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A FM subcarrier method for optical frequency measurement and synthesis
Volume 59, n u m b e r 1
OPTICS COMMUNICATIONS
1 August 1986
A FM S U B C A R R I E R M E T H O D .FOR O P T I C A L F R E Q U E N C Y M E A S U R E M E N T AND S Y N T H E S I S G.M. C U T L E R Department of Physics, Stanford University, Stanford, CA 94305, USA Received 25 March 1986
A simple technique for optical to rf or rf to optical frequency conversion is described. The proposed FM subcarrier m e t h o d uses two frequency locked servo loops to lock a reflecting optical cavity to an optical source, and then an rf oscillator to the optical cavity thus converting optical to rf frequency or vice versa by reversing the servo loops.
This paper proposes and demonstrates a new FM subcarrier method for locking an rf oscillator to a submultiple of the frequency of an optical source or vice versa, which facilitates accurate laser frequency measurement or synthesis over a broad spectral range. Optical frequency synthesis with an accuracy of ~10 -10 would simplify optical spectroscopic measurements which presently rely on comparison with some nearby well documented atomic transition for calibration. Also a simple technique for optical frequency synthesis with an accuracy of ~ 10 -8 is needed to make coherent optical data transmission and reception practical, which coul d ultimately lead to hundreds of high bandwidth channels on a single multimode fiber. The method demonstrated is similar to the dual frequency modulation method [1] (DFM) of DeVoe and Brewer in that it is a FM servo loop method that makes a two step comparison between the optical frequency and rf frequency using an optical cavity as the intermediate step. It differs from the DFM technique in that the optical source is not double frequency modulated. It is, as shown in fig. 1, phase modulated by a single subcarrier which is itself frequency modulated thus requiring only one optical modulator. The modulated optical beam is reflected off the optical cavity to a photo detector. Synchronous detection of the photo detector signal at the optical modulation frequency gives an error signal proportional to the optical source versus optical cavity detuning while synchronous detection at the rf modulation frequen0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Double r ) ~ w2 Balanced Mixer ( ~op FiRe ) Double Balanced Mixer
oop Filter
Phase Modulator Detector
<],
~
Optical Ring Cavity
Preamp Fig. 1. Diagram for locking a voltage controlled rf oscillator (VCO) to a laser by the FM subcarrier m e t h o d . The optical servo loop locks the optical ring cavity resonances to the odd optical sidebands. The rf servo loop locks the average spacing of odd optical sidebands to the cavity mode spacing.
cy produces an error signal proportional to the detuning of the optical modulation frequency from the optical cavity mode spacing. This technique is simpler in both the optics and electronics than earlier methods [ 1,2]. It also retains the very high immunity, inherent with the DFM method, to sideband imbalance caused by unwanted amplitude modulation or even harmonic modulation. In addition, this technique has high immunity to dispersion effects, which must be separately measured and corrected for with the DFM technique. 17
Volume 59, number 1
OPTICS COMMUNICATIONS
The first servo loop, which locks the optical cavity to the laser (hereafter called the optical loop), uses an error signal derived by synchronously detecting the photo detector signal at the optical modulation frequency co 1 , and works on the same principle as the cavity stabilized laser in ref. [3]. When the optical source is modulated, an electric field with the following form is produced
t£=1-~ ~
t(r=l
with the definitions E n = complex amplitude o f nth sideband, coO = laser frequency, and 6o 1 = optical modulation frequency. If and only if the optical modulator does not phase modulate at even multiples o f col and does not amplitude modulate at odd multiples of col then E n satisfies the following symmetry condition
(2)
The reflected beam from the optical cavity will have an electric field
N-I
N
U
Z
CE I-U hi --I laW
(3)
where r(co) is the complex reflection coefficient o f the optical cavity. If the optical carrier is tuned to a symmetric point of r(co), causing tile relation r(co 0 + nco 1) = r*(co 0 nco l ), then eq. (2) holds for the sidebands of the reflected electric field in eq. (3). Hence there will be no amplitude modulation at odd multiples of col in the reflected electric field. When the optical carrier is detuned from this symmetry point the reflected sidebands become imbalanced (violating eq. (2)) thus any phase modulation at odd multiples of co I will be partially converted into amplitude modulation in the reflected electric field. As shown in fig. 2, the carrier and even sidebands are chosen to lie at reflection maxima while the odd sidebands are chosen to lie at the cavity resonances (reflection minima). Therefore the optical loop will lock the cavity reflection maxima to the optical carrier. At this point, two systematic errors become ap-
(11
E = (-1)nE*_n.
{r(co 0 + ricoI )E e x p [ - i ( c o 0 + n c o l ) t ]
+ r*(~O + n W l ) E 2 exp [i(co 0 + n c o l ) t ] } ,
{E nexp[_i(coO +ncol)t]
+ E * exp[i(co 0 + n c o l ) t ] ) ,
~
1 August 1986
N+I
#
N+2
hJ
>t-C~ U
tD Izl Z rr m hi rm
li I
t"° 0 - 3 u° 1 / rr t.) t--
i I Ii
I
P I
c° 0 - 2 c ~
OJO-~ 1
i, i,
! ¢'~0
t.~ O+t-,a I
I I t~ O+2t~ I ¢"aO + 3 t'~° l
O
Fig. 2. Solid lines show the optical carrier position and average positions of the optical sidebands relative to the cavity resonances for the case where the optical modulation frequency is equal to halt"the free spectral range of the cavity corresponding to M = 0 in eq. (10). Dashed and dotted lines show sideband positions during even and odd half cycles of the rf modulation frequency respectively. 18
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OPTICS COMMUNICATIONS
parent. If the optical modulator produces any phase modulation at even multiples of coI or any amplitude modulation at odd multiples of col then there will be a violation of eq. (2) (hereafter called sideband imbalance) for the sidebands of the incident electric field. Under worst case phasing conditions there will be a shift in the lock point of the optical loop that is proportional to sideband imbalance. The second servo loop, which locks the average positions of odd sidebands to the optical cavity resonances (hereafter called the rf loop), derives its error signal by square wave modulating co1 at a low frequency 0o2 and synchronously detecting the photo detector signal at this rf modification frequency co2- The frequency modulation could be done with any signal containing only odd harmonics, but slow square wave modulation eliminates effective even harmonic generation from nonlinearity of the voltage controlled oscillator. The photo detector signal, averaged over many cycles of col, is simply the sum of the powers in each sideband multiplied by its respective cavity reflection coefficient. Therefore under proper locking conditions as shown in fig. 2, each sideband position will oscillate between two symmetrically located positions on opposite sides of symmetric reflection minima (for odd sidebands) or reflection maxima (for even sidebands) thus producing no am signal at frequency 002 . When the optical modulation frequency is shifted, the sidebands will oscillate between positions with different reflection coefficients thus producing a net amplitude modulation at frequency co2This analysis shows that this rf error signal vanishes when the average sideband positions coincide with symmetric minima or maxima of the cavity reflection coefficient leaving the zero crossing of the rf error signal (independent of the amplitudes and phases of the sidebands. Hence sideband imbalance will not shift the lock point of the rf loop given that the optical loop has no shift. There is, however, a linear cross coupling of the error signals that is of first order in sideband imbalance. This cross coupling and the first order shift due to sideband imbalance in the optical loop will result in a shift of the lock point of the rf loop that is of second order in sideband imbalance. Since the optical frequency is a factor of ~ 10 6 greater than the free spectral range of the optical cavity, the first order shift of the optical loop due to sideband imbalance will represent a negligible fractional
1 August 1986
error. And since the shift due to sideband imbalance in the rf loop is of second order, it will also represent a negligible fractional error. The intensity seen by the detector, Er2 averaged over many cycles of coO, is oo
I=½ ~ gl =
--
~ oo
Ol
=--
EnE* r(coo+ncol) ~
X r*(co 0 + m c o l ) e x p [ i ( m - n ) c o 1 t ] .
(4)
So the amplitude of intensity modulation at frequency col will be r(co0+nco 1) L I = t / -~- - ~ [EnE* rt+l X r*(co 0 + (n + 1)COl] .
(5)
The average intensity will be Iav=l
~
IEnr(coO+ncol)l2.
(6)
n=_oo
The cavity reflectivity is given by r(co) = 1 - (1 - K ) [ F +
1 -Fexp(-ico/fc)]-l,
(7)
with the definitions fc = free spectral range, rrF = finesse, and K - 2 = intensity reflection contrast ratio. Assuming that F is large enough that r(co 0 + ncol) is very flat for the carrier and even harmonics as shown in fig. 2, then eq. (7) can be approximated by the following r(co 0 + ncol) = 1,
for n even,
r(co 0 + ncol) = 1 - (1 - K)
X [l+iF(coo+ncol)/Jc]-1,
fornodd.
(8)
The frequencies coO and col expressed in terms of the tuning errors and square wave frequency modulation are
coO = [ ( 2 N + 1)rr + 80]Jc, col = [(2M+
1)rr+(a/F)S(co2t)+fil]fc,
(9) (10)
where 8 0 f c = optical tuning error (rad/s), 8 1 f c = rf tuning error (rad/s), afc/F= depth of frequency modulation (rad/s), co2 = rf modulation frequency, N = optical mode number, M = effective rf mode number, 19
Volume 59, number 1
OPTICS COMMUNICATIONS
and where S(x) is the square wave function with unity amplitude and 21r period. With substitution of eqs. (2), (8), (9)~ and (10) into eq. (5), and averaging over one cycle of 6o2, the resulting error signal to linear order in 6 0 and 61 is e0 = 2 F ( 1 - K ) 6
0 ,~--0 [ 1 - ( 2 n +
× [1 + (2n + 1)2a 2] -2(Ezn E~n+l - EZn+l E~n+2 ) . (11) With substitution of eqs. (2), (8), (9), and (10) into eq. (6), multiplying by S(60 2 t) and averaging over one cycle of 6o2, the resulting error signal to linear order in 6 0 and 6 1 is e 1 = 2aF(1 - K2)61 n~__0(2n + 1) 2 × [1 + (2n + 1)2a 2]-21E2n+l12.
(12)
Thus using first or higher order servo loops to drive the above error signals to zero will cause the following relation
600 = ( X + 1/2)6ol/(M+ 1/2).
(13)
In the case of phase modulation by Z sin 601 t, the sideband amplitudes are simply
E --EJ (Z) In the limit of low Z the error signals are simply e 0 = 2E2F(1 - K ) ( 1 - a2)(1 + a 2 ) - 2
× J1 (z) [g0(z) - s2(z)] 60, e 1 = 2E2aF(1 - K2)(1 + a2)-2J~(Z)61.
(14) (15)
The optical mode number N can be determined by first making a measurement of 601, then changing the mode number to N + P where P is known (by fringe counting while changing the optical cavity size), and then measuring 60'1 (the new subcarrier frequency). Then 600 may be determined by the relation
P[6oO = (M+ 1/2)(1/6o'1 - 1/6ol). Dispersion cancels out to first order in eq. (16). This is because to first order, dispersion does not distort information modulated onto a carrier. It causes the information to travel at the group velocity while the 20
carrier at the phase velocity. Thus after one trip around the cavity, the rf modulation signals will suffer a time delay or advance relative to the optical carrier which must be added to (N + 1/2)/6o 0. Hence first order dispersion changes eq. (13) to
(N + 1/2)/w 0 + T= (M + 1/2)/601,
1)2a 2 ]
1 August 1986
(17)
where T is independent of N if changing N does not add or delete any dispersive material in the optical cavity. Hence the constant T will subtract off in determining 600" It is also possible to calculate the diffraction phase shifts to an accuracy of ~ 10 -3 [4], thus allowing for diffraction correction. This correction would result in a fractional accuracy o f ~ 10 - 9 for N ~ 10 6. Reversing the servo loops by usir~ the error signal e 1 to lock the optical cavity to the rf source and using e 0 to lock the optical source to the optical cavity allows for a method of optical frequency synthesis. If the system is first used in the frequency counting mode to determine N and set it, then the servo loops could be reserved and the optical source could be set at a programmed frequency. An initial prototype using a 4-reflector ring cavity configuration with afc = 600 MHz free spectral range, a finesse of 60 (TrF), and a contrast ratio of 4 (K - 2 ) has been built to demonstrate the technique. The cavity is different from the one in fig. 1 in that it uses the mode defining lens as the element to couple the light into the cavity. The lens is slightly tilted and reflection off the fiat side is used to couple the incident light into the cavity. The transmitted beam and the cavity light reflected off the flat side of the lens is sent to the detector and is equivalent to the detected signal shown in fig. 1. This configuration is used because the reflection losses of the mode defining lens are the dominant losses in the cavity thus the coupling loss is approximately half the total cavity loss which provides optimum cavity coupling and a good contrast ratio. The cavity shown in fig. 1 has non-optimal coupling and a poor contrast ratio if all mirrors are the same. The optical source is a 50/JW stabilized HeNe laser at 6328 ~, accurate to 10 - 7 . A lithium tantalate modulator is used for the optical phase modulation at a frequency of 300 MHz (601/27r)- The modulation index is Z = 0.5. The optical error signal is derived by first down converting the preamplified detector signal component at 300 MHz down to 8 MHz, then syn-
Volume 59, number 1
OPTICS COMMUNICATIONS
Fig. 3. Optical cavity reflectance (top) and optical error signal (bottom) corresponding to eq. (11), recorded by sweeping the optical cavity size with both servo loops open.
chronously detecting the amplified 8 MHz signal with a double balanced mixer. This is more complicated than in fig. 1, but amplifier design at 8 MHz is simpler than at 300 MHz. Fig. 3 shows how this optical error signal and the cavity reflectance varies as the cavity size is swept. The slope of the zero crossing of this error signal agrees well with eq. (14). The rf modulation frequency is 6.25 kHz (~2/2rr). The depth of modulation is 675 kHz (a = 0.14). The rf error signal is derived as shown in fig. I except that the preamplified detector signal is filtered with a 300 Hz bandwidth bandpass filter before going to the dou !ble balanced mixer. In fig. 4 this rf error signal and
Fig. 4. Rf error signal corresponding to eq. (12) (top) and rf detuning corresponding to 81 in eq. (12) (bottom). The optical servo loop is closed, the rf servo loop is open, and a sinusoidal signal is added to the VCO input.
1 August 1986
Fig. 5. Rf error signal (top) and optical error signal (bottom) with both servo loops closed.
the rf detuning is shown while the optical loop is closed with a 600 Hz bandwidth, the rf loop is open and a sinusoidal signal is used to vary the rf detuning. The size of this error signal agrees well with eq. (15) when a correction factor of 2/7r due to the bandpass filter is accounted for. The signal to noise ratio of the rf error signal is 9.5 dB lower than the shot noise linit corresponding to a fractional accuracy of 10 - 8 for 3 second averaging for the more optical values o f Z = 1 and a = 0.5. This high noise level is attributed to the laser. Fig. 5 shows both error signals after closing the rf loop with a 10 Hz bandwidth. In summary, feasibility of the FM subcarrier method has been demonstrated with a signal to noise within 10 dB of the shot noise limit. If a higher rf modulation frequency w 2 were chosen to get away from the laser noise, then shot noise limited performance should be possible. If a large optical cavity (N ~ 107 and M 103 ) with an ultra high finesse of 104 is used, this method ultimately promises a diffraction correction limited fractional accuracy of ~ 10 -10 with a shot noise limited fractional stability o f ~ 10 -14 for 1 second averaging. The authors thanks T.W. Hansch, L.S. Cutler, and J.L. Hall for suggestions contributing to the understanding of the systematic problems encountered thus far. The assistance of E.A. Hildum, J. Ratcliff, and S.A. Newton in procurement and preparation of the lithium tantalate phase modulator, and P.L. Vella in oscillator design, is greatly appreciated. This work has been supported by the I.B.M. Pre21
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OPTICS COMMUNICATIONS
d o c t o r a l F e l l o w s h i p p r o g r a m , a n d in p a r t b y H e w l e t t P a c k a r d Co.
References [1 ] R.G. DeVoe and R.G. Brewer, Phys. Rev. A 30 (1984) 2827.
22
1 August 1986
[2] Z. Bay, G.G. Luther and J.A. White, Phys. Rev. Lett. 29 (1972) 189. [3] J. Hough, D. ttils, M.D. Rayman, L.-S. Ma, L. ttolberg and J.L. Hall, Appl. Phys. B 33 (1984) 179. [4] H. Kogelnik and T. Li, Proc. IEEE 54 (1966) 1312.
OPTICS COMMUNICATIONS
1 August 1986
A FM S U B C A R R I E R M E T H O D .FOR O P T I C A L F R E Q U E N C Y M E A S U R E M E N T AND S Y N T H E S I S G.M. C U T L E R Department of Physics, Stanford University, Stanford, CA 94305, USA Received 25 March 1986
A simple technique for optical to rf or rf to optical frequency conversion is described. The proposed FM subcarrier m e t h o d uses two frequency locked servo loops to lock a reflecting optical cavity to an optical source, and then an rf oscillator to the optical cavity thus converting optical to rf frequency or vice versa by reversing the servo loops.
This paper proposes and demonstrates a new FM subcarrier method for locking an rf oscillator to a submultiple of the frequency of an optical source or vice versa, which facilitates accurate laser frequency measurement or synthesis over a broad spectral range. Optical frequency synthesis with an accuracy of ~10 -10 would simplify optical spectroscopic measurements which presently rely on comparison with some nearby well documented atomic transition for calibration. Also a simple technique for optical frequency synthesis with an accuracy of ~ 10 -8 is needed to make coherent optical data transmission and reception practical, which coul d ultimately lead to hundreds of high bandwidth channels on a single multimode fiber. The method demonstrated is similar to the dual frequency modulation method [1] (DFM) of DeVoe and Brewer in that it is a FM servo loop method that makes a two step comparison between the optical frequency and rf frequency using an optical cavity as the intermediate step. It differs from the DFM technique in that the optical source is not double frequency modulated. It is, as shown in fig. 1, phase modulated by a single subcarrier which is itself frequency modulated thus requiring only one optical modulator. The modulated optical beam is reflected off the optical cavity to a photo detector. Synchronous detection of the photo detector signal at the optical modulation frequency gives an error signal proportional to the optical source versus optical cavity detuning while synchronous detection at the rf modulation frequen0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Double r ) ~ w2 Balanced Mixer ( ~op FiRe ) Double Balanced Mixer
oop Filter
Phase Modulator Detector
<],
~
Optical Ring Cavity
Preamp Fig. 1. Diagram for locking a voltage controlled rf oscillator (VCO) to a laser by the FM subcarrier m e t h o d . The optical servo loop locks the optical ring cavity resonances to the odd optical sidebands. The rf servo loop locks the average spacing of odd optical sidebands to the cavity mode spacing.
cy produces an error signal proportional to the detuning of the optical modulation frequency from the optical cavity mode spacing. This technique is simpler in both the optics and electronics than earlier methods [ 1,2]. It also retains the very high immunity, inherent with the DFM method, to sideband imbalance caused by unwanted amplitude modulation or even harmonic modulation. In addition, this technique has high immunity to dispersion effects, which must be separately measured and corrected for with the DFM technique. 17
Volume 59, number 1
OPTICS COMMUNICATIONS
The first servo loop, which locks the optical cavity to the laser (hereafter called the optical loop), uses an error signal derived by synchronously detecting the photo detector signal at the optical modulation frequency co 1 , and works on the same principle as the cavity stabilized laser in ref. [3]. When the optical source is modulated, an electric field with the following form is produced
t£=1-~ ~
t(r=l
with the definitions E n = complex amplitude o f nth sideband, coO = laser frequency, and 6o 1 = optical modulation frequency. If and only if the optical modulator does not phase modulate at even multiples o f col and does not amplitude modulate at odd multiples of col then E n satisfies the following symmetry condition
(2)
The reflected beam from the optical cavity will have an electric field
N-I
N
U
Z
CE I-U hi --I laW
(3)
where r(co) is the complex reflection coefficient o f the optical cavity. If the optical carrier is tuned to a symmetric point of r(co), causing tile relation r(co 0 + nco 1) = r*(co 0 nco l ), then eq. (2) holds for the sidebands of the reflected electric field in eq. (3). Hence there will be no amplitude modulation at odd multiples of col in the reflected electric field. When the optical carrier is detuned from this symmetry point the reflected sidebands become imbalanced (violating eq. (2)) thus any phase modulation at odd multiples of co I will be partially converted into amplitude modulation in the reflected electric field. As shown in fig. 2, the carrier and even sidebands are chosen to lie at reflection maxima while the odd sidebands are chosen to lie at the cavity resonances (reflection minima). Therefore the optical loop will lock the cavity reflection maxima to the optical carrier. At this point, two systematic errors become ap-
(11
E = (-1)nE*_n.
{r(co 0 + ricoI )E e x p [ - i ( c o 0 + n c o l ) t ]
+ r*(~O + n W l ) E 2 exp [i(co 0 + n c o l ) t ] } ,
{E nexp[_i(coO +ncol)t]
+ E * exp[i(co 0 + n c o l ) t ] ) ,
~
1 August 1986
N+I
#
N+2
hJ
>t-C~ U
tD Izl Z rr m hi rm
li I
t"° 0 - 3 u° 1 / rr t.) t--
i I Ii
I
P I
c° 0 - 2 c ~
OJO-~ 1
i, i,
! ¢'~0
t.~ O+t-,a I
I I t~ O+2t~ I ¢"aO + 3 t'~° l
O
Fig. 2. Solid lines show the optical carrier position and average positions of the optical sidebands relative to the cavity resonances for the case where the optical modulation frequency is equal to halt"the free spectral range of the cavity corresponding to M = 0 in eq. (10). Dashed and dotted lines show sideband positions during even and odd half cycles of the rf modulation frequency respectively. 18
Volume 59, number 1
OPTICS COMMUNICATIONS
parent. If the optical modulator produces any phase modulation at even multiples of coI or any amplitude modulation at odd multiples of col then there will be a violation of eq. (2) (hereafter called sideband imbalance) for the sidebands of the incident electric field. Under worst case phasing conditions there will be a shift in the lock point of the optical loop that is proportional to sideband imbalance. The second servo loop, which locks the average positions of odd sidebands to the optical cavity resonances (hereafter called the rf loop), derives its error signal by square wave modulating co1 at a low frequency 0o2 and synchronously detecting the photo detector signal at this rf modification frequency co2- The frequency modulation could be done with any signal containing only odd harmonics, but slow square wave modulation eliminates effective even harmonic generation from nonlinearity of the voltage controlled oscillator. The photo detector signal, averaged over many cycles of col, is simply the sum of the powers in each sideband multiplied by its respective cavity reflection coefficient. Therefore under proper locking conditions as shown in fig. 2, each sideband position will oscillate between two symmetrically located positions on opposite sides of symmetric reflection minima (for odd sidebands) or reflection maxima (for even sidebands) thus producing no am signal at frequency 002 . When the optical modulation frequency is shifted, the sidebands will oscillate between positions with different reflection coefficients thus producing a net amplitude modulation at frequency co2This analysis shows that this rf error signal vanishes when the average sideband positions coincide with symmetric minima or maxima of the cavity reflection coefficient leaving the zero crossing of the rf error signal (independent of the amplitudes and phases of the sidebands. Hence sideband imbalance will not shift the lock point of the rf loop given that the optical loop has no shift. There is, however, a linear cross coupling of the error signals that is of first order in sideband imbalance. This cross coupling and the first order shift due to sideband imbalance in the optical loop will result in a shift of the lock point of the rf loop that is of second order in sideband imbalance. Since the optical frequency is a factor of ~ 10 6 greater than the free spectral range of the optical cavity, the first order shift of the optical loop due to sideband imbalance will represent a negligible fractional
1 August 1986
error. And since the shift due to sideband imbalance in the rf loop is of second order, it will also represent a negligible fractional error. The intensity seen by the detector, Er2 averaged over many cycles of coO, is oo
I=½ ~ gl =
--
~ oo
Ol
=--
EnE* r(coo+ncol) ~
X r*(co 0 + m c o l ) e x p [ i ( m - n ) c o 1 t ] .
(4)
So the amplitude of intensity modulation at frequency col will be r(co0+nco 1) L I = t / -~- - ~ [EnE* rt+l X r*(co 0 + (n + 1)COl] .
(5)
The average intensity will be Iav=l
~
IEnr(coO+ncol)l2.
(6)
n=_oo
The cavity reflectivity is given by r(co) = 1 - (1 - K ) [ F +
1 -Fexp(-ico/fc)]-l,
(7)
with the definitions fc = free spectral range, rrF = finesse, and K - 2 = intensity reflection contrast ratio. Assuming that F is large enough that r(co 0 + ncol) is very flat for the carrier and even harmonics as shown in fig. 2, then eq. (7) can be approximated by the following r(co 0 + ncol) = 1,
for n even,
r(co 0 + ncol) = 1 - (1 - K)
X [l+iF(coo+ncol)/Jc]-1,
fornodd.
(8)
The frequencies coO and col expressed in terms of the tuning errors and square wave frequency modulation are
coO = [ ( 2 N + 1)rr + 80]Jc, col = [(2M+
1)rr+(a/F)S(co2t)+fil]fc,
(9) (10)
where 8 0 f c = optical tuning error (rad/s), 8 1 f c = rf tuning error (rad/s), afc/F= depth of frequency modulation (rad/s), co2 = rf modulation frequency, N = optical mode number, M = effective rf mode number, 19
Volume 59, number 1
OPTICS COMMUNICATIONS
and where S(x) is the square wave function with unity amplitude and 21r period. With substitution of eqs. (2), (8), (9)~ and (10) into eq. (5), and averaging over one cycle of 6o2, the resulting error signal to linear order in 6 0 and 61 is e0 = 2 F ( 1 - K ) 6
0 ,~--0 [ 1 - ( 2 n +
× [1 + (2n + 1)2a 2] -2(Ezn E~n+l - EZn+l E~n+2 ) . (11) With substitution of eqs. (2), (8), (9), and (10) into eq. (6), multiplying by S(60 2 t) and averaging over one cycle of 6o2, the resulting error signal to linear order in 6 0 and 6 1 is e 1 = 2aF(1 - K2)61 n~__0(2n + 1) 2 × [1 + (2n + 1)2a 2]-21E2n+l12.
(12)
Thus using first or higher order servo loops to drive the above error signals to zero will cause the following relation
600 = ( X + 1/2)6ol/(M+ 1/2).
(13)
In the case of phase modulation by Z sin 601 t, the sideband amplitudes are simply
E --EJ (Z) In the limit of low Z the error signals are simply e 0 = 2E2F(1 - K ) ( 1 - a2)(1 + a 2 ) - 2
× J1 (z) [g0(z) - s2(z)] 60, e 1 = 2E2aF(1 - K2)(1 + a2)-2J~(Z)61.
(14) (15)
The optical mode number N can be determined by first making a measurement of 601, then changing the mode number to N + P where P is known (by fringe counting while changing the optical cavity size), and then measuring 60'1 (the new subcarrier frequency). Then 600 may be determined by the relation
P[6oO = (M+ 1/2)(1/6o'1 - 1/6ol). Dispersion cancels out to first order in eq. (16). This is because to first order, dispersion does not distort information modulated onto a carrier. It causes the information to travel at the group velocity while the 20
carrier at the phase velocity. Thus after one trip around the cavity, the rf modulation signals will suffer a time delay or advance relative to the optical carrier which must be added to (N + 1/2)/6o 0. Hence first order dispersion changes eq. (13) to
(N + 1/2)/w 0 + T= (M + 1/2)/601,
1)2a 2 ]
1 August 1986
(17)
where T is independent of N if changing N does not add or delete any dispersive material in the optical cavity. Hence the constant T will subtract off in determining 600" It is also possible to calculate the diffraction phase shifts to an accuracy of ~ 10 -3 [4], thus allowing for diffraction correction. This correction would result in a fractional accuracy o f ~ 10 - 9 for N ~ 10 6. Reversing the servo loops by usir~ the error signal e 1 to lock the optical cavity to the rf source and using e 0 to lock the optical source to the optical cavity allows for a method of optical frequency synthesis. If the system is first used in the frequency counting mode to determine N and set it, then the servo loops could be reserved and the optical source could be set at a programmed frequency. An initial prototype using a 4-reflector ring cavity configuration with afc = 600 MHz free spectral range, a finesse of 60 (TrF), and a contrast ratio of 4 (K - 2 ) has been built to demonstrate the technique. The cavity is different from the one in fig. 1 in that it uses the mode defining lens as the element to couple the light into the cavity. The lens is slightly tilted and reflection off the fiat side is used to couple the incident light into the cavity. The transmitted beam and the cavity light reflected off the flat side of the lens is sent to the detector and is equivalent to the detected signal shown in fig. 1. This configuration is used because the reflection losses of the mode defining lens are the dominant losses in the cavity thus the coupling loss is approximately half the total cavity loss which provides optimum cavity coupling and a good contrast ratio. The cavity shown in fig. 1 has non-optimal coupling and a poor contrast ratio if all mirrors are the same. The optical source is a 50/JW stabilized HeNe laser at 6328 ~, accurate to 10 - 7 . A lithium tantalate modulator is used for the optical phase modulation at a frequency of 300 MHz (601/27r)- The modulation index is Z = 0.5. The optical error signal is derived by first down converting the preamplified detector signal component at 300 MHz down to 8 MHz, then syn-
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Fig. 3. Optical cavity reflectance (top) and optical error signal (bottom) corresponding to eq. (11), recorded by sweeping the optical cavity size with both servo loops open.
chronously detecting the amplified 8 MHz signal with a double balanced mixer. This is more complicated than in fig. 1, but amplifier design at 8 MHz is simpler than at 300 MHz. Fig. 3 shows how this optical error signal and the cavity reflectance varies as the cavity size is swept. The slope of the zero crossing of this error signal agrees well with eq. (14). The rf modulation frequency is 6.25 kHz (~2/2rr). The depth of modulation is 675 kHz (a = 0.14). The rf error signal is derived as shown in fig. I except that the preamplified detector signal is filtered with a 300 Hz bandwidth bandpass filter before going to the dou !ble balanced mixer. In fig. 4 this rf error signal and
Fig. 4. Rf error signal corresponding to eq. (12) (top) and rf detuning corresponding to 81 in eq. (12) (bottom). The optical servo loop is closed, the rf servo loop is open, and a sinusoidal signal is added to the VCO input.
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Fig. 5. Rf error signal (top) and optical error signal (bottom) with both servo loops closed.
the rf detuning is shown while the optical loop is closed with a 600 Hz bandwidth, the rf loop is open and a sinusoidal signal is used to vary the rf detuning. The size of this error signal agrees well with eq. (15) when a correction factor of 2/7r due to the bandpass filter is accounted for. The signal to noise ratio of the rf error signal is 9.5 dB lower than the shot noise linit corresponding to a fractional accuracy of 10 - 8 for 3 second averaging for the more optical values o f Z = 1 and a = 0.5. This high noise level is attributed to the laser. Fig. 5 shows both error signals after closing the rf loop with a 10 Hz bandwidth. In summary, feasibility of the FM subcarrier method has been demonstrated with a signal to noise within 10 dB of the shot noise limit. If a higher rf modulation frequency w 2 were chosen to get away from the laser noise, then shot noise limited performance should be possible. If a large optical cavity (N ~ 107 and M 103 ) with an ultra high finesse of 104 is used, this method ultimately promises a diffraction correction limited fractional accuracy of ~ 10 -10 with a shot noise limited fractional stability o f ~ 10 -14 for 1 second averaging. The authors thanks T.W. Hansch, L.S. Cutler, and J.L. Hall for suggestions contributing to the understanding of the systematic problems encountered thus far. The assistance of E.A. Hildum, J. Ratcliff, and S.A. Newton in procurement and preparation of the lithium tantalate phase modulator, and P.L. Vella in oscillator design, is greatly appreciated. This work has been supported by the I.B.M. Pre21
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d o c t o r a l F e l l o w s h i p p r o g r a m , a n d in p a r t b y H e w l e t t P a c k a r d Co.
References [1 ] R.G. DeVoe and R.G. Brewer, Phys. Rev. A 30 (1984) 2827.
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[2] Z. Bay, G.G. Luther and J.A. White, Phys. Rev. Lett. 29 (1972) 189. [3] J. Hough, D. ttils, M.D. Rayman, L.-S. Ma, L. ttolberg and J.L. Hall, Appl. Phys. B 33 (1984) 179. [4] H. Kogelnik and T. Li, Proc. IEEE 54 (1966) 1312.