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A new technique for the measurement of phase retardation
A new technique for the measurement of phase retardation M. SYPEK This paper presents a new and simple technique of phase retardation measurement. It is based on computer-aided analysis of diffraction patterns generated by phase objects. Special flatness of the object substrate is not required for this method. The method is especially suitable for techniques where a phase modulation is obtained by the exposure of a photo-sensitive medium. The method can aid the prediction of the desired phase shift from the exposure. The phase retardation measurement method is described for an efficient kin®form manufactured using a simple, well known photochemical (silver-halide) process. Experimental results are presented. KEYWORDS: phase retardation, diffraction patterns
Introduction A typical optical object in an optical system is characterized by the distribution of light attenuation as well as by the phase shift distribution. The light attenuation distribution of the object can be found simply, but accurate phase shift measurement is rather complicated z. Standard interferometric methods require a special type of object substrate (with flatness A/10-A/20 and high homogeneity) which does not cause any noise and makes accurate measurements possible. For phase retardation measurement with a range over 2n in the interferon.eter, two wavelengths of light source are necessary. The diffraction by the object pattern is caused both by the surface relief and by the changes of refractive index throughout the pattern. Reflective interferometric methods are therefore strongly limited. Because of difficulties connected with preparing and measuring the phase object, especially if the phase shift is greater than 2~, a new method of phase retardation value measurement using both diffraction and computer simulation is proposed.
Method for the measurement of phase retardation A test pattern composed of a few diffraction gratings with different grey levels (different depth of moduThe author is at the Institute of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland. Received 26 April 1990. Revised 10 July 1990.
lation - rectangular shape) was plotted using a plotter. This pattern was photographically reproduced on to a phase photo-sensitive medium. The distribution of the exposure ® for a selected grating (depth modulation, i) on the photo-sensitive medium is given by the formula:
Oi(x) = Z f ( x - nAx)
(1)
n
wheref Ax i
is the fringe shape function (image of rectangular shape with depth modulation i) is the period of the grating is the index corresponding to depth of modulation of the chosen grating pattern (different grey levels)
The fringe shape functionf, for each area (i) was measured using a CCD camera and was stored on floppy disk. Profiles of the fringes - for two given areas are shown in Fig. 1. After chemical processing (for some media, chemical processing is not necessary) we obtain phase diffraction gratings with transmittances
ti(x)=exp(iOi)=exp[i Z gi(x--nAx)]
(2)
I1
where gi is the fringe phase shape function of the grating (i). The aim of the method is to find a relationship
0 0 3 0 - 3 9 9 2 / 9 1 / 0 1 0 0 4 2 - 0 3 © 1991 Butterworth-Heinemann Ltd 42
Optics 8- Laser Technology Vol 23 No 1 1991
This pattern is transformed into a phase object with multilevel phase retardation. We will apply the method described in the previous section to the large scale manufacture of kinoform.
1.0
I
After exposure (1) and developing we obtain the desired optical density distribution ~ on the photoplate. 50 IJm
x
50 pm
x
Fig. 1 The fringe profile for a given diffraction grating for two depths of modulation (i) and (/')
between functions f and gi (between exposure O and phase shift 0 of the media). Each area (i) of the diffraction grating is placed in a laser beam. The intensity distribution in the Fraunhofer diffraction field is measured using the CCD camera and stored on floppy disk. This distribution is expressed by the formula Ti( v )
I-~-{exp( i4h(x ) )} l 2
=
where , ~ v
}
Fourier transformation spatial frequency
(6)
where di is the optical density shape of the fringe. Next the photograph is bleached. For further consideration let us assume that for the silver-halide emulsion, dependence between optical density and phase retardation is linear 2. 0
=
,~t~
(7)
Using (7), the formula (6) is transformed into ~i = ,~¢ Z
(8)
di(x - n Ax)
Using the method described we can find a factor .~1 for a given process and emulsion. Making a kinoform with a desired phase shift in a desired place is possible.
(4)
tci(X) = e x p ( i O c i ( x ) )
Experimental results and discussion
Using the Fast Fourier Transformation (FFT) algorithm we can compute the distribution of the intensity in the Fraunhofer diffraction field for the simulated phase object (4). =
nAx)
tl
(3)
The analogical operation is performed by computer simulation. The measured shape of the fringef, is computed into the phase object with complex transmittance
Tci( V )
6J;i = ~ , , d i ( x -
I.~-{exp( i¢ci(x ) )} l 2
(5)
By scanning the gi function in the computer simulation, repeating the FFT algorithm and comparing the experimental and simulation results we can find the gi function value for a given f- function. This algorithm was repeated for different diffraction gratings (i) with different depth of modulation (differentf values). A chart plotting the method used is shown in Fig. 2.
Application in bleached kinoform formation Fabrication of a multigrey-level pattern is the simplest well-known technique used in kinoform fabrication 2-4.
Figure 3 shows the obtained dependence t~(c_z) for the material used (Agfa Gevaert 10E56). The factor, ,r/= 5.11 _+ 0.41, was found. The error is caused by light-measurement errors, numeric-processing errors and deviation from the linear dependence ~ ( 9 ) for the silver-halide emulsion. The linear approximation (7) is sufficient for the 9 range 0-1.5 which corresponds to a phase retardation of (0-2.8)n. Based on the above dependence, several kinoforms were made: a convergent lens, a divergent lens, and a phase diffraction grating with a saw blade fringe profile. Four steps of phase intervals were used. A test chart composed of four digits was imaged by a convergent lens with focal length f = + 500 mm and diameter d = 10 mm (Fig. 4). A sodium discharge lamp was used as a light source.
o Chemical H process
Diffraction ~ { field measurement
l'i racti°n grating I ~ 'xo°ur profile I pattern
gi function ~]
Fig. 2
H
J lmeasurement
[(simu
FFT
ation)l - I
Chart showing the method
Optics 8- Laser Technology Vol 23 No 1 1991
Results
Comparison
gi for
value
fi
~b] Results I-I Fig. 3
I
I
I
I
I
0.2
o.q
0.6
0.8
I .0
.~
Dependence 0(~) for AGFA GEVAERT 10E56 holoplates
43
described above enables phase distribution measurements on the standard substrate with high nonperiodical phase noise to be made. A univocal phase shift measured over 2rr is possible. These facts allow the manufacture of phase objects in quantity without using high technology. However, there are some disadvantages of the method. The fringe shape measurement is limited by microscope resolution. The optics used for reproduction of the plotted pattern limits the number of zones in the kinoform element. By using high quality lenses (for integrated circuit mask technology) we can reach about 500-1000 lines mm -I, which is enough for many applications, especially for far infra-red radiation. The described method can also be used for some ideas testing. Many kinoform HOEs with a great number of parameters can be produced and tested by this simple, inexpensive and quick method.
Fig. 4
Acknowledgement
Test chart imaged by a kinoform lens
Table 1. Diffraction efficiency 1/ in +1 diffraction order as a function of the number of the quantization intervals, n n 0%
2 40
3 68
4 81
5 87
6 91
7 93
8 95
9 96
10 97
20 99
Diffraction efficiency is not less than 70%. The theoretical diffraction efficiency O as a function of the number of quantization intervals for the saw blade profile is given in Table 1. The phase retardation measurement technique presented in this paper is especially relevant to the problem of Holographic Optical Elements (HOE) manufacture. We can find applications of HOEs in many branches of science 89• . HOEs in the form of efficient kinoforms are especially useful ~'5. Diffraction efficiency for a theoretical kinoform reaches 100%. The phase retardation measurement technique
44
The author would like to thank Barbara Smolinska and Andrzej Kolodzielczyk for the many helpful suggestions that they have contributed towards improving this paper. References 1 Koronkevich, V.P. et al,'Kinoform optical components: Methods of design, manufacturing technology and practical applications', Avtometr~va, (1), (1985) 4-25 2 Jordan, J.A. Jr et al, "Kinoform lenses'. Appl Opt, 9, (8), (1970), 1883 3 Dallas, W.J. 'Kinoform fabrication - a new method'. Opt Comm, 8, (4), (1973), 340 4 Clair, J.J,, 'New methods to synthesise kinoforms', Opt Comm. 6, (2), (1972), 135 5 Koronkevich, V.P. et al, 'Kinoform Optical Elements', Optical Information Processing, Plenum Press, (1975) 6 Poleshchuk, A. et al, "Application of an evaporated photoresist (As2S3) in the production of kinoform optical elements', J Imag Sci, 30, (1986), 132-135 7 Sypek, M. 'Phase Retardation Measurement for Simple Kinoform Technology', Proc SPIE - Holography '89 8 Pappu, S.V. 'Holographic optical elements state-of-the-art review: Part l, Opt Laser Teehnol. 21, (5), (1989), 305 9 Pappu, S.V. 'Holographic optical elements state-of-the-art review: Part II', Opt Laser Technol. 21, (6), (1989), 365
Optics El Laser Technology Vol 23 No 1 1991
Introduction A typical optical object in an optical system is characterized by the distribution of light attenuation as well as by the phase shift distribution. The light attenuation distribution of the object can be found simply, but accurate phase shift measurement is rather complicated z. Standard interferometric methods require a special type of object substrate (with flatness A/10-A/20 and high homogeneity) which does not cause any noise and makes accurate measurements possible. For phase retardation measurement with a range over 2n in the interferon.eter, two wavelengths of light source are necessary. The diffraction by the object pattern is caused both by the surface relief and by the changes of refractive index throughout the pattern. Reflective interferometric methods are therefore strongly limited. Because of difficulties connected with preparing and measuring the phase object, especially if the phase shift is greater than 2~, a new method of phase retardation value measurement using both diffraction and computer simulation is proposed.
Method for the measurement of phase retardation A test pattern composed of a few diffraction gratings with different grey levels (different depth of moduThe author is at the Institute of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland. Received 26 April 1990. Revised 10 July 1990.
lation - rectangular shape) was plotted using a plotter. This pattern was photographically reproduced on to a phase photo-sensitive medium. The distribution of the exposure ® for a selected grating (depth modulation, i) on the photo-sensitive medium is given by the formula:
Oi(x) = Z f ( x - nAx)
(1)
n
wheref Ax i
is the fringe shape function (image of rectangular shape with depth modulation i) is the period of the grating is the index corresponding to depth of modulation of the chosen grating pattern (different grey levels)
The fringe shape functionf, for each area (i) was measured using a CCD camera and was stored on floppy disk. Profiles of the fringes - for two given areas are shown in Fig. 1. After chemical processing (for some media, chemical processing is not necessary) we obtain phase diffraction gratings with transmittances
ti(x)=exp(iOi)=exp[i Z gi(x--nAx)]
(2)
I1
where gi is the fringe phase shape function of the grating (i). The aim of the method is to find a relationship
0 0 3 0 - 3 9 9 2 / 9 1 / 0 1 0 0 4 2 - 0 3 © 1991 Butterworth-Heinemann Ltd 42
Optics 8- Laser Technology Vol 23 No 1 1991
This pattern is transformed into a phase object with multilevel phase retardation. We will apply the method described in the previous section to the large scale manufacture of kinoform.
1.0
I
After exposure (1) and developing we obtain the desired optical density distribution ~ on the photoplate. 50 IJm
x
50 pm
x
Fig. 1 The fringe profile for a given diffraction grating for two depths of modulation (i) and (/')
between functions f and gi (between exposure O and phase shift 0 of the media). Each area (i) of the diffraction grating is placed in a laser beam. The intensity distribution in the Fraunhofer diffraction field is measured using the CCD camera and stored on floppy disk. This distribution is expressed by the formula Ti( v )
I-~-{exp( i4h(x ) )} l 2
=
where , ~ v
}
Fourier transformation spatial frequency
(6)
where di is the optical density shape of the fringe. Next the photograph is bleached. For further consideration let us assume that for the silver-halide emulsion, dependence between optical density and phase retardation is linear 2. 0
=
,~t~
(7)
Using (7), the formula (6) is transformed into ~i = ,~¢ Z
(8)
di(x - n Ax)
Using the method described we can find a factor .~1 for a given process and emulsion. Making a kinoform with a desired phase shift in a desired place is possible.
(4)
tci(X) = e x p ( i O c i ( x ) )
Experimental results and discussion
Using the Fast Fourier Transformation (FFT) algorithm we can compute the distribution of the intensity in the Fraunhofer diffraction field for the simulated phase object (4). =
nAx)
tl
(3)
The analogical operation is performed by computer simulation. The measured shape of the fringef, is computed into the phase object with complex transmittance
Tci( V )
6J;i = ~ , , d i ( x -
I.~-{exp( i¢ci(x ) )} l 2
(5)
By scanning the gi function in the computer simulation, repeating the FFT algorithm and comparing the experimental and simulation results we can find the gi function value for a given f- function. This algorithm was repeated for different diffraction gratings (i) with different depth of modulation (differentf values). A chart plotting the method used is shown in Fig. 2.
Application in bleached kinoform formation Fabrication of a multigrey-level pattern is the simplest well-known technique used in kinoform fabrication 2-4.
Figure 3 shows the obtained dependence t~(c_z) for the material used (Agfa Gevaert 10E56). The factor, ,r/= 5.11 _+ 0.41, was found. The error is caused by light-measurement errors, numeric-processing errors and deviation from the linear dependence ~ ( 9 ) for the silver-halide emulsion. The linear approximation (7) is sufficient for the 9 range 0-1.5 which corresponds to a phase retardation of (0-2.8)n. Based on the above dependence, several kinoforms were made: a convergent lens, a divergent lens, and a phase diffraction grating with a saw blade fringe profile. Four steps of phase intervals were used. A test chart composed of four digits was imaged by a convergent lens with focal length f = + 500 mm and diameter d = 10 mm (Fig. 4). A sodium discharge lamp was used as a light source.
o Chemical H process
Diffraction ~ { field measurement
l'i racti°n grating I ~ 'xo°ur profile I pattern
gi function ~]
Fig. 2
H
J lmeasurement
[(simu
FFT
ation)l - I
Chart showing the method
Optics 8- Laser Technology Vol 23 No 1 1991
Results
Comparison
gi for
value
fi
~b] Results I-I Fig. 3
I
I
I
I
I
0.2
o.q
0.6
0.8
I .0
.~
Dependence 0(~) for AGFA GEVAERT 10E56 holoplates
43
described above enables phase distribution measurements on the standard substrate with high nonperiodical phase noise to be made. A univocal phase shift measured over 2rr is possible. These facts allow the manufacture of phase objects in quantity without using high technology. However, there are some disadvantages of the method. The fringe shape measurement is limited by microscope resolution. The optics used for reproduction of the plotted pattern limits the number of zones in the kinoform element. By using high quality lenses (for integrated circuit mask technology) we can reach about 500-1000 lines mm -I, which is enough for many applications, especially for far infra-red radiation. The described method can also be used for some ideas testing. Many kinoform HOEs with a great number of parameters can be produced and tested by this simple, inexpensive and quick method.
Fig. 4
Acknowledgement
Test chart imaged by a kinoform lens
Table 1. Diffraction efficiency 1/ in +1 diffraction order as a function of the number of the quantization intervals, n n 0%
2 40
3 68
4 81
5 87
6 91
7 93
8 95
9 96
10 97
20 99
Diffraction efficiency is not less than 70%. The theoretical diffraction efficiency O as a function of the number of quantization intervals for the saw blade profile is given in Table 1. The phase retardation measurement technique presented in this paper is especially relevant to the problem of Holographic Optical Elements (HOE) manufacture. We can find applications of HOEs in many branches of science 89• . HOEs in the form of efficient kinoforms are especially useful ~'5. Diffraction efficiency for a theoretical kinoform reaches 100%. The phase retardation measurement technique
44
The author would like to thank Barbara Smolinska and Andrzej Kolodzielczyk for the many helpful suggestions that they have contributed towards improving this paper. References 1 Koronkevich, V.P. et al,'Kinoform optical components: Methods of design, manufacturing technology and practical applications', Avtometr~va, (1), (1985) 4-25 2 Jordan, J.A. Jr et al, "Kinoform lenses'. Appl Opt, 9, (8), (1970), 1883 3 Dallas, W.J. 'Kinoform fabrication - a new method'. Opt Comm, 8, (4), (1973), 340 4 Clair, J.J,, 'New methods to synthesise kinoforms', Opt Comm. 6, (2), (1972), 135 5 Koronkevich, V.P. et al, 'Kinoform Optical Elements', Optical Information Processing, Plenum Press, (1975) 6 Poleshchuk, A. et al, "Application of an evaporated photoresist (As2S3) in the production of kinoform optical elements', J Imag Sci, 30, (1986), 132-135 7 Sypek, M. 'Phase Retardation Measurement for Simple Kinoform Technology', Proc SPIE - Holography '89 8 Pappu, S.V. 'Holographic optical elements state-of-the-art review: Part l, Opt Laser Teehnol. 21, (5), (1989), 305 9 Pappu, S.V. 'Holographic optical elements state-of-the-art review: Part II', Opt Laser Technol. 21, (6), (1989), 365
Optics El Laser Technology Vol 23 No 1 1991