h may be used to detect these images (e.g. by means of MoirC). The total intensity IA(x) is measured by an extended photo-detector as a function of the lateral displacement x of the transparency h. The mathematical result for Zh(x) is easily derived to be: Q(x) =JI&‘)
o-
9(X”)
h (x’) Fig. 8. Comparison experiment (b).
4. Conclusions
of-the Talbot
interferometer
(a) and Lau’s
and possible extensions
As a consequence of the theoretical discussion in the previous sections, it follows that some applications of the Talbot effect should be transferable to the Lau experiment. As was said before, the Talbot effect is applicable to Fourier spectroscopy [3] and interferometry [4]. The setup of the Talbot-Fourier-spectrometer is shown in fig. 8a. Therefore a Lau-Fourier-spectrometer will look like fig. 8b. In addition, the Lau experiment can be modified to visualize phase structures. This will be explained in a subsequent paper. In both cases, i.e. Fourier spectroscopy and interferometry, the use of an extended light source as is present in the Lau experiment might be convenient since it avoids the need of a laser and requires less stable conditions. Yet another application of the Lau effect and the Talbot effect is suggested by the correlation equations (6) and (11): I& = Jhl2 @li12. Notice, neither g nor h had to be periodic. Hence, general images can be correlated in a setup as in fig. 4 (with the lens immediately behind g). Spectral width Ah
Volume 28, number 3
OPTICS COMMUNICATIONS
will have to be restricted since the Fresnel transformation is wavelength dependent. Such a correlator had been suggested previously [6 ] . Finally, we would like to thank H.O. Bartelt for many fruitful discussions and S.K. Case for his editorial assistance.
References
March 1979
[2] W.D. Montgomery, J. Opt. Sot. Am. 58 (1968) 1112. [3] A. Lohmann, Proc. ICO Conf. on Optical instruments, London (1961) p. 58; H. Klages, Journal de Physique 2 (1967) C2-40. [4] A.W. Lohmann and D.A. Silva, Optics Comm. 2 (1971) 413; 4 (1972) 326; S. Yokozeki and T. Suzuki, Appl. Opt. 10 (1971) 1575, 1690; D.A. Silva, Appl. Opt. 11 (1972) 2613. [5] E. Lau, Ann. Phys. 6 (1948) 417; Wiss. Ann. 1 (1952) 43. [6] A.W. Lohmann, IEEE Conf. Opt. Computing, Washington D.C. (1975) p. 142, esp. methods 7,8.
[l] F. Talbot, Phil. Mag. 9 (1936) 401; J.M. Cowley and A.F. Moodie, Proc. Phys. Sot. B70 (1957) 486,497,505; B76 (1960) 378; J. Winthrop and C. Worthington, J. Opt. Sot. Am. 55 (1965) 373.
267