Method for scanning along a continuous scanning trajectory with a scanner system

Abstract

The invention relates to a method for scanning along a continuous scanning trajectory with a scanner system (100) comprising a first pair of acousto-optic deflectors (10) for deflecting a focal spot of an electromagnetic beam generated by a consecutive lens system (200) defining an optical axis (z) in an x-z plane, and a second pair of acousto-optic deflectors (20) for deflecting the focal spot in a y-z plane being substantially perpendicular to the x-z plane, characterized by changing the acoustic frequency sweeps with time continuously in the deflectors (12, 12) of the first pair of deflectors (10) and in the deflectors (22, 22) of the second pair of deflectors (20) so as to cause the focal spot to move continuously along the scanning trajectory.

Claims

1. A method for scanning along a continuous scanning trajectory with an electromagnetic beam, comprising the steps of: providing a first pair of acousto-optic deflectors comprising first and second acousto-optic deflectors for deflecting the electromagnetic beam in a x-z plane; providing a second pair of acousto-optic deflectors comprising third and fourth acousto-optic deflectors for deflecting the electromagnetic beam in a y-z plane; passing the electromagnetic beam through said first and second pairs of acousto-optic deflectors while providing time dependent acoustic frequency chirps wherein each individual frequency chirp changes with time continuously and non-linearly in the first, second, third and fourth deflectors so as to cause a focal spot of the electromagnetic beam to move continuously along the scanning trajectory.

2. The method according to claim 1, characterized by said acoustic frequency chirps in the first and second deflectors of the first pair of deflectors satisfying the functions f.sub.1x=f.sub.10x+a.sub.1x.Math.t and f.sub.2x=f.sub.20x+a.sub.2x.Math.t, respectively, wherein t is time, f.sub.1x is the function describing the acoustic frequency chirp in said first deflector, f.sub.10x is a constant frequency offset, a.sub.1x is a time dependent first slope, f.sub.2x is the function describing the acoustic frequency chirp in said second deflector, f.sub.20x is a constant frequency offset, a.sub.2x is a time dependent second slope, and setting a difference between said first and second slopes a.sub.1x and a.sub.2x so as to move the focal spot along the x axis with a first velocity (v.sub.x); and providing acoustic frequency chirps in the third and fourth deflectors of the second pair of deflectors satisfying the functions f.sub.1y=f.sub.10y+a.sub.1y.Math.t, and f.sub.2y=f.sub.20y+a.sub.2y.Math.t, respectively, wherein t is time, f.sub.1y is the function describing the acoustic frequency chirp in said third deflector, f.sub.10y is a constant frequency offset, a.sub.1y is a time dependent third slope function, f.sub.2y is the function describing the acoustic frequency chirp in said fourth deflector, f.sub.20y is a constant frequency offset, a.sub.2y is a time dependent fourth slope, and continuously setting a difference between said third and fourth slopes so as to move the focal spot along the y axis with a second velocity (v.sub.y) having regard to the first velocity (v.sub.x) in order to move the focal spot continuously along the scanning trajectory.

3. A method for scanning along a continuous scanning trajectory with an electromagnetic beam, comprising the steps of: providing a first pair of acousto-optic deflectors comprising first and second acousto-optic deflectors for deflecting the electromagnetic beam in a x-z plane; providing a second pair of acousto-optic deflectors comprising third and fourth acousto-optic deflectors for deflecting the electromagnetic beam in a y-z plane; passing the electromagnetic beam through said first and second pairs of acousto-optic deflectors; providing time dependent acoustic frequency sweeps that change with time continuously and non-linearly in the first and second deflectors of the first pair of deflectors satisfying the functions f.sub.1x=f.sub.10x+a.sub.1x.Math.t and f.sub.2x=f.sub.20x+a.sub.2x.Math.t, respectively, wherein t is time, f.sub.1x is the function describing the acoustic frequency sweep in said first deflector, f.sub.10x is a constant frequency offset, a.sub.1x is a time dependent first slope, f.sub.2x is the function describing the acoustic frequency sweep in said second deflector, f.sub.20x is a constant frequency offset, a.sub.2x is a time dependent second slope, and setting a difference between said first and second slopes a.sub.1x and a.sub.2x so as to move a focal spot of the electromagnetic beam along the x axis with a first velocity (v.sub.x); and providing time dependent acoustic frequency sweeps that change with time continuously and non-linearly in the third and fourth deflectors of the second pair of deflectors satisfying the functions f.sub.1y=f.sub.10y+a.sub.1y.Math.t, and f.sub.2y=f.sub.20y+a.sub.2y.Math.t, respectively, wherein t is time, f.sub.1y is the function describing the acoustic frequency sweep in said third deflector, f.sub.10y is a constant frequency offset, a.sub.1y is a time dependent third slope function, f.sub.2y is the function describing the acoustic frequency sweep in said fourth deflector, f.sub.20y is a constant frequency offset, a.sub.2y is a time dependent fourth slope, and continuously setting a difference between said third and fourth slopes so as to move the focal spot along the y axis with a second velocity (v.sub.y) having regard to the first velocity (v.sub.x) in order to move the focal spot continuously along the scanning trajectory; and setting a difference between said first and second slopes such that the first and second slopes in the first and second deflectors are shifted non-symmetrically, and setting a difference between said third and fourth slopes such that the third and fourth slopes in the third and fourth deflectors are shifted non-symmetrically so as to cause the focal spot to move along the z axis with a third velocity (v.sub.z) having regard to the first and second velocity (v.sub.x and v.sub.y) in order to move the focal spot continuously along the scanning trajectory.

4. The method according to claim 3, characterized by: setting the first and second slopes in the first and second deflectors according to the function: a.sub.1x=b.sub.1xt+c.sub.1x, and a.sub.2x b.sub.2xt+c.sub.2x respectively, setting the third and fourth slopes in the third and fourth deflectors according to the function: a.sub.1y=b.sub.1yt+c.sub.1y, and a.sub.2y b.sub.2yt+c.sub.2y respectively, wherein b.sub.1x, b.sub.2x, b.sub.1y, b.sub.2y, c.sub.1x, c.sub.2x, c.sub.1y c.sub.2y are constants and choosing the value of b1x, b2x, b1y, b2y, c1x, c2x, c1y, c2y, f.sub.10x, f.sub.20x, f.sub.10y, f.sub.20y such that the two deflector pairs produce the same z coordinates (z=z.sub.x=z.sub.y) and the same third velocities (v.sub.z=v.sub.zx=v.sub.zy) for the focal spot of the electromagnetic beam.

5. The method according to claim 3, characterized by setting the differences between the slopes such that the first and second slopes are shifted so as to satisfy K.sub.2x()a.sub.2x+K.sub.1x()a.sub.1x=const, and the third and fourth slopes are shifted so as to satisfy K.sub.2y()a.sub.2y+K.sub.1y()a.sub.1y=const, wherein is the optical wavelength of the electromagnetic beam and functions K.sub.1x(), K.sub.2x(), K.sub.1y(), K.sub.2y() describe the dependence of a deflection angle of the electromagnetic beam on the acoustic frequency in the first, second, third and fourth deflector, respectively.

Description

(1) Further details of the invention will be apparent from the accompanying figures and exemplary embodiments.

(2) FIG. 1 is a schematic illustration of the basics of beam deflection via a pair of acousto-optic deflectors.

(3) FIG. 2 is a schematic illustration of a prior art scanning system comprising two consecutive pairs of deflectors focusing in the x-z and y-z planes.

(4) FIG. 3 is a schematic illustration of another prior art scanning system.

(5) FIG. 1 is a schematic illustration of the basics of beam deflection via a pair of acousto-optic deflectors 10 comprising a first deflector 12 and a second deflector 12 having counter propagating acoustic waves 13 and 13 for performing scanning in the x-z plane in a known way. The crystal lattice constants of the crystal making up the deflector 12, 12 are slightly modified by the acoustic waves propagating there through, whereby the deflector crystals act as thick optical gratings of modifiable grating constant.

(6) Accordingly, an incident electromagnetic beam 14 (generally a laser beam) is split by the first deflector 12 into an undeflected zero order beam 16, a first order deflected beam 18 and higher order deflected beams which are neglected in the following discussion as generally only the first order beam 16 is of interest. The first order beam 18 deflected (diffracted) by the second deflector 12 will have the same direction as the incident beam 14 and consequently as the zero order beam 16 deflected by the first deflector 12. Therefore this zero order beam 16 must be separated from the twice diffracted first order beam 18 exiting the second deflector 12. There are two commonly applied technologies for separating the zero order beam 16. If the deflectors 12, 12 are made up of anisotropic crystals and use anisotropic Bragg diffraction involving slow shear acoustic waves the polarisation of the first order diffracted beam 18 is rotated by 90 degrees compared to the undiffracted zero order beam 16, thus the zero order beam 16 may simply be filtered out via a polarising filter. According to the second technology the twice diffracted first order beam 18 and the zero order beam 16 are separated spatially: the spacing d between the two deflectors 12, 12 must be greater than that predicted by the beam aperture D of the first deflector 12 and the first order diffraction angle . In practice the required d spacing is approximately d=10*D. This imposes that the two counter-propagating acoustic beams cannot be realized within the same deflector.

(7) FIG. 2 illustrates a prior art scanning system 100 comprising two consecutive pairs of deflectors 10 and 20. The first pair 10 (comprises a first and a second deflector 12, 12 provided for focusing in the x-z plane, while the second pair 20 ( ) comprises a third and a fourth deflector 22, 22 being provided for focusing in the y-z plane.

(8) FIG. 3 illustrates a prior art scanning system 100 containing a different arrangement of deflectors. The deflectors 12, 12 and 22, 22 are now grouped in two consecutive pairs 110 and 120, a drift compensating unit and a z-focusing unit. Both pairs 110 and 120 contain a deflector 12, 12 operating in the x-z plane and a deflector 22, 22 operating in the y-z plane. The two deflector pairs 110, 120 are linked optically with a telecentric imaging system 60. The scanning system 100 is further imaged to the back aperture of an objective or similar lens system 200 via a second telecentric imaging system 60.

(9) In order to compensate for different types of optical aberrations various scanning systems 100 have been proposed as discussed in detail in WO2010076579.

(10) The present invention can be applied in any prior art scanner comprising two pairs of acousto-optic deflectors and in particular with any of acousto-optic deflector systems disclosed in WO2010/076579 The inventive method is suitable for increasing the speed of acousto-optic scanning in two-photon microscope technology and allows both for scanning in 2D (along segments within a given focal plane, i.e. where the z coordinate is constant), and for scanning in 3D (along an arbitrary 3D trajectory within the sample).

(11) The frequency functions in the deflectors 12, 12 of the pair 10 deflecting in the x-z plane can be defined as
f.sub.1x=f.sub.10x+a.sub.1x.Math.t,f.sub.2x=f.sub.20x+a.sub.2x.Math.t

(12) Similarly the frequency functions for the deflectors 22, 22 of the pair 20 deflecting in the y-z plane is:
f.sub.1y=f.sub.10y+a.sub.1y.Math.t,f.sub.2y=f.sub.20y+a.sub.2y.Math.t

2D Scanning

(13) In the more simple embodiment line scans are made possible in 2D by keeping the z coordinate constant and changing only the x and y coordinates. In this case it is possible to make use of the slope mismatch between the acoustic frequency sweeps (i.e. a.sub.1x is not equal to a.sub.2x as in the RAMP operation mode) within the consecutive acousto-optic deflectors 12, 12, or 22, 22 deflecting in the x-z or y-z plane. The velocity of the scanning in a given plane can be set by nearly symmetrically increasing the mismatch between the slopes of the deflectors in the deflector pairs 10 and 20 respectively: this means that a.sub.1xa.sub.2x, a.sub.1ya.sub.2y is no longer zero. If the deflectors 12 and 12 of the first pair 10 are identical, and the deflectors 22 and 22 of the second pair 20 are identical as well, the v.sub.x and v.sub.y velocity of the deflected focus spot does not change, if the slopes in the two deflectors 12, 12 and 22, 22 of a pair 10 and 20 are shifted symmetrically to maintain:
a.sub.x=a.sub.1xa.sub.2x=const, and a.sub.y=a.sub.1ya.sub.2y=const.

(14) Hence a.sub.1x and a.sub.2x and a.sub.1y and a.sub.2y can be chosen as:
a.sub.1x=a.sub.10x+a.sub.x and a.sub.2x=a.sub.10xa.sub.x
a.sub.1y=a.sub.10y+a.sub.y and a.sub.2y=a.sub.10ya.sub.y.

(15) If the deflectors 12, 12 and 22, 22 respectively are not identical, then the following equations can be used.

(16) If the frequency sweeps responsible for the deflection in the x-z plane have the slopes a.sub.1x and a.sub.2x, respectively, than the focal spot will move in the measurement plane along the x axis with the velocity:
v.sub.x=(K.sub.2()a.sub.2xK.sub.1()a.sub.1x).Math.f.sub.obj/M

(17) where K() is the dependence of the deflection angle on the acoustic frequency f in a given deflector: =K()f, being the optical wavelength, The first and second deflectors of a pair may be of different configuration and geometry, therefore K.sub.1 and K.sub.2 are different. M is the magnification of the optical system between the scanner 100 and the objective 200 and f.sub.obj is the effective focal length of the objective, or of any lens system used as an objective 200. The same is valid for the y-z plane:
v.sub.y=(K.sub.2()a.sub.2yK.sub.1()a.sub.1y).Math.f.sub.obj/M

(18) Thus by setting the two velocities both arbitrary drift directions in a given focal plane can be adjusted.

(19) Simulations have shown that the spot itself does not change its parameters significantly during the drift, the Strehl ratio only decreases with the distance from the optimum point (from the middle of the scanned volume) according to the rule valid for stationary focal spots.

(20) The sweep slope mismatch is optimized for minimum astigmatism in any z0 plane, to obtain the best spot size and shape. The minimum astigmatism restriction causes difference in the frequency slopes of the x deflecting and y deflecting deflector pairs: a.sub.1x and a.sub.2x as well as a.sub.1y and a.sub.2y set for the x-z and y-z planes respectively. We use a quite simple method in the algorithm that determines the frequency sweep slopes, which cause the spot to move in a plane at a predetermined z in the wanted direction with a wanted velocity v.

(21) In the scanning system 100 depicted in FIG. 3, the drift compensating unit's 110 deflectors 12 and 22 are imaged onto the deflectors 22 and 12 of the scanner unit 120. The design of the optical system was made for zero astigmatism in the nominal focal plane of the microscope incorporating the scanning system 100, the astigmatism would increase nearly linearly with the z distance from this plane, if the slope values in the x and y deflectors would be equal. These are set however for zero or nearly zero astigmatism for any z=const plane, by experimentally selecting the slopes of each deflector to get the best possible focal spot PSF over the whole scanned volume. The zero astigmatism condition is z.sub.x=z.sub.y. The z value in the x-z or y-z plane can be determined directly from the slopes in the respective deflectors:

(22) $z_x=\frac{M_x^2{f}_{\mathrm{obj}}\left(v_{ac}/\left(K_{1x}{a}_{1x}+K_{2x}{a}_{2x}\right)\right)}{M_x^2\left(v_{ac}/\left(K_{1x}{a}_{1x}+K_{2x}{a}_{2x}\right)\right)+f_{\mathrm{obj}}},z_y=\frac{M_y^2{f}_{\mathrm{obj}}\left(v_{ac}/\left(K_{1y}{a}_{1y}+K_{2y}{a}_{2y}\right)\right)}{M_y^2\left(v_{ac}/\left(K_{1y}{a}_{1y}+K_{2y}{a}_{2y}\right)\right)+f_{\mathrm{obj}}}$

(23) where M.sub.x and M.sub.y are the magnifications of the telecentric system linking the scanning system 100 and the objective 200 in the respective planes.

(24) The slope differences a.sub.x=a.sub.1xa.sub.2x, a.sub.y=a.sub.1ya.sub.2y between the two deflectors 12, 12 and 22, 22 deflecting in the same direction x or y, respectively, are set by the desired spot drift parameters: direction and velocity. The direction can be defined as the angle of the drift direction with respect the x axis. A given direction a and given velocity v can be set by the slope differences determined as:

(25) $K_{2x}\left(\right)a_{2x}-K_{1x}\left(\right)a_{1x}=\frac{v.Math.\cos \left(\right).Math.M_x}{f_{\mathrm{obj}}},K_{2y}\left(\right)a_{2y}-K_{1y}\left(\right)a_{1y}=\frac{v.Math.\sin \left(\right).Math.M_y}{f_{\mathrm{obj}}}.$

(26) The z level of the plane does not change, if the slopes in the two deflectors (12, 12 and 22, 22) of a pair (10 and 20) are shifted symmetrically to maintain K.sub.2x()a.sub.2x+K.sub.1x()a.sub.1x=const and K.sub.2y()a.sub.2y+K.sub.1y()a.sub.1y=const.

3D Scanning

(27) When we do not use linear chirps, instead we use nonlinear chirps, and in the same time do not maintain symmetric shift in the slopes of the different deflectors, we can in principle achieve scanning along arbitrary 3D path, given by the function z=f(x,y), x,y,z being the Cartesian coordinates of the sample volume, e.g. with the origin at the point defined by intersection of the optical axis z and the nominal focal plane of the objective. The basic equations for the velocities using the well known a.sub.1x etc slope values are:

(28) $v_{z_x}=2M_x{f}_{\mathrm{obj}}\frac{K_{2x}{K}_{1x}\left({\stackrel{.}{a}}_{2x}{a}_{1x}-{\stackrel{.}{a}}_{1x}{a}_{2x}\right)}{{\left[M_x\left(K_{2x}{a}_{2x}-K_{1x}{a}_{1x}\right)+\left(K_{2x}{a}_{2x}-K_{1x}{a}_{1x}\right)\right]}^2}$$v_{z_y}=2M_y{f}_{\mathrm{obj}}\frac{K_{2y}{K}_{1y}\left({\stackrel{.}{a}}_{2y}{a}_{1y}-{\stackrel{.}{a}}_{1y}{a}_{2y}\right)}{{\left[M_y\left(K_{2y}{a}_{2y}-K_{1y}{a}_{1y}\right)+\left(K_{2y}{a}_{2y}-K_{1y}{a}_{1y}\right)\right]}^2}$

(29) But to have the spot not spread out in space z.sub.x=z.sub.y and v.sub.zx=v.sub.zy must always be fulfilled. These give restrictions on the possible values of the slopes and their temporal derivatives, .sub.1x, .sub.2x etc.:

(30) $M_x\frac{K_{2x}{a}_{2x}-K_{1x}{a}_{1x}}{K_{2x}{a}_{2x}+K_{1x}{a}_{1x}}=M_y\frac{K_{2y}{a}_{2y}-K_{1y}{a}_{1y}}{K_{2y}{a}_{2y}+K_{1y}{a}_{1y}}\mathrm{and}$$2M_x{f}_{\mathrm{obj}}\frac{K_{2x}{K}_{1x}\left({\stackrel{.}{a}}_{2x}{a}_{1x}-{\stackrel{.}{a}}_{1x}{a}_{2x}\right)}{{\left[M_x\left(K_{2x}{a}_{2x}-K_{1x}{a}_{1x}\right)+\left(K_{2x}{a}_{2x}-K_{1x}{a}_{1x}\right)\right]}^2}==2M_y{f}_{\mathrm{obj}}\frac{K_{2y}{K}_{1y}\left({\stackrel{.}{a}}_{2y}{a}_{1y}-{\stackrel{.}{a}}_{1y}{a}_{2y}\right)}{{\left[M_y\left(K_{2y}{a}_{2y}-K_{1y}{a}_{1y}\right)+\left(K_{2y}{a}_{2y}-K_{1y}{a}_{1y}\right)\right]}^2}$

(31) The coordinate z can be generally expressed as:

(32) $z=f_{\mathrm{obj}}-\frac{f_{\mathrm{obj}}}{M_x\frac{K_{2x}{a}_{2x}-K_{1x}{a}_{1x}}{K_{2x}{a}_{2x}+K_{1x}{a}_{1x}}+1}=f_{\mathrm{obj}}-\frac{f_{\mathrm{obj}}}{M_y\frac{K_{2y}{a}_{2y}-K_{1y}{a}_{1y}}{K_{2y}{a}_{2y}+K_{1y}{a}_{1y}}+1}$

EXAMPLE 1

(33) In an exemplary setting a.sub.1x and a.sub.2x are controlled according to the equations:
a.sub.1x=b.sub.1xt+c.sub.1x, and a.sub.2x=b.sub.2xt.
In this case:
{dot over (a)}.sub.1x=b.sub.1x,{dot over (a)}.sub.2x=b.sub.2x,

(34) Furthermore, taking a scanning system 100 wherein the deflectors 12, 12 are identical: K.sub.1x=K.sub.2x=K.sub.x thus:

(35) $z_x=f_{\mathrm{obj}}-\frac{f_{\mathrm{obj}}}{M_x\frac{\left(b_{1x}-b_{2x}\right)t+c_{1x}}{\left(b_{1x}+b_{2x}\right)t+c_{1x}}+1},v_{z_x}=2M_x{f}_{\mathrm{obj}}\frac{c_{1x}{b}_{2x}}{{\left[M_x\left(b_{1x}t-b_{2x}t+c_{1x}\right)+\left(b_{2x}t+b_{1x}t+c_{1x}\right)\right]}^2}.$

(36) Using these values the frequencies in a given deflector pair 10, e.g. x are:
f.sub.1x=f.sub.10x+b.sub.1xt.sup.2+c.sub.1xt, and f.sub.2x=f.sub.20x+b.sub.2xt.sup.2

(37) With these the x coordinate can be determined:
x=f.sub.objK.sub.xM.sub.x((b.sub.1xb.sub.2x)t.sup.2+c.sub.1xt+(f.sub.10xf.sub.20x)).

(38) The above considerations can be applied for controlling the y direction scanning similarly:
a.sub.1y=b.sub.1yt+c.sub.1y, and a.sub.2yt and K.sub.1y=K.sub.2y=K.sub.y

(39) from which the y coordinate can be determined in the same manner:
y=f.sub.objK.sub.yM.sub.y((b.sub.1yb.sub.2y)t.sup.2+c.sub.1yt+(f.sub.10yf.sub.20y)).

(40) Using the constraints set for z (z.sub.x=z.sub.y, and v.sub.zx=v.sub.zy) constraints can be found between b.sub.1x, b.sub.2x, b.sub.1y, b.sub.2y, c.sub.1x, c.sub.1y, f.sub.10x, f.sub.20x, f.sub.10y, f.sub.20y.

(41) In order to render the equations more simple f.sub.10x, f.sub.20x are chosen as: f.sub.10x=d.sub.x, and f.sub.20x=0 and f.sub.10y=d.sub.y, and f.sub.20y=0.

(42) In this case the above constraints result in:

(43) $d_y=\frac{y-x}{f_{\mathrm{obj}}{M}_y{K}_y}+\frac{M_x}{M_y}{d}_x.$

(44) A further constraint can be set by requiring that the velocity along z be constant with t, meaning that the t dependent terms in the expression of v.sub.z must have zero coefficients.

(45) This puts the constraint to the b coefficients:

(46) $b_{1x}=\frac{\left(-1+M_x\right)}{\left(1+M_x\right)}{b}_{2x}.$

(47) Applying this to the formula of x (and y symmetrically), we get:

(48) $x=M_x{K}_x{f}_{\mathrm{obj}}\left(\frac{-2b_{1x}}{M_x-1}{t}^2+c_{1x}t+d_{1x}\right)$

(49) and the velocity along x accordingly:

(50) 0$v_x=\left(\frac{-4b_{1x}}{M_x-1}t+c_{1x}\right)$

(51) the expression of z:

(52) $z=\left(\frac{M_x{f}_{\mathrm{obj}}{c}_{1x}\left(M_x+1\right)-2M_x{b}_{2x}t}{\left(M_x+1\right)\left(M_x+1\right)c_{1x}}\right).$

(53) To avoid t dependence of v.sub.x (and t.sup.2 dependence of x) b.sub.1x can set as b.sub.1x=0. This immediately implies b.sub.2x=0 that would result in v.sub.zx=0, unless the magnification between the cells and the objective in the x-z plane is 1: M.sub.x=1.

(54) If b.sub.1x=0 and M.sub.x=1 simultaneously, very simple formulas arise for the coordinates, since only the slope of the frequency in the second deflectors x 12 and y 22 must vary with time. For this both magnifications M.sub.x=M.sub.y=1 should be constrained. This can be nearly satisfied with a setup involving long focal length (compared to the distances between the x and y deflectors 12, 12, 22, 22 and the deflector sizes I) lenses, or exactly satisfied with specially designed cylinder astigmatic lenses. In this simple case the coordinates are:

(55) $x=K_x{f}_{\mathrm{obj}}\left(c_{1x}t+d_{1x}\right),y=K_y{f}_{\mathrm{obj}}\left(c_{1y}t+d_{1y}\right),z_x=\left(\frac{f_{\mathrm{obj}}{c}_{1x}-2b_{2x}t}{2c_{1x}}\right),z_y=\left(\frac{f_{\mathrm{obj}}{c}_{1y}-2b_{2y}t}{2c_{1y}}\right)$

(56) and the velocities:

(57) $v_x=c_{1x},v_y=c_{1y},v_z=f_{\mathrm{obj}}\frac{b_{2x}}{2c_{1x}}$

(58) But from the z constraints

(59) $\left(z_x=z_y\right)\frac{b_{2x}}{c_{1x}}=\frac{b_{2y}}{c_{1y}},\mathrm{and}{d}_y=\frac{y-x}{f_{\mathrm{obj}}{K}_y}+d_x$

(60) When wishing to scan along an arbitrary continuous trajectory the above equations allow to set c.sub.1x, c.sub.1y and d.sub.yd.sub.x so as to determine the desired x, y coordinates, whereas the z coordinate can be set accordingly by setting b.sub.2x, and b.sub.2y, using c.sub.1x and c.sub.1y determined from x and y.

(61) The above-described embodiments are intended only as illustrating examples and are not to be considered as limiting the invention. Various modifications will be apparent to a person skilled in the art without departing from the scope of protection determined by the attached claims.