JP4580592B2 - Optical disc master exposure system - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to an optical disk master exposure equipment, related to the optical disk master exposure apparatus which performs the optical disc master manufacturing with a pitch spiral information track such linear velocity or the like using a light beam or electron beam in detail, the control the feed system The present invention relates to a method and apparatus for correcting the influence of errors and turntable deflection and enabling high-density information track formation.
[0002]
[Prior art]
In recent years, development of an ultra-high-density mastering device for realizing an optical disc system having a recording density (100 Gbit / in ² ) about 30 times that of a DVD as a large-capacity optical disc medium has been studied. On the other hand, according to the paper “O61“ High-resolution NRRO detection in ultra-high density mastering equipment ”(Pioneer Hiroaki Kitahara, Yoshiaki Kojima, Yasumitsu Wada)” With a high recording density, both the shortest pit length and the pit width are expected to be 70 nm or less. At this time, the recording position accuracy required for the mastering apparatus is estimated to be on the order of sub-nanometers. In this document, it is difficult to achieve the target accuracy of σ0.5nm by mechanical positioning. Therefore, positioning errors between the rotary stage and the linear motion stage are detected with high resolution and corrected by electron beam deflection. The relative position accuracy is improved by using the rotational asynchronous component of the angular error (rotational unevenness) which is a problem as the positioning error of the rotary stage and the non-repetitive axial runout (NRRO). The study results are described. Through the above studies, in the ultra-high density mastering apparatus, it is considered that the correction of the non-synchronous component and the non-synchronous axial runout component of the rotation unevenness is an important issue. Note that the synchronization component does not affect the track pitch accuracy.
[0003]
In response to this problem, Japanese Patent Laid-Open No. 2000-20964 (hereinafter referred to as a conventional example) relates to correction of the beam irradiation position in the feed direction, and based on the shake information in the feed direction according to the rotation angle of the turntable, Alternatively, the beam irradiation position is corrected based on the shake information in the feed direction according to the rotation angle of the current turntable with reference to the shake information in the feed direction according to the rotation angle of the turntable for a predetermined period. .
[0004]
[Problems to be solved by the invention]
However, in the above conventional example, in order to form a constant linear velocity / equal track pitch spiral information track on the optical disc master, in addition to the conventional method, the response to the feed position error of the feed system drive control and the CLV drive command pulse There is a problem that needs to cope with accuracy. Further, when the recording format of the optical disc master to be recorded is CLV (Constant Linear Velocity), the laser is used to change the rotation speed of the spindle motor in accordance with the radial position of the irradiation spot on the optical disc master. Since the detection signal generated from the length measuring device is also supplied to the reference signal generator 13, the generation of the CLV drive command pulse is a feed control error or an accuracy error of the drive command pulse of the feed system in the control state. It is generated based on the radial position information including. Therefore, even if the beam irradiation position is corrected according to the above-described conventional method, there is no guarantee that an equal track pitch spiral information track will be formed.
[0005]
The present invention has been made to solve these problems, and compensates for the effects of feed control errors and turntable synchronous and asynchronous vibrations on the basis of strictly accurate CLV drive command pulse trains. It is another object of the present invention to provide a method and an apparatus for forming a high-density information track having a high-precision uniform linear velocity / equal track pitch spiral shape.
[0006]
[Means for Solving the Problems]
In order to solve the above problems, a turntable on which an optical disc master is mounted, a rotation driving means for rotating the turntable, and a recording light beam or an electron beam are converged based on an information recording signal mounted as an information track. Converging irradiating means for irradiating, and transverse feed driving means for relatively moving a turntable mounted with the converging irradiating means or the optical disc master in a plane parallel to the optical disc master, the rotational driving means and the lateral feed An optical disc master exposure apparatus that exposes by CLV driving to form an information track so that the linear density is constant on the optical disc master by controlling the driving means, and drives a drive command pulse train for the rotational drive means and the lateral feed drive means to CLV drive Generate and output the period length of each pulse from the start of operation while changing it at regular intervals. A lock generating circuit; a frequency dividing circuit for dividing a reference clock; a frequency dividing data latch circuit for latching frequency-divided data; and a delay circuit for inputting a frequency-divided pulse from the frequency-dividing circuit and outputting a plurality of delay pulses. A pulse selection circuit that selectively outputs a target delay pulse from a plurality of delay pulses output from the delay circuit, and a delay pulse selection data latch circuit that latches delay pulse selection data that is input to the pulse selection circuit to perform pulse selection A period data calculation circuit that outputs frequency-divided data and delay pulse selection data that are set according to the CLV drive command pulse to be output, and the period data is output for each CLV drive command pulse. In the optical disk master exposure apparatus set in the delay pulse selection data latch circuit, the starting radius position of the CLV drive is R, line The spiral radius position whose length is to be measured is r, the spiral track pitch is P, the number of tracks is n (n is a positive integer), the linear velocity is VL, one turn of the turntable, or one pitch length for lateral feed When the number of CLV drive command pulses is Nc, a pulse train having a cycle satisfying (CLV drive command pulse cycle) = 2πR ₀ / (Nc · VL) + (2n−1) πP / (Nc ² · VL) is generated. An optical disc master exposure apparatus characterized by the above. Therefore, when performing CLV driving in an optical disc master exposure apparatus using a turntable moving type or optical system moving type light beam or electron beam, a turntable that can serve as a reference for correcting the exposure beam position and an accurate CLV for feeding. A drive command pulse train generation method can be provided, and further, a beam irradiation position error can be obtained with reference to the CLV drive command pulse train.
[0018]
DETAILED DESCRIPTION OF THE INVENTION
Ro
In the optical disk master exposure apparatus of the present invention, the starting radius position of CLV drive is R, the spiral radius position where the line length is to be measured is r, the spiral track pitch is P, the number of tracks is n (n is a positive integer), the line When the speed is VL and the number of CLV drive command pulses for one turn of the turntable or one pitch length is Nc, (CLV drive command pulse cycle) = 2πR ₀ / (Nc · VL) + (2n− 1) A pulse train having a period satisfying πP / (Nc ² · VL) is generated.
[0019]
【Example】
First, the principle of the present invention will be described.
In a spiral track of equal pitch formed by CLV drive, the total track line length (L) is expressed by the following equation (1).
L = π (r ² −R ₀ ² ) / P (1)
r = R ₀ + nP (2)
Here, R ₀ represents the starting radius position of CLV driving, r represents the spiral radius position where the line length is to be measured, P represents the spiral track pitch, and n represents the number of tracks. N is 1, 2, 3,.
From Formula (1) and Formula (2), L = 2πR ₀ n + n ² πP (3)
Further, the line length (Ln) of each track is expressed by the following equation (4) from the equation (3).
Ln = 2πR ₀ + (2n−1) πP (4)
[0020]
From equation (4), it can be seen that the line length difference between adjacent tracks is constant as follows.
(Line length difference between adjacent tracks) = 2πP = constant (5)
[0021]
Expression (2) expresses a spiral locus to be obtained in units of tracks, that is, a radial position every time the rotation angle of the locus goes around 2π from the center of the spiral along the track. As a result, the track line length increases by 2πP for each track. This means that in the CLV drive, if the linear velocity is VL, the time required to go around one track by 2πP / VL for each track increases. 9 and 10 show how the frequency of the CLV drive command pulse train changes at this time. FIG. 9 shows a case where the frequency is updated for each track, and FIG. 10 shows a case where the frequency is updated by dividing the track into a plurality of parts. In either case, a pulse train having no error can be generated with respect to an ideal CLV drive command pulse train when viewed within a long time. However, looking at the time within the track or the time when the track was divided into a plurality of times, there was an error in the speed command of the drive command pulse generated every moment.
[0022]
Therefore, in the CLV drive command pulse train generation method of the present invention, the cycle is updated for each CLV drive command pulse, and an error from the ideal CLV drive command pulse train is eliminated in the drive command pulse generated every moment. FIG. 1 shows a change in the period of a pulse train generated by the CLV drive command pulse train generating method according to the present invention. Now, if the number of CLV drive command pulses for one turn of the turntable or one pitch length is Nc, the line length (Lp) corresponding to n CLV drive command pulses is Lp = 2πR ₀ n / ^{Nc + (n / Nc) 2} πP (6)
Further, from equation (6), the line length (Lpn) corresponding to each CLV drive command pulse is expressed as the following equation (7).
Lpn = 2πR ₀ / Nc + (2n−1) πP / Nc ² (7)
[0023]
From equation (7), it can be seen that the line length difference corresponding to the adjacent CLV drive command pulse is constant as follows.
(Line length difference equivalent to adjacent CLV drive command pulse)
= 2πP / Nc ² = constant (8)
[0024]
Here, the linear velocity VL is constant (difference in cycle length between adjacent CLV drive command pulses).
= 2πP / (Nc ² · VL) = constant (9)
It becomes.
Therefore,
(CLV drive command pulse cycle)
= 2πR ₀ / (Nc · VL) + (2n−1) πP / (Nc ² · VL) (10)
Generation of an ideal CLV drive command pulse can be realized by generating a pulse train having a cycle that satisfies the above.
[0025]
The generation time for each pulse obtained by such a method is the CLV for forming a constant linear velocity / equal pitch spiral information track on the optical disc master while changing the driving speed, that is, the cycle of the CLV driving command pulse every moment. It can be a standard for guaranteeing the driving state.
[0026]
FIG. 2 is a block diagram showing a configuration of a CLV drive command pulse train generation device according to another embodiment of the present invention. A feature of the present invention is that, in accordance with the above-described CLV drive command pulse train generation method, frequency-divided data and pulse selection data are updated for each pulse to obtain a strictly accurate CLV drive command pulse train output. In FIG. 2, the CLV drive command pulse train generation apparatus of the present embodiment has a delay circuit 21 that generates a plurality of equal delay time interval delay pulse groups within a basic clock cycle (Ta), and this minimum division delay time is a unit. From the cycle given by equation (10), the frequency division number and delay number of the basic clock of the next CLV drive command pulse are calculated, and the frequency division data and pulse selection are taken into account by considering the pulse selection data of the current output pulse. Data is obtained and set in the latch circuits 22 and 23, respectively. The calculation is performed with sufficient effective digits so that the influence of the calculation error does not appear in the output pulse accuracy. In addition, when the accurate period setting cannot be performed even with the minimum resolution of the period setting determined by the delay time, the nearest pulse is set as the output pulse. However, the calculation is performed accurately to the last, and pulse output is continuously performed in the vicinity of the strictly correct CLV drive command pulse generation time.
[0027]
By using a strictly accurate feed drive command pulse train obtained by the above invention as a reference, a beam irradiation position error accompanying a feed drive control system control error and a turntable runout in the feed direction as described below The beam irradiation position error is detected at the same time.
[0028]
FIG. 3 is a block diagram showing a configuration of a beam irradiation position correction circuit according to the first embodiment of another invention. In the figure, a deviation counter 31 receives a CLV drive command pulse (Pg) serving as a reference and a feed position detection pulse (return pulse) (Ls) output from the feed position detector, and feed control error of the feed system. Is calculated and output. The data conversion circuit 32 outputs position error data associated with the shake according to the magnitude of the turntable shake detection signal, that is, according to the output of the A / D conversion circuit 33 of the turntable shake detection signal (Vd1). The adder / subtracter circuit 34 calculates based on the feed position error data of the feed system from the deviation counter 31 and the position error data accompanying the turntable swing from the data conversion circuit 32, and the current beam based on Pg. An irradiation position error is obtained, and the D / A conversion circuit 35 outputs a beam irradiation position correction signal (Bc) after D / A conversion, and outputs it to a beam irradiation position correction means (not shown). The feed position information for the beam position correction circuit may use the output of the first feed position detector prepared for driving control of the feed motor, or the second feed position detector exclusively for use. May be prepared. In general, the deviation data of the command pulse train and the return pulse output from the first feed position detector generated in the feed drive control system may be directly incorporated into the beam position correction circuit.
[0029]
Next, by using the strictly accurate feed drive command pulse train obtained by the above invention as a reference, a beam irradiation position error associated with a feed drive control system control error and a beam irradiation position error associated with a turntable deflection in the feed direction. Then, as will be described next, a beam irradiation position error is detected by adding roundness error information on the turntable shake detection surface.
[0030]
FIG. 4 is a block diagram showing a configuration of a beam irradiation position correction circuit according to the second embodiment of another invention. In the figure, the same reference numerals as those in FIG. 3 denote the same components. As a different component, the addition / subtraction circuit 34 calculates the roundness error information of the turntable shake detection surface together. In other words, the roundness error output circuit 41 stores roundness error information according to the rotation angle, which is evaluated in advance by the turntable alone, according to the rotation of the turntable of the rotational position output circuit 42. Roundness error information is output based on the output turntable rotation pulse (Tp) and rotation position data (Tz) obtained from the origin pulse input indicating the rotation origin.
[0031]
FIG. 5 is a block diagram showing a configuration of a beam irradiation position correction circuit according to a third embodiment of another invention. In the figure, the same reference numerals as those in FIG. 4 denote the same components. First, the operation of the present embodiment will be described. When shake information obtained from the turntable shake amount detector is V (θ),
V (θ) = R (θ) + dR (θ) + NR (θ) (11)
[0032]
Here, R (θ) is the rotational shaft runout amount synchronized with the turntable rotation, dR (θ) is the runout amount due to the roundness error of the shake detection surface, and NR (θ) is asynchronous with the turntable rotation. Each representing the amount of rotational shaft runout. Further, θ is a rotation angle obtained from a rotation angle detecting means (a pulse is output according to the rotation angle) attached to the turntable rotation shaft.
[0033]
In this embodiment, the asynchronous component NR (θ) is averaged by a plurality of measurements or removed by an appropriate low-pass filter circuit as in the conventional example, and the synchronous axis runout R (θ) is a sine waveform, A × It is assumed that sin (θ) (= R (θ)) (assuming that the locus of the synchronous axis runout draws a circle around the true center of rotation). By doing so, the roundness error dR (θ) is
dR (θ) = V (θ) −A × sin (θ−φ) (12)
[0035]
As required. Here, A and φ are set so that the integral amount over one period of dR (θ) is minimized. DR (θ) obtained in this way is used as roundness error information. The roundness error information dR (θ) is obtained by processing by software in a system controller (not shown) having the shake amount and turntable rotation angle information as inputs. As shown in FIG. 5, the obtained roundness error information is set in the roundness error data storage circuit 51 according to the position information by the system controller. Therefore, it is not necessary to obtain roundness error information on the turntable shake measurement surface in advance, and it is possible to repeatedly measure as necessary. Data processing is performed by software and automated from measurement to setting of roundness error information in the roundness error data storage circuit 51, and a highly versatile beam irradiation position correction system can be configured.
[0036]
FIG. 6 is a schematic plan view showing a state in which the second turntable shake detector is arranged in a direction perpendicular to the feed direction. As can be seen from the figure, since the turntable runout itself is minute, there is no problem even if the feed direction and the runout in the direction perpendicular thereto are detected and handled as runouts in independent directions. The optical disk master exposure apparatus is equipped with a second beam deflecting unit corresponding to the second shake detection direction, and independently performs beam irradiation position correction.
[0037]
FIG. 7 is a schematic diagram showing the configuration of an optical disc master exposure apparatus according to the first embodiment of another invention. In the figure, in an electron beam column 71, an electron beam emitted from an electron beam source is blanked by a signal from an information track signal generation circuit 73 via a blanking electrode 72. Further, the electron beam is generated by the first deflection means 75 (in the case of an electron beam exposure machine, a deflection electrode, a light beam) by a beam irradiation position correction signal (Bc) from a beam irradiation position correction circuit 74 having the configuration shown in FIG. In the case of an exposure machine, an AO deflector is used) and the electron beam irradiation position on the optical disk master 76 is corrected. In the vacuum vessel 77, an optical disc master 76, a turntable 78 on which the optical disc master 76 is mounted, a turntable rotary motor 79 for rotating the turntable 78, and a coaxial with the turntable 78, a rotation pulse and an origin pulse are sent according to the rotation angle. The output turntable rotation pulse generator 80 to output, the first shake detector 81 for detecting the deflection in the feed direction of the turntable 78, the feed screw 82 for laterally moving them, and the second for detecting the feed position. The feed position detector 83 is included. A feed motor 84 that rotates the feed screw 82 is installed outside the vacuum vessel 77.
[0038]
According to the optical disc master exposure apparatus having such a configuration, the CLV drive command pulse train (Ps, Pt) is sent from the CLV drive command pulse train generation device 85 having the configuration shown in FIG. Is output. The feed motor drive control system 86 and the turntable rotation drive control system 87 include first feed position detection information from the first feed position detector 88 and return pulses (Tp, Tz) from the turntable rotation pulse generator 80. Is used to perform feed drive and rotational drive following the CLV drive command pulse. The electron beam irradiation position correction operation includes the feed system CLV drive command pulse (Ps), the position information of the turntable obtained from the second feed position detector 83, and the first shake detector 81 in the feed direction. From the turntable shake information obtained from the above, the deflection electrode of the first deflecting means 75 uses the beam irradiation position correction signal (Bc) output by the beam irradiation position correction circuit 74 having the configuration shown in FIG. Beam irradiation position correction is performed. When the beam irradiation position correction circuit having the configuration shown in FIGS. 4 and 5 is mounted, the turntable rotation pulse (Tp) and the rotation origin pulse (Tz) obtained from the turntable rotation pulse generator 80 are This is input to the beam irradiation position correction circuit 74.
[0039]
FIG. 8 is a schematic diagram showing the configuration of an optical disc master exposure apparatus according to a second embodiment of another invention. The difference from the optical disk master exposure apparatus of the first embodiment shown in FIG. 7 is that a second shake detector 89 and an electron beam are used to detect the shake in the direction perpendicular to the feed direction of the turntable. The second deflecting means 90 that can be deflected in the direction perpendicular to the feeding direction is added. At the same time as the first embodiment, the electron beam irradiation position correction is performed using the second deflection means 90 in accordance with the second shake detection information detected by the second shake detector 89. Is doing. In the drawing, a processing circuit for converting the beam irradiation position correction signal (Bc) according to the second shake detection information detected by the second shake detector 89 is necessary, but is omitted. Since the operation of the drive system and the electron beam irradiation position correction operation in the feed direction are the same as those in the first embodiment, the details are omitted.
[0040]
Next, using the optical disk master exposure apparatus shown in FIG. 7 and FIG. 8, it is possible to manufacture an optical disk master equipped with a high density linear velocity / equal pitch spiral information track, and to produce a uniform linear velocity / equal track pitch spiral height. A high density optical disc stamper and an optical disc medium having a density information track can be produced.
[0041]
In addition, this invention is not limited to the said Example, It cannot be overemphasized that various deformation | transformation and substitution are possible if it is description in a claim.
[0042]
【The invention's effect】
As described above, the turntable on which the optical disc master is mounted, the rotation driving means for rotating the turntable, and the convergence for convergently irradiating the recording light beam or electron beam based on the information recording signal mounted as the information track. An irradiation means; and a transverse feed driving means for moving the turntable mounted with the convergent irradiation means or the optical disc master relatively in a plane parallel to the optical disc master. An optical disc master exposure apparatus that exposes by CLV driving to form an information track so that the linear density is constant on the optical disc master by control, and a drive command pulse train for the rotation driving means and the transverse feed driving means is sent at the start of the CLV driving operation. Generates and outputs the reference clock while changing the period length for each pulse from a certain time. A raw circuit, a frequency dividing circuit that divides the reference clock, a frequency division data latch circuit that latches the frequency-divided data, a delay circuit that receives the frequency division pulse from the frequency division circuit and outputs a plurality of delay pulses, A pulse selection circuit that selectively outputs a target delay pulse from a plurality of delay pulses output from the delay circuit; a delay pulse selection data latch circuit that latches delay pulse selection data that is input to the pulse selection circuit and performs pulse selection; A period data calculation circuit for outputting frequency division data and delay pulse selection data set according to the CLV drive command pulse to be output, the period data being output for each CLV drive command pulse, In the optical disk master exposure apparatus set in the delay pulse selection data latch circuit, the starting radius position of CLV drive is R and the line length is measured. The spiral radius position to be set is r, the spiral track pitch is P, the number of tracks is n (n is a positive integer), the linear velocity is VL, the turntable is rotated once or a CLV drive command for one pitch length of lateral feed When the number of pulses is Nc, a pulse train having a cycle satisfying (CLV drive command pulse cycle) = 2πR ₀ / (Nc · VL) + (2n−1) πP / (Nc ² · VL) is generated. This is an optical disc master exposure apparatus. Therefore, when performing CLV driving in an optical disc master exposure apparatus using a turntable moving type or optical system moving type light beam or electron beam, a turntable that can serve as a reference for correcting the exposure beam position and an accurate CLV for feeding. A drive command pulse train generation method can be provided, and further, a beam irradiation position error can be obtained with reference to the CLV drive command pulse train.
[Brief description of the drawings]
FIG. 1 is a diagram showing a change in the cycle of a CLV drive command pulse of the present invention.
FIG. 2 is a block diagram showing a configuration of a CLV drive command pulse train generation circuit according to another embodiment of the present invention.
FIG. 3 is a block diagram showing a configuration of a beam irradiation position correction circuit according to a first embodiment of another invention.
FIG. 4 is a block diagram showing a configuration of a beam irradiation position correction circuit according to a second embodiment of another invention.
FIG. 5 is a block diagram showing a configuration of a beam irradiation position correction circuit according to a third embodiment of another invention.
FIG. 6 is a schematic view showing the arrangement of turntable shake detectors.
FIG. 7 is a schematic diagram showing the configuration of an optical disc master exposure apparatus according to a first embodiment of another invention.
FIG. 8 is a schematic diagram showing the configuration of an optical disc master exposure apparatus according to a second embodiment of another invention.
FIG. 9 is a diagram showing how the frequency of the CLV drive command pulse train changes when the frequency is updated for each track in the conventional CLV drive command pulse train method.
FIG. 10 is a diagram showing a change in frequency of a CLV drive command pulse train when a frequency is updated by dividing a track into a plurality of portions in a conventional CLV drive command pulse train method.
[Explanation of symbols]
21; delay circuit; 22, 23; latch circuit; 31; deviation counter;
32; data conversion circuit, 33; A / D conversion circuit, 34; addition / subtraction circuit,
35; D / A conversion circuit; 41; roundness error output circuit;
42; rotational position output circuit; 51; roundness error data storage circuit;
71; electron beam column; 72; blanking electrode;
73; Information track signal generation circuit; 74; Beam irradiation position correction circuit;
75; first deflecting means; 76; master optical disc;
77; vacuum vessel, 78; turntable,
79; turntable rotation motor,
80; turntable rotation pulse generator,
81; first shake detector; 82; lead screw;
83; second feed position detector; 84; feed motor;
85; CLV drive command pulse train generator,
86; Feed motor drive control system, 87; Turntable rotation drive control system,
88; first feed position detector; 89; second shake detector;
90: Second deflection means.

Claims (1)

A turntable on which an optical disc master is mounted, a rotation driving means for rotating the turntable, a convergence irradiation means for convergently irradiating a recording light beam or an electron beam based on an information recording signal mounted as an information track, and the convergence The turntable having the irradiation means or the optical disc master is provided with a lateral feed drive means for relatively laterally moving in a plane parallel to the optical disc master, and is controlled on the optical disc master by the control of the rotation drive means and the lateral feed drive means. An optical disk master exposure apparatus that exposes by CLV driving to form an information track so that the linear density is constant, and that sends a drive command pulse train to the rotation driving means and the lateral feed driving means for each pulse from the start of the CLV driving operation. the cycle length to produce output while varying increments a certain time, a reference clock generating circuit, a reference black A frequency dividing circuit that divides the clock, a frequency-dividing data latch circuit that latches the frequency-divided data, a delay circuit that receives a frequency-divided pulse from the frequency-dividing circuit and outputs a plurality of delay pulses, and an output of the delay circuit A pulse selection circuit that selectively outputs a target delay pulse from a plurality of delay pulses, a delay pulse selection data latch circuit that latches delay pulse selection data that is input to the pulse selection circuit and performs pulse selection, and an output CLV drive command A period data calculation circuit that outputs frequency division data and delay pulse selection data set according to the pulse, and the period data is output for each CLV drive command pulse, and the frequency division data latch circuit and delay pulse selection data latch In an optical disc master exposure apparatus set in a circuit,
The starting radius position of the CLV drive is R, the spiral radius position where the line length is to be measured is r, the spiral track pitch is P, the number of tracks is n (n is a positive integer), the linear velocity is VL, the turntable is rotated once or When the number of CLV drive command pulses for lateral feed of one pitch length is Nc,
(CLV drive command pulse cycle) = 2πR ₀ / (Nc · VL) + (2n−1) πP / (Nc ² · VL)
An optical disk master exposure apparatus that generates a pulse train having a period satisfying

Images