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Effective cavity length in mode-locked Nd: glass oscilattors due to intra-cavity etalons
Effective cavity length in modelocked Nd:gla& oscillators due to intra-cavity etalons 6. SOOM, J.E. BALMER The insertion of Fabry-Perot etalons into actively mode-locked laser oscillators for pulse width selection brings about a readjustment of the cavity length if the modulator drive frequency is kept constant. Values are given, for practical use, of the total change in cavity length which is a combination of two effects: (i) the change in optical path length within the cavity when the etalon is inserted and, (ii) the change resulting from the phase shift generated by the etalon. KEYWORDS: lasers, laser oscillators, FabryPerot etalons. cavities
Introduction Actively mode-locked/Q-switched Nd-YAG and Ndglass laser oscillators have been widely used to generate single, high-power, nearly transform-limited pulses of sub-nanosecond duration.‘-’ Owing to its broad linewidth (>20 nm), N&glass is theoretically capable of producing pulses of sub-picosecond duration.’ Insertion of Fabry-Perot etalons into the resonator cavity provides a convenient means for covering the complete range of pulse durations in such an oscillator between approximately 50 ps and 1 ns. Since stability requirements often make a fmed frequency operation of the mode-locking modulator preferable, the insertion of etalon plates enforces a change in the mechanical length of the cavity that may reach lo-20 mm for a particular etalon. This is usually accounted for by mounting one of the resonator mirrors on a precision translation stage. For practical use it thus seems desirable to know the amount of mirror translation required for a particular etalon thickness and reflectivity. It is the purpose of this paper to give exact values for this change in cavity length which has previously been shown6 to be a combination of two effects, namely (i) the change in optical length of the resonator cavity due to the refractive index of the etalon material and, (ii) the increase in round-trip time of the pulse in the cavity due to the phase shift generated in the etalon. The calculated values are then compared to experimental results from an existing N&glass oscillator.
experiments have shown that phosphate glasses, for example, exhibit only a minor inhomogeneous broadening,’ In a separate paper. we present pulsewidth/bandwidth product measurements for an actively mode-locked/Q-switched Nd:phosphate glass oscillator which are in excellent agreement with the homogeneous theory? For our calculations, here, we essentially adopt the time-domain analysis presented by Kuizenga and Siegman6 for a homogeneously broadened laser, in which the gain, modulator, and etalon transfer functions are all approximated by Gaussians. In particular, this is done here for the amplitude transmission of a lossless Fabry-Perot plate of thickness’h atid refractive index n, given byP r, = (1 - R)l( 1 - Rei8) with 6 = (47r/X) &co&
(2)
where R = rr’ is the product of the amplitude reflectivities of both reflecting surfaces (r = J in most cases), and 8 is the tilt angle of the etalon. The (round-trip) transmission function of the etalon in a Fabry-Perot type laser cavity is approximated near a particular transmission peak, o,, by the Gaussian r, = exp[-A(o
- w,)‘] exp[iB(o
- o,)]
A = 4/A&,
B = 4(R)‘f2/Aw e
where Aw, is the effective bandwidth given by
(34 of the etalon,
Aw, = (c/h’) (1 - R)/(R)“’ The authors are at the Institute of Applied Physics, University of Bern, CH-3012 Bern. Switzerland. Received 27 October 1987.
0030-3992/88/010036-03/$03.00 36
(3)
with
Theory The linewidth of the N&glass lasing transition is basically subject to inhomogeneous broadening mechanismsP However, fluorescence line narrowing
(1)
(4)
with h’ = nhcos6 (8 =O for most practical 01988
Butterworth
Et Co (Publishers)
cases). From
Ltd
OPTICS AND LASER TECHNOLOGY
LHG-8 3OW
r-5m
100%
7
-c-a
I
1,
output
e
Q- Switch (28 MHz)
mirrw r---
* * *
7
I
T’
1
Ethfl
Apwture
Fig. 1
Schematic diagram of the sctiwly mode-locked/Q-switched
M@de-locker 450 MHz) Nd:glass oscillator
(3) it is apparent that the etalon transmission function affects the spectrum in such a way as to introduce a phase shift proportional to B. We can now consider a Gaussian pulse (without frequency chirp) passing through the etalon. Its timedependent field amplitude is described by E(t) = EO exp(-at
‘) exp(-io,l)
(5)
where or is the mean optical frequency of the pulse and a is real and determines the Gaussian envelope of the pulse. The pulse spectrum is obtained from (5) by Fourier transformation as E(w) = EO/(2a)‘/’ exp[-(w
- orJ2/40]
(6)
Multiplying (3) and (6) and transforming back into the time domain, we obtain the time-dependent field amplitude, E’(r), after one round-trip through the etalon (assuming that or = 0,): E’(r) = (l +
2,1,2exp[
- (l;L-f)&]
exp(-iW)
considerations, the phasei shift (in frequency space) from (3) is seen to result n a delay B in the time domain. This can be und rstood if we remember that the pulse transmitted thr ugh the etalon is built up by constructive interference f the multiple reflections between the etalon surfac s. As a consequence, its peak is delayed in time by a li ite amount which depends on both the thickness an reflectivity of the etalon. This gives rise to an incre1 se in cavity round-trip time or. equivalently, a decreas(e in modulation frequency/,. For the case of amplitude1 modulation without etalon, fm is given by6 fmo = [4@‘c + g/A@]-
where L is the (optical) length of the cavity, g is the saturated round-trip gain t line centre, and Ao is the linewidth of the gain med t urn. For most practical cases, glAo < UC, so that f,,,, = 4/4L. The corresponding self-consistent solution for the circulation pulse in a resonator with etalon leads to the expressionrO
(7)
fm = [4(Uc+ g/Aw + 2(R)“‘/Aw,]-
The effect the etalon has on the pulse is seen to be two-fold. Firstly, it increased the duration of the circulating pulse by an amount which is determined by A and. consequently, by the effective bandwidth of the etalon. Secondly, and more interestingly for our
P
.E
AR=30%
-
l
which for an uncoated
R=25%
1
(9)
If now we wish to hold then modulation frequency at a fixed value. fmo. the cavity length has to be shortened to a value L’ = L - SL,, such! that&,,,(L) = f&L’). The resulting change in cavity liength (assuming g constant. which is not exactly trueiO)‘is then given by 6L, = ~c(R)“YAc+
-
(8)
’
= 2@h’/(l-R) etalon
(10)
becomes
6L,* = (n - 1)%‘/2h
(11)
In addition, due to its refractive index, the etalon causes a change in optical liength of the cavity given by 6L” = (n-1)h’ln
(12)
SO that the total correction for the cavity length due to insertion of the etalon amolints to
6L,,, = 6L, + SL, = 2Rh’V( 1-R) 0.1
1.0
and for the uncoated
EtaIon thickness h (mm) Fig. 2 Total change in cavity length against etalon thickness for two different surface reflsctivitiis and 8mO. Solid lines era obtained from
VOL 20 NO 1 FEBRUARY 1988
+ (n-l)h’/n
(13)
10.0
(14)
etalon
SL,,, = sL,* + SL” = (n*-l)/7’/2n
(14)
37
Experiment A schematic diagram of the actively mode-locked/ Qswitched N&glass oscillator used in these experiments is shown in Fig. 1. The active medium is a 6.35 mm diameter by 76 mm long LHG-8 phosphate glass rod with end faces angled at 3”. Brewster-angled acousto-optic devices are used for both mode-locking and Q-switching. The mode-locker driver frequency, fm. is 50 Mhz and the corresponding resonator length is 1.5 m. The output mirror (R = 70%) is mounted on a precision translation stage. A set of coated BK-7 etalons with thicknesses between 90 pm and 15 mm was used to obtain transformlimited pulses with pulsewidths ranging from 70 ps up to 1 ns. More details of the oscillator performance will be given elsewhere.’ For each etalon the resonator length was adjusted for optimum output performance. and the exact mirror position read from the translation stage. Fig. 2 shows the measured change in cavity length, c&,~. as a function of the etalon thickness, h, for different etalon reflectivities. All the etalons were inserted nearly perpendicularly to the resonator axis (19.e 0). Also shown for comparison is the calculated change in cavity length due to the increased optical path length alone (dashed line). It is seen that the experimental values are in excellent agreement with our model (solid lines). Noticeable deviations are seen only for the two extremal measurements (!I0 pm/2596 and 15 mm/30%). In the case of the thin etalon this is most likely explained by experimental uncertainties in the measurement of distances of order 10 pm. On the other hand, with the thickest etalon we approach the operational limit of our oscillator with respect to stable mode-locking,’ so that the slight drift-off of the
30
experimental effect
data point may be attributed
to this
Conclusions The change in cavity length caused by insertion of Fabry-Perot etalons into a mode-locked laser oscillator has been investigated in detail. Experimental data are shown to be in very good agreement with a model which describes the total length change as the sum of a phase shift in the etalon and the increase in optical path length due to the refractive index of the etalon. Such data can be useful, for example, for implementing automatic pulsewidth selection in short pulse laser systems.
Acknowledgements The authors are grateful to W. Lampart and R. Weber for helpful discussions and to the Swiss National Science Foundation for the funding of this work.
References Magnante, P.C. J Appl Phys 40 (1969) 4437-4440 Kuizenga, D.J. IEEE J Quant Electron 17 (1981) 1694-1708 Roschger, E.W., Schwarzenbach, A.P., Balmer, J.E., Weber, H.P. IEEE J Quant Electron. 21 (1985) 465-469 Albrecht, G.F., Cnmeisen, M.T., Smith, D. IEEE J Quant
Ekctron 21 (1985) 1189-1193 Koechner, W. ‘Solid-State Laser Engineering’, Springer, New York, (1976) Kuizenga, D.J., Siegman, A.E. IEEE J Quant Electron 6 (1970) 694-708 Hall, D.W., Weber, M.J. IEEE J Quant Elecrron 20 (1984) 831-834 Soom, B., Balmer, J.E. (to be published) Born. M.. Wolf. E. ‘Princinles of Ootics’ Peraamon Press. New.York (1973) _ Soom, B. Diploma thesis, University of Bern, 1987 (unpublished)
OPTICS AND LASER TECHNOLOGY
Introduction Actively mode-locked/Q-switched Nd-YAG and Ndglass laser oscillators have been widely used to generate single, high-power, nearly transform-limited pulses of sub-nanosecond duration.‘-’ Owing to its broad linewidth (>20 nm), N&glass is theoretically capable of producing pulses of sub-picosecond duration.’ Insertion of Fabry-Perot etalons into the resonator cavity provides a convenient means for covering the complete range of pulse durations in such an oscillator between approximately 50 ps and 1 ns. Since stability requirements often make a fmed frequency operation of the mode-locking modulator preferable, the insertion of etalon plates enforces a change in the mechanical length of the cavity that may reach lo-20 mm for a particular etalon. This is usually accounted for by mounting one of the resonator mirrors on a precision translation stage. For practical use it thus seems desirable to know the amount of mirror translation required for a particular etalon thickness and reflectivity. It is the purpose of this paper to give exact values for this change in cavity length which has previously been shown6 to be a combination of two effects, namely (i) the change in optical length of the resonator cavity due to the refractive index of the etalon material and, (ii) the increase in round-trip time of the pulse in the cavity due to the phase shift generated in the etalon. The calculated values are then compared to experimental results from an existing N&glass oscillator.
experiments have shown that phosphate glasses, for example, exhibit only a minor inhomogeneous broadening,’ In a separate paper. we present pulsewidth/bandwidth product measurements for an actively mode-locked/Q-switched Nd:phosphate glass oscillator which are in excellent agreement with the homogeneous theory? For our calculations, here, we essentially adopt the time-domain analysis presented by Kuizenga and Siegman6 for a homogeneously broadened laser, in which the gain, modulator, and etalon transfer functions are all approximated by Gaussians. In particular, this is done here for the amplitude transmission of a lossless Fabry-Perot plate of thickness’h atid refractive index n, given byP r, = (1 - R)l( 1 - Rei8) with 6 = (47r/X) &co&
(2)
where R = rr’ is the product of the amplitude reflectivities of both reflecting surfaces (r = J in most cases), and 8 is the tilt angle of the etalon. The (round-trip) transmission function of the etalon in a Fabry-Perot type laser cavity is approximated near a particular transmission peak, o,, by the Gaussian r, = exp[-A(o
- w,)‘] exp[iB(o
- o,)]
A = 4/A&,
B = 4(R)‘f2/Aw e
where Aw, is the effective bandwidth given by
(34 of the etalon,
Aw, = (c/h’) (1 - R)/(R)“’ The authors are at the Institute of Applied Physics, University of Bern, CH-3012 Bern. Switzerland. Received 27 October 1987.
0030-3992/88/010036-03/$03.00 36
(3)
with
Theory The linewidth of the N&glass lasing transition is basically subject to inhomogeneous broadening mechanismsP However, fluorescence line narrowing
(1)
(4)
with h’ = nhcos6 (8 =O for most practical 01988
Butterworth
Et Co (Publishers)
cases). From
Ltd
OPTICS AND LASER TECHNOLOGY
LHG-8 3OW
r-5m
100%
7
-c-a
I
1,
output
e
Q- Switch (28 MHz)
mirrw r---
* * *
7
I
T’
1
Ethfl
Apwture
Fig. 1
Schematic diagram of the sctiwly mode-locked/Q-switched
M@de-locker 450 MHz) Nd:glass oscillator
(3) it is apparent that the etalon transmission function affects the spectrum in such a way as to introduce a phase shift proportional to B. We can now consider a Gaussian pulse (without frequency chirp) passing through the etalon. Its timedependent field amplitude is described by E(t) = EO exp(-at
‘) exp(-io,l)
(5)
where or is the mean optical frequency of the pulse and a is real and determines the Gaussian envelope of the pulse. The pulse spectrum is obtained from (5) by Fourier transformation as E(w) = EO/(2a)‘/’ exp[-(w
- orJ2/40]
(6)
Multiplying (3) and (6) and transforming back into the time domain, we obtain the time-dependent field amplitude, E’(r), after one round-trip through the etalon (assuming that or = 0,): E’(r) = (l +
2,1,2exp[
- (l;L-f)&]
exp(-iW)
considerations, the phasei shift (in frequency space) from (3) is seen to result n a delay B in the time domain. This can be und rstood if we remember that the pulse transmitted thr ugh the etalon is built up by constructive interference f the multiple reflections between the etalon surfac s. As a consequence, its peak is delayed in time by a li ite amount which depends on both the thickness an reflectivity of the etalon. This gives rise to an incre1 se in cavity round-trip time or. equivalently, a decreas(e in modulation frequency/,. For the case of amplitude1 modulation without etalon, fm is given by6 fmo = [4@‘c + g/A@]-
where L is the (optical) length of the cavity, g is the saturated round-trip gain t line centre, and Ao is the linewidth of the gain med t urn. For most practical cases, glAo < UC, so that f,,,, = 4/4L. The corresponding self-consistent solution for the circulation pulse in a resonator with etalon leads to the expressionrO
(7)
fm = [4(Uc+ g/Aw + 2(R)“‘/Aw,]-
The effect the etalon has on the pulse is seen to be two-fold. Firstly, it increased the duration of the circulating pulse by an amount which is determined by A and. consequently, by the effective bandwidth of the etalon. Secondly, and more interestingly for our
P
.E
AR=30%
-
l
which for an uncoated
R=25%
1
(9)
If now we wish to hold then modulation frequency at a fixed value. fmo. the cavity length has to be shortened to a value L’ = L - SL,, such! that&,,,(L) = f&L’). The resulting change in cavity liength (assuming g constant. which is not exactly trueiO)‘is then given by 6L, = ~c(R)“YAc+
-
(8)
’
= 2@h’/(l-R) etalon
(10)
becomes
6L,* = (n - 1)%‘/2h
(11)
In addition, due to its refractive index, the etalon causes a change in optical liength of the cavity given by 6L” = (n-1)h’ln
(12)
SO that the total correction for the cavity length due to insertion of the etalon amolints to
6L,,, = 6L, + SL, = 2Rh’V( 1-R) 0.1
1.0
and for the uncoated
EtaIon thickness h (mm) Fig. 2 Total change in cavity length against etalon thickness for two different surface reflsctivitiis and 8mO. Solid lines era obtained from
VOL 20 NO 1 FEBRUARY 1988
+ (n-l)h’/n
(13)
10.0
(14)
etalon
SL,,, = sL,* + SL” = (n*-l)/7’/2n
(14)
37
Experiment A schematic diagram of the actively mode-locked/ Qswitched N&glass oscillator used in these experiments is shown in Fig. 1. The active medium is a 6.35 mm diameter by 76 mm long LHG-8 phosphate glass rod with end faces angled at 3”. Brewster-angled acousto-optic devices are used for both mode-locking and Q-switching. The mode-locker driver frequency, fm. is 50 Mhz and the corresponding resonator length is 1.5 m. The output mirror (R = 70%) is mounted on a precision translation stage. A set of coated BK-7 etalons with thicknesses between 90 pm and 15 mm was used to obtain transformlimited pulses with pulsewidths ranging from 70 ps up to 1 ns. More details of the oscillator performance will be given elsewhere.’ For each etalon the resonator length was adjusted for optimum output performance. and the exact mirror position read from the translation stage. Fig. 2 shows the measured change in cavity length, c&,~. as a function of the etalon thickness, h, for different etalon reflectivities. All the etalons were inserted nearly perpendicularly to the resonator axis (19.e 0). Also shown for comparison is the calculated change in cavity length due to the increased optical path length alone (dashed line). It is seen that the experimental values are in excellent agreement with our model (solid lines). Noticeable deviations are seen only for the two extremal measurements (!I0 pm/2596 and 15 mm/30%). In the case of the thin etalon this is most likely explained by experimental uncertainties in the measurement of distances of order 10 pm. On the other hand, with the thickest etalon we approach the operational limit of our oscillator with respect to stable mode-locking,’ so that the slight drift-off of the
30
experimental effect
data point may be attributed
to this
Conclusions The change in cavity length caused by insertion of Fabry-Perot etalons into a mode-locked laser oscillator has been investigated in detail. Experimental data are shown to be in very good agreement with a model which describes the total length change as the sum of a phase shift in the etalon and the increase in optical path length due to the refractive index of the etalon. Such data can be useful, for example, for implementing automatic pulsewidth selection in short pulse laser systems.
Acknowledgements The authors are grateful to W. Lampart and R. Weber for helpful discussions and to the Swiss National Science Foundation for the funding of this work.
References Magnante, P.C. J Appl Phys 40 (1969) 4437-4440 Kuizenga, D.J. IEEE J Quant Electron 17 (1981) 1694-1708 Roschger, E.W., Schwarzenbach, A.P., Balmer, J.E., Weber, H.P. IEEE J Quant Electron. 21 (1985) 465-469 Albrecht, G.F., Cnmeisen, M.T., Smith, D. IEEE J Quant
Ekctron 21 (1985) 1189-1193 Koechner, W. ‘Solid-State Laser Engineering’, Springer, New York, (1976) Kuizenga, D.J., Siegman, A.E. IEEE J Quant Electron 6 (1970) 694-708 Hall, D.W., Weber, M.J. IEEE J Quant Elecrron 20 (1984) 831-834 Soom, B., Balmer, J.E. (to be published) Born. M.. Wolf. E. ‘Princinles of Ootics’ Peraamon Press. New.York (1973) _ Soom, B. Diploma thesis, University of Bern, 1987 (unpublished)
OPTICS AND LASER TECHNOLOGY