Nov 1, 1992 - cavity that can account for a bias. ... than 2%. A bias remains a cause of error in the .... the tape was such that the backscattered pulse reentered ...
November 1, 1992 / Vol. 17, No. 21 / OPTICS LETTERS
1535
Nonreciprocal measurements in femtosecond ring lasers Ming Lai, Jean-Claude Diels, and Michael L. Dennis Department of Physics and Astronomy and Centerfor High TechnologyMaterials, The University of New Mexico, Albuquerque, New Mexico 87131 Received May 19, 1992
Sensitive gyroscopic response has been demonstrated with a passively mode-locked ring dye laser. Rapid change in the phase of scattering has been identified as an important mechanism responsible for the extremely low lock-in threshold of this laser gyro. Coupling between counterpropagating pulses has been confirmed to take place only in the overlapping region of the pulses. Intracavity measurements of the electro-optical effect are demonstrated with fields as small as 20 mV/cm.
cidence on curved optics, hence strong astigmatism. Choosing the absorber jet as the origin, we calculated
We have investigated the gyroscopic response of a femtosecond ring dye laser operating on a rotating platform. The expected advantages of a modelocked laser gyro over a conventional cw laser gyro are a smaller coupling between the two counterpropagating waves and the absence of mode competition in a homogeneously broadened gain medium. Both features are a result of the simple fact that the two counterpropagating pulses in a passively modelocked laser cavity only meet at two points, both outside the gain medium. However, in a passively mode-locked laser, it is a strong coupling between two counterpropagating waves at the absorber that
the complete round-trip ABCD matrix of the ring
using the algebraic manipulation language MACSYMA. The expressions for the matrix elements are used in a FORTRAN optimization program, which scans all possible values of the folding angles and intermirror
distances to determine the stability ranges (distances between pairs of curved mirrors) and the optimum condition for which a round focal spot is obtained in the absorber,2 by taking into account self-lensing effects in the absorber jet.3 This program enables us also to select cavity configurations with an uneven number of foci, which ensures reversal of the (spatially dispersed) beam at each round trip. The resulting cavity, shown in Fig. 1, has an area of 1.42 m2 and a perimeter of 5.09 m. The
establishes the crossing point between the two
pulses. It is therefore reasonable to expect that this same coupling may result in a lock-in of the gyro response. We demonstrate that, despite this coupling, the femtosecond dye laser has a gyroscopic response with no measurable dead band. The gyroscopic response or scale factor R of a ring laser is the ratio of the beat note Av (in hertz) between the counterpropagating modes in the laser to the rotation rate Qt (in radians per second) of the laser support. In the case of a femtosecond laser, accurate delay lines have to be used to make the pulse trains corresponding to the two opposite senses of rotation interfere on the detector. A beat note between the two femtosecond pulse trains is readily understood in the frequency domain. Each pulse train corresponds to a comb of equally spaced spectral lines (laser modes). The spectral combs of the two pulse trains are identical but shifted with re-
scale factor Qkis 0.79915 MHz/(rad/s) or 31.40 kHz/(deg/s).
The gyro response of this mode-locked dye laser operating at 620 nm with a mixture of Rhodamine 6G and Sulforhodamine 640 as gain (this energy transfer mixture has a longer recovery time than Rhodamine 6G) and DODCI as saturable absorber is shown in Fig. 2. The pulse duration is -100 fs. The beat frequency is obtained by overlapping the two pulse trains temporally and spatially on a slow photodiode by means of an extracavity delay line, as shown in Fig. 1. The data confirm the absence of a measurable dead band noted on measure-
ment of Fresnel drag.4 There are numerous nonreciprocal nonlinear mechanisms in the laser
cavity that can account for a bias. Our observations point to saturation-induced changes in index in the gain jet. While the intensities are nearly balanced at the absorber jet (as expected because of the mu-
spect to each other by a small amount that is due, for
instance, to a cavity rotation. The beat note is a measure of the relative shift between the two spectral combs. If A is the ring laser area, P is its perimeter, and A is the wavelength, the gyroscopic
tual coupling through the transient grating in the
absorber), there can be as much as a 20% imbalance between the two intensities at the gain jet. The imbalance can be minimized by adjusting the output coupling at mirrors Ml and M2 (Fig. 1). The residual beat note can be eliminated up to the uncertainty that is due to the Earth's rotation rate (0.035 deg/s) by longitudinal translation of the absorber jet. This
response is given by1
AV4A l
R.
PA
Since a large ratio of area to perimeter is desirable, a large square cavity femtosecond laser was designed. If the number of flat mirrors is to be kept to a minimum, such a shape implies large angles of in0146-9592/92/211535-03$5.00/0
longitudinal positioning affects the intensities of ©
1992 Optical Society of America
1536
OPTICS LETTERS
/ Vol. 17, No. 21 / November 1, 1992 DELAY LINE
M2
M3
LiGAIN-Y-
Fig. 1. Schematic of the laser cavity and the detection interferometer. M's, mirrors; P's, prisms; PD, photodiode. 6
I
4 2
0~
a) 0 a) LL m -2 a)
-4 -6 '-0.18 -0.12 -0.06
0
0.06
0.12
0.18
Rotation Rate (deg/s)
Fig. 2. Plot of the beat frequency versus the rotation rate of the ring laser.
the pulses in the gain jet, because of the strong saturation lensing in the absorber jet.3 While the bias was changed from negative to positive by adjusting the cavity alignment and the absorber jet position, the laser average outputs did not change by more than 2%. A bias remains a cause of error in the measurement rather than a convenience,' so long as it cannot be compensated for in a systematic fashion (i.e., independently of the knowledge of the rotation rate). The slope of the data corresponds exactly to the predicted scale factor. No departure from linearity can be observed within the accuracy of the measurement. The error bars in the abscissa are due to inaccuracies of the drive mechanism at the lowest speeds. The main causes of error in the data are vibrations of the components of the measuring interferometer and, to a lesser extent, small vibration of the laser base or components. Let us evaluate the noise introduced by mechanical vibrations of amplitude dmcch and frequency Vmcch of a cavity mirror.
Fluctuations in path length difference AP-
result in a broadening of the beat note V = VyAP/P = Vmechdmech/A(where Trt is the cavity round-trip time and v and A are the frequency and wavelength of the light). An identical contribution comes from the Doppler shift that is due to vibradmechvmechTrt
tions.
With the mirror velocity being
Vmechdmech, the Doppler shift is
Vmech= Av = (Vmech/C)V
Vmechdmech/A, which is the same expression as for the cavity-length fluctuation. The 100-Hz linewidth of
the beat frequency is thus due to vibrations of the components of this unstabilized laser. If we assume that both the scale factor and the mechanical vibration amplitudes of the cavity are proportional to the size of the cavity, the performances of this gyro are not sensitive to its size. The waveform of the beat note is approximately sinusoidal down to the lowest frequencies observable. Even at the lowest frequencies, we do not observe any modulation in the amplitude of any of the beams. These two observations concur to indicate a low threshold for lock-in. In order to be able to observe the lock-in characteristics of this laser, we introduced a scattering ele-
ment (antireflection-coated glass window) at the
pulse crossing point opposite to the absorber jet. The amount of coupling was adjusted by translating the glass scatterer through the crossing point, which resulted in a conventional dead band in the gyro response. At beat frequencies close to the lock-in threshold, the waveform is distorted, and there is strong amplitude modulation of each of the counterrotating pulse train. The width of the dead band as a function of delay is-as expected-of the order of the pulse length (50 ,um). By feeding an output pulse back to the laser cavity with an adjustable delay, we have confirmed that, for different pulse durations, the width of the dead band is indeed equal to the pulse length.4 One expects for a laser gyro that, at low rotation rates, the gyroscopic response vanishes because of scattering of one beam into the other. If r is the scattering coefficient for the field of one of the laser
beams into the counterpropagating laser field, a general upper limit for the lock-in rotation rate £l, is given by6
al C=rcA
(2)
2A
a
b
C
d
e
f
Fig. 3. Elimination of the dead band by the motion of scatters. Trace a, no scatters; trace b, scatters at rest; traces c-f, scatters in increasing magnitudes of motion. In trace f, the beat note resumed its shape as if there were no scatters.
November 1, 1992 / Vol. 17, No. 21 / OPTICS LETTERS
beam in a time much shorter than the period of the beat note to be observed. This is what happens in the dye laser. The transient time of the scatters in
4
I
3
the absorber jet is -1 ps, much shorter than the
0 2
typical beat note period of 1 ms. A great advantage of a mode-locked laser gyro is the feasibility of introducing intracavity elements without disturbing its gyroscopic response, provided
a)
Ce IL m3 0
a)
that the inserted elements are not located at the
-1
-2 -0.2
I
I
0
0.2
0.4
0.6
0.8
1
1.2
Applied Voltage (V)
Fig. 4. Beat note versus the peak voltage of the detected pulse.
To establish a lower limit for the scattering coefficient r introduced by the saturable absorber jet, we performed
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a simple measurement
on a jet of pure
ethylene glycol. The beam of an argon-ion laser is focused onto the jet by a 2.5-cm focal-length lens. The geometry and aperture sizes are carefully adjusted to eliminate backscattering from all surfaces (lens and jet). The minimum backscattering radiation from the jet is 1.5 x 10-6 of the incident intensity. This value corresponds to a field backscattering coefficient of r = 1.2 X 10-3. According to Eq. (2), such a value for the backscattering
coeffi-
cient corresponds to a lock-in rotation rate of 5 deg/s for a ring of 1.4-M2 area, at a wavelength of 620 nm.
The actual lock-in rate should be much larger, in view of the additional contribution from the absorber dye.
In order to interpret the extremely low lock-in threshold in the presence of large scattering from the absorber jet, we investigated the effect of transverse motion of scatters. One of the laser outputs was focused onto reflective tape. The location of
the tape was such that the backscattered pulse
reentered the laser cavity and overlapped with the pulse in the opposite direction. The tape was attached to a speaker so that the scattering particles of the tape could move transversely to the laser beam. When the tape was at rest, strong lock-in was observed.
As the driving voltage (at 50 Hz) to
the speaker was increased, the beat signal resumed its sinusoidal shape as if no scattering was present (Fig. 3). The maximum amplitude of the 50-Hz vibration was 120 ,um.
The observation of Fig. 3 can be readily understood by examining the equation of motion of the relative phase T between a pair of the beating modes,
P = 2irQQfl+ a sin(T + e),
(3)
where Q.,a, and e, are, respectively, the mode splitting (beat note frequency) due to cavity rotation or bias, the coupling coefficient of the modes, and the phase shift of the backscattering. When the phase shift e changes much faster than Q.,the second term in Eq. (3) vanishes on average, and the two laser modes experience no coupling. Physically, the requirement for eliminating the dead band appears to be that the scattering particles move through the
pulse crossing points. We have taken advantage of this to study the low-voltage response of a Pockels cell. The Pockels cell (3 kV for A/2 rotation at 620 nm) is inserted into the cavity near the gain jet. An electrical pulse (pulse duration of 1 ns) from a fast avalanche photodiode detecting one of the laser outputs is applied to the cell. With a proper delay line, the electrical pulses coincide at the cell with one of the cavity pulses. Because the other cavity
pulse always reaches the cell between electrical
pulses, the two counterpropagating cavity pulses experience different indices in the cell. Therefore the bias of the beat frequency of the ring laser is modulated by the amplitude and polarity of the electrical pulse. Figure 4 (the points) is a plot of the bias versus the applied pulse voltage, in good agreement with the calculated value from the electro-optical coefficient of the cell (the solid line). In conclusion, we have investigated the gyro response of a passively mode-locked ring dye laser. The investigated laser system is basically a sensitive instrument for anisotropic measurement. Because of the feasibility of adding intracavity optics, it is suitable for many intracavity experiments that are not possible with cw ring lasers. When a sample is located away from the pulse crossing points, any index change induced by, for instance, an electrical or optical pulse train synchronized with one of the cavity pulses can be measured by means of the beat frequency. With a pulse duration of 100 fs, the above measurement can have a high time resolution. Intracavity pump-probe experiments can be straightforward. In these experiments, one cavity pulse is used as a pump pulse and the other is used as a probe pulse; the relative delay can be obtained simply by moving the sample away from or toward a pulse crossing point. This research was supported by the U.S. Office of Naval Research.
References 1. E. 0. Schulz-Dubois, IEEE J. Quantum Electron. QE-2, 299 (1966).
2. J.-C. Diels, Dye Lasers Principles: With Applications (Academic, Boston, Mass., 1990), pp. 41-132. 3. X. M. Zhao and J. C. Diels, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Di-
gest Series (Optical Society of America, Washington,
D.C., 1992), paper CWG34. 4. M. L. Dennis, J.-C. Diels, and M. Lai, Opt. Lett. 529 (1991).
16,
5. D. Gnass, N. P. Ernsting, and E P. Schaefer, Appl. Phys. B 53, 119 (1991).
6. J. R. Wilkinson, Ring Lasers (Pergamon, New York, 1987), pp. 1-103.
