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Diffraction from phase modulated apertures
Diffraction from phase modulated apertures M. KUITTINEN, T. JAASKELAINEN We apply the Rayleigh-Sommerfeld diffraction integral to solve the diffracted fields of periodically phase-modulated apertures. We show that the diffraction of small phase modulated apertures (with few refractive index modulation periods) behaves as for phase gratings, but the sharp diffraction peaks may already appear in the Fresnel region behind the aperture. Thus, these components may serve as lensless miniature interconnection gratings in micro-optics. KEYWORDS: apertures, phase modulation, diffraction, gratings
Introduction
Methods
The design of interconnection gratings has been investigated by many different groups using a variety of approximations 1-4. Most of these papers consider far-field diffraction of infinite beams from an infinite grating. However, in practice, both the grating and the beam are finite and, in order, to design lensless devices with small physical dimensions, the output plane should be close to the grating.
According to the general Rayleigh-Sommerfeld diffraction formula T the wave field u2(x, y) at a point of plane 2 (see Fig. 1) may be solved from
Bragg diffraction of finite beams by thick gratings has been extensively studied 5, but many interesting interconnection gratings are thin and these can be analysed by amplitude transmittance theory. Latimer has investigated the interference effects of Talbot plane patterns 6. His work considers amplitude modulation (Ronchi grating) of a plane wave. Near-field diffraction of thin gratings is described by the Rayleigh-Sommerfeld diffraction formula. Instead of the Rayleigh-Sommerfeld diffraction formula, the Fourier transform (Fraunhofer diffraction formula) is generally used to design interconnection gratings.
x
z l ~ ejk~ de dfl
(1)
where u~-(e, fl) is the incident wavefield, r(e, fl) is the complex transmittance function of the aperture, k = 2rt/2, where 2 is the wavelength, z12 is the distance of the two planes and rlz is the distance between a pair of points at planes 1 and 2. Equation (1) is valid throughout the entire space to the right of the diffracting aperture. Assuming that r~2 ~ 2 and
y
In this paper we investigate the diffraction properties of phase modulated apertures (i.e. the diffraction of finite beams). We show that these devices offer new possibilities for the design of miniature beam-splitters or optical interconnection gratings. We consider structures with a very small number of periods. The calculation is based on the Rayleigh-Sommerfeld diffraction formula. As far as we know our approach is new, although the method itself is well-known.
X
,,2
The authors are in the V~iis~il~i Laboratory, Department of Physics, University of Joensuu, PO Box 111, FIN-80101 Joensuu, Finland. Received 31 May 1993. Revised 15 November 1993.
Z=Z 1
Fig. 1
Z~Z
Illustration of the notation
0030-3992/94/0201 09-05 © 1994 Butterworth-Heinemann Ltd Optics & Laser Technology Vol 26 No 2 1994
109
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen writing r12 in the form r12 = E(z2 - zl) 2 + (x - ~)2 + (y -/~)231/2
= z12 l +
- \ z12 /
+
(2)
\ z12 / 3
Equation (1) states
exp kZl2 1+
×
\ Z12 ,/
+ --
\ ZI2 ,/ J
)
dot dfl
JSz12[ 1 + ( X z ~ f f ) Z + ( Y ' ~zxe f l ~ 2/ qd\ (3)
where u~(x, y) = u((7, fl)'c(~, fl). In fact, (3) is a convolution integral and presents convolution between the functions u~(x, y) and the impulse response
ha2(x,y) =
~ d0~dfl
i.e.
Uz(X, y) = u~ ® h12 where ® stands for convolution. We stress that it is possible to calculate diffracted fields on the Fresnel and Fraunhofer regions by using (5). The only restriction is that r12 >~ 2, which means only that the distance between the aperture plane and the observation plane must be at least 10 wavelengths. With the aid of Fourier theory it is possible to solve u2 by taking FTs from Ul and h~2, multiplying them together and taking the inverse FT of the product. The only thing of concern in numerical calculations is to make the vectors long enough so that the whole diffracted field from the aperture is gathered on plane 2. In this work we consider one-dimensional cases only. We assume that the aperture has periodic phase modulation. Hence, in the sinusoidal case, the transmittance function of the aperture can be expressed in the form
z(x) = exp ( - j 22n A n c o s ( 2 ~ x ) ) d
(6)
where An is the amplitude of the refractive index modulation, A is the period of the modulation and d is the thickness of the phase modulation area. In calculations we use 2 as a metric unit and we have defined 'modulation strength' ~" from (6) as
7" = 2hAnd~2. Results First we consider an aperture splitting an incoming plane wave into three equal intense outputs. As an example of such a device we constructed an aperture with four modulation periods (A = 2n2) and with
11 0
If we measure the directions of diffraction orders - I and +1 from the simple trigonometric equations we achieve diffraction angle 0" = 9.3 °. On the other hand, the diffraction angle should be solved from the equation 9 ;t n 3 sin O'i'= n 1 sin 0' -- i
exp(jkz1211+(z~2)2+(z~2)21u2) j,;~z12ii+(z~z)2+(z~2)21
modulation strength 7" = 1.43. The value of 7" is same as that known to produce three equal intense beams in the infinite grating diffraction s . The diffraction patterns at various distances from the aperture are shown in Fig. 2. At the distance of 5002 (Fig. 2(b)) the three peaks have equal intensity and they are clearly separate. The size of this kind of beam-splitter element is very small: for example at the wavelength of 1.5 pm the width of the aperture is ~ 38/~m and the size of the diffraction pattern is 750/Lm at a distance of 750 pm. This means that it is possible to pack multiple beam-splitters onto a very small device by making separate apertures of suitable distances from each other.
(7)
A
where n o w n l = n 3 = 1 , 0 ' = 0 ° , i = 1 and A = 2n/2. With these values 0~ is 9.2 °. So the diffraction angle produced by the small phase modulated aperture differs only slightly from the diffraction angle of the corresponding infinite phase grating. Next, we consider the effect of the starting point of the phase modulation on the diffraction pattern. In the previous case, the phase modulation was fully symmetric compared with the middle point of the aperture. Using the same aperture parameters as above we noticed a distortion in the diffraction pattern that depended on the lack of symmetry in the phase modulation of the aperture. For example, if the starting phase changes 3n/4 units, then the diffraction efficiencies of orders - 1, 0 and + 1 changes from a value of 26.6% of all orders to values of 26.0%, 27.0% and 26.8% respectively. This means that to achieve the same diffraction distribution as that of a phase grating the phase pattern on the aperture must be exactly symmetric. From Fig. 2 we know that if the aperture contains only a few phase-modulation periods, separate peaks are achieved very close behind the aperture. But what happens if we have, for example, 15 or 150 periods in the aperture? The answer is that we must go much further away from the aperture to achieve separate peaks. Figure 3 shows two cases: in (a) and (b) the number of periods is 15 and the distance from the aperture is (a) 10002 and (b) 20002, while in (c) and (d) the number of periods is 150 and the distances are 50002 and 100002, respectively. The correspondence of aperture size and distance is clearly seen. As a rule of thumb, we may conclude that if the size of the aperture is doubled (together with the number of phase modulation periods in it) we have to go four times further to achieve the same kind of diffraction pattern as with the smaller aperture. By considering the intensity distribution of the zeroth order from the Optics & Laser Technology Vol 26 No 2 1994
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen 1.0
1.0
0.8
0.8
>,0.6
,_>,,0.6
69
c 0.4
0.4
c
c
•-
"- 0.2 0.0 -400
0.2 0.0
-240
-80
a
80
= (x)
240
400
-400
-240
-80
80
=(x)
b
1.0
1.0
0.8
0.8
240
400
0.6
>,0.6 (/)
[o
c- 0.4
0.4 c
0.2
"- 0 . 2
-400
-240
-80
80 x
c
240
/'~AA 0.0 -400 -240
400
(X)
d
Jk.-./~ -80
80
=(x)
240
400
Fig. 2 Diffraction patterns at various distances from an aperture which contains four phase modulation periods and modulation strength 7" = 1.43. In (a) the distance is 250).; in (b) 500).; in (c) 750). and in (d) 1000).
1.0
1.0
0.8
0.8
>-,0.6
..,_,~0.6
(y)
c(D 0.4
0.4
(.-
-
c
' - 0.2
0.2
0.0 -1 000
-500
a
0
500
1000
x(;~)
-1000 b
0
5OO
1 000
2200
4400
=(;k)
1.0
1.0
0.8
0.8
>-,0.6
..,..,~0.6 69
c- 0.4
0.4
C
-
-500
c
0.2
0.0 -4400
"- 0.2
-2200
0
2200
4400
O0 -4400
J ~• --J -2200 0
O Fig. 3 Diffraction patterns from an aperture with 15 modulation periods in (a) and (b). In (c) and (d) diffraction is from an aperture with 150 modulation periods. The modulation strength is, in all cases, 7" = 1.43 and distances are in (a) 1000).; in (b) 2000).; in (c) 5000).; and in (d) 10000). Optics & Laser Technology Vol 26 No 2 1994
111
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen aperture with 150 periods we noticed that the distribution approaches the sinc 2 function when the distance from the aperture increases. Sinc 2 is also the form of the diffraction pattern of the pure aperture at the Fraunhofer region. Magnusson and Gaylord 8 have solved the diffraction efficiencies of various types of thin gratings. According to their results 8 the diffraction efficiencies ql of sinusoidal gratings behave like Ji2(7), ,, where J/ stands for the Bessel function of order i and, on the other hand, i also means the diffraction order. For the square wave they presented the following results I 0 os2 (7") sin 2 = (~")
when i = 0 when i = even when i = odd
We also calculated the relative diffraction
efficiencies as a function of 7"from apertures which contain only five periods of square wave or sinusoidal refraction index modulation. We made the calculations at the distance of 10002 and the values from zero to 2~ were given for the modulation strength. We found out that the relative diffraction efficiencies behave in exactly the same manner as in the case of infinite phase gratings. However, in the case of the square wave, where the symmetry requirement of phase modulation is not fulfilled, there is distortion in the diffraction efficiencies of orders - 1 and + 1. If we consider the diffraction of gratings in the Fourier plane, then all distances are assumed to be equal and optimized gratings can be thought to be planned with this assumption. However, if we consider the diffraction from a small aperture, which has phase modulation strongly diffracting the higher diffraction orders, the meaning of the distances involved must be taken into account. With a grating period of A = 2n2, the diffraction angle of the fourth order is 39.5 °. At the distance of 60002 this means that the central point of the fourth order is 49532 away from the middle point of the diffraction pattern and the distance from the middle point of the aperture to that point is 77802. This separation of distances (17802) causes distortion of the diffracted intensities. As an example, we calculated the diffraction pattern from an aperture which contains eight periods of a kinoform. The selected kinoform produces nine equal intensity beams with efficiency of 10.4% (see Ref. 10). The calculated diffraction pattern with the relative intensities of peaks at a distance of 60002 is shown in Fig. 4. It is noticeable that although the middle peak is 2.8 times higher than the lowest ones, the power of the middle peak is only 1.3 times higher than the powers of the peaks nearest the edge. This is because the peaks nearest the edge are wider than the middle ones. The differences in peak intensities at the observation plane can be corrected by making the diffraction pattern narrower, whereupon the effect of distance decreases. The narrowing of the diffraction pattern is possible by increasing the length of phase modulation period A. However, at the same time, the separation of peaks also decreases.
1 12
1,0 0.8 e0.6 q)
-~0.4 E
0.2
-6000
-5000
0 z(X)
5000
6000
Fig. 4 Diffraction pattern from an aperture which contains eight periods of kinoform. The kinoform is planned to produce nine equal intense beams with an efficiency of 10.4% 1°. In the figure, the powers of peaks are, from left to right, 8.6%, 9.7%, 10.5%, 11.1%, 11.2%, 11.0%, 10.6%, 9.8% and 8.6% from the total diffracted power. The distance from the aperture is 60002
So far, we have studied the diffraction from small phase modulated apertures illuminated by plane waves. On the other hand, the situation can be thought of such that we illuminate a large grating with a very narrow beam when actually only few periods are illuminated. Because narrow beams with constant amplitude are difficult to make, we calculated the diffraction pattern in the case where the grating is illuminated with a narrow laser beam. The amplitude distribution of a laser beam is Gaussian, and it can be expressed in the form
A(x)
= Ao e-t~x-x°)/x°]2
(9)
where A o is the amplitude in the middle of the beam, Xo is the coordinate of the middle point and x is the distance from the middle point. The phase of the laser beam is assumed to be constant. Because we wanted only to test the effect of the symmetric amplitude distribution we decided to split the laser beam so that, at the edge of the illumination region, the amplitude has the value Ao/e. As expected, the symmetric amplitude distribution has no effect on the form of the diffraction pattern (see Fig. 5) and only the .0
,
4
r
0.8 "~0.6
~ 0.4
""
0.2 0.0
' -~
260
,
,' ~
-ls6
/~
-52
s2
260
Fig. 5 Diffraction patterns from an aperture with five periods of cosine phase modulation. The modulation strength is 2/' = 1.43. The continuous curve presents plane-wave illumination and the dashed curve the laser beam illumination Optics & Laser Technology Vol 26 No 2 1994
Diffraction from phase modulated apertures: M. Kuittinen and 7-. Jaaskelainen total intensity decreases. It is also remarkable that the small modulation between peaks disappears almost perfectly. This is because, at the edges of the aperture that damps the edge effect, the amplitude is small.
Conclusions Our calculations show that a small phase modulated aperture with a few periods diffracts like an infinite grating. The main difference between an aperture diffraction and that of a grating is that the separate peaks are produced only a few hundred wavelengths away from the aperture. To achieve exactly the same kind of diffraction as gratings, the phase modulation on the aperture must be fully symmetric. In the case of higher-order diffraction (kinoform example in the results section) our calculations show that the effect of distance on the diffracted intensity is remarkable. If exact uniformity is needed the grating structure should be re-optimized. The optimization task in the Fresnel region is very complicated and time consuming, but we suggest that the optimization could be performed with traditional methods in a Fourier plane by taking into account the required corrections. The diffraction properties of apertures remain when a few phase modulated periods are illuminated with a wave which has a symmetric amplitude distribution. This means that it is possible to achieve grating-like diffraction by illuminating a grating with a narrow laser beam. Although we have considered one-dimensional cases only, the results are valid in two dimensions. The description of the aperture must be performed in two
dimensions and then the only thing to do is to change all FTs into two-dimensional FTs in the calculations. Phase modulated apertures could have many applications in integrated optics. When the propagating wave has a symmetric amplitude distribution and constant phase, the phase modulated aperture can be used as a beam-splitter. This means that this kind of beam-splitter could be added directly to the output of a diode laser. Also, small devices which contain several beam-splitters can be fabricated by making phase modulating apertures at suitable distances from each other.
References 1 Dammann, H., G6rtler, K. High-efficiency in-line multiple imaging by means of multiple phase holograms, Opt Commun, 3 (1971) 312-315 2 Kraekhard, U., Streibel, N. Design of Dammann-gratings for array generation, Opt Commun, 74 (1989) 31-36 3 Turunen, J., Vasara, A., Westerholm, J. Kinoform phase relief synthesis: a stochastic method, Opt Eng, 28 (1989) 1162-1167 4 Jaaskelainen, T., Kuittinea, M. Planar interconnection gratings, Opt Comp & Proc, 2 (1992) 29-38 5 Moharam, M.G., Gaylord, T.K., Magnusson, R. Bragg diffraction of finite beams by thick gratings, J Opt Soc Am, 70 (1980) 300-304 6 Latimer, P. Talbot plane patterns: grating images or interference effects? Appl Opt, 32 (1993) 1078-1083 7 Gaskill, J.D. Linear Systems, Fourier Transforms, and Optics, Wiley, New York (1978) 361-390 8 Magnusson, R., Gaylord, T.K. Diffraction efficiencies of thin phase gratings with arbitrary grating shape, J Opt Soc Am, 68 (1978) 806-809 9 Gaylord, T.K., Moharam, M.G. Analysis and applications of optical diffraction by gratings, Proc IEEE, 73 (1985) 894-937 l0 Turunen, J., Fagerholm, J., Vasara, A., Taghlzadeh, M.R. Detour-phase kinoform interconnections: the concept and fabrication considerations, J Opt Soc Am A, 7 (1990) 1202-1208
Optics & Laser Technology
Optics & Laser Technology Vol 26 No 2 1994
11 3
Introduction
Methods
The design of interconnection gratings has been investigated by many different groups using a variety of approximations 1-4. Most of these papers consider far-field diffraction of infinite beams from an infinite grating. However, in practice, both the grating and the beam are finite and, in order, to design lensless devices with small physical dimensions, the output plane should be close to the grating.
According to the general Rayleigh-Sommerfeld diffraction formula T the wave field u2(x, y) at a point of plane 2 (see Fig. 1) may be solved from
Bragg diffraction of finite beams by thick gratings has been extensively studied 5, but many interesting interconnection gratings are thin and these can be analysed by amplitude transmittance theory. Latimer has investigated the interference effects of Talbot plane patterns 6. His work considers amplitude modulation (Ronchi grating) of a plane wave. Near-field diffraction of thin gratings is described by the Rayleigh-Sommerfeld diffraction formula. Instead of the Rayleigh-Sommerfeld diffraction formula, the Fourier transform (Fraunhofer diffraction formula) is generally used to design interconnection gratings.
x
z l ~ ejk~ de dfl
(1)
where u~-(e, fl) is the incident wavefield, r(e, fl) is the complex transmittance function of the aperture, k = 2rt/2, where 2 is the wavelength, z12 is the distance of the two planes and rlz is the distance between a pair of points at planes 1 and 2. Equation (1) is valid throughout the entire space to the right of the diffracting aperture. Assuming that r~2 ~ 2 and
y
In this paper we investigate the diffraction properties of phase modulated apertures (i.e. the diffraction of finite beams). We show that these devices offer new possibilities for the design of miniature beam-splitters or optical interconnection gratings. We consider structures with a very small number of periods. The calculation is based on the Rayleigh-Sommerfeld diffraction formula. As far as we know our approach is new, although the method itself is well-known.
X
,,2
The authors are in the V~iis~il~i Laboratory, Department of Physics, University of Joensuu, PO Box 111, FIN-80101 Joensuu, Finland. Received 31 May 1993. Revised 15 November 1993.
Z=Z 1
Fig. 1
Z~Z
Illustration of the notation
0030-3992/94/0201 09-05 © 1994 Butterworth-Heinemann Ltd Optics & Laser Technology Vol 26 No 2 1994
109
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen writing r12 in the form r12 = E(z2 - zl) 2 + (x - ~)2 + (y -/~)231/2
= z12 l +
- \ z12 /
+
(2)
\ z12 / 3
Equation (1) states
exp kZl2 1+
×
\ Z12 ,/
+ --
\ ZI2 ,/ J
)
dot dfl
JSz12[ 1 + ( X z ~ f f ) Z + ( Y ' ~zxe f l ~ 2/ qd\ (3)
where u~(x, y) = u((7, fl)'c(~, fl). In fact, (3) is a convolution integral and presents convolution between the functions u~(x, y) and the impulse response
ha2(x,y) =
~ d0~dfl
i.e.
Uz(X, y) = u~ ® h12 where ® stands for convolution. We stress that it is possible to calculate diffracted fields on the Fresnel and Fraunhofer regions by using (5). The only restriction is that r12 >~ 2, which means only that the distance between the aperture plane and the observation plane must be at least 10 wavelengths. With the aid of Fourier theory it is possible to solve u2 by taking FTs from Ul and h~2, multiplying them together and taking the inverse FT of the product. The only thing of concern in numerical calculations is to make the vectors long enough so that the whole diffracted field from the aperture is gathered on plane 2. In this work we consider one-dimensional cases only. We assume that the aperture has periodic phase modulation. Hence, in the sinusoidal case, the transmittance function of the aperture can be expressed in the form
z(x) = exp ( - j 22n A n c o s ( 2 ~ x ) ) d
(6)
where An is the amplitude of the refractive index modulation, A is the period of the modulation and d is the thickness of the phase modulation area. In calculations we use 2 as a metric unit and we have defined 'modulation strength' ~" from (6) as
7" = 2hAnd~2. Results First we consider an aperture splitting an incoming plane wave into three equal intense outputs. As an example of such a device we constructed an aperture with four modulation periods (A = 2n2) and with
11 0
If we measure the directions of diffraction orders - I and +1 from the simple trigonometric equations we achieve diffraction angle 0" = 9.3 °. On the other hand, the diffraction angle should be solved from the equation 9 ;t n 3 sin O'i'= n 1 sin 0' -- i
exp(jkz1211+(z~2)2+(z~2)21u2) j,;~z12ii+(z~z)2+(z~2)21
modulation strength 7" = 1.43. The value of 7" is same as that known to produce three equal intense beams in the infinite grating diffraction s . The diffraction patterns at various distances from the aperture are shown in Fig. 2. At the distance of 5002 (Fig. 2(b)) the three peaks have equal intensity and they are clearly separate. The size of this kind of beam-splitter element is very small: for example at the wavelength of 1.5 pm the width of the aperture is ~ 38/~m and the size of the diffraction pattern is 750/Lm at a distance of 750 pm. This means that it is possible to pack multiple beam-splitters onto a very small device by making separate apertures of suitable distances from each other.
(7)
A
where n o w n l = n 3 = 1 , 0 ' = 0 ° , i = 1 and A = 2n/2. With these values 0~ is 9.2 °. So the diffraction angle produced by the small phase modulated aperture differs only slightly from the diffraction angle of the corresponding infinite phase grating. Next, we consider the effect of the starting point of the phase modulation on the diffraction pattern. In the previous case, the phase modulation was fully symmetric compared with the middle point of the aperture. Using the same aperture parameters as above we noticed a distortion in the diffraction pattern that depended on the lack of symmetry in the phase modulation of the aperture. For example, if the starting phase changes 3n/4 units, then the diffraction efficiencies of orders - 1, 0 and + 1 changes from a value of 26.6% of all orders to values of 26.0%, 27.0% and 26.8% respectively. This means that to achieve the same diffraction distribution as that of a phase grating the phase pattern on the aperture must be exactly symmetric. From Fig. 2 we know that if the aperture contains only a few phase-modulation periods, separate peaks are achieved very close behind the aperture. But what happens if we have, for example, 15 or 150 periods in the aperture? The answer is that we must go much further away from the aperture to achieve separate peaks. Figure 3 shows two cases: in (a) and (b) the number of periods is 15 and the distance from the aperture is (a) 10002 and (b) 20002, while in (c) and (d) the number of periods is 150 and the distances are 50002 and 100002, respectively. The correspondence of aperture size and distance is clearly seen. As a rule of thumb, we may conclude that if the size of the aperture is doubled (together with the number of phase modulation periods in it) we have to go four times further to achieve the same kind of diffraction pattern as with the smaller aperture. By considering the intensity distribution of the zeroth order from the Optics & Laser Technology Vol 26 No 2 1994
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen 1.0
1.0
0.8
0.8
>,0.6
,_>,,0.6
69
c 0.4
0.4
c
c
•-
"- 0.2 0.0 -400
0.2 0.0
-240
-80
a
80
= (x)
240
400
-400
-240
-80
80
=(x)
b
1.0
1.0
0.8
0.8
240
400
0.6
>,0.6 (/)
[o
c- 0.4
0.4 c
0.2
"- 0 . 2
-400
-240
-80
80 x
c
240
/'~AA 0.0 -400 -240
400
(X)
d
Jk.-./~ -80
80
=(x)
240
400
Fig. 2 Diffraction patterns at various distances from an aperture which contains four phase modulation periods and modulation strength 7" = 1.43. In (a) the distance is 250).; in (b) 500).; in (c) 750). and in (d) 1000).
1.0
1.0
0.8
0.8
>-,0.6
..,_,~0.6
(y)
c(D 0.4
0.4
(.-
-
c
' - 0.2
0.2
0.0 -1 000
-500
a
0
500
1000
x(;~)
-1000 b
0
5OO
1 000
2200
4400
=(;k)
1.0
1.0
0.8
0.8
>-,0.6
..,..,~0.6 69
c- 0.4
0.4
C
-
-500
c
0.2
0.0 -4400
"- 0.2
-2200
0
2200
4400
O0 -4400
J ~• --J -2200 0
O Fig. 3 Diffraction patterns from an aperture with 15 modulation periods in (a) and (b). In (c) and (d) diffraction is from an aperture with 150 modulation periods. The modulation strength is, in all cases, 7" = 1.43 and distances are in (a) 1000).; in (b) 2000).; in (c) 5000).; and in (d) 10000). Optics & Laser Technology Vol 26 No 2 1994
111
Diffraction from phase modulated apertures: M. Kuittinen and T. Jaaskelainen aperture with 150 periods we noticed that the distribution approaches the sinc 2 function when the distance from the aperture increases. Sinc 2 is also the form of the diffraction pattern of the pure aperture at the Fraunhofer region. Magnusson and Gaylord 8 have solved the diffraction efficiencies of various types of thin gratings. According to their results 8 the diffraction efficiencies ql of sinusoidal gratings behave like Ji2(7), ,, where J/ stands for the Bessel function of order i and, on the other hand, i also means the diffraction order. For the square wave they presented the following results I 0 os2 (7") sin 2 = (~")
when i = 0 when i = even when i = odd
We also calculated the relative diffraction
efficiencies as a function of 7"from apertures which contain only five periods of square wave or sinusoidal refraction index modulation. We made the calculations at the distance of 10002 and the values from zero to 2~ were given for the modulation strength. We found out that the relative diffraction efficiencies behave in exactly the same manner as in the case of infinite phase gratings. However, in the case of the square wave, where the symmetry requirement of phase modulation is not fulfilled, there is distortion in the diffraction efficiencies of orders - 1 and + 1. If we consider the diffraction of gratings in the Fourier plane, then all distances are assumed to be equal and optimized gratings can be thought to be planned with this assumption. However, if we consider the diffraction from a small aperture, which has phase modulation strongly diffracting the higher diffraction orders, the meaning of the distances involved must be taken into account. With a grating period of A = 2n2, the diffraction angle of the fourth order is 39.5 °. At the distance of 60002 this means that the central point of the fourth order is 49532 away from the middle point of the diffraction pattern and the distance from the middle point of the aperture to that point is 77802. This separation of distances (17802) causes distortion of the diffracted intensities. As an example, we calculated the diffraction pattern from an aperture which contains eight periods of a kinoform. The selected kinoform produces nine equal intensity beams with efficiency of 10.4% (see Ref. 10). The calculated diffraction pattern with the relative intensities of peaks at a distance of 60002 is shown in Fig. 4. It is noticeable that although the middle peak is 2.8 times higher than the lowest ones, the power of the middle peak is only 1.3 times higher than the powers of the peaks nearest the edge. This is because the peaks nearest the edge are wider than the middle ones. The differences in peak intensities at the observation plane can be corrected by making the diffraction pattern narrower, whereupon the effect of distance decreases. The narrowing of the diffraction pattern is possible by increasing the length of phase modulation period A. However, at the same time, the separation of peaks also decreases.
1 12
1,0 0.8 e0.6 q)
-~0.4 E
0.2
-6000
-5000
0 z(X)
5000
6000
Fig. 4 Diffraction pattern from an aperture which contains eight periods of kinoform. The kinoform is planned to produce nine equal intense beams with an efficiency of 10.4% 1°. In the figure, the powers of peaks are, from left to right, 8.6%, 9.7%, 10.5%, 11.1%, 11.2%, 11.0%, 10.6%, 9.8% and 8.6% from the total diffracted power. The distance from the aperture is 60002
So far, we have studied the diffraction from small phase modulated apertures illuminated by plane waves. On the other hand, the situation can be thought of such that we illuminate a large grating with a very narrow beam when actually only few periods are illuminated. Because narrow beams with constant amplitude are difficult to make, we calculated the diffraction pattern in the case where the grating is illuminated with a narrow laser beam. The amplitude distribution of a laser beam is Gaussian, and it can be expressed in the form
A(x)
= Ao e-t~x-x°)/x°]2
(9)
where A o is the amplitude in the middle of the beam, Xo is the coordinate of the middle point and x is the distance from the middle point. The phase of the laser beam is assumed to be constant. Because we wanted only to test the effect of the symmetric amplitude distribution we decided to split the laser beam so that, at the edge of the illumination region, the amplitude has the value Ao/e. As expected, the symmetric amplitude distribution has no effect on the form of the diffraction pattern (see Fig. 5) and only the .0
,
4
r
0.8 "~0.6
~ 0.4
""
0.2 0.0
' -~
260
,
,' ~
-ls6
/~
-52
s2
260
Fig. 5 Diffraction patterns from an aperture with five periods of cosine phase modulation. The modulation strength is 2/' = 1.43. The continuous curve presents plane-wave illumination and the dashed curve the laser beam illumination Optics & Laser Technology Vol 26 No 2 1994
Diffraction from phase modulated apertures: M. Kuittinen and 7-. Jaaskelainen total intensity decreases. It is also remarkable that the small modulation between peaks disappears almost perfectly. This is because, at the edges of the aperture that damps the edge effect, the amplitude is small.
Conclusions Our calculations show that a small phase modulated aperture with a few periods diffracts like an infinite grating. The main difference between an aperture diffraction and that of a grating is that the separate peaks are produced only a few hundred wavelengths away from the aperture. To achieve exactly the same kind of diffraction as gratings, the phase modulation on the aperture must be fully symmetric. In the case of higher-order diffraction (kinoform example in the results section) our calculations show that the effect of distance on the diffracted intensity is remarkable. If exact uniformity is needed the grating structure should be re-optimized. The optimization task in the Fresnel region is very complicated and time consuming, but we suggest that the optimization could be performed with traditional methods in a Fourier plane by taking into account the required corrections. The diffraction properties of apertures remain when a few phase modulated periods are illuminated with a wave which has a symmetric amplitude distribution. This means that it is possible to achieve grating-like diffraction by illuminating a grating with a narrow laser beam. Although we have considered one-dimensional cases only, the results are valid in two dimensions. The description of the aperture must be performed in two
dimensions and then the only thing to do is to change all FTs into two-dimensional FTs in the calculations. Phase modulated apertures could have many applications in integrated optics. When the propagating wave has a symmetric amplitude distribution and constant phase, the phase modulated aperture can be used as a beam-splitter. This means that this kind of beam-splitter could be added directly to the output of a diode laser. Also, small devices which contain several beam-splitters can be fabricated by making phase modulating apertures at suitable distances from each other.
References 1 Dammann, H., G6rtler, K. High-efficiency in-line multiple imaging by means of multiple phase holograms, Opt Commun, 3 (1971) 312-315 2 Kraekhard, U., Streibel, N. Design of Dammann-gratings for array generation, Opt Commun, 74 (1989) 31-36 3 Turunen, J., Vasara, A., Westerholm, J. Kinoform phase relief synthesis: a stochastic method, Opt Eng, 28 (1989) 1162-1167 4 Jaaskelainen, T., Kuittinea, M. Planar interconnection gratings, Opt Comp & Proc, 2 (1992) 29-38 5 Moharam, M.G., Gaylord, T.K., Magnusson, R. Bragg diffraction of finite beams by thick gratings, J Opt Soc Am, 70 (1980) 300-304 6 Latimer, P. Talbot plane patterns: grating images or interference effects? Appl Opt, 32 (1993) 1078-1083 7 Gaskill, J.D. Linear Systems, Fourier Transforms, and Optics, Wiley, New York (1978) 361-390 8 Magnusson, R., Gaylord, T.K. Diffraction efficiencies of thin phase gratings with arbitrary grating shape, J Opt Soc Am, 68 (1978) 806-809 9 Gaylord, T.K., Moharam, M.G. Analysis and applications of optical diffraction by gratings, Proc IEEE, 73 (1985) 894-937 l0 Turunen, J., Fagerholm, J., Vasara, A., Taghlzadeh, M.R. Detour-phase kinoform interconnections: the concept and fabrication considerations, J Opt Soc Am A, 7 (1990) 1202-1208
Optics & Laser Technology
Optics & Laser Technology Vol 26 No 2 1994
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