Patent Publication Number: US-2022221767-A1

Title: Control Method of Optical Deflector, and Optical Deflection Device

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a national phase entry of PCT Application No. PCT/JP2020/015813, filed on Apr. 8, 2020, which claims priority to Japanese Application No. 2019-082043, filed on Apr. 23, 2019, which applications are hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to a method for controlling an optical deflector and an optical deflection device. 
     BACKGROUND 
     Optical deflection units that change the direction of travel of light by applying an alternating voltage such as a sine wave to a dielectric (electro-optical material) in a paraelectric phase are used in various fields such as laser printers, wavelength sweep light sources, and the like. For example, a wavelength sweep light source in which an electro-optical crystal in a paraelectric phase (dielectric crystal of a paraelectric phase) is disposed in an optical resonator has been proposed (see PTL 1). The wavelength sweep light source superimposes a DC voltage to fill a trap in electro-optical crystals with electrons as a bias voltage in applying an AC voltage for fast deflection to the electro-optical crystal. The wavelength sweep light source configured in this manner can suppress fluctuations in the light output, the sweep wavelength band, and the coherence length over an extended period of time, and has excellent long-term stability. 
     Techniques have been proposed for applying an AC drive voltage in which a DC voltage is superimposed as a bias voltage while irradiating the electro-optical crystal with light from an optical irradiation unit (see PTL 2). In this proposal, it is believed that electron injection into the trap within the electro-optical crystal can reduce the time to reach a steady state. It is also believed that optical deflectors using electro-optical materials will also be applied in the future in laser processing and the like other than the fields described above. 
     KTN (KTa 1-x Nb x O 3 ) or KLTN (K 1-y Li y Ta 1-x Nb x O 3 ) having a large electro-optical effect are known as electro-optical crystals having the above-described characteristics. Hereinafter, in a case where there is no need to distinguish between KTN and KLTN, these are collectively referred to as KTN. Furthermore, in a case where Ti or Cr material is used as an electrode, the charge can be injected into the KTN, and the internal electric field generated by the injected charge can be used to achieve a high speed and wide angle optical deflector. 
       FIG. 11  illustrates a configuration of a conventional optical deflector using KTN crystals (see PTL 3).  FIG. 11  illustrates a configuration viewed from the incident direction of light. The KTN crystal  11  is sandwiched, at the top surface and the bottom surface, and is held by two metal blocks (electrode blocks)  13   a  and  13   b , between a graphite sheet  12   a  and a graphite sheet  12   b . An electrode pair (not illustrated) including a positive electrode and a negative electrode for applying a control voltage is formed on the opposing upper and lower surfaces of the KTN crystal  11 , and are electrically connected to a control voltage source via the graphite sheets  12   a  and  12   b , and the metal blocks  13   a  and  13   b  in contact. The application of a voltage from a control voltage source not illustrated, and electron injection into the KTN crystal  11  generate an electric field within the KTN crystal  11  and generate a refractive index distribution within the KTN crystal  11 . An optical axis of incident light is set orthogonal to the direction of the electric field, and a voltage is applied between the pair of electrodes to deflect incident light. 
     The graphite sheets  12   a  and  1213 , are inserted to prevent the KTN crystal  11  from breaking due to vibration in a case of applying a high frequency control voltage to the KTN crystal it Aluminum nitrides (AlNs)  14   a  and  14   b  are inserted on both sides of the KTN crystal  11 . The function of the AlNs  14   a  and  14   b  is to serve as a heat transfer member for positioning the KTN crystal  11  and for maintaining the temperature of the two metal blocks  13   a  and  13   b  uniform. A Peltier element  16  is disposed between the metal block  13   a  and a support plate  15 , and thermistors (temperature detection units)  17   a  and  17   b , are embedded within the metal blocks  13   a  and  13   b , respectively. 
     A temperature control device  18  detects temperature by the thermistors  17   a  and  17   b , and heats or cools the metal block  13   a  by the Peltier element  16  to maintain the KTN crystal  11  at the appropriate set temperature (constant temperature). The temperature control device  18  detects the temperature by measuring the resistance value of the thermistor  17   a  and the thermistor  17   b , connected in series, and keeps the dielectric constant of the KTN crystal  11  constant. 
     CITATION LIST 
     Patent Literature 
     
         
         PTL 1: JP 6193773 B 
         PTL 2: JP 2017-219732 A 
         PTL 3: JP 2017-203847 A. 
       
    
     SUMMARY 
     Technical Problem 
     Generally, light has an intensity distribution in a cross section perpendicular to the direction of travel, but hereinafter the centroid of this intensity distribution will be referred to as the centroid position of the light or simply the position of the light. Depending on the field of application of the optical deflector, it is required that the centroid position of the light deflected by the optical deflector have desired time dependency. For example, two optical deflectors can be connected in series to deflect the light in a circular manner. In this way, in a case of deflecting the light along a circle of radius A, the centroid position of the light deflected by each optical deflector is required to have time dependency such as x=A cos(ω+B), y=A sin(ω+B). Here, each of x and y is an orthogonal coordinate axes, ω is the angular frequency, t is the time, and B is the initial phase. 
     As described in Patent Literatures 1, 2, and 3, it is known to apply an AC voltage (alternating voltage) to an electro-optical crystal. However, even in a case where a sine wave voltage is applied as a voltage, it is not clear whether the time dependency of the position of the deflected light beam is represented by a sine wave (shows a simple harmonic motion). 
     Embodiments of the present invention are made to solve the problems described above, and an object of embodiments of the present invention is to make the position (trajectory) of light deflected by an optical deflector to have desired time dependency. 
     Means for Solving the Problem 
     A method for controlling an optical deflector according to embodiments of the present invention is a method for controlling an optical deflector that changes a deflection angle depending on a voltage to be applied, the method for controlling the optical deflector deriving a goal voltage V=g goal (t) for providing a deflection angle θ with goal time dependency θ=θ goal (t) when time is denoted by t, the method including: an initial step in which in a case where θ is a periodic function of time, a period is denoted by T, in a case where θ is not a periodic function of time, a duration is denoted by T and a start time of the deflection angle with the goal time dependency θ=θ goal (t) is set as t=0, and when time in a period 0≤t&lt;T is considered, the number of repetitions is set as n=0; an increment step of setting n=n+1; a step A n  of applying a voltage g n (t) with the period T or the duration T to the optical deflector; a step B n  of checking whether θ=θ goal (t) is satisfied with a goal accuracy, next to the step A n ; a step C n  of setting g n (t) as a goal voltage; a step D n  of deriving θ=f rise, n (V) indicating voltage dependency of a deflection angle when a voltage applied to the optical deflector rises, and voltage dependency θ=f all, n (V) of a deflection angle when a voltage applied to the optical deflector falls; a step E n  of setting g rise, n+1 (t)=f rise, n   −1 {θ goal (t)}, setting g fall, n+1 (t)=f fall, n   −1 =f fall, n   −1 (θ goal (t)), and configuring g n+1 (t) from g rise, n+1 (t) and g fall, n+1 (t), next to the step D n ; and a step F n  of performing the increment step next to the step E n , wherein the step A n  is subsequent to the increment step, in the step B n , in a case where θ=θ goal (t) is not satisfied with the goal accuracy, the step D n  is performed following the step B n , in the step B n , in a case where θ=θ goal (t) is satisfied with the goal accuracy, the step C n  is performed following the step B n  and a process is ended, g n+1 (t) is equal to g rise, n+1 (t) when the voltage applied to the optical deflector rises, and is equal to g fall, n+1 (t) when the voltage applied to the optical deflector falls, and the voltage dependency of the deflection angle when the voltage applied to the optical deflector rises is different from the voltage dependency of the deflection angle when the voltage applied to the optical deflector falls. 
     In one configuration example of the method for controlling the optical deflector described above, the optical deflector is constituted of an electro-optical material with a trap in a paraelectric phase and being for accumulating charge within the material, and an optical axis of incident light on the optical deflector is set orthogonal to a direction of an electric field of the voltage applied to the optical deflector, and the voltage is applied to the optical deflector to deflect incident light incident on the optical deflector. 
     In one configuration example of the method for controlling the optical deflector described above, the electro-optical material is either KTN [KTa 1-α Nb α O 3  (0&lt;α&lt;1)] crystals or KLTN [K 1-β Li β Ta 1-α Nb α O 3  (0&lt;α&lt;1, 0&lt;β&lt;1)] crystals with lithium being added. 
     An optical deflection device according to embodiments of the present invention includes: an optical deflection unit configured to change a deflection angle depending on a voltage to be applied; a voltage control unit configured to apply the voltage to the optical deflection unit; and a storage unit configured to store a value of a voltage output by the voltage control unit, wherein the storage unit stores a goal voltage for providing a deflection angle with goal time dependency, and the goal voltage is determined by the method according to claim  1 . 
     In one configuration example of the optical deflection device described above, the optical deflector is constituted of an electro-optical material with a trap in a paraelectric phase and being for accumulating charge within the material, and an optical axis of incident light on the optical deflector is set orthogonal to a direction of an electric field of the voltage applied to the optical deflector, and the voltage is applied to the optical deflector to deflect incident light incident on the optical deflector. 
     In one configuration example of the optical deflection device described above, the electro-optical material is either KTN [KTa 1-α Nb α O 3  (0&lt;α&lt;1)] crystals or KLTN [K 1-β Li β Ta 1-α Nb α O 3  (0&lt;α&lt;1, 0&lt;β&lt;1)] crystals with lithium being added. 
     Effects of Embodiments of the Invention 
     As described above, according to embodiments of the present invention, the step of applying g n+1 (t) consisting of g rise, n+1 (t)=f rise, n   −1  (θ goal (t)), and g fall, n+1 (t)=f fall, n   −1  (θ goal (t)) to the optical deflector to achieve θ=θ goal (t) is repeated until θ=θ goal (t) indicating the time dependency of the deflection angle is satisfied with the goal accuracy. so that the position (trajectory) of the light deflected by the optical deflector has the desired time dependency. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a configuration diagram illustrating a configuration of an optical deflection device according to an embodiment of the present invention. 
         FIG. 2  is a flowchart for describing a method for controlling an optical deflector according to an embodiment of the present invention. 
         FIG. 3  is an explanatory diagram for describing a position x of an output destination of light incident on the optical system  104  output from the optical deflector  101 , and output from the optical system  104 . 
         FIG. 4A  is a configuration diagram illustrating a configuration in a case of a space where the distance from a pivot iota to the position x of the light output destination is L, as the optical system  104  illustrated in  FIG. 3 . 
         FIG. 4B  is a configuration diagram illustrating another configuration in a case of a space where the distance from the pivot  101   a  to the position x of the light output destination is L, as the optical system  104  illustrated in  FIG. 3 . 
         FIG. 5A  is a characteristic diagram illustrating an example of a voltage V which a voltage control unit  102  applies (outputs) to the optical deflector  101 . 
         FIG. 5B  is a characteristic diagram illustrating a change in a deflection angle θ obtained as a result of the voltage control unit  102  applying the voltage V illustrated in  FIG. 5A  to the optical deflector  101 . 
         FIG. 5C  is a characteristic diagram illustrating the relationship between the instantaneous voltage V and the deflection angle θ in a case where a change in the deflection angle θ illustrated in  FIG. 5B  is obtained as a result of the voltage control unit  102  applying the voltage V illustrated in  FIG. 5A  to the optical deflector  101 . 
         FIG. 6  is an explanatory diagram for describing different characteristics of an electro-optical crystal at a voltage rising time θ=f rise (V) and a falling time θ=f fall (V). 
         FIG. 7  is a characteristic diagram illustrating an example of a voltage waveform of a period T. 
         FIG. 8A  is a characteristic diagram illustrating an example of θ=f 0 (V). 
         FIG. 8B  is a characteristic diagram illustrating an inverse function V=f 0   −1 (θ) of θ=f 0 (V). 
         FIG. 9A  is a characteristic diagram illustrating the time dependency of a goal applied voltage determined in Example 1. 
         FIG. 9B  is a characteristic diagram illustrating the time dependency of the deflection angle obtained in a case of applying the applied voltage illustrated in  FIG. 9A . 
         FIG. 10A  is a characteristic diagram illustrating the time dependency of the goal applied voltage determined in Example 2. 
         FIG. 10B  is a characteristic diagram illustrating the time dependency of the deflection angle in Example 2. 
         FIG. 11  is a configuration diagram illustrating a configuration of a conventional optical deflector using KTN crystals. 
     
    
    
     DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS 
     Hereinafter, an optical deflection device according to an embodiment of the present invention will be described with reference to  FIG. 1 . The optical deflection device includes an optical deflector  101  that changes the deflection angle depending on an applied voltage, a voltage control unit  102  that applies a voltage to the optical deflector  101 , and a storage unit  103  that stores a value of a voltage to be output by the voltage control unit  102 . The voltage control unit  102  outputs the voltage of the value stored in the storage unit  103  to the optical deflector  101 . The storage unit  103  stores a voltage V=g goal (t) such that the trajectory of the output light has desired time dependency. Details will be described below. The voltage control unit  102  is connected to an electrode block (not illustrated) of the optical deflector  101 . The trajectory of the light output from the optical deflector  101  has the desired time dependency, by driving at the voltage from the voltage control unit  102 . 
     The optical deflector  101  is composed of an electro-optical material with a trap in a paraelectric phase and being for accumulating charge within the material. The optical axis of the incident light on the optical deflector  101  is set orthogonal to the direction of the electric field of the voltage applied to the optical deflector  101 . A voltage is applied to the optical deflector  101  to deflect incident light incident on the optical deflector  101 . Examples of such an electro-optical material include KTN [KTa 1-α Nb α O 3  (0&lt;α&lt;1)] crystals or KLTN [K 1-β Li β Ta 1-α Nb α O 3  (0&lt;α&lt;1, 0&lt;β&lt;1)] crystals with lithium. 
     The storage unit  103  stores a goal voltage V=g goal (t), which provides a deflection angle θ with the goal time dependency θ=θ goal (t). 
     Hereinafter, a method for deriving a goal voltage g goal (t) (a method for controlling the optical deflector) stored by the storage unit  103  will be described with reference to the flowchart of  FIG. 2 . This method for derivation (the method for controlling the optical deflector) is to derive the voltage V=g goal (t), which provides the time dependency θ=θ goal (t) of the goal deflection angle. With the deflection angle θ and the voltage V as a periodic function of time with a certain period T, and in evaluating whether the deflection angle has changed in the desired time and in deriving an inverse function, 0≤t&lt;T is considered as a definition region oft. 
     First, in step S 101 , the number of repetitions n=0 is set (initial step). Next, at step S 102 , a voltage g n (t) having a period T is applied to the optical deflector  101  (step A n ). Here, g n (t) is a voltage signal, e.g., “DC bias voltage+sine wave voltage” when n=0. When n≥1, g n (t) is a voltage obtained in step S 106  described below. 
     Next, in step S 103 , it is confirmed whether θ=θ goal (t) is satisfied with goal accuracy (step B n ). Here, an evaluation is made whether the deflected light has a trajectory with the desired time dependency. In other words, an evaluation is made whether the deflection angle has made a desired time change represented by Equation (5) below. 
     The evaluation of whether the deflection angle has made the desired time change represented by Equation (5) can be achieved by a variety of methods. For example, the evaluation can be made by plotting the experimental results of the time dependency of the deflection angle on a graph and visually determining by humans from the resulting graph. The evaluation described above can be made, for example, depending on whether the maximum value of the absolute value of the difference at each time of θ goal (t) with the experimental results of the time dependency of the deflection angle is less than or equal to a desired threshold value appropriately defined. The evaluation described above can also be made, for example, depending on whether the average value of the absolute value of the difference at each time of θ goal (t) with the experimental results of the time dependency of the deflection angle is less than or equal to a desired threshold value appropriately defined. 
     In a case where, in step S 103 , θ=θ goal (t) is satisfied with the goal accuracy (yes in step S 103 ), then in step S 104 , g n (t) is set to the goal voltage g goal (t) (step C n ) and the process ends. 
     On the other hand, in step S 103 , in a case where θ=θ goal (t) is not satisfied with the goal accuracy (no in step S 103 ), then in step S 105 , θ=f rise, n (V) indicating the voltage dependency of the deflection angle when (during) the voltage applied to the optical deflector  101  rises and the voltage dependency θ=f fall, n (V) of the deflection angle when (during) the voltage applied to the optical deflector  101  falls are derived (step D n ). Next, in step S 106 , g rise, n+1 (t)=f rise, n   −1  {θ goal (t)}, and g fall, n+1 (t)=f fall, n   −1  (θ goal (t)) are set, and g n+1 (t) is configured from g rise, n+1 (t) and g fall, n+1 (t) (step E n ). Next, in step S 107 , n=n+1 is set as an increment step (step F n ) and the process returns to step S 102 . Each of the steps described above continues until θ=θ goal (t) is satisfied with the goal accuracy. 
     Hereinafter, more details will be described.  FIG. 3  is a diagram for describing a position x of an output destination of light incident on an optical system  104  output from the optical deflector  101  and output from the optical system  104 . Suppose that the light emitted from the center of rotation (pivot)  101   a  of the optical deflector  101  is required to indicate, after passing through the optical system  104  composed of optical components such as lenses, mirrors, and the like, the time dependency described by the function h(t) of the period T such as “x=h(t) . . . (i)” on the x-axis. 
     For example, in a case where x is required to show simple harmonic motion, h(t) can be expressed as “h(t)=x 0  sin(ωt+γ)+x 2  . . . (2)”. Note that in Equation (2), x 0  is the amplitude, ω is the angular frequency, and γ is the initial phase. x 2  is a constant, and in this case, is the central position of the simple harmonic motion. The period T is 2π/ω. 
     For example, in a case where x is required to show a straight trajectory at an equal speed of speed v, h(t) can be expressed as “h(t)=vt+x 2  . . . (3)”. In Equation (3), x 2  is a constant, and in this case, is a position at time t=0. 
     Meanwhile, the position at which the light beam reaches on the x-axis is determined by the incident angle θ on the optical system  104 . In other words, x is a function of θ, which is expressed as x=e(θ). Then, assuming that the time dependency of the deflection angle θ that shows the desired trajectory x=h(t) is θ=θ goal (t), the equation “h(t)=e {θ goal (t)} . . . (4)” is satisfied. Solving Equation (4) inversely holds “θ goal (t)=e −1  {h(t)} . . . (5)”. 
     The function e may be known because it is determined by the characteristics and the arrangement of the optical components that are used. Thus, if the desired trajectory h(t) is determined, θ goal (t) is determined from Equation (5). It is not critical to select what type of optical system  104  is to be selected, and in one optical system  104 , a voltage may be determined to achieve the desired deflection angle θ=θ goal (t). 
     Thus, as the optical system  104 , a method for determining a voltage such that θ=θ goal (t) assuming the space where the distance from the pivot iota illustrated in  FIG. 4A  to the x-axis is L. Note that the origin of the deflection angle is set as desired, and thus, as illustrated in  FIG. 4B , it is possible to set the deviation from the state illustrated in  FIG. 4A . 
     When the deflection angle θ of the optical deflector  101  is θ=0, the position of the transmitted light is set to x=0. By means of the optical deflector, the position of the light at which the deflection angle θ is provided can be expressed as “x=e(θ)=L tan θ . . . (6)”. 
     In Equation (6), in a case where θ is sufficiently small, it can be approximated as “x=e(θ)≅Lθ . . . (7)”. For example, when θ=7.5 degrees (=0.1309 rad=130.9 mrad), tan θ=0.1317, so tan θ/θ=1.006, and thus Equation (7) may be satisfied if around 0.6% of error is permitted. When θ=10 degrees (=0.1745 rad=174.5 mrad), tan θ=0.1763, so tan θ/θ=1.01, and thus Equation (7) may be satisfied if around 1% of error is permitted. When θ=15 degrees (=0.2618 rad=261.8 mrad), tan θ=0.2679, so tan θ/θ=1.02, and thus Equation (7) may be satisfied if around 2% of error is permitted. As illustrated in  FIG. 4B , setting the origin of the deflection angle θ to be near the center of the deflection angle that changes in time allows the maximum value of the absolute value of the deflection angle to be reduced. As a result, the accuracy of the approximation can be increased. 
     Hereinafter, Equation (7) is considered to be satisfied. The deflection angle θ giving Equation (1) is θ goal (t), so by rearranging with the relationship where Equation (1) and Equation (7) are equal, θ goa l(t) can be expressed as “θ goal (t)=h(t)/L . . . (8)”. Comparing Equation (5) with Equation (8), it can be seen that in this case, function e −1  is 1/L. 
     Consider the case where, for example, the DC bias voltage V DC +the sine wave voltage (period T) is applied as the voltage V which the voltage control unit  102  applies (outputs) to the optical deflector  101  as illustrated in  FIG. 5A , and the resulting deflection angle θ has changed as in  FIG. 5B . From  FIGS. 5A and 5B , the relationship between voltage and deflection angle can be graphed by plotting (instantaneous) voltage V on the horizontal axis and (instantaneous) deflection angle θ on the vertical axis. In this case, the relationship between the voltage and the deflection angle is as illustrated in  FIG. 5C . In other words, the deflection angle θ can be expressed as a function of the instantaneous voltage V as θ=f(V). 
     Depending on the applied conditions of the voltage, the distribution of charge injected into the electro-optical crystals may vary. In other words, the distribution of the charge at time t depends on the history of how the voltage has been applied until the time t. Thus, even if the (instantaneous) voltages are equal at V A  at certain times t A  and t B , the distribution of the charge at the times t A  and t B  is not generally said to be equal. Because the deflection angle depends on the charge distribution, θ=f(V) can be generally different at the voltage rising time θ=f rise (V) and the falling time θ=f fall (V), as illustrated in  FIG. 5C . 
     The characteristics described above are described with reference to  FIG. 6 .  FIG. 6( a )  is the time dependency of the voltage, in which a time zone when the voltage is rising, that is, a time zone in which the voltage is rising in time, is named “rise”, a time zone when the voltage is falling, that is, a time zone in which the voltage is falling in time, is named “fall”, and the instantaneous voltages at the times t A  and t B  are V A .  FIG. 6( b )  is the time dependency of the deflection angle, and illustrates a case in which and the deflection angles are different at times t A  and t B .  FIG. 6( c )  is the voltage dependency of the deflection angle. It can be seen from  FIG. 6  that the voltage dependency of the deflection angle varies at the time zone “rise” and the time zone “fall” as a result of different deflection angles at the times t A  and t B  where instantaneous voltages are the same. 
     Here, consider the voltage waveform of the period T as illustrated in  FIG. 7 . In other words, consider a voltage waveform in which the voltage rises from time t=0 to t=T 1 , the voltage is constant from time t=T 1  to t=T 2  (V=V B ), and the voltage falls from time t=T 2  to t=T. In such a case, T 3  satisfying T 1 ≤T 3 ≤T 2  is selected, and 0≤t&lt;T 3  is considered as a time zone “rise” in which the voltage primarily rises in time. T 3 ≤t&lt;T is considered as a time zone “fall” in which the voltage primarily falls in time. In other words, in the time zone “rise” in which the voltage rises in time, the voltage does not need to be constantly rising in time. The time zone “rise” indicates a time zone in which the voltage is primarily rising in time when a voltage signal of period T is applied. Similarly, in the time zone “fall” in which the voltage falls in time, the voltage does not need to be constantly falling in time. The time zone “fall” indicates a time zone in which the voltage is primarily falling in time when a voltage signal of the period T is applied. 
     As described above, if the voltage signal V=g(t) is provided, by measuring the deflection angle, the deflection angle θ can be expressed as a function of the instantaneous voltage V as θ=f rise (V), θ=f fall (V). Thus, the deflection angle θ can be generally expressed as “θ=f rise  {g(t)}, θ=f fall  {g(t)} . . . (9)”, as a function of the time t. Here, the domain of θ=f rise  {g(t)} is a time zone in which 0≤t&lt;T and the voltage rises in time. The domain of θ=f fall  {g(t)} is a time zone in which 0≤t&lt;T and the voltage falls in time. 
     Suppose that when the relationship θ=f rise, 0 (V), θ=f fall, 0 (V) derived at step D n=0  is satisfied, voltages V exist such that the deflection angle is as Equation (8). Assuming that the voltages are denoted by g rise, 1 (t), g fall, 1 (t), then “θ goal (t)=f rise, 0  {g rise, 1 (t)}, θ goal (t)=f fall, 0  {g fall, 1 (t)} . . . (10)” is satisfied. In other words, “g rise, 1 (t)=f rise, 0   −1  {θ goal (t)}, g fall, 1 (t)=f fall, 0   −1  {θ goal (t)} . . . (11)” is satisfied. 
     In other words, in a case of assuming that θ=f rise, 0 (V), θ fall, 0 (V) is not dependent on the history of the voltage applied, by substituting θ goal (t) of Equation (8) as θ in V=f rise, 0   −1 (θ), V=f fall, 0   −1 (θ) obtained by inversely solving θ= 0 (V), θ fall, 0 (V) experimentally determined, the voltages (t) g rise, 1 (t), g fall, 1 (t) in which x shows a trajectory with desired time dependency can be determined (step E n =0). 
       FIGS. 8A and 8B  illustrate an example of θ=f 0 (V) and V=f 0   −1 (θ). By rotating the entire graph of θ=f 0 (V) illustrated in  FIG. 8A  by 90 degrees in a counterclockwise direction, and by inverting the θ axis, V=f 0   −1 (θ) illustrated in  FIG. 8B  is obtained. 
     At time 0≤t&lt;T, a function for periodically repeating a function in which the voltage at the time of voltage rise is g rise, n (t), and the voltage at the time of voltage fall is g fall, n (t) is defined as g n (t). 
     As previously mentioned, generally θ=f(V) depends on the history of applied voltage. However, if the history of applied voltage is similar, the shape of θ=f(V) should also be similar. Thus, by repeating the actual applying of the voltage g n+1 (t) constituted by the voltages g rise, n+1 (t), g fall, n+1 (t) obtained in step E n , and the deriving of θ=f rise, n+1 (V), θ=f fall, n+1(V)  at the next incremented step D n+1 , g rise, n (t), g fall, n (t) and f rise, n (t), f fall, n (t) are converged, and the deflection angle θ satisfies Equation (8), and as a result, a voltage condition in which x has the desired time dependency can be obtained. Note that, because it is not practical to repeat indefinitely (n→∞), the procedure is repeated until the θ actually indicates the desired time dependency with the desired accuracy. 
     The voltage g 1 (t) obtained at step E n=0  is actually applied at the next incremented step A n=1 . Thereafter, at step B n=1 , an evaluation is made as to whether the time dependency of the centroid position of the deflection light is desired. In a case where the time dependency of the centroid position of the deflection light is desired, the voltage g 1 (t) obtained in step E n=0  is the voltage signal V=g goal (t) (C n=1 ) to be determined. In a case where the time dependency of the centroid position of the deflection light is not desired, “θ=f rise, 1 (V), θ=f fall, 1 (V)” indicating the instantaneous voltage dependency of the deflection angle θ is derived (step D n=1 ). Here, suppose that there are voltages V such that the deflection angle is Equation (8) under the conditions of θ=f rise, 1 (V), θ=f fall, 1 (V). Assuming that these are g rise, 2 (t), g fall, 2 (t), “g rise, 2 (t)=f rise, 1   −1  {θ goal (t)}, g fall, 2 (t)=f fall, 1   −1  {θ goal (t)} . . . (12)” is satisfied (step E n=1 ). 
     By repeating the steps described above, the voltage condition g n  can be expressed as “g rise, n (t)=f rise, n−1   −1  {θ goal (t)}, g fall, n (t)=f fall, n−1   −1  {θ goal (t)} . . . (13)” (step E n−1 ). 
     Hereinafter, more details will be described using examples. 
     Example 1 
     First, Example 1 will be described. When the voltage illustrated in  FIG. 5A  is applied to the optical deflector  101 , the optical deflector  101  has the characteristics of  FIGS. 5B and 5C . The applied voltage is determined in which the deflection light output from the optical deflector  101  having such characteristics shows the simple harmonic motion described in Equation (2). The deflection angle at that time can be described as “θ goal (t)=h(t)/L={x 0  sin(ωt+γ)+x 2 }/L=(x 0 /L)sin(ωt+γ)+x 2 /L . . . (14)”, substituting Equation (2) into Equation (8). 
     The maximum value of the absolute value of the deflection angle represented by Equation (14) is max|±x 0 /L+x 2 /L|. For example, when x 0 &gt;0, L&gt;0, x 2 &lt;0, max|±x 0 /L+x 2 /L|=|−x 0 /L+x 2 /L|. In a case where this value is sufficiently small, Equation (7) can be applied. For example, with γ=−π/2, g rise, n (t), g fall, n (t) are determined by the method for controlling the optical deflector of the embodiment. The voltages g rise, n (t), g fall, n (t) obtained in step E n  can be expressed as “g rise, n (t)=f rise, n−1   −1  {(x 0 /L)sin(ωt−π/2)+x 2 /L}, g fall, n (t)=f fall, n−1   −1  {(x 0 /L)sin(ωt−π/2)+x 2 /L} . . . (15)”, with γ=−π/2, and then substituting Equation (14) into Equation (13). 
     At time 0≤t&lt;T, a function for periodically repeating a function in which the voltage at the time of voltage rise is g rise, n (t), and the voltage at the time of voltage fall is g fall, n (t) is defined as g n (t), and g goal (t)=g n (t) can be determined by repeating the procedure until θ indicates the desired time dependency with the desired accuracy. The applied voltage obtained in this way is illustrated in  FIG. 9A , and when this voltage is applied, the deflection angle has the time dependency of sine wave as illustrated in  FIG. 9B . 
     Example 2 
     Next, Example 2 is described. In Example 2 as well, when the voltage illustrated in  FIG. 5A  is applied to the optical deflector  101 , the optical deflector  101  has the characteristics of  FIGS. 5B and 5C . The applied voltage is determined in which the deflection light output from the optical deflector  101  having such characteristics shows a uniform velocity reciprocating motion of the following equation at period T. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           h 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                         
                           
                             vt 
                             - 
                             
                               
                                 
                                   ( 
                                   vT 
                                   ) 
                                 
                                 / 
                                 4 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               o 
                             
                           
                           ≤ 
                           t 
                           &lt; 
                           
                             T 
                             / 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                 vt 
                               
                               / 
                               L 
                             
                             + 
                             
                               
                                 
                                   ( 
                                   
                                     3 
                                     ⁢ 
                                     vT 
                                   
                                   ) 
                                 
                                 / 
                                 4 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 / 
                                 2 
                               
                             
                           
                           ≤ 
                           t 
                           &lt; 
                           T 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The deflection angle in this case can be described by the following equation substituting Equation (16) into Equation (8). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             θ 
                             goal 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                         
                           
                             h 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           / 
                           L 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ( 
                                 vt 
                                 ) 
                               
                               / 
                               L 
                             
                             - 
                             
                               
                                 
                                   ( 
                                   vT 
                                   ) 
                                 
                                 / 
                                 4 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               o 
                             
                           
                           ≤ 
                           t 
                           &lt; 
                           
                             T 
                             / 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 - 
                                 
                                   ( 
                                   vt 
                                   ) 
                                 
                               
                               / 
                               L 
                             
                             + 
                             
                               
                                 
                                   ( 
                                   
                                     3 
                                     ⁢ 
                                     vT 
                                   
                                   ) 
                                 
                                 / 
                                 4 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 / 
                                 2 
                               
                             
                           
                           ≤ 
                           t 
                           &lt; 
                           T 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The voltages g rise, n , g fall, n  obtained in step E n−1  can be expressed as “g rise, n (t)=f rise, n−   −1  {(v/L)t−(vt/4L)}, g fall, n (t)=f fall, n−1   −1 {(v/L)t+(3vT/4L)} . . . (18)” by substituting Equation (17) into Equation (13). 
     At time 0≤t&lt;T, a function for periodically repeating a function in which the voltage at the time of voltage rise is g rise, n (t), and the voltage at the time of voltage fall is g fall, n (t) is defined as g n (t), and g goal (t)=g n (t) can be determined by repeating the procedure until θ indicates the desired time dependency with the desired accuracy. The applied voltage obtained in this way is illustrated in  FIG. 10A , and when this voltage is applied, the deflection angle has the linear time dependency illustrated in  FIG. 10B . 
     Note that in Example 2, the period of the voltage rise and the period of the voltage fall are both equal with the voltage T/2, but one may be longer (shorter) than the other. 
     If the deflection angle θ is sufficiently small, Equation (7) is satisfied, but a case in which Equation (6) is used without approximation will now be described. The deflection angle θ that satisfies Equation (1) is θ goal (t), so by rearranging with the relationship where Equation (1) and Equation (6) are equal, “θ goal (t)=tan −1  {h(t)/L} . . . (19)” is satisfied as an equation corresponding to Equation (8). Other configurations are similar to the above. 
     Note that, in the case of a typical optical system, Equation (5) can be used as an equation corresponding to Equation (8). Other configurations are similar to the above. 
     Note that the optical deflector can also be configured with an electro-optical crystal (KTN crystal) by which an optical deflection phenomenon occurs and a light irradiation unit that irradiates light to the optical crystal. Furthermore, the optical deflector is not limited to an electro-optical crystal, and the optical deflector can be configured such that the voltage dependency θ=f rise (V) of the deflection angle when the voltage rises and the voltage dependency θ=f fall (V) of the deflection angle when the voltage falls are different. Although the above describes a periodic function of time as the deflection angle and voltage, when the desired deflection is started and ended after a finite period of time, without repeating periodically, the control method described with respect to the periodic function can be applied by setting the start time of the desired deflection to t=0, and the end time to t=T, with T as the duration. 
     As described above, according to embodiments of the present invention, the step of applying g n+1 (t) consisting of g rise, n+1 (t)=f rise, n   −1  (θ goal (t)), and g fall, n+1 (t)=f fall, n   −1  (θ goal (t)) to the optical deflector to achieve θ=θ goal (t) is repeated until θ=θ goal (t) indicating the time dependency of the deflection angle is satisfied with the goal accuracy, so that the position (trajectory) of the light deflected by the optical deflector has the desired time dependency. 
     Note that, the present invention is not limited to the embodiments described above, and it is obvious that many modifications and combinations can be implemented by a person having ordinary knowledge in the field within the technical spirit of the present invention. 
     REFERENCE SIGNS LIST 
     
         
         
           
               101  Optical deflector 
               102  Voltage control unit 
               103  Storage unit.