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Nonlinear optical effects induced in ion beams during transport
1096
Nuclear
Instruments
and Methods in Physics Research
B56f57
(1991) 1096-1098 North-Holland
Nonlinear optical effects induced in ion beams during transport
The main origin of nonlinear beam distortions is discussed and the influences of fringing fields are reviewed. Also discussed ways to reduce those distortions by advantageous shapes of first-order beam envetopes and to correct only the left-over distortions multipole elements.
1. Introduction In designing beam transport and analyzing systems one typically uses a Gaussian or first-order approximation for describing the beam. Higher-order effects usually are regarded as detrimental and are corrected later by strategically positioned multipole elements, i.e., hexapoles, octopoles, etc. [l]. Since an &h-order multipole affects only nth- and higher-order aberrations and leaves aberrations of f n - I) and lower-order alone, one can correct higher- and higher-order aberrations by adding multipoles of higher and higher order without destroying the already achieved corrections to lower order. However, since any n th-order multipole affects also higher than nth-order aberrations, the magnitudes of these higher-order aberrations typically increases drastically with the correction of severe aberrations of n th order 111. Because of this it is advisable to design the ion-optical system at the beginning not only to fulfill the first-order optical needs together with geometric requirements, but also to see to it that its higher-order aberrations If stay within limits and that 2) the teft-over aberrations are correctable by only very few weak mdtipole elements. This is commonly achieved by using exclusively rather long and weak lenses and by keeping the beam envelopes as small as possible wherever higher-order phase-space distortions occur. Since such spots are often related to element boundaries, much of the following discussion wil1 be concentrated on the discussion of fringing fields.
2. ‘i%e design of quadrupk
channels
Though fringing fields are typically discussed in connection with sector fields, one still assumes in most calculations for quadrupoles that their field gradients 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
are by
rise and fall abruptly at the element boundaries in a non-Maxwellian fashion. Though quadrupoles have no geometric second-order effects this procedure neglects strong effects of third-order. Additionally to neglecting third-order aberrations this procedure also neglects all fringing-field effects on the first-order optics [2]. Though these first-order effects modify the x- and y-focus lengths of an individual quadrupole only by several tenths of a percent, the added effects of many quadrupoles can change the beam envelopes noticeably (see fig. 1). Thus in reality a beam envelope may look somewhat different from the one calculated by a code [3] that neglects these fringing-fields effects. Though in most cases it is possible at the end to obtain a beam envelope similar to the one seen in the fringing-field-free calculation, such an adjustment procedure requires elaborate quadrupote tuning and the existence of good beam diagnostics throughout the ion-optical system. Furthermore it requires a certain overdesign of all quadrupoles so that the power in an individual one can - in bad cases - be
Fig. 1. Beam envelopes in the xz-plane are shown for a quadrupofe channel designed with and without taking the fringing fields into account. In the upper part the trajectories are shown as expected from fining-~e~d-f~ quadrupofes whife in the lower part trajectories are shown that would result if fringing fields would be assumed to exist in all quadrupoles, a case which is much closer to reality.
B.V. (North-Holland)
1097
H. Wollnik / Nonlinear optical effects
raised permanently by 10% of its design value. Thus it seems advisable to include these fringing-field effects from the beginning [4-63. Even if at the time of the system design the then assumed fringing-field distributions are only approximate, the experimentally observed beam envelope at the end typically agrees [7] with the calculated one within the measuring accuracy. There are only a few quadrupole channels in which higher-order aberrations are of importance. However, if quadrupoles are used in fine-focus systems or in combinations with high-resolution particle spectrometers, it is worthwhile to note that typically - 90% of the geometric third-order aberrations of the system are due to the existence of the fringing field and only - 10% are due to the existence of the main field region of the quadrupole. Thus designing a quadrupole arrangement for which higher-order aberrations are important without including fringing-field effects is only of limited use in most cases. The third-order aberrations in a real quadrupole channel can be corrected by octopoles placed at positions [1,3] at which the beam envelopes have different sizes in x- and y-directions. However, one should note that the geometric aberrations of third-order can be corrected by only 3 octopoles since there are many cross relations between the aberration coefficients due to the condition of symplecticity [1,9-111. Chromatic aberrations, which result from the fact that the focal lengths of quadrupoles vary with the ion energy, can only be corrected by using hexapole elements in connection with sector fields.
3. The design of beam transport systems sector magnets
that include
For dispersive systems that include sector fields it is important to design the system such that the resolving power is maximized or achromaticity is achieved. Such an optimization should involve the concept of flux integrals 18) or of Q-values 11,121. Both these concepts pay attention to beam envelopes in the plane of deflection, maximizing the beam areas in the sector fields to achieve high resolving power. However, if a wide beam is achieved in the plane of deflection usually also a large magnet gap is required which, because of its extended f~nging-field, causes large aberrations. In order to achieve high resolving powers one usually tries to reduce the aberrations in the x-direction. Besides chromatic aberrations, which in dispersive systems usually are of secondary interest, there are geometric aberrations at the final image position rend. These geometric aberrations consist of terms (x 1xx)xt + . f ’ as well as of terms (x ] yy)yf + . . . with x1 and y, describing the beam half diameters at some position z, along the beam axis. Placing a hexapole at position zr
Fig. 2. Ion trajectories are shown for a stigmatic focusing homogeneous sector field that uses inclined field boundaries. This sector field is assumed to be preceded by a quadrupole quadruplet. Ion trajectories are shown as projected onto the plane of deflection (x-plane) and the perpendicular yz-plane. Note that the scale in the y-direction is only 10% of the scale in x-direction. This quadrupole channel is perfectly focusing in the x-direction but not in the y-direction at the positions of the entrance or exit slits of the sector field anaiyser. Thus the beam width in the magnet stays the same as when the quadrupoies were not used. However, there is a much smaller beam envelope at the positions of the fringe fields of the sector field which in turn reduces the requirements on the magnet air gap as well as an some of the aberrations in the x-direction.
will increase the coefficient (x ] yy) by about the same amount it decreases the coefficient (x 1xx). Thus a simultaneous correction of both aberrations is impossible if the beam cross section at t, is approximately circular. The same is true if one looks at third-order aberrations (x 1xxx)x: and (x 1xyy)x, _yf as well as of higher order. To design a well correctable system thus requires the beam cross section at z, to considerably deviate from a circle with the x to y ratio being for instance 5 : 1. Thus it is advisable to confine the firstorder beam envelope in the y-direction throughout a sector field but especially at its entrance and exit. This asymmetry in the beam cross section can be achieved by adding to the system y-focusing quadrupole lenses [12-141. However, care must be taken that these quadrupole lenses are large enough to allow passage for the beam and precise enough that they do not introduce other higher-order aberrations. A way out is not to use a particle source that illuminates similar x,px and y.p, phase-space areas but one in which the beam starts from a narrow region in the x-direction under large angles and a wide but almost parallel beam in the y-direction (see fig. 2). This situation can always be achieved by a quadrupole transfer system between the ion source and the system’s entrance slit. In this way three advantages are achieved simultaneously. 1) The magnet air gap can remain small. This reduces the stored magnetic energy and thus the magnet XIV. ACCELERATOR
TECHNOLOGY
casts considerably. In turn this also reduces the magnet pale width since this width is the x-beam width plus lateral fringing-field regions at both sides with both these regions being about equal to the magnet air gap. final The aberrations (x I vv)v:? 2) (x I VY)X,& -. . are much smaller than (x 1xx)xf, (x ( xXx)x:, . -. from the beginning Furthermore, because of the large x to y ratio of the beam cross section a correction by multipole elements in the magnet sector - or shortly before or after it - is effective, i.e. such a correction can be achieved by relatively weak muftipoles so that the higher-order aberrations remain unchanged. 3) At the position of an 0.1 mm wide entrance sht the beam is usually stifl - 100 times wider in the y-dire&on. Thus space-charge effects are reduced > 100 times as compared to a pointlike image of about 0.1 mm diameter at the entrance slit. Thus the use of a quadrupoie transfer system seems to be an advantageous and flexible approach to allow most any dispersive system to have only small aberrations at its final image.
Acknawkxtgement Fox financial support I am grateful to the German Bundesminister fur Forschung und Tecbnologie.
References
PI H. Wollnik, Optics of Charged
Particles (Academic Press, Orlando, 1987). PI H. Matsuda and H. Wollnik, Nucl. Instr. and Meth. 103 (1972) 117. [31 K.L. Brown, D.C. Carey, C. Iselin and F. Rothacker, CERN-Report 80.04 (1980). (41 M. Bern, H. Wollnik, J. Brezina and W. Wendel, AMCO-7, GSI-Rep. THD-26 (1984) p. 679. 151 S. Kowalski and H. Enge, Tech. Rep., MIT, Cambridge, MA (198s). and M. Ben, AIP Conf, Proc. t4 H. Wollnik, B. Hartmann 177 (1988) 74. 171 LJ. Rohrer, private communication. 181 K.L. Brown, Rev. Sci. Instr. 31(1960) 556; SLAC Rep. 91 (1979). [91 A.J. Drag& AIP Conf. Proc. 87 (1982). [IO1 H. Wollnik and M. Berz, Nucl. Instr. and Meth. 321 (1985) 127. III1 H. Rose, Nucl. Instr. and Meth. A258 (1987) 374. [I21 H. Wollnik, Nucl. Instr. and Meth. 95 (1971) 453. [I31 H. Wollnik, Proc. EMIS Conf., Marburg, eds. H. Wagner and W. Walcher, BMBW-Fo~chungs~r K 70-28 (1970) p. 282. “I. Matsuo, Y. Fujita, T. Sakurai and I, 1141 H. Matsuda, Ran&use, Xnt. J. Mass Speetr. Ion Proc. 91 (1989) I.
Nuclear
Instruments
and Methods in Physics Research
B56f57
(1991) 1096-1098 North-Holland
Nonlinear optical effects induced in ion beams during transport
The main origin of nonlinear beam distortions is discussed and the influences of fringing fields are reviewed. Also discussed ways to reduce those distortions by advantageous shapes of first-order beam envetopes and to correct only the left-over distortions multipole elements.
1. Introduction In designing beam transport and analyzing systems one typically uses a Gaussian or first-order approximation for describing the beam. Higher-order effects usually are regarded as detrimental and are corrected later by strategically positioned multipole elements, i.e., hexapoles, octopoles, etc. [l]. Since an &h-order multipole affects only nth- and higher-order aberrations and leaves aberrations of f n - I) and lower-order alone, one can correct higher- and higher-order aberrations by adding multipoles of higher and higher order without destroying the already achieved corrections to lower order. However, since any n th-order multipole affects also higher than nth-order aberrations, the magnitudes of these higher-order aberrations typically increases drastically with the correction of severe aberrations of n th order 111. Because of this it is advisable to design the ion-optical system at the beginning not only to fulfill the first-order optical needs together with geometric requirements, but also to see to it that its higher-order aberrations If stay within limits and that 2) the teft-over aberrations are correctable by only very few weak mdtipole elements. This is commonly achieved by using exclusively rather long and weak lenses and by keeping the beam envelopes as small as possible wherever higher-order phase-space distortions occur. Since such spots are often related to element boundaries, much of the following discussion wil1 be concentrated on the discussion of fringing fields.
2. ‘i%e design of quadrupk
channels
Though fringing fields are typically discussed in connection with sector fields, one still assumes in most calculations for quadrupoles that their field gradients 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
are by
rise and fall abruptly at the element boundaries in a non-Maxwellian fashion. Though quadrupoles have no geometric second-order effects this procedure neglects strong effects of third-order. Additionally to neglecting third-order aberrations this procedure also neglects all fringing-field effects on the first-order optics [2]. Though these first-order effects modify the x- and y-focus lengths of an individual quadrupole only by several tenths of a percent, the added effects of many quadrupoles can change the beam envelopes noticeably (see fig. 1). Thus in reality a beam envelope may look somewhat different from the one calculated by a code [3] that neglects these fringing-fields effects. Though in most cases it is possible at the end to obtain a beam envelope similar to the one seen in the fringing-field-free calculation, such an adjustment procedure requires elaborate quadrupote tuning and the existence of good beam diagnostics throughout the ion-optical system. Furthermore it requires a certain overdesign of all quadrupoles so that the power in an individual one can - in bad cases - be
Fig. 1. Beam envelopes in the xz-plane are shown for a quadrupofe channel designed with and without taking the fringing fields into account. In the upper part the trajectories are shown as expected from fining-~e~d-f~ quadrupofes whife in the lower part trajectories are shown that would result if fringing fields would be assumed to exist in all quadrupoles, a case which is much closer to reality.
B.V. (North-Holland)
1097
H. Wollnik / Nonlinear optical effects
raised permanently by 10% of its design value. Thus it seems advisable to include these fringing-field effects from the beginning [4-63. Even if at the time of the system design the then assumed fringing-field distributions are only approximate, the experimentally observed beam envelope at the end typically agrees [7] with the calculated one within the measuring accuracy. There are only a few quadrupole channels in which higher-order aberrations are of importance. However, if quadrupoles are used in fine-focus systems or in combinations with high-resolution particle spectrometers, it is worthwhile to note that typically - 90% of the geometric third-order aberrations of the system are due to the existence of the fringing field and only - 10% are due to the existence of the main field region of the quadrupole. Thus designing a quadrupole arrangement for which higher-order aberrations are important without including fringing-field effects is only of limited use in most cases. The third-order aberrations in a real quadrupole channel can be corrected by octopoles placed at positions [1,3] at which the beam envelopes have different sizes in x- and y-directions. However, one should note that the geometric aberrations of third-order can be corrected by only 3 octopoles since there are many cross relations between the aberration coefficients due to the condition of symplecticity [1,9-111. Chromatic aberrations, which result from the fact that the focal lengths of quadrupoles vary with the ion energy, can only be corrected by using hexapole elements in connection with sector fields.
3. The design of beam transport systems sector magnets
that include
For dispersive systems that include sector fields it is important to design the system such that the resolving power is maximized or achromaticity is achieved. Such an optimization should involve the concept of flux integrals 18) or of Q-values 11,121. Both these concepts pay attention to beam envelopes in the plane of deflection, maximizing the beam areas in the sector fields to achieve high resolving power. However, if a wide beam is achieved in the plane of deflection usually also a large magnet gap is required which, because of its extended f~nging-field, causes large aberrations. In order to achieve high resolving powers one usually tries to reduce the aberrations in the x-direction. Besides chromatic aberrations, which in dispersive systems usually are of secondary interest, there are geometric aberrations at the final image position rend. These geometric aberrations consist of terms (x 1xx)xt + . f ’ as well as of terms (x ] yy)yf + . . . with x1 and y, describing the beam half diameters at some position z, along the beam axis. Placing a hexapole at position zr
Fig. 2. Ion trajectories are shown for a stigmatic focusing homogeneous sector field that uses inclined field boundaries. This sector field is assumed to be preceded by a quadrupole quadruplet. Ion trajectories are shown as projected onto the plane of deflection (x-plane) and the perpendicular yz-plane. Note that the scale in the y-direction is only 10% of the scale in x-direction. This quadrupole channel is perfectly focusing in the x-direction but not in the y-direction at the positions of the entrance or exit slits of the sector field anaiyser. Thus the beam width in the magnet stays the same as when the quadrupoies were not used. However, there is a much smaller beam envelope at the positions of the fringe fields of the sector field which in turn reduces the requirements on the magnet air gap as well as an some of the aberrations in the x-direction.
will increase the coefficient (x ] yy) by about the same amount it decreases the coefficient (x 1xx). Thus a simultaneous correction of both aberrations is impossible if the beam cross section at t, is approximately circular. The same is true if one looks at third-order aberrations (x 1xxx)x: and (x 1xyy)x, _yf as well as of higher order. To design a well correctable system thus requires the beam cross section at z, to considerably deviate from a circle with the x to y ratio being for instance 5 : 1. Thus it is advisable to confine the firstorder beam envelope in the y-direction throughout a sector field but especially at its entrance and exit. This asymmetry in the beam cross section can be achieved by adding to the system y-focusing quadrupole lenses [12-141. However, care must be taken that these quadrupole lenses are large enough to allow passage for the beam and precise enough that they do not introduce other higher-order aberrations. A way out is not to use a particle source that illuminates similar x,px and y.p, phase-space areas but one in which the beam starts from a narrow region in the x-direction under large angles and a wide but almost parallel beam in the y-direction (see fig. 2). This situation can always be achieved by a quadrupole transfer system between the ion source and the system’s entrance slit. In this way three advantages are achieved simultaneously. 1) The magnet air gap can remain small. This reduces the stored magnetic energy and thus the magnet XIV. ACCELERATOR
TECHNOLOGY
casts considerably. In turn this also reduces the magnet pale width since this width is the x-beam width plus lateral fringing-field regions at both sides with both these regions being about equal to the magnet air gap. final The aberrations (x I vv)v:? 2) (x I VY)X,& -. . are much smaller than (x 1xx)xf, (x ( xXx)x:, . -. from the beginning Furthermore, because of the large x to y ratio of the beam cross section a correction by multipole elements in the magnet sector - or shortly before or after it - is effective, i.e. such a correction can be achieved by relatively weak muftipoles so that the higher-order aberrations remain unchanged. 3) At the position of an 0.1 mm wide entrance sht the beam is usually stifl - 100 times wider in the y-dire&on. Thus space-charge effects are reduced > 100 times as compared to a pointlike image of about 0.1 mm diameter at the entrance slit. Thus the use of a quadrupoie transfer system seems to be an advantageous and flexible approach to allow most any dispersive system to have only small aberrations at its final image.
Acknawkxtgement Fox financial support I am grateful to the German Bundesminister fur Forschung und Tecbnologie.
References
PI H. Wollnik, Optics of Charged
Particles (Academic Press, Orlando, 1987). PI H. Matsuda and H. Wollnik, Nucl. Instr. and Meth. 103 (1972) 117. [31 K.L. Brown, D.C. Carey, C. Iselin and F. Rothacker, CERN-Report 80.04 (1980). (41 M. Bern, H. Wollnik, J. Brezina and W. Wendel, AMCO-7, GSI-Rep. THD-26 (1984) p. 679. 151 S. Kowalski and H. Enge, Tech. Rep., MIT, Cambridge, MA (198s). and M. Ben, AIP Conf, Proc. t4 H. Wollnik, B. Hartmann 177 (1988) 74. 171 LJ. Rohrer, private communication. 181 K.L. Brown, Rev. Sci. Instr. 31(1960) 556; SLAC Rep. 91 (1979). [91 A.J. Drag& AIP Conf. Proc. 87 (1982). [IO1 H. Wollnik and M. Berz, Nucl. Instr. and Meth. 321 (1985) 127. III1 H. Rose, Nucl. Instr. and Meth. A258 (1987) 374. [I21 H. Wollnik, Nucl. Instr. and Meth. 95 (1971) 453. [I31 H. Wollnik, Proc. EMIS Conf., Marburg, eds. H. Wagner and W. Walcher, BMBW-Fo~chungs~r K 70-28 (1970) p. 282. “I. Matsuo, Y. Fujita, T. Sakurai and I, 1141 H. Matsuda, Ran&use, Xnt. J. Mass Speetr. Ion Proc. 91 (1989) I.