THEORETICAL & APPLIED MECHANICS LETTERS 2, 043005 (2012)
Command functions of open loop galvanometer scanners with optimized duty cycles V. F. Dumaa) 3OM Optomechatronics Group, Faculty of Engineering, Aurel Vlaicu University of Arad, Arad 310130, Romania
(Received 4 June 2012; accepted 23 June 2012; published online 10 July 2012) Abstract The paper approaches the problem of the command functions of galvanometer-based scanners (GS) that are necessary to produce the linear plus parabolic scanning function of the GS, which we have proved previously to produce the highest possible duty cycle (i.e., time efficiency) of the device. We have completed this theoretical aspect (which contradicted what has been stated previously in the literature, where it has been considered that the linear plus sinusoidal scanning function was the best) with the experimental study of the most used scanning functions of the GSs (sawtooth, sinusoidal and triangular), with applications in biomedical imaging, in particular in optical coherence tomography, demonstrating that the triangular function is always the best one to be applied, from both an optical and a mechanical point of view. In the present study the input voltage/command function which should be applied to the GS to produce the desired triangular scanning function (with controlled non-linearity for the fastest possible stop-and-turn portions) was determined analytically, in relationship with the active torque that drives the device. This command function is analyzed with regard to the specific, respectively required parameters of the GS: natural frequency and damping factor, respectively scan speed and amplitude. The modeling in an open loop control structure of the GS is finally discussed as a trade-off between using the highest possible c 2012 The Chinese Society duty cycle and minimizing the maximum peaks of the input voltage. of Theoretical and Applied Mechanics. [doi:10.1063/2.1204305] Keywords scanners, galvanometer scanner, optomechatronics, scanning functions, command functions, duty cycle, driving torque Galvanometer scanner (GS)1,2 is one of the most used scanning devices, for a wide range of applications, from commercial (e.g., barcode scanning) to industrial (in metrology3 and manufacturing) and to highend applications (i.e., medical imaging, such as confocal microscopy (CM) and optical coherence tomography (OCT).4–8 ) The uni-dimensional (1D) scanners are the basis for 2D (plane)9,10 , 3D (spatial), and 4D (timeincluded, for real time in vivo imaging)8 scanning systems. The issues of GS are therefore, due to their numerous applications, a high priority for the optics and optomechatronics communities. Their technological aspects,11 such as magneto-electric circuits, mobile elements, bearings (including non-contact), pivots, heat sinkers, control,12 and testing13 have been the subject of extensive work, especially in the last two decades; until now an almost standard design14 exists in what regards the basic construction of GSs. The sources of errors of GSs (for both 1D and 2D devices) and their methods of correction, scanning functions,15,16 techniques17 and protocols18 are still opened directions of research. The fundamental advantage of GSs is their high positioning precision. Their main issue is their lower scan speed with regard to polygon mirror scanners (PMs),1,19 their main competitor, and this has re-imposed PMs6 in certain applications, such as swept laser sources (SS) for a) Corresponding
author. Email:
[email protected].
OCT20 in the last decade, despite the inherent sources of errors that characterize the PM scanner. The present paper will approach a fundamental problem of the GSs: their necessary driving functions for the optimized linear plus parabolic scanning function that we have demonstrated21 to produce the highest duty cycle/time efficiency of the device, necessary especially in high-end applications such as biomedical imaging (e.g., CM and OCT). This result has been contrary to what has been previously stated in the literature,14 where it was being considered (but not demonstrated) that the linear plus sinusoidal scanning functions were the best from this point of view. The scope of this research is to deduce these command functions,22 to study and to optimize them with regard to the parameters of the device and of the process. A galvoscanner (Fig. 1) has a mobile element (magnet or coil) on which the oscillatory mirror is placed. The functional parameters of the scanner are indicated in Fig. 1(a): O. A., optical axis; θa , angular scan amplitude; 2H, total scan amplitude; 2xa , linear scan amplitude; θ, scan angle; x(t), scanning function; L, distance from the galvomirror axis to the lens. The parameters of the device (Fig. 1(b)) are: J, the axial mass inertia moment of the mobile element; c, the damping coefficient, and k, the elastic coefficient. The scanning is not uni-directional (as for PMs), but bi-directional, and that is usually a drawback, especially for biomedical imaging. The worst issue (in that concerns the duty cycle of the process) is the fact that in a raster scan-
043005-2 V. F. Duma
Theor. Appl. Mech. Lett. 2, 043005 (2012) Mobile element (with galvomirror) N
Lens Galvomirror (J)
S
(O.A.) x(θ)
θ
± θa/2 Magnetic
N
S
(O.A.) 2xa
2H
S
N
Damper (c)
L
circuit
θ
Spring (k)
(b) Mobile element and fix magnet
(a) Scanning system
Fig. 1. Galvanometer scanner (GS). x
0.7T
H xa
Dt
2 T
0 xa H
ta
2τ
x
0.4T H xa
t
T2
τ
1 xa H
0.33T
t
3T/2
T
2T
(b) Linear on the active portions, with fast stop-and-turn parts (optimized in Ref. 21)
(a) Sawtooth (1) and sinusoidal (2)
Fig. 2. Three of the most used scanning function of the GSs (state-of-the-art).
ning regime the galvomirror needs time for stop-andturn (while PMs achieve high scan rates in a continuous rotation, although with several errors, of which the most important is the non-linearity produced by the eccentricity of the scanning facet with regard to the pivot of the polygon.1–3,6,19 ). The most used scanning functions x(t), defined (Fig. 1(a)) as the current position of the beam (considered perfectly collimated) with regard to the optical axis (O.A.) are sawtooth, sinusoidal and triangular (Fig. 2). We have demonstrated experimentally,7 with a high-speed OCT setup23 that sawtooth and sinusoidal (Fig. 2(a)) are less competitive; the triangular regime is the one that offers the largest artifacts-free images. However, the pure triangular input signal (with sharp stop-and-turn portions) does not give an identical output; at higher scan frequencies (fs > 50 Hz) and also for large scan ampli⎧ vt2 ⎪ ⎪ − + H, ⎪ ⎪ 2τ ⎪ ⎪ ⎪ ⎪ ⎪ −v(t − T /4), ⎪ ⎪ ⎪ ⎪ ⎨ 2 vt vT vT 2 x(t) = − t − H + , ⎪ 2τ 2 8τ ⎪ ⎪ ⎪ ⎪ ⎪ v(t − 3T /4), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ vT vT 2 ⎪ vt2 ⎩ + t+H − , − 2τ 2 8τ
tudes the triangular signals turns progressively into a sinusoidal, as we have analyzed.7 This is why it is better14 to design the input functions with linear plus non-linear portions (the latter for the return parts) (Fig. 2(b)), to obtain, within certain limits of the fs an output signal/scanning function θ(t) identical to the input signal. The critical aspect in this respect is to minimize the duration τ of the non-linear part in order to maximize the duty cycle (time efficiency) of the process η = ta /(ta + 2τ ),
(1)
defined as the ratio between the duration of its active, linear part 2ta and of its period T = 1/fs = 2(ta + 2τ ). As mentioned above, the linear plus parabolic scanning function is the best one from this point of view; its equations are,21 on the time intervals in Fig. 2(b)
t ∈ (0, τ ) t ∈ (τ, T /2 − τ ) T T t∈ − τ, + τ , 2 2 t ∈ [T /2 + τ, T − τ ] t ∈ [T − τ, T ]
where |v| = x(t) ˙ = 2xa /ta = constant is the value of the
scan speed of the linear portion of the function.
(2)
043005-3 Command functions of open loop galvanometer scanners
The dynamical equation of the mobile element of the GS in open loop is of the classical type J θ¨ + cθ˙ + kθ = T (t) ⇒ θ¨ + 2ξω0 θ˙ + ω02 θ = T (t) /J, (3) √ where ω0 = k/J and ξ = c/(2 kJ) are the natural pulsation and the unitless damping coefficient of the system, respectively. Ta (t), the reduced active magneto-electric torque necessary to produce the oscillatory movement, is ˙ and θ(t) ¨ derived21 from Eq. (3) using the functions θ(t) obtained from the expression of the movement law of the spot, x(θ) = L tan 2θ (Fig. 1), Ta (t) = T (t) /J =
1 L3
xx˙ 2 1 + (x/L)
2
2 +
1 2ξω0 x˙ − x ¨ ω02 x + arctan . 2 2L 1 + (x/L) 2 L
(4)
We have previously pointed out Ta (t)22 for low scan frequencies fs , suitable for industrial applications (e.g., dimensional measurements).3,24 For high scan frequencies its graphs for each of the five time intervals (i– v) in Eq. (2) are obtained in Fig. 3 using Eqs. (2) and (4) for three of the most significant values of fs : f1s = 100 Hz (the level for which GSs are usually tested), f2s = 200 Hz (a common desirable value of the scanner), and f3s = 300 Hz (the nowadays upper limit of lateral scanning with GSs in OCT). The time interval in Fig. 3 has been extended to the entire period of Ta (0.01 s, for the first function) to study the entire graphs, although smaller time intervals, according to Eq. (1) have to be considered for each portion of the function Ta (t) ∝ i(t), the current applied in the magneto-electric circuit (i.e., the command function). The parameters considered for the scanning process are: L = 100 mm, H = 15 mm, and xa = 14 mm, therefore the characteristic coefficient r = xa /H is 0.93. The designing equation of the linear plus parabolic scanning function21 provides from the definition (1) the expression of the duty cycle for this scanning function vτ = 2(H − xa ) ⇒ η = r/(2 − r),
(5)
therefore η = 86.92%, and for the three frequencies, τ1 = (1 − η)/(4fs ) = 327 μs, τ2 = 163.5 μs and τ3 = 109 μs, while v1 = 6.116 m/s, v2 = 12.232 m/s and v3 = 18.348 m/s. The values ω0 = 20 rad/s and ξ = 1 (critical damping) were also considered (according to the analysis in Fig. 4). The main issue in this discussion concerns the peaks of Ta (t), proportional to the peaks of the input voltage for the GS, significant for the non-linear portions of the scanning function, which are two orders of magnitude higher than those for the linear portions. From Fig. 3 these peak values also increase by a factor of two for the row of scan frequencies considered (from 100 Hz to 200 Hz and to 300 Hz). One has to remark that this
Theor. Appl. Mech. Lett. 2, 043005 (2012)
increase is more significant for lower fs : the peaks also grow by a factor of two for each increase of the frequency from 10 Hz to 20 Hz, rather than to 50 Hz and to 100 Hz. To decrease these peaks to a reasonable value is a necessity if one aims to obtain devices that are feasible from respectively an electrical and a mechanical point of view, as the GSs can whistand only limited values of the input voltage and of Ta (t), respectively. There are several ways to approach this issue and thus to avoid voltage peaks that would actually affect the circuits of the GS, while theoretically they provide an ideal scanning function. From the study of the active torque with regard to ω0 (Fig. 4) or ξ one may see that little improvement can be obtained by altering these parameters. For ξ, due to its closed values and minor influence on the torque there are actually no differences. From Eq. (3), ω0 depends on J, therefore it is a direct function of the size and mass of the mirror, imposed by the specific application, that is by the neccesary aperture of the mirror and by the power of the laser that is used with the scanning system. Therefore, a significant increase of the peaks occurs with regard to ω0 only for very high values, therefore for the small inertia that characterizes the light, low dimensions mirrors of MEMS (micro-electro-mechanical systems) scanners. The two methods that need considering are therefore: (1) to give up some of the duty cycle, and thus the voltage peaks are reduced roughly to a half with every 0.1 decrease in r = xa /H, and with every more than 10% decrease in η (Fig. 5); (2) to preserve the duty cycle, but to increase the linear scanning domain H (Fig. 6), which has quite a similar effect. As this problem actually addresses only the stop-and-turn portions (i), (iii) and (v), for which (Fig. 3) the peaks are significantly higher, in Figs. 4 and 5 only the graphs for the (i) time interval are presented; the situation for the (iii) and (iv) time intervals is similar. For method (1), from the study performed for fs = 300 Hz in Fig. 5 as r, therefore η is decreased, the torque is brought to more acceptable levels: from η1 to η2 the peaks decrease to less than half the value, while from η2 to η3 and further on, the decrease is less spectacular, so η2 is for these parameters the right choice, and such an analysis may be performed for every set of required parameters. While approaching this issue we may thus determine what is the actual η a user may reach - even when using the theoretically optimum scanning function (linear plus parabolic) demonstrated. For method (2), as H is higher (Fig. 6), the peaks of Ta (t) decrease significantly (again roughly by a factor of two, for each decrease of H, in the example), as for the same η, the oscillating mirror actually has more space to stop-and-turn. The problem is that H is usually imposed by the application, and that the maximum possible scan amplitude θa (Fig. 1(a)) is limited, especially for high scan frequencies to have stable oscillations.7 However, one should have this in mind in the designing stage, in choosing the distance L (Fig. 1), as it may thus be possible to increase H for a given value of the
043005-4 V. F. Duma
1
Ta1 (t) Ta2 (t) Ta3 (t)
0
0 ts (a)
T N. m
1
0
-1 -0.01
0.01
1
Ta1 (t) Ta2 (t) Ta3 (t)
0.01
Τ 104
-1 -0.01
Ta1 (t) Ta2 (t) Ta3 (t)
0 ts (d)
0
0005 ts (c)
001
Ta1 (t) Ta2 (t) Ta3 (t)
0
-1
0.01
Ta1 (t) Ta2 (t) Ta3 (t)
Τ 106
1
0
Τ 106
0
-1 0 ts (b)
T N. m
-1 -0.01
Τ 104
T N. m
Τ 106
T N. m
T N. m
1
Theor. Appl. Mech. Lett. 2, 043005 (2012)
−
0
0005 ts (e)
001
Fig. 3. The reduced active torque Ta (t) for each time interval (i–v) of the scanning function for the scan frequencies: f1s = 100 Hz, f2s = 200 Hz and f3s = 300 Hz (the functions are considered for the entire period T = 0.01 s at 100 Hz).
Τ 106
2
T N. m
T N. m
2
0 Ta1 (t) Ta2 (t) Ta3 (t)
-2 -2
2
angular scanning amplitude, and to still use the necessary value of xa with high η and with lower peaks of the input voltage. A similar study can be made with regard to the distance L or to the ratio H/L. To conclude the study we have to determine the actual value of η that may be reached for a certain scanning velocity v, in order to have but a maximum allowed value of the active torque Ta (t). From Eqs. (1)–(5) Ta
max
=
Jv
1 , 1 − r 4HL[1 + (H/L) ] 2
·
Ta1 (t) Ta2 (t) Ta3 (t)
Τ 10-3 0
2
ts
Fig. 4. The reduced active torque for fs = 300 Hz for three natural frequencies of the mobile element of the GS: ω01 = 10 rad/s, ω02 = 102 rad/s, ω03 = 103 rad/s.
2
0
-2 -2
Τ 10-3 0 ts
Τ 106
(6)
from which the maximum reachable duty cycle of the
Fig. 5. The reduced active torque for the scan frequency fs = 300 Hz for three decreasing ratios r, therefore for three decreasing duty cycles: from r1 = 0.93 (η3 = 86.92%) to r2 = 0.85 (η2 = 70.09%), and finally to r3 = 0.73 (η3 = 57.48%).
GS is ηmax =
Ta Ta
max /Cv
2
max /Cv
2
−1 , +1
(7)
where C=
J 2
4HL[1 + (H/L) ]
.
In order to demonstrate that this is indeed ηmax , in Fig. 7 it is shown for two representative fs (therefore scan speeds v-Eq. (5)) that indeed the inertia torque
043005-5 Command functions of open loop galvanometer scanners Τ 106
T N. m
2.5
0
-2 -2
Ta1 (t) Ta2 (t) Ta3 (t)
0 ts
Τ 10-3
Τ 104 T1 (t) T2 (t) T3 (t)
0
-2.5 -0.02
2
Fig. 6. The reduced active torque for the scan frequency fs = 300 Hz for three total scanning domains: from H1 = 7.5 mm to H2 = 15 mm and H3 = 30 mm — for the same parameters of the device and a ratio r = 0.93 (therefore η = 86.92%); the linear (active) part of the scanning domain is therefore respectively: x1a = 7 mm, x2a = 14 mm and x3a = 28 mm.
Ta max is the major component of the active torque for the (i) time interval—for which the peaks are precisely in the (0, τ ) interval. It can be seen that the elastic and the damping components of the active torque may be neglected (especially for a higher fs ) with regard to the inertia component. Equation (7) therefore evaluates with a good approximation ηmax with regard to Ta max and v. We present our researches regarding the command functions of 1D galvanometer scanners (GS). We determine and study them for the optimal scanning functions of a GS, linear on its active parts and with parabolic fast stop-and-turn portions that we have demonstrated theoretically21 to produce the maximum duty cycle of the device in open loop. We discussed in the paper the methods to decrease the peaks of the input voltage that have to be reduced without a major loss in terms of duty cycle to make the device feasible. While this study referred to open loop GSs, our ongoing researches target close loop devices, for which we have already proved experimentally7 that the triangular input signals (for which linear plus parabolic are a particular case) are the best to produce artifact-free images in OCT. This direction of research has to be continued with modeling, simulation and experiments on GSs with various feed-backs to optimize their different functional aspects. Future work also addresses 2D (with double axis GSs or with PM plus GS assemblies) and 3D systems (e.g., with GS and Risley prisms25 ) for fast accurate scanning especially in high-end applications such as real time biomedical imaging. This paper was presented at the 11th Conference on Dynamical Systems—Theory and Applications, December 5-8, 2011, Lodz, Poland. This work was supported by the Romanian Education and Research Ministry through the National University Research Council (NURC/CNCS) IDEAS Grant
1
T N. m
T N. m
2
Theor. Appl. Mech. Lett. 2, 043005 (2012)
0 ts (a) fs=50 Hz
0.02
Τ 106 T1 (t) T2 (t) T3 (t)
0
-1 -3.3
0 ts (b) fs=300 Hz
Τ 10-3 3.3
Fig. 7. Comparison of the three components of the active torque: T1 , inertia torque; T2 , damping torque; T3 , elastic torque for two scan frequencies. The specific time periods are considered; the different orders of magnitude of the amplitudes of the peaks can be remarked.
1896/2008. As part of this work has begun in 2009-2010, the support of the US Department of State through Fulbright Scholar Grant 474/2009 is also gratefully acknowledged.
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