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Electron holography in the study of the electrostatic fields: the case of charged microtips
Ultramicroscopy45 (1992) 77-83 North-Holland
Electron holography in the study of the electrostatic fields: the case of charged microtips G. M a t t e u c c i , G.F. Missiroli, M. M u c c i n i Laboratorio di Microscopia Elettronica, Dipartimento di Fisica, Universith di Bologna, Via Irnerio 46, 40126 Bologna, Italy
and G. Pozzi Dipartimento di Scienza dei Materiali, Universitgt di Lecce, Via Arnesano, 73100 Lecce, Italy
Received 18 February 1992
The use of electron holographyhas been tested to map the electrostatic field around a charged microtip. Theoretical and experimental results show that whenever a modulated reference wave is used the equiphase lines observed in the final maps are not directly or simply related to the potential distribution. A set of double-exposure holograms has been used to circumvent the limited brightness and coherence of the field emission gun so as to obtain a larger useful field.
I. Introduction Up to now electron holography has been demonstrated to be a successful technique capable of investigating, among other problems [1], the trend of electric [2-6] and magnetic [7,8] microfields arising from specimens made transparent to electrons. Such fields are not always confined within the specimen but can propagate around it; therefore, it is tempting to try to map such leakage fields because of their importance in a variety of applications. In a number of papers [3-6] we called attention to the difficulties which can be encountered in such an attempt, since the reference wave is no longer a plane wave but is modulated by the long-range tail of the field under investigation. The results of this analysis can be summarized as follows: (1) the most reliable contour map can only be obtained by recording a double-exposure hologram directly in the electron microscope, (2) the information stored in the resulting contour map is that of a fictitious object whose phase is given by the difference between the object phase and that of the modulated reference beam, (3) to extract the maximum information from these holograms prior knowledge of the field is necessary in order to achieve the best fit for theoretical and experimental results. With these premises, we have undertaken experiments in order to obtain the holographic mapping of the electric field arising from a charged tungsten tip which has so far been studied only by means of interference electron microscopy [9], with the final aim of investigating the newly developed monatomic point sources [10]. The results presented in this work, which complete some preliminary observations [11], are an intermediate but important step towards this goal. With the aid of a theoretical model for the field around the tip which gives an analytical result for the projected potential, it has been possible to predict the general trend of the experimental contour map. 0304-39~91/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
78
G. Matteucci et al. / Electron holography in the study of electrostatic fields
"
Moreover, the comparison between theoretical and experimental results is m a d e m o r e accurate by the possibility, offered by the double-exposure technique, of making a m o n t a g e of reconstructed c o n t o u r maps of adjacent regions. In this way, it has been possible to partly overcome the experimental limitations due to the r e d u c e d width of the interference field of a single hologram.
2. The field model
T h e theoretical analysis of the electrostatic field in the o u t e r space of a c h a r g e d tip first considers the simple model m a d e by two linear segments, each of length 2c and whose centers are 2h distant, placed along the y axis in a symmetric position with respect to the x z plane of an xyz coordinate system (fig. 1). E a c h segment has a constant and opposite charge density ~r. T h e analytical expression of the potential distribution V ( x , y, z ) can be obtained [12] by integrating the formula: 1 V(x,
y, z ) =
4~'eo
+c f_
o"
c ~/x2+(y-h-t)2+z
1 dt + - - f _ 2 4~'eo
+c
-~r ---
c ~/x2+(y+h_t)2+z
2
dt,
(1)
where Eo is the dielectric constant of vacuum. T h e integration leads to: V( x, y, z ) = - -
sinh- 1
+ s i n h -1
tc
- sinh- l
2+z2
y+h,)tc - sinh -1
.....
x
2
+z2
,
(2)
and it can be seen that the potential distribution has a rotational symmetry and is zero when y = 0. N e a r and a r o u n d the extremities of the two charged lines, the equipotential surfaces behave approximately as a family of hyperboloids of rotation.
hi Eh
X
Fig. 1. Theoretical model to calculate the field near a charged microtip. The free parameters h and c are shown together with the equipotential surfaces near each charged segment of length 2c.
79
G. Matteucci et al. / Electron holography in the study of electrostatic fields
Therefore it is reasonable to assume that the field described by eqs. (1) or (2) may be used to represent, at least in a first approximation, the field produced by a charged tip in front of a conducting plane (y --- 0). The distance between the tip vertex and the conducting plane (y = 0) was 15 /zm. The charge density or was chosen in order to obtain the equipotential surface that represents the tip shape at about 10 V. This procedure is illustrated in fig. 2, where an IBM P C / A T equipped with a video board able to display 512 × 512 pixels at 256 grey levels was used to simulate the equipotential lines around the tip in the specimen plane (z = 0). In order to display such a distribution a set of equipotential surfaces was chosen with a constant potential difference. The region T inside the equipotential surface (which more closely resembles the tip) was darkened. From the analytical expression of the potential, the phase ~p(x, y ) can be calculated by performing the integral: 37" I + °°
p(x, y) = --~ j_~ V(x, y, z) dz,
(3)
where A and E are the electron wavelength and energy respectively. The integration leads again to an analytical expression: ~p(x, y)
¢ {[-c
AE 4~-%
+ l x l sin -1
+ (y-h)]
lnCx2 + [ c - ( y - h ) ] 2
-c+(y-h)
q-[-g-(y-h)]
-Ixl sin-l(?x 2+ [ c + ( y - h ) ] 2 c+(y+h)
+]xl sin-'( ¢x2+ [ c + ( y + h ) ] 2 2
2
+ [c+(y+h)] lnCx2 + [c+(y+h)] 2 )
+[c-(y+h)]
-c_+(y+h))}.
-Ix[ sin-'( CxZ+ [ c - ( y + h ) ]
lnCx2-.[ - [c-l- ( y - h ) ]
InCx2+[c-(y+h)]
2
(4)
The holographic method reveals the loci of points with constant phase shift as a set of curves with a phase difference of 2 rr between two successive dark and white ones. Fig. 3 shows the computer simulation of the equiphase lines obtained by the coherent superposition of the objeCt wave (eq. (4)) and a plane reference wave. While the trend of the potential in the (x, y, z = 0) plane is easy to guess (fig. 2), the interpretation of fig. 3, where the equiphase lines seem to enter the tip shadow T, is less intuitive because the phase shift, suffered by electrons along their trajectories, is related to the potential distribution around the tip integrated along the z axis. However, when experimental observations are made of the field close to the tip apex, we must consider that also the reference beam is modulated by the field of the tip which extends a few microns away from the tip itself. Therefore the final contour map will show the loci of constant phase difference between the perturbed reference wave and the object wave and does not exactly represent the object phase variations. It is worthwhile naming "proper contour maps" those maps of the field obtained by
80
G. Matteucci et al. / Electron holography in the study of electrostatic fields
Fig. 2. Computer simulation of equipotential lines, in the xy plane, of a charged microtip.
using a plane reference wave (i.e. an unperturbed reference wave), while as "improper contour maps" those in which the reference wave is modulated by the fringing field. In our case by using formula (4) and by taking into account the distance between the interfering points (in the electron microscope), the perturbed reference wave can be calculated and the "improper contour map" displayed directly in the computer. In the following section this theoretical result will be compared with the experimental one.
Fig. 3. Computer simulation of the equiphase lines around the tip T.
G. Matteucci et al. / Electron holography in the study of electrostatic fields
81
3. Experimental results Using a standard electrolytical thinning process, the tips were obtained from a polycrystalline tungsten wire (0.25 mm diameter), in a cell with 2% N a O H solution and by applying 2 V, 50 Hz alternating voltage [13]. One was mounted in the center of a 2 mm aperture, which was inserted on a special specimen holder equipped with electrical contacts connected to an external voltage supply. The aperture and the tip, electrically insulated from the microscope, could then be biassed and, by rotating the aperture, it was possible to arrange the tip and the biprism axis in a mutually perpendicular position. A voltage of the order of ten volts was applied to the tip. A Philips EM 400T microscope equipped with a field emission gun and a M611enstedt-Diiker biprism [14] operated in the diffraction mode with the objective lens switched off to focus the tip the diffraction lens was used. The final magnification was in the range 1000-2000 x according to the different values of the camera length. The electron interferometer operated at the selected area plane. Double-exposure holograms were recorded with an interference distance of about 5/zm. Fig. 4 shows a double-exposure electron hologram in which the reference wave is perturbed. The dark regions represent the equiphase lines in the area near the tip T when it was held at 7.5 V. In preceding papers [4-6] dealing with p - n junctions and charged latex spheres, we showed that the equiphase lines were strictly related to and also displayed the trend of the projected equipotential surfaces. On the contrary, the present case is the first in which the equiphase lines observed in the final interferogram cannot be related simply to the equipotential surface shape. Since the investigated area around the tip is fairly limited due to the relatively poor brightness of our field emission gun, the overall trend of these lines cannot be displayed in a large enough region. In order to follow their trend around the tip on a wider area, three double-exposure holograms were taken from parallel and adjacent regions and then mounted together. It is important to note that the success of this procedure is linked to the fact that double-exposure holograms are recorded, so that the internal reference plane wave for the contour mapping is provided by the hologram without object. Fig. 5a shows a montage of these three regions (labelled as 1, 2 and 3) in which the useful interference field extending along the tip axis is about 15 ~m. The three strips are of different width since the overlapping regions were removed. It can be noted that in this "improper hologram" the equiphase lines circle around the vertex of the tip T and then join the tip itself, behaviour that cannot be inferred previously. Fig. 5b reports the computer simulation obtained by the coherent superposition of the object wave (eq. (4)) and a perturbed reference wave and adjusting the parameters h, c and the charge density or in
Fig. 4. "Improper hologram" displaying the equiphase lines near the apex of a charged microtip T.
82
G. Matteucci et al. / Electron holography in the study of electrostatic fields
a
Fig. 5. (a) Collage of three (1, 2, 3) double-exposure holograms taken from adjacent regions showing the trend of the equiphase lines in a wider area. (b) Computer simulation of the equiphase lines obtained with a perturbed reference wave.
order to fit with the experimental data. The number of equiphase lines is the same as what would be obtained by a double-exposure electron hologram performed with a perturbed reference wave passing 5 txm distant from the object wave and with the same relative orientation of the biprism and the tip as that shown by the electron holograms of fig. 5a. The satisfactory agreement between experimental and theoretical results is evident. The comparison between figs. 3 and 5b clearly shows the difference between the trend of the phase distribution displayed by a hologram recorded with an unperturbed reference wave instead of a perturbed one. 4. Conclusions
The above results concerning the study of the field around the tip are the first example, to our knowledge, where a striking difference between equipotential and equiphase lines occurs. Experimental
G. Matteucci et al. / Electron holography in the study of electrostatic fields
83
e v i d e n c e c o n f i r m s t h e conclusions o f o u r p r e v i o u s w o r k [6] on t h e investigation o f l o n g - r a n g e e l e c t r o m a g netic fields by m e a n s o f e l e c t r o n h o l o g r a p h y . G e n e r a l l y s p e a k i n g , it t u r n s o u t that, w h e n an e l e c t r o n h o l o g r a m is r e c o r d e d with a m o d u l a t e d r e f e r e n c e wave ( " i m p r o p e r h o l o g r a m " ) , a careful t h e o r e t i c a l a n d e x p e r i m e n t a l analysis of t h e w h o l e p r o b l e m m u s t b e m a d e b e f o r e claiming t h e o b s e r v e d e q u i p h a s e lines as r e p r e s e n t a t i v e o f t h e t r e n d o f t h e field distribution. M o r e o v e r , it has also b e e n shown that, f r o m t h e e x p e r i m e n t a l p o i n t o f view, an a d d i t i o n a l m e r i t of t h e d o u b l e - e x p o s u r e m e t h o d is t h a t d i f f e r e n t h o l o g r a m s can be a s s e m b l e d t o g e t h e r to o b t a i n a l a r g e r field o f view o f the " i m p r o p e r c o n t o u r m a p " , thus i n c r e a s i n g t h e accuracy o f the c o m p a r i s o n b e t w e e n experim e n t a l a n d t h e o r e t i c a l results.
Acknowledgements T h e skillful t e c h n i c a l assistance o f Mr. S. P a t u e l l i is gratefully a c k n o w l e d g e d . This w o r k has b e e n s u p p o r t e d by funds f r o m M U R S T c o o r d i n a t e d by C o n s o r z i o I N F M a n d C N R - G N S M .
References [1] A. Tonomura, Rev. Mod. Phys. 59 (1987) 639. [2] S. Frabboni, G. Matteucci and G. Pozzi, Phys. Rev. Lett. 55 (1985) 2196. [3] S. Frabboni, G. Matteucci and G. Pozzi, Ultramicroscopy 23 (1987) 29. [4] J.W. Chen, G. Matteucci, A. Migliori, G.F. Missiroli, E. Nichelatti, G. Pozzi and M. Vanzi, Phys. Rev. A 40 (1989) 3136. [5] G. Matteucci, G.F. Missiroli, J.W. Chen and G. Pozzi, Appl. Phys. Lett. 52 (1988) 176. [6] G. Matteucci, G.F. Missiroli, E. Nichelatti, A. Migliori, M. Vanzi and G. Pozzi, J. Appl. Phys. 69 (1991) 1835. [7] G. Matteucci, G.F. Missiroli and G. Pozzi, IEEE Trans. Magn. MAG-20 (1984) 1970. [8] A. Tonomura, J. Appl. Phys. 61 (1987) 4297. [9] Yu.A. Kulyupin, S.A. Nepijko, N.N. Sedov and V.G. Shamonya, Optik 52 (1978 79) 101. [10] H.W. Fink, Phys. Scripta 38 (1988) 260. [11] G.F. Missiroli, M. Muccini and G. Pozzi, Atti XVIII Congr. Italiano Micr. Elettr. Supplemento Microscopia Eletronica n. 2 (1991) p. 409. [12] E. Durand, Electrostatique: Vol. 2 (Masson, Paris, 1966) p. 173. [13] W.P. Dyke and W.W. Dolan, Adv. Electron. Electron Phys. 8 (1956) 89. [14] G. M/511enstedt and H. Diiker, Z. Phys. 145 (1957) 377.
Electron holography in the study of the electrostatic fields: the case of charged microtips G. M a t t e u c c i , G.F. Missiroli, M. M u c c i n i Laboratorio di Microscopia Elettronica, Dipartimento di Fisica, Universith di Bologna, Via Irnerio 46, 40126 Bologna, Italy
and G. Pozzi Dipartimento di Scienza dei Materiali, Universitgt di Lecce, Via Arnesano, 73100 Lecce, Italy
Received 18 February 1992
The use of electron holographyhas been tested to map the electrostatic field around a charged microtip. Theoretical and experimental results show that whenever a modulated reference wave is used the equiphase lines observed in the final maps are not directly or simply related to the potential distribution. A set of double-exposure holograms has been used to circumvent the limited brightness and coherence of the field emission gun so as to obtain a larger useful field.
I. Introduction Up to now electron holography has been demonstrated to be a successful technique capable of investigating, among other problems [1], the trend of electric [2-6] and magnetic [7,8] microfields arising from specimens made transparent to electrons. Such fields are not always confined within the specimen but can propagate around it; therefore, it is tempting to try to map such leakage fields because of their importance in a variety of applications. In a number of papers [3-6] we called attention to the difficulties which can be encountered in such an attempt, since the reference wave is no longer a plane wave but is modulated by the long-range tail of the field under investigation. The results of this analysis can be summarized as follows: (1) the most reliable contour map can only be obtained by recording a double-exposure hologram directly in the electron microscope, (2) the information stored in the resulting contour map is that of a fictitious object whose phase is given by the difference between the object phase and that of the modulated reference beam, (3) to extract the maximum information from these holograms prior knowledge of the field is necessary in order to achieve the best fit for theoretical and experimental results. With these premises, we have undertaken experiments in order to obtain the holographic mapping of the electric field arising from a charged tungsten tip which has so far been studied only by means of interference electron microscopy [9], with the final aim of investigating the newly developed monatomic point sources [10]. The results presented in this work, which complete some preliminary observations [11], are an intermediate but important step towards this goal. With the aid of a theoretical model for the field around the tip which gives an analytical result for the projected potential, it has been possible to predict the general trend of the experimental contour map. 0304-39~91/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
78
G. Matteucci et al. / Electron holography in the study of electrostatic fields
"
Moreover, the comparison between theoretical and experimental results is m a d e m o r e accurate by the possibility, offered by the double-exposure technique, of making a m o n t a g e of reconstructed c o n t o u r maps of adjacent regions. In this way, it has been possible to partly overcome the experimental limitations due to the r e d u c e d width of the interference field of a single hologram.
2. The field model
T h e theoretical analysis of the electrostatic field in the o u t e r space of a c h a r g e d tip first considers the simple model m a d e by two linear segments, each of length 2c and whose centers are 2h distant, placed along the y axis in a symmetric position with respect to the x z plane of an xyz coordinate system (fig. 1). E a c h segment has a constant and opposite charge density ~r. T h e analytical expression of the potential distribution V ( x , y, z ) can be obtained [12] by integrating the formula: 1 V(x,
y, z ) =
4~'eo
+c f_
o"
c ~/x2+(y-h-t)2+z
1 dt + - - f _ 2 4~'eo
+c
-~r ---
c ~/x2+(y+h_t)2+z
2
dt,
(1)
where Eo is the dielectric constant of vacuum. T h e integration leads to: V( x, y, z ) = - -
sinh- 1
+ s i n h -1
tc
- sinh- l
2+z2
y+h,)tc - sinh -1
.....
x
2
+z2
,
(2)
and it can be seen that the potential distribution has a rotational symmetry and is zero when y = 0. N e a r and a r o u n d the extremities of the two charged lines, the equipotential surfaces behave approximately as a family of hyperboloids of rotation.
hi Eh
X
Fig. 1. Theoretical model to calculate the field near a charged microtip. The free parameters h and c are shown together with the equipotential surfaces near each charged segment of length 2c.
79
G. Matteucci et al. / Electron holography in the study of electrostatic fields
Therefore it is reasonable to assume that the field described by eqs. (1) or (2) may be used to represent, at least in a first approximation, the field produced by a charged tip in front of a conducting plane (y --- 0). The distance between the tip vertex and the conducting plane (y = 0) was 15 /zm. The charge density or was chosen in order to obtain the equipotential surface that represents the tip shape at about 10 V. This procedure is illustrated in fig. 2, where an IBM P C / A T equipped with a video board able to display 512 × 512 pixels at 256 grey levels was used to simulate the equipotential lines around the tip in the specimen plane (z = 0). In order to display such a distribution a set of equipotential surfaces was chosen with a constant potential difference. The region T inside the equipotential surface (which more closely resembles the tip) was darkened. From the analytical expression of the potential, the phase ~p(x, y ) can be calculated by performing the integral: 37" I + °°
p(x, y) = --~ j_~ V(x, y, z) dz,
(3)
where A and E are the electron wavelength and energy respectively. The integration leads again to an analytical expression: ~p(x, y)
¢ {[-c
AE 4~-%
+ l x l sin -1
+ (y-h)]
lnCx2 + [ c - ( y - h ) ] 2
-c+(y-h)
q-[-g-(y-h)]
-Ixl sin-l(?x 2+ [ c + ( y - h ) ] 2 c+(y+h)
+]xl sin-'( ¢x2+ [ c + ( y + h ) ] 2 2
2
+ [c+(y+h)] lnCx2 + [c+(y+h)] 2 )
+[c-(y+h)]
-c_+(y+h))}.
-Ix[ sin-'( CxZ+ [ c - ( y + h ) ]
lnCx2-.[ - [c-l- ( y - h ) ]
InCx2+[c-(y+h)]
2
(4)
The holographic method reveals the loci of points with constant phase shift as a set of curves with a phase difference of 2 rr between two successive dark and white ones. Fig. 3 shows the computer simulation of the equiphase lines obtained by the coherent superposition of the objeCt wave (eq. (4)) and a plane reference wave. While the trend of the potential in the (x, y, z = 0) plane is easy to guess (fig. 2), the interpretation of fig. 3, where the equiphase lines seem to enter the tip shadow T, is less intuitive because the phase shift, suffered by electrons along their trajectories, is related to the potential distribution around the tip integrated along the z axis. However, when experimental observations are made of the field close to the tip apex, we must consider that also the reference beam is modulated by the field of the tip which extends a few microns away from the tip itself. Therefore the final contour map will show the loci of constant phase difference between the perturbed reference wave and the object wave and does not exactly represent the object phase variations. It is worthwhile naming "proper contour maps" those maps of the field obtained by
80
G. Matteucci et al. / Electron holography in the study of electrostatic fields
Fig. 2. Computer simulation of equipotential lines, in the xy plane, of a charged microtip.
using a plane reference wave (i.e. an unperturbed reference wave), while as "improper contour maps" those in which the reference wave is modulated by the fringing field. In our case by using formula (4) and by taking into account the distance between the interfering points (in the electron microscope), the perturbed reference wave can be calculated and the "improper contour map" displayed directly in the computer. In the following section this theoretical result will be compared with the experimental one.
Fig. 3. Computer simulation of the equiphase lines around the tip T.
G. Matteucci et al. / Electron holography in the study of electrostatic fields
81
3. Experimental results Using a standard electrolytical thinning process, the tips were obtained from a polycrystalline tungsten wire (0.25 mm diameter), in a cell with 2% N a O H solution and by applying 2 V, 50 Hz alternating voltage [13]. One was mounted in the center of a 2 mm aperture, which was inserted on a special specimen holder equipped with electrical contacts connected to an external voltage supply. The aperture and the tip, electrically insulated from the microscope, could then be biassed and, by rotating the aperture, it was possible to arrange the tip and the biprism axis in a mutually perpendicular position. A voltage of the order of ten volts was applied to the tip. A Philips EM 400T microscope equipped with a field emission gun and a M611enstedt-Diiker biprism [14] operated in the diffraction mode with the objective lens switched off to focus the tip the diffraction lens was used. The final magnification was in the range 1000-2000 x according to the different values of the camera length. The electron interferometer operated at the selected area plane. Double-exposure holograms were recorded with an interference distance of about 5/zm. Fig. 4 shows a double-exposure electron hologram in which the reference wave is perturbed. The dark regions represent the equiphase lines in the area near the tip T when it was held at 7.5 V. In preceding papers [4-6] dealing with p - n junctions and charged latex spheres, we showed that the equiphase lines were strictly related to and also displayed the trend of the projected equipotential surfaces. On the contrary, the present case is the first in which the equiphase lines observed in the final interferogram cannot be related simply to the equipotential surface shape. Since the investigated area around the tip is fairly limited due to the relatively poor brightness of our field emission gun, the overall trend of these lines cannot be displayed in a large enough region. In order to follow their trend around the tip on a wider area, three double-exposure holograms were taken from parallel and adjacent regions and then mounted together. It is important to note that the success of this procedure is linked to the fact that double-exposure holograms are recorded, so that the internal reference plane wave for the contour mapping is provided by the hologram without object. Fig. 5a shows a montage of these three regions (labelled as 1, 2 and 3) in which the useful interference field extending along the tip axis is about 15 ~m. The three strips are of different width since the overlapping regions were removed. It can be noted that in this "improper hologram" the equiphase lines circle around the vertex of the tip T and then join the tip itself, behaviour that cannot be inferred previously. Fig. 5b reports the computer simulation obtained by the coherent superposition of the object wave (eq. (4)) and a perturbed reference wave and adjusting the parameters h, c and the charge density or in
Fig. 4. "Improper hologram" displaying the equiphase lines near the apex of a charged microtip T.
82
G. Matteucci et al. / Electron holography in the study of electrostatic fields
a
Fig. 5. (a) Collage of three (1, 2, 3) double-exposure holograms taken from adjacent regions showing the trend of the equiphase lines in a wider area. (b) Computer simulation of the equiphase lines obtained with a perturbed reference wave.
order to fit with the experimental data. The number of equiphase lines is the same as what would be obtained by a double-exposure electron hologram performed with a perturbed reference wave passing 5 txm distant from the object wave and with the same relative orientation of the biprism and the tip as that shown by the electron holograms of fig. 5a. The satisfactory agreement between experimental and theoretical results is evident. The comparison between figs. 3 and 5b clearly shows the difference between the trend of the phase distribution displayed by a hologram recorded with an unperturbed reference wave instead of a perturbed one. 4. Conclusions
The above results concerning the study of the field around the tip are the first example, to our knowledge, where a striking difference between equipotential and equiphase lines occurs. Experimental
G. Matteucci et al. / Electron holography in the study of electrostatic fields
83
e v i d e n c e c o n f i r m s t h e conclusions o f o u r p r e v i o u s w o r k [6] on t h e investigation o f l o n g - r a n g e e l e c t r o m a g netic fields by m e a n s o f e l e c t r o n h o l o g r a p h y . G e n e r a l l y s p e a k i n g , it t u r n s o u t that, w h e n an e l e c t r o n h o l o g r a m is r e c o r d e d with a m o d u l a t e d r e f e r e n c e wave ( " i m p r o p e r h o l o g r a m " ) , a careful t h e o r e t i c a l a n d e x p e r i m e n t a l analysis of t h e w h o l e p r o b l e m m u s t b e m a d e b e f o r e claiming t h e o b s e r v e d e q u i p h a s e lines as r e p r e s e n t a t i v e o f t h e t r e n d o f t h e field distribution. M o r e o v e r , it has also b e e n shown that, f r o m t h e e x p e r i m e n t a l p o i n t o f view, an a d d i t i o n a l m e r i t of t h e d o u b l e - e x p o s u r e m e t h o d is t h a t d i f f e r e n t h o l o g r a m s can be a s s e m b l e d t o g e t h e r to o b t a i n a l a r g e r field o f view o f the " i m p r o p e r c o n t o u r m a p " , thus i n c r e a s i n g t h e accuracy o f the c o m p a r i s o n b e t w e e n experim e n t a l a n d t h e o r e t i c a l results.
Acknowledgements T h e skillful t e c h n i c a l assistance o f Mr. S. P a t u e l l i is gratefully a c k n o w l e d g e d . This w o r k has b e e n s u p p o r t e d by funds f r o m M U R S T c o o r d i n a t e d by C o n s o r z i o I N F M a n d C N R - G N S M .
References [1] A. Tonomura, Rev. Mod. Phys. 59 (1987) 639. [2] S. Frabboni, G. Matteucci and G. Pozzi, Phys. Rev. Lett. 55 (1985) 2196. [3] S. Frabboni, G. Matteucci and G. Pozzi, Ultramicroscopy 23 (1987) 29. [4] J.W. Chen, G. Matteucci, A. Migliori, G.F. Missiroli, E. Nichelatti, G. Pozzi and M. Vanzi, Phys. Rev. A 40 (1989) 3136. [5] G. Matteucci, G.F. Missiroli, J.W. Chen and G. Pozzi, Appl. Phys. Lett. 52 (1988) 176. [6] G. Matteucci, G.F. Missiroli, E. Nichelatti, A. Migliori, M. Vanzi and G. Pozzi, J. Appl. Phys. 69 (1991) 1835. [7] G. Matteucci, G.F. Missiroli and G. Pozzi, IEEE Trans. Magn. MAG-20 (1984) 1970. [8] A. Tonomura, J. Appl. Phys. 61 (1987) 4297. [9] Yu.A. Kulyupin, S.A. Nepijko, N.N. Sedov and V.G. Shamonya, Optik 52 (1978 79) 101. [10] H.W. Fink, Phys. Scripta 38 (1988) 260. [11] G.F. Missiroli, M. Muccini and G. Pozzi, Atti XVIII Congr. Italiano Micr. Elettr. Supplemento Microscopia Eletronica n. 2 (1991) p. 409. [12] E. Durand, Electrostatique: Vol. 2 (Masson, Paris, 1966) p. 173. [13] W.P. Dyke and W.W. Dolan, Adv. Electron. Electron Phys. 8 (1956) 89. [14] G. M/511enstedt and H. Diiker, Z. Phys. 145 (1957) 377.