Configuration Study of Pre-Mode Cleaner and Reference Cavity in the 40m PSL System

Transcription

1 ASER INTERFEROMETER GRAVITATIONA WAVE OBSERVATORY -IGO- CAIFORNIA INSTITUTE OF TECHNOOGY MASSACHUSETTS INSTITUTE OF TECHNOOGY Technical Note IGO-T R 07/29/03 Configuration Study of Pre-Mode Cleaner and Reference Cavity in the 40m PS System Seiji Kawamura This is an internal working note of the IGO Project. California Institute of Technology IGO Project MS Pasadena CA Phone (626) Fax (626) info@ligo.caltech.edu WWW: Massachusetts Institute of Technology IGO Project MS 20B-145 Cambridge, MA Phone (617) Fax (617) E- mail: info@ligo.mit.edu File /home/blacke/documents/t9000xx.ps printed November xx, 1999

2 ABSTRACT The IGO Pre-Stabilized aser (PS) system consists of a aser, a reference cavity and its servo system, and a pre-mode cleaner and its servo system. In the IGO PS system the light is picked off for the reference cavity before the pre-mode cleaner. An alternative way is to pick off the light for the reference cavity after the pre-mode cleaner. These two configurations are compared in terms of the loop gain, the frequency stabilization, the effect of the resonant frequency of the cavities, and the effect of the noise in the cavity servo systems. It was found that it is advantageous to pick off the light after the pre-mode cleaner to suppress the effect of the variation in the resonant frequency of the pre-mode cleaner on the frequency of the light coming out of the PS system. No obvious disadvantages were found in this configuration.

3 1. Introduction The IGO Pre-Stabilized aser (PS) system consists of a aser, a reference cavity () and its servo system, and a pre-mode cleaner () and its servo system. In the IGO PS system the light is picked off for the reference cavity before the pre-mode cleaner. The error signal obtained in the readout system is filter-amplified and fed back to the aser. The error signal obtained in the readout system is filter-amplified and fed back to the. Recently it was found at the 40m prototype at Caltech that the variation in the resonant frequency of the caused by the vibration of the was limiting the sensitivity in the frequency of the light coming out of the PS. It was suggested by O. Miyakawa that an alternative configuration, that is, to pick off the light after the, could reduce the noise. Actually C. Mow-owry has been preparing the change of the configuration at the 40m. In this brief report these two configurations, picking off the light for the before and after the, are compared in terms of the loop gain of the servo systems, the frequency stabilization, the effect of the variation in the resonant frequency of the cavities on the frequency of the light coming out of the PS, and the effect of the noise in the cavity servo systems on the frequency of the light. 2. Block Diagram Figure 1 shows the simplified schematic diagrams of the PS system with the two configurations: the light picked off for the reference cavity (a) before the pre-mode cleaner and (b) after the pre-mode cleaner. PD Pre-Mode Cleaner PD Pre-Mode Cleaner aser aser PD Reference Cavity PD Reference Cavity (a) (b) Fig. 1 Configurations of the pre-stabilized aser with the light picked off for the reference cavity (a) before the pre-mode cleaner and (b) after the pre-mode cleaner. The block diagrams of the systems with the two configurations are shown in Fig. 2. Here each symbol represents the following physical quantity: : Frequency of the aser : Resonant frequency of the determined by the cavity length of the : Resonant frequency of the determined by the cavity length of the out : Frequency of the light coming out of the PS : ow pass filter due to the cavity pole of the

4 ( ω s + ω, s: aplace variable, ω : cavity pole frequency of the ) H : High pass filter due to the cavity pole of the s ( H, s: aplace variable, ω : cavity pole frequency of the ) s + ω : ow pass filter due to the cavity pole of the ω (, s: aplace variable, ω : cavity pole frequency of the ) s + ω A : Gain of the sensor/filter/amplifier/actuator of the servo in the configuration (a) A : Gain of the sensor/filter/amplifier/actuator of the servo in the configuration (a) B : Gain of the sensor/filter/amplifier/actuator of the servo in the configuration (b) B : Gain of the sensor/filter/amplifier/actuator of the servo in the configuration (b) A triangle with + and represents a discriminator. Note that and H has the following relationship: + H 1. This relationship will be used very often in the following calculations. out(a) H H A B A B (a) (b) Fig. 2 Block diagrams of the pre-stabilized aser with the light picked off for the reference cavity (a) before the pre-mode cleaner and (b) after the pre-mode cleaner. 3. oop Gain We will first compare the loop gains for both servos in the configuration (a) and (b). The loop gains 1 of the servo and the servo in the configuration (a), G (a), G (a), are, respectively G A (a) 1 In this report a loop gain of a servo system is defined to include -1 as a negative feedback.

5 G A. (a) The loop gains of the servo and the servo in the configuration (b), G (b), G (b), are obtained by calculating the servo suppression ratio 2 for and, respectively. B(1 + B) G(b) 1+ B G (b) B ( 1+ B 1 + B Note that the following plausible equations are not exactly correct: G B (b) (b) G B, ) because of the combined servo loops existing in the configuration (b). The IGO PS system for the 40m prototype has the configuration (a) with the following parameters: Table 1 Parameters of the IGO PS Reference Cavity Servo Pre-Mode Cleaner Servo Cavity Pole Frequency, ω Unity Gain Frequency, f UG- Cavity Pole Frequency, ω Unity Gain Frequency, f UG- 40 khz 500 khz 200 khz 1.6 khz In the following discussion, we assume that the system with the configuration (b) should also give the same performance as this system, namely: f << ω << ω f UG < UG Under this 40m PS condition, we can easily prove that G B (b) G(b) B ( f < f UG ) are correct 3. Therefore in order to make the loop gain of both servos on the configuration (b) the same as those on the configuration (a) within the bandwidth of each servo system, we must satisfy the following conditions for B and B : A B B A We will call these equations the conditions for equivalence. 4. Frequency Stabilization Here we will compare the frequency stabilization for the configuration (a) and (b). The transfer function from to out(a) in the configuration (a) is 2 In this report a servo suppression ratio is defined to be 1/(1-Gloop ), where G loop is the loop gain of the servo system. Therefore the loop gain, G loop, can be calculated from the servo suppression ratio. 3 Note that 1 (f < f UG- ).

6 out(a) (1 + A (1 + A) )( 1 + A Under the 40m PS condition, this can be approximated to out(a) ( 1+ A) This indicates that the frequency of the light is stabilized by the loop gain of the servo, G (a) A, and low-pass-filtered by the,. The transfer function from to in the configuration (b) is (1 + B) 1 B + (1 + B) B Under the 40m PS condition and also with the conditions for equivalence, this can be approximated to ( 1+ A) This indicates that the frequency of the light is stabilized by the loop gain of the servo, G (a) A, and low-pass-filtered by the,. Therefore the frequency of the light is stabilized exactly in the same manner for the configuration (a) and (b). 5. Effect of the Resonant Frequency of the Cavities We will compare the effect of the resonant frequency of the cavities for the configuration (a) and (b). ) (1) The transfer function from to out(a) in the configuration (a) is out(a) H 1+ A This indicates that the effect of the resonant frequency of the on out is suppressed by the loop gain, G (a) - A, and high-pass-filtered by the, H. The transfer function from to in the configuration (b) is H 1 B + (1 + B ) B Under the 40m PS condition and also with the conditions for equivalence, this can be approximated to H A (1 + A ) 1 This indicates that the effect of the resonant frequency of the on out is suppressed by the equivalent loop gain A (1+A )and high-pass-filtered by the, H. Therefore the configuration (b) gives more suppression to the effect of the resonant frequency of the on out by the equivalent loop gain of G A than the configuration (a). (2)

7 The transfer function from to out(a) in the configuration (a) is out(a) (1 + A) A (1 + A )( 1+ A ) Under the 40m PS condition, this can be approximated to out(a) A 1 A This indicates that the effect of the resonant frequency of the on out is low-pass-filtered by the, below the unity gain frequency of the servo, and in addition decreases with the loop gain of the, G A, above the unity gain frequency of the servo. The transfer function from to in the configuration (b) is (1 + B ) B 1 B + (1 + B ) B Under the 40m PS condition and also with the conditions for equivalence, this can be approximated to A 1 A This indicates that the effect of the resonant frequency of the on out is direct below the unity gain frequency of the servo, and decreases with the loop gain of the, G A, above the unity gain frequency of the servo. Therefore the configuration (a) gives more suppression to the effect of the resonant frequency of the on out by the low pass filter of the,, than the configuration (b). 6. Effect of Noise in the Servos In this section we will compare the effect of the noise in the servos for the configuration (a) and (b). Shot noise and electronic noise existing in the cavity locking servo systems can be treated as noise, N and N, injected right after the discriminator and the cavity low pass filter as shown in Fig. 3. out(a) H H A N N B N N A B (a) (b)

8 Fig. 3 Noise existing in the cavity servo systems. (1) Ν The transfer function from Ν to out(a) in the configuration (a) is out(a) AH N 1+ A Under the 40m PS condition, this can be approximated to out(a) AH N 1+ A This indicates that the effect of the noise of the servo on out(a) is high-pass-filtered by the, H and decreases with A above the unity gain frequency of the servo. The transfer function from Ν to in the configuration (b) is BH N 1 B + (1 + B ) B Under the 40m PS condition and also with the conditions for equivalence, this can be approximated to AH N 1+ A )(1 A ) ( This indicates that the effect of the noise of the servo on out(a) is high-pass-filtered by the, H, and suppressed by the loop gain G A, and in addition decreases with A above the unity frequency of the servo. Therefore the configuration (b) gives more suppression to the effect of noise of the on out by the loop gain of G A than the configuration (a). (2) Ν The transfer function from Ν to out(a) in the configuration (a) is out(a) (1 + A) A N (1 + A )( 1+ A ) Under the 40m PS condition, this can be approximated to out(a) A N 1 A This indicates that the effect of the Noise on the servo on out is low-pass-filtered by the,, and increases with 1/ below the unity gain frequency of the servo, and decreases with A above the unity gain frequency of the servo. The transfer function from Ν to in the configuration (b) is (1 + B ) B N 1 B + (1 + B) B Under the 40m PS condition and also with the conditions for equivalence, this can be approximated to A N 1 A

9 This indicates that the effect of the Noise on the servo on out increases with 1/ below the unity gain frequency of the servo, and decreases with A above the unity gain frequency of the servo. Therefore the configuration (a) gives more suppression by the low pass filter of the, 7. Summary The results obtained by the above calculations are summarized in Table 2. Table 2 Summary of the comparison of the various performances between configuration (a) and (b). oop gain of oop gain of Frequency stabilization* Effect of resonant frequency of * Effect of resonant frequency of * Effect of noise of servo* Effect of noise of servo* Configuration (a) Configuration (b) Comments G A G B B (a) (b) ( f < f ) UG G (a) A G(b) B out(a) out(a) out(a) ( 1+ A) H 1+ A A 1 A out(a) AH N 1+ A out(a) N A 1 A 40m PS conditions are assumed. * Conditions of equivalence are assumed. N N ( 1+ A) H A (1 + A 1 A 1 A AH ( 1+ A)(1 A) A 1 A ) A (conditions for equivalence) A B (conditions for equivalence) Same (b) better than (a) by G A (a) better than (b) by (b) better than (a) by G A (a) better than (b) by The outstanding difference between the configuration (a) and (b) is the effect of the resonant frequency of as expected. The configuration (b) gives significantly more suppression to the effect than the configuration (a). In addition the effect of the noise of the servo is also better in the configuration (b) than (a) by G A, although the effect of this noise might be negligible even with the configuration (a). All the advantages the configuration (a) has are only effective above the cavity pole frequency of the ; thus they are insignificant. 7. Conclusions There are no significant disadvantages for the configuration (b). We should go ahead and change the configuration from (a) to (b).

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