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Miao Zhu* and John L. Hall. Joint Institute for Laboratory Astrophysics, University of Colorado and. National Institute of Standards and Technology, Boulder, ...
802

J. Opt. Soc. Am. B/Vol. 10, No. 5/May 1993

M. Zhu and J. L. Hall

Stabilization of optical phase/frequency of a laser system: application to a commercial dye laser with an

external stabilizer Miao Zhu* and John L. Hall JointInstitute for LaboratoryAstrophysics, Universityof Coloradoand National Institute of Standards and Technology,Boulder, Colorado80309-0440 Received July 28, 1992; revised manuscript received October 8, 1992

We present a comprehensive and quasi-tutorial review of the theory for analyzing the optical power spectrum of an optical field that has noise modulations of both the amplitude and the phase. We also present experimental results of the frequency stabilization of a commercial dye laser to a high-finesse Fabry-Perot cavity (0.49-Hz resulting full linewidth) and of the optical phase locking of the dye laser to a second reference laser (putting 97% of the optical power into the carrier) using an external stabilizer scheme. This external optical phase/frequency stabilization technique can be applied to virtually any cw laser system.

1. INTRODUCTION AND MOTIVATION FOR THE STUDY Time and frequency are, at present, the most accurately measurable physical quantities,",2 so that measurements of the other physical quantities, if possible, are reduced to measurements of time and frequency, e.g., the definition

of the speed of light' and the definition of the volt.4

Nevertheless, improvements in the current atomic frequency standard are already required by some applications, including the precision measurements of the periods and the changes in the periods of pulsars. These measurements reveal the only experimental evidence of the existence of gravity waves.5 7 In fact the stable pulsar PSR1937+21 could eventually be used as a time standard over long periods.78' Therefore it is of great interest to produce a better atomic frequency standard, especially one in the optical frequency domain, where, by virtue of the high transition frequency, one can obtain potentially higher Q values than those in the present microwave frequency standards. There are two main requirements for making a highperformance optical frequency standard. The first is to choose a suitable atomic (or ionic) transition as a passive reference. This reference transition should have a narrow natural linewidth, for example, Q - 1 in the Yb+ ion,9 and its center frequency should be relatively insensi-

tive to the environmental disturbances.

Since the

Doppler effect is more prominent in the optical frequency domain, the quantum absorbers should have small velocities in the laboratory frame. This goal can be realized by using laser cooling-trapping techniques,"' 2 based on the great amount of recent progress in this area.' 2 The second requirement for the optical frequency standard is a suitable coherent optical source (a laser) as the local oscillator, whose frequency can be stabilized to the center frequency of the preselected atomic reference transition. Obviously the laser phase/frequency noise plays 0740-3224/93/050802-15$05.00

an important role in the optical frequency standard. It contributes to the system's short-term instability directly, as discussed in connection with microwave frequency

standards. 3

4

It also degrades the long-term stability

since it may broaden the apparent linewidth in the atomic reference transition, thus reducing the achievable line Q value for the reference transition. Furthermore, when one uses modulation techniques, laser frequency noise at

the modulation frequency is particularly unwelcome. Other laser applications, for example, high-resolution spectroscopy and other precision measurements, also require laser fields with low phase/frequency noise. In order to serve as the local oscillator for an optical frequency standard, the laser has to fulfill two preliminary

requirements: (1) its tunable frequency range must include the atomic reference transition and (2) its phase/frequency noise has to be low so that its short-term linewidth is much narrower than that of the atomic reference transition. Unfortunately, lasers with low intrinsic phase/frequency noise, for example, diode-laser-pumped solid state lasers, often do not cover the spectral range of interest. Therefore we are usually forced to use noisier types of lasers, such as dye lasers and diode lasers. After a laser has been chosen, the next task is to design and build a servo system to reduce the laser's intrinsic technical phase/frequency noise, ideally to the level limited by the measurement noise (quantum noise). In this paper we focus on this task, i.e., using a servo system to reduce laser phase/frequency noise so that it can be used as the local oscillator for the optical frequency standard or as the optical source for some other demanding precision measurement applications. In the experiment reported here, we chose to stabilize the optical phase/frequency of a commercial continuous wave (cw) organic dye laser system for several reasons. First, the cw dye laser is used widely because it can cover a large fraction of the optical spectrum-from the near infrared to the blue-producing relatively high output pow© 1993 Optical Society of America

Vol. 10, No. 5/May 1993/J. Opt. Soc. Am. B

M. Zhu and J. L. Hall

ers (hundreds of milliwatts) by using a variety of dyes. The range can be extended into the near UV at moderate power levels with second-harmonic generation in nonlinear optical crystals. Second, the cw dye laser normally is considered to be one of the noisiest types of laser. The frequency and intensity noise on the dye laser often limit the useful attainable precision in the experiments to levels orders of magnitude degraded from those set by fundamental limits. Thus one of the most important-basically essential-prerequisites of precision laser measurements is the reduction of the optical phase/frequency noise (or linewidth) of the dye laser. Third, it is useful to stabilize a noisy laser system externally in order to demonstrate the potential application of this powerful technique. The generality of the technique in reducing intrinsic noise of basically any cw laser is made clear by success in removing the noise of a commercially available dye laser by external means. In the following sections we review the theory that is appropriate for analysis of the optical power spectral density when the optical field has noise modulations on both the amplitude and the phase (Section 2). In Section 3 we describe the general requirements for stabilization of the laser phase/frequency and discuss in some detail the question of practical references for optical phase or frequency and the transducers for controlling the optical phase or frequency. We also compare different configurations for the laser servo system. In Sections 4 and 5 we describe our experimental setup and present our experimental results of using external techniques to stabilize a commercial dye laser.

In the conclusion (Section 6) we briefly

discuss applications to other laser systems. 2.

SPECTRAL ANALYSIS

OF OPTICAL

FIELDS: A TUTORIAL EXPOSITION To characterize a frequency standard system, or other stable oscillators, we can measure the system either in the time domain (Allan variance) or in the frequency domain (power spectral density).3 4 For an atomic clock in the microwave frequency range the linewidth of the local oscillator (ordinarily a quartz-crystal-based oscillator) is much narrower than the measured linewidth of the atomic reference transition, which usually is limited by the transit time broadening. One finds that time domain measurement is well adapted to this spectral range, especially after the local oscillator's frequency is stabilized to the atomic reference transition. However, in the optical frequency range the situation is often the opposite: the linewidth of a free-running laser is typically much broader than a decent atomic reference transition, for example, 5 Thus the intercombination transition in the Ca atom. phase/frequency optical the in the process of reducing

803

quency is finally stabilized to the atomic reference transition, we will again use the time domain measurement to characterize this optical frequency standard system. In discussing the local oscillator of the optical frequency standard system, as well as for the other precision measurements involving a laser system, knowledge of the optical power spectral density of a noise-modulated laser output is of fundamental importance. To use the parlance of statistical communication theory 6 -20 developed for the radio frequency (rf) range, we are interested in analyzing a stochastic-process-modulated carrier. The same methods are also applicable in the optical frequency domain, where we consider the optical electric field rather than output voltage or current of an rf oscillator. Since 2 many stochastic processes display Gaussian statistics, mainly because of the central-limit theorem, in the follow-

ing discussion we will consider only the zero mean Gaussian stochastic process unless otherwise noted explicitly. We further assume that all the stochastic processes involved are ergodic, so that a time average is equivalent to an ensemble average (i.e., we do not have to build a large number of stabilized laser systems to do the ensemble average). A.

General Theory

The output electric field of an ideal cw single-mode laser can be expressed as

E(t) = E(t)= E0 exp(-iwot - 40,

(1)

where e is a unit vector indicating the polarization and Eo, too, and qpoare the constant amplitude, angular frequency, and initial phase of the electric field, respectively. Since the polarization of the electric field usually is fixed by the polarization-sensitive optical elements in the laser cavity, for example, a Brewster-angled dye jet stream or Brewster windows, we direct our attention to the scalar part of the optical electric field E(t). In the presence of random noise modulations the scalar part of the electric field can be written as

E(t) = E(t)exp[-ioot - 40N(t)],

(2)

where 5oN(t)is the (noise) phase modulation and E(t) is the instantaneous amplitude of the field. For the systems of interest, both (PN(t) and e(t) can be assumed to be slowly varying relative to )o0. A useful concept is the instantaneous angular frequency: o(t)c= °+

doN(t)

dt

(3)

Limitations on this picture have been discussed by Mandel.2 2 The instantaneous amplitude E(t) can be written in terms of the (noise) amplitude modulation VN(t):

noise of the local oscillator, which will be used to interro-

gate the atomic reference transition, we are initially more interested in the optical power spectral density. We are interested especially in the fractional power in the optical carrier or within a narrow bandwidth around the carrier. In practice, to enhance the short-term stability and reduce the laser linewidth it is more useful to adjust the servo parameters while watching the power spectral density as the criterion. Therefore we limit our discussion here to the frequency domain. Of course, when the laser's fre-

2-J2fp e(t) = _-Eo _

ei;[1

{Eo[1 + VN(t)] 0

+ VN(t)]} VN(t)

-1

VN(t)< -1

(4)

where the contour C of the integration in the ; plane is indicated in Fig. 1 for the cases VN(t) Ž -1 and VN(t) < -1 (over modulation). It is useful to write e(t) in the integral 9 2 form when the characteristic function is calculated.'1 '

804

J. Opt. Soc. Am. B/Vol. 10, No. 5/May 1993

I -

M. Zhu and J. L. Hall

where C* is the contour conjugate to C and

plane

ID,,

oC;T) = (exp[i;VN(t) + iqVN(t + T) + iN(t)

+ iqN(t

+ T)]D (8)

E

-

l

l

-

-

f

--

-

w

is (by definition) the characteristic function. 2 ' To account for a partial correlation between the amplitude and phase modulation processes, we write

b

l

VN(t) = KN(t

(a)

+ to) + UN(t),

(9)

where K is a scale factor for the correlation, to represents any time delay (to < 0) or time advance (to > 0) in amplitude modulation relative to phase modulation, and UN(t) represents that part of the amplitude noise uncorrelated

c- plane

with

(PN(t).

Combining Eqs. (8) and (9), we obtain

(D -q7,, 0-; T) =

(exp[i;VN(t) + iVN(t + ) + iON(t) + iqN(t + )])

=

(exp[i;UN(t) + i UN(t +

T)]) (exp[i;KqN(t

+ to)

+ i7K9ON(t + to + T) + iON(t) + iVN(t + )])- (10) (b) Fig. 1.

Contours for Eq. (4). Closure is (a) in the upper halfplane for VN(t) -1 and. (b) in the lower half-plane for

VN(t)< -1.

See text for details.

In the general case (K 0) one can simplify this characteristic function further by using the properties of the Gaussian process,9-2 and the power spectral density exhibits an asymmetrical feature around the carrier frequency

In general one can define the autocorrelation function Rx(X) for the time-varying quantity X(t) as

Rx()

(X(t)X*(t + )),

(5)

where the angle brackets denote an ensemble average, which is equal to a time average in the ergodic stochastic process. Then the power spectral density of the electric

field can be obtained using the Wiener-Khintchine theorem 24 :

f R(T)exp-icor)dr.

PE(w =-

20 Cwo.

When the mechanisms of the amplitude modulation and the phase modulation are uncorrelated (K = 0) and the variance of VN(t) is much smaller than unity (i.e., we ignore the small probability of overmodulation), Eq. (7) can be written as RE(T) =

E exp(icooT){1+

X (exp[ipN(t + ) -

Rv(T)= R,()

by Eqs. (2) and (4) is given by RE(T) =

exp{i[1 +

12

VN(t)]

(VN(t)VN(t + T)),

= (N(tXoN(t + ))

(12)

(13)

we obtain

(exp[iqoN(t + )-

= E, exp(iooT)Kf d~- d7l

(11)

and invoking the moment theorem of Gaussian processes,

(E(t)E*(t+ T))

PN(t)])-

Defining the autocorrelation functions for the modulation noise sources

(6)

The autocorrelation function of the electric field described

(VN(t)VN(t + ))}

pN(t)]) =

exp[R*,(T) - R(0)], (14)

and Eq. (11) takes the form RE(T) = E exp(icoT)exp{-fl,(T)}

-in[1 + VN(t + )] - 4N(t) + 0N(t + T) =

Eo

exp(i;

7r exp(icolT)Ld;L d77 X (exp[iVX(t)

0e 2

i

+

QO()=R(O) - R(T)

- i'qVN(t + T) - i4N(t)

2 exp(icoT)

X V>(,-71,

,1;

d T),

(15)

where

n

+ 4N(t + T)]) =

Eo exp(icoT)Rv(r)exp[-f'1,()],

dq exP(; 2- i-X-) ( 7)

(16)

The restriction on VN(t) as a Gaussian process is not required in Eq. (15). Since the Fourier transformation is a linear process, we see from Eqs. (6) and (15) that the power spectral density of the electric field is the sum of two terms. The first term contains information about the phase fluctuations

Vol. 10, No. 5/May 1993/J. Opt. Soc. Am. B

M. Zhu and J. L. Hall

only, while the second contains information about both phase and amplitude fluctuations. The total power, including both the phase and the amplitude modulations, can be obtained by integrating the power spectral density over all frequencies. Using the properties of the Fourier transform, we obtain

which indicates that the fraction of the optical power that remains in the carrier is exp(- r'ms). Separating the carrier from the noise modulation sidebands, from Eq. (15) we have RE(T) =

E' exp(iwor)exp(-p'rms) + E 2 exp(icooi) + Rv(&)]exp[Rp(T)]

x exp(-_o 2){[1 Ptotai=

PE())d(o =

RE(O)

= Eo{

+ Rv(0)}.

(17)

Thus the phase modulation does not add extra power to the electric field, while the amplitude modulation does. The ratio of this excess power (because of the amplitude modulation) to the power in the absence of modulation is Rv(O), which is simply the variance of

VN(t).

As mentioned above, we are often interested in the fraction of the laser output power that falls within a narrow bandwidth around the carrier frequency or that in the carrier itself. In this regard, it can be seen from the frame72 52 6 that work of the frequency modulation (FM) theory the low-frequency components of the FM process play an important role since the effective modulation index ,6 of a given Fourier component is inversely proportional

to its

frequency for a fixed maximum frequency excursion. The fraction of power remaining in the carrier itself is J0Q3), which decreases to zero as the modulation index ap-

proaches infinity. In general the nonzero power fraction that remains in a discrete spectral component (e.g., in the carrier) corresponds to the nonvanishing harmonic component with the same frequency in the autocorrelation func.27,28 Assuming that the amplitude tion in the limit T modulation is a stochastic modulation process without long-term coherence, we have the property = 0,

Rv(),.

(18)

which implies that the second term in Eq. (15) vanishes as T -x o. Thus the behavior of RE(T) is determined purely by the phase term, fl,(T), for a large delay time r. Accordingly, it is useful to examine fl,(T) in more detail. Using the Wiener-Khintchine theorem [Eq. (6)], we obtain - R,(T)

l,(T)--R,(O)

=f

805

-

1}, (22)

where the first term represents the carrier and the second