A study of mode-locking in a ruby laser operating near 77⁰K
Item Type text; Thesis-Reproduction (electronic)
Authors Osmundsen, James Frederick, 1944-
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Download date 27/09/2021 16:34:02
Link to Item http://hdl.handle.net/10150/554791 A STUDY OF MODE-LOCKING IN A RUBY LASER
OPERATING NEAR 77°K
; by
James Frederick Osmundseii
A Thesis Submitted to the Faculty of the
COMMITTEE ON OPTICAL SCIENCES (GRADUATE)
In Partial Fulfillment of the Requirements • For the Degree of
MASTER OF SCIENCE
In the Graduate College •
THE UNIVERSITY OF ARIZONA
1 9 7 f , STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfill ment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowl edgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the inter ests of scholarship. In all other instances, however, permission must be obtained from the author.
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
FREDERIC A. HOPy Dat< Assistant Professor of Optical Sciences ACKNOWLEDGMENTS
This thesis is dedicated to Pat for her unfailing encouragement and understanding.
iii TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS ...... V
ABSTRACT ...... vi
CHAPTER
1, INTRODUCTION ...... 1
2. COMPLETE MODE-LOCKING IN A LASER WITH A SATURABLE ABSORBER ...... 5
The Letokhov Model 11 Conditions Required for Complete Mode- Locking in a Laser with a Saturable A b s o r b e r ...... 17 Theory for Experimental Determination of the Parameter A ...... 27
3. THE EXPERIMENT ...... 31
The Alignment Procedure ...... 36 Mode-Locking the Laser ...... 39 Measurement of the Lasing Linewidth .... 46 Determination of the Parameter A and the Degree of Mode-tLocking ...... 60
4, CONCLUSION...... 72
LIST OF REFERENCES „ . 75
iv LIST OF ILLUSTRATIONS
Figure Page
1. Shape of a Pulse for Large N „ 8
2. Relationship Between N and p for Complete Mode^Locking ...... 26
3. . The Laser Components and Their Relative Positions in the Laser ...... 32
4. Flashlamp Power Supply ...... 35
5. An Output Pulse Train Recorded with the H.P. 183A Oscilloscope ...... 42
6. Mode-Locked Pulse Trains from a Ruby Laser Operating Near 77°K . . , ...... 45
7. The Fabry Perot Interferometer ...... 47
8. Some Interference Rings ...... 49
9. Schematic Diagram of Interference Rings. .... 51
10. Plot of Pumping Energy and Population Inversion Versus Saturable Absorber Absorbence . 63
11. Relationship Between t^ and A . 65
12. Plot of In KtdA/dt^) Versus t^ , 66
V ABSTRACT
This thesis describes an experiment demonstrating passive mode-locking in a ruby laser operating near 77°K.
First, the theory is reviewed and the Letokhov model is used to define the conditions required for complete passive mode-locking to occur in a laser using a saturable absorber„
Also included is a review of the theory required for experimental determination of the laser parameters which determine the conditions for passive mode-locking with a saturable absorber. Next, a description of the actual experiment is given. The laser and the procedure for its proper alignment are described, Then a description is given of the actual procedure that was followed to produce mode- locking, to measure the lasing linewidth, and to determine the degree of mode-locking.
vi CHAPTER 1
INTRODUCTION
Mode-locking in ruby lasers at room temperature has been observed by a number of investigators (Mocker and
Collins 1965; Sacchi, Sancini, and Svelto 1967; Mack 1968;
Bradley, Morrow, and Petty 1970)„ At room temperature the linewidt'h of the ruby R^-line is about 360 GHz. (Schawlow
1961), and mode-locked ruby pulses as short as 2 ps. have been observed (Mack 1968) .
As the temperature of a ruby gain medium is reduced from room temperature, the characteristics of the R^-line are seen to change. The homogeneously broadened line is t seen to narrow down to about 20 GHz. at 100QK« Below this temperature the single line is seen to split into a doublet with a 12 GHz. separation; and at 77°K the components of the doublet are completely inhomogeneously broadened and completely resolved, each having a linewidth of about 3 GHz. in a high quality single crystal ruby. This splitting is just the crystal field splitting of the ruby ground state into two doubly degenerate levels which are unresolved at room temperature because the width of the R^-line is so large. 2
The positions of the energy levels of ruby are also strongly temperature-dependent„ Between 300°K and 7?°K the
R^-line is displaced from 6943 A to 6934 A, Also, with the large reduction of the linewidth with decreasing tempera ture, a very large increase in the gain at line center is seen to occur„
Because of these changes in the R^-line, it is of interest to see if they would lead to any fundamental changes in the mode-locking process in a ruby laser operated at 77°K.
One change is an observed increase in the minimum pulse duration because of the decreased linewidth of the lasing transition. And one theory (Letokhov 1969) indi cates that this reduced linewidth and the reduced number of oscillating modes should make mode-locking more difficult to achieve. However, according to the analysis presented by
Kuznetsova (197 0) , the narrower linewidth and fewer oscillating modes should instead make mode locking easier to achieve.
Another effect of the reduced linewidth is enhanced frequency pulling of the lasing modes. At room temperature this effect is negligible in a ruby laser; but at 77°K the frequency pulling effect, in the linear approximation, is as large as 4% (Slegman 1971, p. 408), The modes should be pulled toward the center of the line; the separation between modes should be reduced; and the actual lasing linewidth should be at least 4% less than that expected
neglecting frequency pulling. In addition, there should be
a small nonlinear frequency pulling effect because of the
inhomogeneous groadening (Maitland and Dunn 1969, p. 246).
This nonlinear frequency pulling should prevent the lasing modes from being equidistant in frequency and might have an effect upon the quality of the mode-locking if it is large enough.
Another change that also should be looked for results from the inhomogeneous broadening of the R^-line,
This is a possible increase in the pulse duration toward
the end of the pulse train (Kafri, Kimel, and Shamir 1972),
This results because in an inhomogeneously broadened medium
the population inversion is frequency dependent. Thus, over the duration of a pulse train the inversion responsible for
the modes in the wings of the line is depleted sooner than
that at the center of the line, and the lasing linewidth gets narrower as the outer modes cease oscillating.
Finally, it is reasonable to expect, since the Fe
line splits into a doublet, that lasing and mode—locking might actually occur simultaneously at two different wave
lengths, Evidence to support this possibility will be
shown.
Chapter 2 discusses the theory of complete mode-
locking in a laser with a saturable absorber. The Letokhov model is used to determine the conditions required in the laser for complete mode-locking to occur„ Then the theory for experimental determination of the degree of mode- locking in a given laser is discussed.
Chapter 3 describes the actual mode-locking of a ruby laser operating near 77°K and applies the theory to actual experimental determination of the degree of mode- locking in this laser. CHAPTER 2
COMPLETE MODE-LOCKING IN A LASER WITH A SATURABLE ABSORBER
The output spectrum of a free running ruby laser can be described as a superposition of contributions from all of the oscillating resonant cavity modes. If the laser is considered to be a linear device in which the oscillating modes are viewed to be completely independent, it is possible to describe the field inside a resonator of length
L in terms of a superposition of the separate oscillating modes. The simplest case is when only the axial modes of the resonator are favored; and the nth mode has the form
E (x, t) = e sin [ (m+n) ^-] sin [ (w tnm) t+(f> ] . (1) ii i_j cj n
The number m is an arbitrary integer such that m+n = 2L/A.
The frequency w is the frequency separation between modes and is equal to ttc/L , , the frequency at line center, is equal to m ttc/L. The amplitude of the nth mode is en , which is determined by the line shape; and cf>n is the phase of the nth mode. If the number of lasing modes is N , then the total field is 6
At some arbitrary point in space, for example at the output mirror, the total electric field as a function of time for a free-running laser can be written as
E (t) = £ enexp{i [ (a)o+na)) t+(}>n] } . (3) N
Note that if the phases ^ are fixed and the modes are equally spaced in frequency, then (3) is periodic in the period T = 2tt/u) = 2L/c, which is the round-trip transit time of the resonator. That is,
E (t+T) = £ enexp{ i [ (wo+nm) (t + ^-)+(}>n] } N
0) = £ e exp{i [ (m +na)) t+<|) ] }exp{i [2it (— + n) ] } Xl C3 Xl CO
= E(t) (4) since w^/w is an integer, and therefore, w exp [ 2TTi (— + n) ] = 1, (5)
When a laser is mode-locked, all the phases are equal and constant (or vary by an amount which is proportional to n). Thus, the phases may be set equal to zero if a proper shift of the time reference is made. And if for simplicity
it is also assumed that all N oscillating modes have equal amplitudes of e^ = 1, then for a mode-locked laser (5) can be written as 7
(N-l) 2 sin(Nwt/2) E (t) = Z exp [i (to +n(jo) t] = exp [iw t] o sin (a)t/2) — (N-l) ° 2
Since the average laser output power (averaged over a period 2tt/o)o < t < 2w/w) is proportional to E (t) * E* (t) , the output power is given by
sin (Nwt/2) (7) sin (wt/2)
From (7) it can be seen that:
1, For a large number of locked modes N the laser
output is in the form of a pulse train with a
period T = 2t t / w = 2L/c.
2, The peak power of a pulse is N times the average
power.
3, The peak field amplitude is equal to N times the
amplitude of a single mode,
4, The pulse duration, defined as the time from the
peak to the first zero (which for large N is also
approximately the pulse width at the half-power
points) is t = T/N,
The number of oscillating modes is just the lasing line- width divided by c/2L. Figure 1 shows the shape of a pulse for large N, In a real laser the mode amplitudes are determined by the lineshape of the gain medium and therefore they are not equal. The result is that in real life the - -*-4 TIN XV N 3T//V t
L
Figure 1. Shape of a Pulse for Large N pulse shape is approximately the same except that in a completely mode-locked pulse train there are not any secondary pulses between the major pulses.
Up to this point the field distribution in the resonator has been treated in terms of standing waves and the discussion of mode-locking has been limited to con sideration of the time dependence of the field at a given point in space. However, it is also illuminating to con sider the spatial consequences of mode-locking. Since a standing wave can be considered as the sum of two waves traveling in opposite directions, it is possible, after some trigonometric manipulation, to write the total field in the resonator as a sum of traveling waves.
N/2 „ E (x, t) = 1/2 1 e {cos [ (m+n) — (x-ct)-<(> ] -N/2 n L n
- cos [ (m+n) jr(x+ct) +
Setting the 's equal to zero, the field in a mode-locked laser may be written as:
N/2 E (x, t) = [ E e exp [ (in^) (x-ct) ] ] cos^(x-ct) -N/2 n L L
N/2 + [ £ e expl (inj-) (x+ct) ] ] cos^fx+ct) (9) -N/2 " L L
At any given time t the radiation distribution outside the boundaries of the resonator consists of a family of pulses 10 having a width (in space) of approximately L/N and spaced by 2L which are traveling to the right at the velocity of light, A second such family is traveling to the left.
Inside the resonator a single pulse of width L/N bounces back and forth between the mirrors at the velocity of light.
Assume that a saturable absorber is placed inside the resonator. Such an absorber will then favor the super position of modes represented by (9) over other possible radiation distributions. This is because (9) corresponds to the shortest possible pulses and thus to the highest possible energy density. Consequently, they will suffer the smallest loss in the saturable absorber (_Garmire and
Yariv 1967), Once some mechanism causes all of the phases to be equal the laser tends to oscillate with the modes locked.
Actually, it is possible to have more than one light pulse in the resonator. If the saturable absorber is just placed arbitrarily at some point in the resonator, one must consider the possibility that, instead of only one light pulse, there can actually be two which meet at the saturable absorber, forming a standing wave of higher amplitude, and consequently an absorption low enough to allow two pulses of lower than usual amplitude to pass simultaneously. However, in each, complete period each light pulse must also go through the saturable absorber once by itself, resulting in additional absorption, And, although it. is possible to obtain two families of pulses in the resonator, a single family is ordinarily preferred unless the saturable absorber is placed at the center of the resonator, In this case the two pulses never have to go through the saturable absorber alone, and pulses separated by L/c rather than 2L/c are clearly preferred. If two families of pulses are able to occur as a result of this mechanism it is clear that they will be separated in time by 2d/c where d is the distance from the saturable absorber to the end of the resonator.
The Letokhov Model
Up to this point the discussion has only dealt with what is observed once mode-locking has occurred in a laser with a saturable absorber, No mention has been made as to how the mode-locked condition occurs initially, There are a number of theoretical descriptions of passive mode- locking (Smith 1970), However, it is most likely that the
Letokhov (19 69) model gives the most accurate description of passive mode-locking in solid state lasers such as ruby and neodymium glass.
In many descriptions the model commonly used is the
"Cutler model.'1 In this model the saturable absorber is treated as a nonlinear element called an "expander" (Cutler
1955) which produces a periodic array of "sidebands" or
"combination tones" to which the laser modes become locked.
The pulses become shorter and shorter until they reach 12 minimum duration as a result of the "expander" action which creates "sidebands" successively farther and farther apart.
In the Letokhov model random fluctuations or intensity bursts in the initial laser radiation, resulting from the random interference between the radiation in a large number of modes, are ultimately responsible for the train of pulses in the laser output. An intensity burst in which the energy is concentrated in,a short time comparable to the relaxation time of the saturable absorber ultimately appears in the initial laser radiation in the resonator and bleaches the saturable absorber, allowing radiation to pass unattenuated for a short time. Intensity bursts with smaller energy are completely absorbed. The result is a single intensity burst that builds up and bounces back and forth in the resonator to produce the observed output pulse train. During build-up, as the intensity burst interacts with the saturable absorber, only the peak portion of the burst which concentrates the greatest energy in the absorber relaxation time is allowed to pass unattenuated. Thus, with each successive pass through the absorber the burst is shortened slightly in duration until a limiting duration is reached. In contrast to the "Cutler model.," however, the pulses start with approximately the same short duration that they ultimately attain. This action of the saturable absorber tends to stabilize the length Gin space 1 of the pulse to about c/'frv f 13 where Av is the lasing linewidth. As has been shown, such a pulse length corresponds to the case when all the phases
Also, experimental evidence tends to support the
Letokhov model, Duguay, Shapiro, and Rentzepis (1967) have done experiments in which they observed spontaneous pico^ second pulses in the output radiation of neodymium glass and ruby lasers at room temperature operating in the Q-switched regime without a saturable absorber and also when operating in the free-running regime.
The analyses of Letokhov (1969) , Kuznetsova (1970), and Hausherr, Mathieu, and Weber (1973) show that the saturable absorber placed at one end of the resonator serves two functions:
1. It serves to select one ultrashort light pulse per
resonator round-trip time from the initial random
noise pulses„ ^
2. It compresses the pulses to the minimum duration
allowed by the gain medium linewidth,
According to Letokhov (196 6) and Mandel and Wolf
(1965) f up until the instant laser oscillation occurs the field amplitude and the phase of the radiation in each axial resonator mode are random functions of time, determined by the statistical properties of the spontaneous emission in the mode. In the classical limit, when the average number of photons in the mode is (nQ ) >> 1, the field in each mode en (t) is a narrow band Gaussian noise whose random amplitude obeys the Rayleigh distribution, and the phase (t) is uniformly distributed in the interval (0, 2t t ) . As laser oscillation begins, the field in the mode is transformed into a coherent oscillation, with a relatively stable amplitude. However, if there are no mechanisms to cause phasing of the modes, then the phases of the different modes are independent and are randomly distributed in the interval
(0 , 2t t ) . Therefore the total field in a large number of these modes remains as before a Gaussian noise, regardless of the coherence of the field in each mode. Consequently, the fluctuations of the amplitude A (t) of the total field obey the Rayleigh distribution:
W (A) = — -y- exp ( (10) W(I) - exp (- ^yy) • (11) If the width of the frequency band occupied by the axial modes of the laser is Av, then the average duration of an intensity fluctuation is: The probability P(3) of a fluctuation spike with amplitude 3 times larger than the average amplitude fluctuation is given by the relation: P (3) = f W (I) dl = e 3 (13) 3(1) The intensity fluctuations will occur on the average at a frequency of the order of Av. Consequently, a fluctuation spike with amplitude 3(1) occurs on the average within a time interval t (3) given by the expression t (3) (14) If the axial modes are strictly equidistant in frequency, then the random intensity I(t) is a periodic function with period T, The deviation of the modes from being strictly equidistant and the drift of the phase of the field in the modes prevents the spontaneous intensity fluctuations from being exactly periodic. An appreciable change in the random function I(t) will occur during the resonator round-trip time T since the frequency 6v is not small compared to 1/T, where 6v is the average deviation of the difference between the field frequencies of the neighboring axial modes because of nonlinear frequency pulling and because of random drift of the phases, In order for a spontaneous fluctuation spike having an ampli tude of 3(1) to occur in every resonator round-trip time T the following relationship must hold 16 t (3) S T (15) Then the maximum excess 3 of the amplitude of the fluctua tion spikes over the average value is from (14) and (15), 3 * In N (16) In neodymium glass and ruby lasers operated at room temperature with resonators of reasonable length the number of excited axial modes can reach approximately 10^ to 105. Whereas, in a similar ruby laser at 77°K this number is of the order of 10. In a neodymium glass or a ruby laser at room temperature the most intense fluctuation spikes can be as large as 10times the average fluctuation amplitude. In a rubylaser at 77°K the mostintense fluctuation spikes are only about 2 or 3 times the average fluctuation amplitude. Although the amplitude of the largest spontaneous fluctuation spikes in some lasers can be many times larger than average, the main radiation energy is concentrated in the background of average amplitude spikes. This follows because the total number of spikes in the interval T is of the order of N and therefore it is much larger than 3, A saturable absorber must discriminate against these back^ ground pulses and ensure the predominant development of the most intense fluctuation spikes. 17 Conditions Required for Complete Mode-Locking in a Laser with a Saturable Absorber Assume: (1) that the resonator round-trip time T is much longer than the absorber relaxation time so that after the passage of a pulse the absorber coefficient is able to return to the low intensity value x q before the passage of the next pulse; (2) that the pulse duration t is much greater than the relaxation time of the absorber so that the absorption is determined by the instantaneous value of the intensity in the pulse I (t); and (3) that during the first stages in the development of a pulse, the pulse energy is small so that it is possible to neglect gain saturation. Making these assumptions, the following equation, derived by Letokhov (196 9), can be used to describe the evolution of an ultrashort pulse in a laser from the initial bursts of noise fluctuations. (17) (f)K (t) is the normalized intensity of the pulse after the Kth pass and is equal to I (t)/Is , is the saturation intensity of the absorber, is the gain in the active medium in the Kth pass, and y is the loss per pass in the resonator without the absorber, The dependence of the gain on k is due to the pumping of the active medium during the delay time , from 18 the instant when the gain reaches its threshold value to the instant when the absorber becomes saturated. This delay time is given by the expression 2t I , a T 1//2 T1 = Tt^ ln(:r - / 2^ )] <18> o o o p where t^ is the time constant characterizing the growth rate of the gain as a result of pumping at the instant that the lasing threshold is exceeded, (19) t = t o Assuming that the growth rate of the gain is small, the gain in the Kth pass can be written in the form 01 = “o + TK (20) (K) ft t = t where a is the threshold gain o J ao = a(to ) = Y + xo (21) During the time the gain a is greater than the threshold gain a and thus the gain is greater than the loss. Averaging over this delay time one obtains an average gain a a = “o + It I1 = ao(1 + (22) Assuming that this average gain is constant during the 19 period T., Equation (17) can be integrated to obtain an implicit solution for the normalized intensity of an output pulse 4>K (t) (p (t) +A A-l exp[(a-xo-Y )-K] = (23) where A is the parameter (a-x -y) A = — -7)-- (24> Equation (23) describes the evolution of the pulse during the time T-^, where #^(t) is the normalized spontaneous noise intensity in the laser and is equal to (t)/Is . The parameter A is the parameter determining this evolution. The evolution of the initial random noise to an intensity of (f>K = 1 is called the linear stage. This requires kg passages, where k is equal to t/t^ and t^ is equal to T/2. Thus, _ 21n(l/ Defining an average gain coefficient a as the gain coeffi cient after k^/2 passages by ks tL a = ao + y * — (26) P Equation (25) can be written as 20 l n d / 6 ) ks = = — • (27) a - y - xq Using Equations (25), (26), and (27) it can then be seen that 2 (a-y-x ) , t 1/2 2- = — • 12 -r— In (1/(1) )) . (28) xo xo t p ° This result is useful in determining the proper laser operating regime and the low intensity absorption coeffi cient required of the saturable absorber so that the saturable absorber discriminates against all but the single most intense pulse in the resonator round-trip time, Selection of this single most intense pulse constitutes complete mode locking of the lasing modes. One begins by calculating the probability W(y) for the two largest pulses in the time T to have an intensity ratio between v = ^ an<^ 1 at the end of the linear stage. I^ is the intensity of the largest pulse and I2 is the intensity of the next largest pulse. This probability is obtained by integrating (Papoulis 1965, p. 196) dW = N 2 • p(I1) ' p(I2) • dl1 • dl 2 (29) over the values of I^ and I2 that satisfy the conditions 21 I max < i1 < -», i.max (I) In N (30) The probabilities p(I^) and p(I2) are given by Equation (11). The actual integration then leads to the following expression for the probability W(v). W(v) = 1/2 [t^ - • N (1 V)-l] (31) The fact that W(v) can be larger than one just means that it is possible for more than two pulses with an intensity ratio between v and 1 to occur in the round trip time T . Note that the probability of having two pulses in the interval T is smallest when the two pulses have nearly the same intensity, that is when v is nearly equal to one. It is this region of small W(v) that is of interest since the saturable absorber can easily discriminate between large and small pulses. What is of most interest is to determine the parameters required in order for the saturable absorber to discriminate between two pulses with nearly identical intensities. When the two pulses have nearly the same intensity so that v is nearly equal to 1, the probability W(v) can be approximated by W(v) = 1/2 [N ^1 V ^-l] 1 - v << 1 (32) 22 During the linear phase before the pulses become intense enough to bleach the absorber the intensity ratio v is constant. Since N is the number of oscillating axial resonator modes at the end of the linear stage, W(v) repre sents the probability at the end of the linear stage. However, once the pulses reach the saturation intensity for the absorber this intensity ratio is modified by the absorber. The nonlinearity of the absorber changes the statistical behavior of the radiation from that of quasi- thermal light to that of mode-locked pulses Griitter, Weber, and Dandliker, 1969), At the end of the nonlinear stage the absorber is bleached and there is a modified intensity ratio D. By using Equation (23) it can be shown that this modified intensity ratio is related to the original intensity ratior by the relation D = v 1/A, (33) where A is given by 2 (ct-y-x ) 2 (ot-y-x ) A = — ^ ------X - <34> Using Equation (28) it follows that A = r- ’ [2 ’ In (l/4> ) ] (3 5) xo Lp 23 The final probability for the appearance of two pulses in the time interval T having an intensity ratio between D and one is W(D) = 1/2 [N (1’~D )_i] (36) Like W(v), W(D) can also be larger than one. However, for complete mode-locking the absorber must select a single pulse, and therefore the probability W(D) must be made as small as possible. It can be seen that in order to have complete mode- locking with the shortest pulse duration, the parameter A and the number of lasing modes must be kept small. To keep A small, the low intensity absorption coefficient x q should be as large as possible, depending upon the maximum value of the gain coefficient a , At the same time the nonsaturable or linear losses y in the resonator must be kept small. Also, the gain medium should be pumped so that the laser oscillates just above threshold. This maximizes the parameter t^ and therefore the average gain a is then only slightly larger than the total loss per pass, y + x q . Thus, a large number of passages are required for the pulses to build up from the initial noise, A large number of passages enhances the compressing and discriminating effectiveness of the saturable absorber, It is desirable to reduce the number of oscillating modes as much as possible, but without decreasing the lasing 24 line width Av. This is accomplished by keeping the resonator as short as possible. A qualitative explanation can be given for why N should be kept as small as possible by noting the fact that a large number of oscillating modes increases the proba bility of several commensurable high^intensity spikes appearing in the initial field and therefore a higher order nonlinearity (larger value of 1/A) is necessary to reliably select out the most intense single spike, A quantitative and enlightening relationship between the number of oscillating resonator modes and the required order of nonlinearity required for complete mode locking has been derived by Kuznetsova (197 0), The sufficient condition for complete mode-locking to occur, with a probability greater than n and with the most intense pulse having an energy M times larger than the energy emitted in the period T in all the remaining pulses is expressed by the relation -r- ' ^ T N v i )+ 1 1 i n C37) a = (M(N-l) J 1//p (38) P = 1/A (39) If N is 10 or larger this condition can be expressed more simply in the form of a dependence of p on N: P > -~1- N)in+(];/n^ ln M (40) 25 In Figure 2 the curve corresponds to the equality in (37) with n = 0.5 and M = 9. The region above the curve represents those values of p and N for which one can expect with a probability greater than 0.5 that in a round-trip time T only a single intense pulse will be seen. And the energy of this pulse will exceed by at least a factor of 9 times the energy in all the remaining pulses, Figure 2 clearly demonstrates the fact that the larger the number N of oscillating modes the more difficult the achievement of complete mode-locking„ For a large N complete mode-locking requires larger values of the non linear transformation index p. Thus, one would be required to reduce the laser gain a closer to the threshold value or increase the initial absorption of the saturable absorber. In any given laser p can be increased only to a certain limit. This limit is imposed by the restraints upon the maximum possible value t . t is limited by the requirement that the delay time t^ be smaller than the duration of the pumping and smaller than the relaxation time of the gain medium. The figure also shows that the mode-locking condi tion is met to excess in lasers with few oscillating modes„ On the other hand in lasers with a large number of oscillating modes and a limited value for p a slight varia tion in the initial parameters can, by reducing the p index, disrupt the conditions for complete mode-locking. There fore, these lasers require stricter control of the pump 26 loo 10 I o loo N Figure 2, Relationship Between N and p for Complete Mode^ Locking power and the ratio of linear and nonlinear losses in the resonator than do lasers with fewer oscillating modes. Theory for Experimental Determination of the Parameter ^ From experimental measurements and Equation (35) it is possible to determine the parameter A for a given laser. Spectroscopic measurement of the lasing linewidth enables one to determine the number of lasing modes. It can be shown from the results of She and Tan (1966) that as a result of pumping the normalized inversion in the active medium below and near threshold is given by l 2t n'(t) = [l-2e p] (41) a nt where n^ is the normalized threshold inversion for the laser without an absorber and where t^ is the pumping time constant, If the pump is turned on at t = 0, a time t ^ , is required for the inversion to reach its threshold value. When the laser is at threshold, t = t^ and th n , 2t n 1 = n ' = --- = — [l-2e P] (42) a a n t n t The normalized threshold inversion n^_ is given by 28 Where & is the length of the gain medium and aaQ is the absorption per unit length in the unpumped gain medium. A Also it can be shown that n ^ is related to the concentra tion of the saturable absorber by the relation n (44) where o s is the cross section for the absorber molecules and Ns is the number density of the absorber molecules. o^ and are the cross section and number density of the ions in the gain medium respectively. Experimentally it is observed that, as the dye concentration in the absorber cell is increased, the pumping energy (measured in terms of power supply voltage) required to just reach oscillation threshold also increases. It is observed that this relationship is linear; and, since n 1. is proportional to the pumping energy (at least in a dilute system), a plot of pumping energy versus absorber molecule concentration is also a plot for n '. versus ai absorber molecule concentration. The plot is a straight line having the form n (45) ai KNS + 1. The zero-absorber-concentration intercept is necessarily equal to one, and K can be determined by measuring the actual slope of the line, 29 Experimentally, one can also choose a constant pump energy and then plot the relationship between the time t^ and the absorber concentration. A plot of these data essentially represents the following equation fcd 2t 1 - 2e P = rr [KN +1] , (46) From this equation and the experimental plot it is then possible to determine the value of the quantity nfc and the pumping time constant t^ for the given power supply voltage. Taking the derivative of Equation (47) with respect to t , one has d td dN 9 , 2t KdtT = K 7 (2F-)e P - H?) d t p Then taking the natural logarithm one has dN 9 t , I n [K-ttt—] = ln[±-(^)] - o T ~ ’(48) d n t p p Thus for each experimental value found for t^, one can measure from the experimental plot a value for dN^/dt^. Then another plot of the natural logarithm of K (dNs/dt^) versus t^ on semi-log paper should yield a straight line as indicated by Equation (48), The measured slope of this line gives the value for the quantity l/2t^ and the extrapolated intercept gives the value for the quantity 30 )]. Then nt and can be determined from 2 -(-slope) = exp (intercept) (49) and — = (-slope) (50) P The parameter is also needed in order to deter mine A and can also be determined by experimental methods. The saturation intensity could probably be measured for a desired saturable absorber by illuminating it with a ruby laser which was operating at 77°K and measuring the laser intensity required to just bleach the absorber. The initial noise intensity in the laser could be measured with a sensitive detector by removing the saturable absorber and the output mirror and measuring the intensity of the light emitted spontaneously by the gain medium into the solid angle occupied by the resonator modes which lase ordinarily, CHAPTER 3 THE EXPERIMENT Figure 3 illustrates the laser components and their relative positions in the laser. The laser rod used was a high quality, pink ruby 3+ having a Cr concentration of 0.05%. The boule from which it was cut was grown by the Czochralski method, and the ruby rod itself was of good scatter free optical quality. The length of the rod was 7 cm and its diameter was 1 cm. The ends were cut at Brewster's angle and the rod was oriented so that the radiation was horizontally polarized. The c-axis of the rod was oriented so that it was in a vertical plane at an angle of 60 degreees to the rod axis. In order to cool the rod it was held inside a vacuum chamber by a copper cold finger which was attached to the bottom of a liquid, nitrogen reservoir. Although the temperature of the ruby rod was not monitored during the experiment, it is still reasonable to assume that its temperature was in a range near 77 °K. Pumping caused the temperature of the rod to rise. However, after pumping, the rod again cooled off rapidly because of the large thermal conductivity of ruby at these reduced temperatures. The rod 31 ALIGNMENT LASER OUTPUT TRANSVERSE MO RUBY SELECTOR ROD DYE _ CELL VACUUM _ CHAMBER Figure 3. The Laser Components and Their Relative Positions in the Laser 33 was allowed to cool sufficiently by firing the laser at three minute intervals. Two water cooled flash lamps were used to pump the ruby rod. This was accomplished by enclosing the rod and the flash lamps in a silvered rectangular box. Each flash lamp was 3 cm from the ruby rod. A rectangular box is not a very efficient way to couple energy from the flash lamps into a laser rod, but it works in this case because a large population is not required. Because of the narrow line width of the lasing transition at low temperatures, the gain for a given per cent inversion is higher than that at room temperature. The ruby rod, the flash lamps, and the flash lamp reflector were all enclosed in the vacuum chamber. Laser radiation entered and left this chamber through high quality glass windows. The front and back surfaces of these windows were not parallel but instead formed a wedge. The reason for this was to eliminate all parallel reflecting surfaces from the resonator cavity aside from the two end mirrors. Parallel reflecting surfaces in the resonator cavity act as second resonator cavity and thus act as a filter to limit' the number of lasing modes and thus the lasing linewidth. The laser resonant cavity was 125 cm long and was bounded by two plane and parallel mirrors. The output mirror had a reflectivity of 74.5% at the ruby wavelength. The back sufrace of this mirror was tilted at a wedge angle 34 of several degrees with respect to the front surface to• prevent mode selection. A wedge was not required on the 100% mirror, and the back surface of this mirror was approximately parallel to the front surface. The dye cell, which contained the saturable absorber dye, was constructed from two high quality glass wedges separated by a 2 mm thick spacer. The space between the wedges contained the dye. The dye was injected into the cell through a small stoppered hole in the spacer and was replaced after every five or six laser shots. The purpose of the transverse, mode selector was to restrict laser oscillation to the lowest order transverse mode or, TEMo q mode. It consisted simply of a variable aperture. It was found that the most reliable TEM mode oo operation occurred when the diameter of the aperture was set equal to 3 mm. The flash lamp power supply was designed so that the flash lamp output approximated as close as possible a square pulse with a duration of about 1.5 ms. Figure 4 shows the circuitry used for the flash lamp power supply. The high voltage power supply used to chargethe capacitor bank was of the type used to power photomultiplier tubes. Its output voltage was adjusted by a digital thumb wheel switch, and the voltage could be set to within about one volt of the desired voltage. An output current meter Figure 4. Flashlamp Power Supply 36 indicated when the capacitor bank was charged to the desired voltage. * The laser output was detected with a fast response (1.0ns) PIN photodiode (Motorola type MRD510) and a Hewlett- Packard HP 183A oscilloscope (250 MHz bandwidth). A small fraction of the laser output beam (7.6%) was reflected out of the main beam with a glass-plate beam splitter. This light was collected by a lens and focused onto the photo diode. A lens with a focal length of 50 mm had to be used \ in order to focus the entire beam onto the small photodiode surface. A spike filter for the ruby, wavelength having a transmission of 30% was used in front of the photodiode to filter out stray light from the flash lamps, and a neutral density filter with neutral density of 1.9 was used in front of the spike filter to prevent saturation of the photodiode. Calculations indicate that the peak power of the most intense mode-locked pulses seen was about 700 to 800 watts. The Alignment Procedure The first step in the procedure for aligning the. laser was to properly align the laser rod using a Ne-Ne alignment laser as a horizontal line of reference. The laser rod was first loosely clamped into the cold finger and then rotated about its axis so that the Brewster's angle end-faces were in a vertical, plane. This was accomplished by observing the laser beam reflected from one of the end 37 faces, making certain that this reflected beam was horizontal. The cold finger was then clamped tight around the laser rod. A small amount of silicone grease was applied to the cold finger before tightening to provide good thermal contact between it and the laser rod. Next the vacuum chamber cover minus windows was attached to the bottom of the dewar. Then the entire dewar and vacuum chamber were rotated and translated so that the alignment laser beam entered the center of one laser rod end-face and exited at the center of the other end-face. Next, plane parallel vacuum chamber windows were installed, and the dewar was rotated so that the alignment laser again entered and exited at the centers of the laser rod end faces. The vacuum chamber was designed so that both the laser rod end-faces and the windows would be at their respective Brewster's angles. The next step was to replace the plane parallel windows with the wedged windows. The total effect of this was to shift the angle at which the alignment beam entered the laser rod since a wedged window acts like a prism. The dewar was again rotated until the alignment laser beam entered and exited at the centers of the laser rod end-faces. However, the vacuum chamber windows were no longer at Brewster's angle; but the small departure from the correct angle was not significant and did not affect laser operation. 38 Next, the dye cell was filled with acetone and placed in position so that the alignment laser beam passed through the center tif the cell windows. The output mirror was then positioned so that the alignment beam, after leaving the dye cell, hit the mirror center. The output mirror was then adjusted so that the alignment beam was reflected exactly back upon itself. Finally the 100% mirror was placed in position and adjusted so that it too reflected the alignment beam exactly back upon itself. At this point the dewar was filled with liquid nitrogen and the cold finger and laser rod were allowed to cool down for about 15 minutes. Final alignment of the mirrors was then accomplished by looking at the alignment beam projected upon the wall. When the mirrors were parallel a well defined and well centered Fabry-Perot ring pattern was seen. Finally, the transverse mode selector was placed in position so that it was centered upon the alignment beam and the aperture was adjusted to 3 mm. As a final alignment check the laser was fired without a saturable dye in the dye cell, just acetone, and the output was observed on the wall. No lasing occurred until the power supply voltage had been increased to 2170 volts. When lasing occurred one could see a quite distinct red spot. Lasing did not occur when the power supply voltage was reduced below this voltage by as little as 5 39 volts. It was impossible to reduce this threshold voltage by making minor adjustments to the mirror alignment; and thus, it was assumed that the laser was properly aligned. Mode-Locking the Laser It was not known what dye concentration would be required to produce mode-locking. Determination of this parameter was the next step. Cryptocyanine in acetone was chosen as the saturable absorber. The basis for this decision was that Mack (1968) used cryptocyanine in acetone to produce reliable mode- locking in a ruby laser at room temperature. First, a working solution was made. It consisted of 2.667 mg of cryptocyanine dissolved in 50 ml of electronic grade acetone. The absorbence of this solution was measured in a 2 mm thick cell (the same thickness as the dye cell used in the laser) with a Cary-14 spectrophotometer. The absorbence was measured to be 3.57. Absorbence is defined by the relationship A = 0N&, (51) where a is the cross section for the absorption process, N is the number density of absorbers, and £ is the thickness of the absorbing medium. Thus the absorbence is linearly related to the dye concentration and the thickness of the dye cell. The transmission of a dye cell is related to its absorbence by the relation A = log^Q — . (52) Dye solutions having transmissions of approximately 30%, 25%, 20%, 15%, 10% for a thickness of 2 mm were made by appropriate dilution of portions of the working solution with acetone. The operation of the laser was then examined using these separate dye samples as saturable absorbers, in the laser dye cell. With the 30% solution, no mode-locking effects were seen in the laser output. The power supply voltage required to reach lasing threshold was increased only slightly over th t required with no saturable absorber. The laser output, as seen from the oscilloscope trace, looked much the same as that for laser oscillation without a saturable absorber. The only observed effect upon the laser output was a decrease in the number of relaxation oscillation spikes, •A with a corresponding increase in the time between spikes. The amplitude of the spikes was also increased over that of the spikes with no saturable absorber. With decreasing saturable absorber transmision of from 30% to 25% and then to 20%, the number of relaxation oscillation spikes decreased; the time between the spikqs increased; and the amplitude of the spikes increased. ' 41 When the saturable absorber transmission was reduced to 15%, the laser output showed evidence of mode- locking. The power supply voltage required to just produce laser output was 2 350 volts. The laser was being operated just above threshold, and Figure 5 is an accurate drawing of one of the better pulse trains produced by the laser. A drawing had to be substituted for an actual reproduction of the photograph of the oscilloscope trace because the writing speed of the oscilloscope, camera, and Polaroid type 410 film combination was not sufficient to properly record a reproducible image. The pulse train was approximately 100ns long, and the peak output power was approximately 700 watts. The pulses were regularly spaced with a repetition time of approximately 9ns, which is equal to the resonant cavity round trip time 2L/C for a resonant cavity having a length of 12 5 cm. The duration of each pulse measured at the half power points appeared to be approximately 3ns. Since the rise time of the HP18 3A oscilloscope and photodiode was at least 1.5ns, the actual pulse duration was probably shorter than 3 ns. No attempt was made to use the 10% solution to produce mode-locking except to observe that the required power supply voltage was above 2400 volts. This was near the upper limit for safe operation of the power supply and 42 Figure 5. An Output Pulse Train Recorded with the H.P. 183A Oscilloscope . 43 flash lamps, and working at a voltage any higher than this was deemed unsafe. « The spot.on the wall appeared to be round without any lobes, and when the beam was examined at a distance of 10 meters the beam cross-section still consisted of one circular spot. This tends to indicate that the laser was oscillating in the TEMq o mode. A burn pattern would have been more conclusive, but the output power was not great enough to produce a burn pattern. The pulse trains seen were of two characteristic types. In some pulse trains the laser output between pulses was negligible just as one would expect from a well behaved mode-locked laser oscillating in a single transverse mode. Other pulse trains that were observed were actually a superposition of two pulse trains. Initially there would be just a single pulse train. However, a second pulse train would start to build up after about one-third of the first pulse train. Also, there appears to have been a definite time relationship between the two pulse trains, the pulses in the second pulse train always followed those in the first pulse train by approximately 3 to 4ns. It turned out that the oscilloscope traces recorded with the PIN photodiode and the H.P. 183A oscilloscope were response time limited and that the pulses were actually shorter than 3ns. This fact was verified by using a Tektronix 519 oscilloscope and an ITT type FW114A vacuum 44 planar photodiode to record the pulse trains. The rise time of a Tektronix 519 oscilloscope is specified to be 0.35ns, and the rise time of an ITT type FW114A vacuum planar photodiode is specified to be 0.5ns. The actual rise time of a mode-locked pulse is then (MeAbel, 1969, p. 12) 2 2 9 t pulse = / (t trace) - (t^ scope) - (t diode) = / (t trace)2 - 0.37. (53) Assuming that the pulses had a triangular shape, that they were symmetric, and that the rise times and fall times of the oscilloscope and the photodiode were reason ably symmetric, it was concluded from the Tektronix 519 oscilloscope traces that the shortest pulses had a duration of approximately 0.8ns at the half-power points. The oscilloscope traces used for these experiments could not be reproduced, again because of insufficient writing speed. However, a good slow-sweep trace (lOOns/cm) showing the two pulse trains is reproduced in Figure 6. This trace indi cates very clearly by the periodic regularity of the pulses that it is possible to achieve complete mode-locking in a ruby laser cooled to liquid nitrogen temperatures. Figure 6. Mode-Locked Pulse Trains from a Ruby Laser Operating Near 77°K 46 Measurement of the Lasing Linewidth The lasing linewidth was also measured. From the observed lasing linewidth one can infer a limit for the minimum possible pulse duration. A Fabry-Perot etalon was used to measure the line width of the laser output. This interferometer was chosen because of its large resolving power and its simplicity of construction. It consisted of two plane mirrors aligned parallel to each other. One mirror had a reflectivity of 99.8% and the other had a reflectivity of 95%. Both were dielectric multilayer mirrors and were optimized for operation at 6943JL The absorption losses of the two mirrors were negligibly small. The mirrors were held in separate mirror mounts and were separated by a distance of 5 cm. The mirror mounts were glued down to a 12 inch diameter, 2 inch thick "Cervit" mirror blank, which served as the base for the interferometer. The "Cervit" base provided both the required mechanical and thermal stability. Figure 7 is a schematic diagram of the inter ferometer. The full laser output, which was assumed to be more or less collimated, was directed through a lens, , into the interferometer. A second lens, , was then used to form an image of the interference rings on photographic film. The lens is a field lens whose function is to distribute light into the desired area in the film plane P . p Figure 7. The Fabry-Perot Interferometer 48 Its focal length determines the number of interference rings that are recorded. This number is determined by the relation N = 5 ! " * ,54) 4f ^ A where D = diameter of the incident laser beam n = index of refraction of the material between the mirrors (in this case n = 1) H = separation of the mirrors A = vacuum wavelength of the laser output. The laser beam had a diameter of about 5 mm, and the focal length of the field lens was 40 cm. In agreement with (54), three circular interference rings were recordable. A few of the recorded interferograms used to determine the lasing linewidth are reproduced in Figure 8. A circular aperture having a diameter of 1 cm was placed directly behind mirror #2 of the interferometer. Its purpose was to limit the aperture of the interferometer so that the surface quality of the mirrors could be used to maximum advantage. The reason why a shorter focal length field lens was not used is that although more than three interference rings would then be allowed, there was not enough energy in the incident beam to adequately record more than just the first 49 Figure 8. Some Interference Rings three. The type of photographic film being used was high speed Polaroid type 410 film. This type film was used simply for the convenience of having an immediate picture after each laser shot and is not as sensitive to the ruby wavelength as other types of photographic film that might have been used with less convenience.. However, since in a Fabry-Perot interferometer the first interference ring corresponds to the highest order of interference, the first few interference rings were sufficient for measurement purposes. The lasing linewidth was determined by measuring the widths of the interference rings and the separation between the rings. The interference ring patterns that were recorded were interpreted with the aid of Figure 9. It represents schematically the widths of the rings and the width of the spacing between rings for four orders of interference. The width of the rings corresponds to the lasing linewidth, and the distance between the edge of one ring and the corresponding edge of the next successive ring corresponds to an order of interference or the free spectral range of the interferometer, which is known. Thus, by measuring the relative positions of the inner and outer ring edges in the first few central interference orders it was possible to determine the fraction of an order of inter ference corresponding to the lasing linewidth. The fraction is given to good approximation (Tolansky, 1947, p. 53) by Figure 9. Schematic Diagram of Interference Rings 52 (55) Then the lasing linewidth is given in units of cm ^ by 2 ilcoscj) (56) where £ is the interferometer mirror separation and (}> is the small interferometer angle corresponding to each ring. Since cp is small for the first few rings, cos(f) can be set equal to unity. Finally (57) The ring widths and the distances corresponding to A^a, A^b, A^a, A^b etc., were measured from the photographic film using a low power microscope with a calibrated movable stage. From these measurements the lasing linewidth was determined for each of several laser shots. Equation (57) enables one to determine the lasing linewidth with good precision only if one can assume that the resolving power of the interferometer is very large. Another way to say the same thing is to say that the instru ment width of the interferometer must be signficantly smaller than the linewidth one wishes to measure. Un fortunately, however, the instrument width of the inter ferometer was probably not very small. 53 It is probable that several conditions worked to degrade the resolving power considerably even though the mirror reflectivity was very high. These include an in ability to align and maintain the mirrors perfectly parallel. Also, the flatness of the mirrors is not known. However, it is not likely that they were of any better quality that 2/20 over the area that was actually used. A smaller area could not be used to improve the performance of the mirrors because diffraction was then seen to degrade the resolving power. The minimum theoretical band width that may be resolved by a Fabry-Perot interferometer, that is the instrument width (assuming perfectly parallel and plane mirrors of infinite extent), is given by (Born and Wolf, 1959, p. 328) “ i-■ ^ , o r , = 1 It can be shown that an rms deviation in mirror surface flatness of an amount A£ will give rise to an apparent spectral width for a monochromatic line of M l = (59) where i is the mirror spacing. From this it can be seen that the theoretical resolving power given by Ro = a/ - (60) m m 54 is reduced by a factor of two, in reality, if the deviation in mirror surface flatness is " i ' > « > The implication of this is that it is not sufficient to have mirrors with just high reflectivities in a Fabry- Perot interferometer. It is also necessary that the mirrors be very flat. They should be very carefully aligned, and they should not be mechancially distorted. Only with excellent mirror quality and alignment is it possible to take full advantage of high mirror reflectivities. In fact, if the mirror quality and alignment are not extremely good it is better to avoid extremely high reflectivities. Since, as it was assumed previously, the mirror flatness and alignment were most probably no better than A£ = A/2 0 it is, unfortunately, quite probable that the instrument width of the interferometer due to just the plate quality and alignment alone was at least 300 MHz. Indeed, poor mirror quality and alignment were the major factors limiting the resolving power. It was found that 1 cm was the optimum diameter for the limiting aperture of the inter ferometer; and it was found that only if the aperture was made smaller than this, did diffraction begin to limit the resolving power. An unfortunate result of the poor mirror quality and inexact alignment was that the interfermoter was not 55 able to resolve the discrete mode structure in the laser output. These modes were separated by approximately 120 MHz, and since the instrument width of the interferometer was approximately 300 MHz, it was impossible to resolve them. The interferometer was able to just resolve the modes in the He-Ne alignment laser. The modes in this laser were separated by approximately 500 MHz. The resolving power of the interferometer could have been doubled by doubling the mirror separation from 5 cm to 10 cm. However, the resolving power still would not have been large enough; and further, this would have reduced the interferometer free spectral range by a factor of two from 3 GHz down to 1.5 GHz. Reducing the interferometer free spectral range was undesirable since the expected laser linewidth was in the range from 1 GHz to 3 GHz. There would have been an overlap of the interference rings corresponding to successive orders of interference, and it would have been impossible to determine the laser linewidth. Because the instrument width of the interferometer was approximately 300 MHz, the measured linewidth was approximately 300 MHz wider than the actual laser linewidth. The lasing linewidth was found to vary from shot to shot, and taking into account the instrument width, the minimum linewidth observed was 1.3 GHz and the maximum was 2.2 GHz. 56 The laser output was also monitored simultaneously with the H.P. 183A oscilloscope. The traces were of too poor quality to reproduce. But it was noted that the trace corresponding to the linewidth of 1.3 GHz consisted of a single pulse train while the traces with two pulse trains had linewidths of nearly 2.2 GHz. The reason why two pulse trains were usually seen with a definite time relationship is explainable by the fact that the transition in ruby splits into two separate transitions at low temperatures (D'Haenens and Asawa, 1962). The two transitions result from distortion of the cubic AlgOg crystal field which splits the fourfold degenerate 4 3+ Ag ground level of the Cr ion into a ± 1/2 ( A^) level. The two transitions share the same excited state, and are identified as ;E (^E) -H; 1/2 (^A2 ) and E (2E) -hi 3/4 (^Ag). At room temperature the width of the line is so large that the two transitions are unresolved. But at temperatures below 100°K the two transitions form a doublet. The spectral splitting is 11.4 GHz and at 77°K in a good single crystal ruby each transition has a linewidth of about 3 GHz (Bean and Izatt, 1973). The coupling between the two ground levels is a sensitive function of temperature. The relaxation time T^ -7 — 1 varies from about 10 sec at 300°K to about 10 sec at 4 4.2°K. At temperatures near 4.2°K the A^ levels are essentially uncoupled, and each transition has its own 57 characteristic lasing threshold. At higher temperatures the . relative populations of the 4A^ levels and hence the oscillation threshold requirements for the separate transi tions are determined by the relaxation time . If the relaxation rate is comparable to or fast compared to the _ 9 pumping rate into the E( E) level, then determines the lasing threshold for the E (^E)->-+1/2 (4A 2) transition in the ' 2 4 presence of stimulated emission in the E ( E) ->-+3/2 ( A2) transition. This is because for o polarization (the electric field perpendicular to the c-axis of the ruby), the transition probability for the E (^E) ->+3/2 (4A 2 ) transition is 1.5 times that for the E"(^E) ->+1/2 ( ^ 2) transition. The T^ relaxation populates the terminal.±1/ 2 ( ^ 2 ) level of the E (^E)-H;l/2 (4A 2 ) transition from the terminal ±3/2 (^A^) level of the E (^E) ^-+3/2 (4A 2) transition. Thus, this reduces the inversion and increases the oscillation threshold for the 2 4 E ( E) ->±1/2 ( a 2) transition. .This assumes, of course, that the stimulated emission rate is fast compared to the pumping rate. This is the situation in ruby as verified by the observation of relaxation oscillations. At 78PK the pumping level required to produce lasing in both transitions should be about 1.1 to 1.2 times that required to just produce 2 4 lasing in the E( E)->+3/2 ( A^) transition (D'Haenens and Asawa, 1962). Thus each of the two pulse trains corresponds to a separate lasing transition. Laser oscillation and then mode locking first occurs in the E (2E)->-±3/2 (4A2) transition. With continued pumping this is followed by laser oscillation and mode-locking in the E (2E)->-±1/2 (4A2) transition. The first pulse train has a larger amplitude than the second pulse train because of a larger transition probability. Also, because the two transitions share the same initial excited state and have ground states which are coupled, the first pulse train causes a modulation of the population inversion in both transitions. As a result, when laser 2 4 oscillation occurs in the E ( E) ->-±1/2 ( A2 ) transition its gain is actively modulated at the frequency corresponding to the resonant cavity round trip time. This causes mode- locking in this transition also. The time relationship between the two pulse trains is then determined by the position of the ruby rod in the resonant cavity. At 77°K the two ruby transitions are largely in- homogeneous ly broadened. This is due to the fact that at such low temperatures lattice vibrations or thermal motions are small and the only remaining significant broadening mechanisms are those associated with localized crystal defects such as dislocations, interstitial foreign atoms, missing atoms, and lattice strains. As in Doppler broadening the effect of these localized defects is to cause the 3+ resonant frequencies of the separate Cr ions throughout the crystal to be slightly different so that the overall response of the collection of ions is smeared out in 59 frequency. An applied signal interacts only with the atoms in an inhomogeneously broadened medium that happens to be in resonance and, thus, with only a small fraction of the atoms contributing to the entire linewidth. This is why in an inhomogeneously broadened medium the population inversion is frequency dependent. In a laster that has an inhomogeneously broadened gain medium the modes can oscillate independently; and the lasing linewidth is determined by the number of resonant cavity modes having sufficient gain to oscillate. For this reason the lasing linewidth is not always the same width as the fluorescence linewidth. In fact, if the gain medium is only pumped slightly above threshold, the lasing linewidth is considerably narrower than the fluorescence linewidth since only a small number of resonant cavity modes have sufficient gain to oscillate. This explains why the pulses in the second pulse train are longer in duration than those in the first pulse train and also why the pulse durations observed corresponded to linewidths much narrower than 3 GHz. With a given pumping rate the E(^E) ±1/2{^ A ^ ) transition gain is always — 2 4 less above threshold than is the gain of the E ( E) ±3/2( A^) transition. Thus, the number of modes able to oscillate in -—2 4 the E ( E) ±1/2( Ag) transition is always less than that in the E(^E) ±3/2 ( ^ 2 ) transition. 60 Another reason why the lasing linewidths were con siderably narrower than 3 GHz could have been that in lasers having saturable absorbers it has been observed that the saturable absorber tends to narrow the lasing linewidth by allowing only the most intense modes to oscillate (Sooy, 1965) . There remains only one final aspect of the experi mental data that must be explained. This is to answer why, if the two lasing ruby transitions are separated by 11.1 GHz, are they not resolved by the interferometer? The answer is, of course, that since the free spectral range of the interferometer was 3 GHz, two spectral lines having a frequency separation of 12 GHz (an exact multiple of 3 GHz) would form interference rings which exactly overlap. The (n+4)-th order of interference for the shorter wavelength line would exactly correspond to the n-th order of inter ference for the other. Consequently the two lines having a frequency of 11.1 GHz would produce interference rings, which would very nearly overlap. Since both lines are wider than 0.9 GHz, they were not resolved. Determination of the Parameter A and • the Degree of Mode-Locking As was explained in the theory, the degree of mode- locking is determined by the ability of the saturable absorber to select out the single most intense fluctuation spike in the initial laser radiation. It was shown that 61 this ability is dependent upon laser operating parameters such as the pumping parameter t , the single pass time t^, the number of lasing modes N, the ratio Recall that 1/t^ is the growth rate of the gain in the active medium at the lasing threshold as the result of pumping at a given power supply voltage. The theory explains how to experimentally determine this parameter and another useful parameter, n^_, which is the normalized threshold population inversion required for lasing without 62 a saturable absorber. These two parameters were determined as follows. The dye cell was initially filled with acetone and the average power supply voltage required to just produce lasing was recorded. The voltage was 2170 volts. The dye cell was then emptied and dried and then successively filled, emptied, dried, and then refilled with five dilute dye solutions of increasing concentration. The average power supply voltage required to just produce lasing was recorded for each dye concentration. This was all done without removing the dye cell from the resonant cavity or disturbing its position or orientation. . Figure 10 is a plot of the square of the power supply voltage versus the measured absorbance of each of the five dye concentrations. Since a pink ruby can be assumed to be a dilute system, the population inversion is directly pro portional to the pumping energy or the square of the power supply voltage. Thus, Figure 10 is also a plot of the relationship between n 1. and A. The straight line is the straight line which best fits the data and is assumed to represent a p ot of Equation (45). The constant K can be determined from measuring the slope of the line and was found to be equal to 1.88. The error bars indicate the variation in the power supply voltage required for lasing. The variation was always about ± 5.volts. iue 0 Po o Pmig nry n Pplto Inversion Population and Energy Pumping of Plot 10. Figure Power supply voltage squared ess aual Asre Absorbence Absorber Saturable Versus 63 64 Next, the power supply voltage was set to 2350 volts and the delay time t^ required for lasing to occur after the flash lamp ignition was recorded for the no-dye case and for each of the five dye solutions. Five oscilloscope pictures of the laser output and flash lamp output were taken for each case. The delay time t^ was read directly from these photographs. The relationship between the average delay time and A is plotted in Figure 11. As explained previously, this plot essentially represents Equation (42) with n^ = n 1^ and t = t ,. d Next, the slope of the curve in Figure 11 was measured at a number of delay times. Figure 12 is a plot of the natural nogarithm of K(dA/dt^) versus t^, and the plot is a straight line, as it should be. The slope and extra polated intercept of this line were measured graphically, and the experimental values of n^ and t^ were then deter mined using Equations (49) and (50). The values obtained were n t = .01 (62) tp = 6.25 x 10~2 ms. (63) The experimental number found for n^ is in good agreement with theoretical calculations for this number. From theory the population inversion per unit volume re quired for laser oscillation of just the highest gain mode 65 44 il! t, (milliseconds) Figure 11. Relationship Between t^ and A In K(dA/dt iue 2 Po o I Kd/t) Versus K(dA/dt^) In of Plot 12. Figure 66 67 when a transition is inhomogeneously broadened is given by M - N = (1/2) ^ 4 ^4^ • (64) 1 /HF C3 tc The required inversion is directly proportional to the broadening of the natural linewidth. The natural linewidth Av q is related to the spontaneous lifetime by t A v o = H • ( 6 5 ) At 77°K the two transitions are broadened by crystal strains so that Av is equal to 3 GHz. The spontaneous lifetime is equal to about 3 x 10 sec. T h u s , the line width is broadened beyond the natural linewidth by the factor 2 ttt A v , which is equal to 5.7 x 107 . At room tempera ture this factor is about 100 times larger. The time tc is the average residence time of the quanta in the active medium and is given by tc = cT n(l/RY • (66) The length £ is the length of the active medium. The veolcity of light c in the medium is equal to c^/p. R is the effective reflection coefficient of the resonant cavity per transit and includes the resonant cavity loss per transit due to absorption, scattering, and diffraction. The factor of 1/2 in Equation (64) must be included because laser oscillation occurs only in those modes in which the 68 radiation is linearly polarized perpendicular to the c-axis of the ruby rod. The predominant loss in the laser cavity was due to near normal reflection from the surfaces of the vacuum chamber windows and from the dye cell windows. The total loss per transit due to these reflections was estimated to be 25%. Since the output mirror had a reflectivity of about 75%, the total loss per transit was approximately 50%. Thus R was equal to 0.5. With v = 4.3 x 1 0 ^ Hz y = 1.78 & = 7 cm R = 0.5 N = 1.6 x 10"*"^ cm ^ o the calculated value for n^_ is N -Nf nt = 'v 0.0048. (67) o Multiplying this value by 2, because there are actually two transitions involved, gives a value for n^_ which is very nearly equal to the observed experimental value. Since t^ was determined from the same graph as nfc, it is surmized that the measured value for this parameter is also reasonably accurate. The final parameter needed in order to establish the degree of mode-locking is in Equation (35). Un fortunately it was not possible to obtain an experimental 69 value for this parameter. The saturation intensity of the saturable absorber I and the noise intensity I were s J o obtained from theoretical calculation. The saturation intensity is given by 1s = i5T- ■ <6 8 > S -19 For ruby hv is 2.9 x 10 joules. The cross-section a for cryptocyanine in acetone was determined from the molecular weight and the absorption coefficient at 6934A to be -15 2 3.6 x 10 cm . The relaxation time t is less than s -12 20 x 10 sec (Mack, 1968). The saturation intensity was calculated to be I = 4 x 106 wa t ^ - . (69) cm Since the beam diameter in the resonant cavity was approximately 3 mm, the beam power required for saturation was P 2.8 x 105 w a tts. s The noise intensity Iq was taken to be the maximum expected spontaneous emission from the active medium in the solid angle defined by the laser output beam. The maximum spontaneous emission per unit volume of the active medium that could be expected in a 4 it steradians is given by spontaneous emission _ No 0.94 v mew A 7 ------3----- "2---- TaT" (70) 4 tt steradian-cm for an inhomogeneously broadened medium. Since the Q fluorescent linewidth was 3x10 Hz for each transition and 70 since the effective transition rate was approximately 2/t the total spontaneously emitted power per unit volume of the -4 3 active medium was 1.74 x 10 watts/cm . Thus the noise power in the beam inside the resonant cavity before entering the saturable absorber was Pq = 1.7 x 10 4 watts. (71) The calculated value for A is then P -9 (j> = p^- ^ 6 x 10 (72) s and 2n (-t) = 21.6. (73) _2 Using Equation (35), a value of 2.8 x 10 was obtained for A. The probability W(D) is then calculated from Equation (36). Assuming that laser oscullation was occurring in ten modes, and that D = 0.1, the value of the probability W(D) is W(D) = 0.077. (74) This means that the probability to have at the end of the nonlinear phase two pulses with an intensity ratio as large as 0.1 within the resonant cavity round trip time was 0.077 and that the laser was reliably mode-locked. Snnce I/A is equal to p in Equation (40) it can be seen that p for the laser was approximately 35. 71 The curve in Figure 2 represents, the relationship between the number of lasing modes N and the nonl’inear transformation index p of the saturable absorber so that conditions are just sufficiently met in a laser for complete mode-locking. For values of N and p in the region above the curve the conditions for complete mode-locking are more than sufficiently met. Thus, our laser, having only 10 lasing modes and having p equal to 35, was completely mode-locked by a wide margin. CHAPTER 4 CONCLUSION Figure 6 provides excellent proof that it is. possible to obtain passive mode-blocking in a ruby laser operating near 77°K„ Further, it appears probable that laser oscillation and mode-locking occurs in both the E( 2E)->-± 3/2 (^A2) and the E( 2E)->-± 1/2(_^A2) transitions „ Passive mode^-locking definitely occurs in the former transit tion with ten modes completely mode^-locked; and there are several indications that the latter transition is also lasing and that mode-locking occurs in this transition also. It was at first believed that the second pulse train was just the result of lasing in higher order transverse modes, but this possibility is largely ruled out because the transverse mode selector probably allowed only the TEM qq modes to oscillate. From Figure 10 it can be seen that the population inversion was 1,16 times the population inversion just 2 ^ required for lasing in the E (. E) ->± 3/2 C A^) transition, ' 2 Thus, it was also sufficient for lasing in the Ej.->± 1/2C^A2) transition according to D ’Haenens and Asawa C1962). Further, there is the definite time relationship between the two pulse trains which was always the same from 72 73 one laser shot to the next. The possibility that the second pulse train was just a train of so-called "satellite pulses" caused by reflections from laser components in the resonator or as a result of the fact that the dye cell was not in direct contact with the output mirror was also ruled out. This was possible because the dye cell was close enough to the output mirror so that if there were two pulse trains overlapping in the dye cell they could not have been resolved in any case. Also, care was exercised in the alignment of the laser to direct all reflected beams from reflecting surfaces out of the resonator. On the other hand the ruby rod was approximately the correct distance from the 100% mirror to account for the possibility that mode-locked pulses due to one transition would affect the gain medium at the proper time delay to produce the second train of pulses. The fact that two transitions were actually lasing was observable from the spectroscopic measurements. It was found that the conditions required for complete mode^locking were met to excess in the laser due to the narrow lasing linewidth, However, the other effects that might be expected because of the narrow linewidth (except for longer pulse duration) such as linear and nonlinear frequency pulling were not observable. The 4% linear frequency pulling effect and its effect upon pulse repetition rate was too small to be resolvable (T should have been increased by'about 0,3ns), And apparently the nonlinear frequency pulling did not affect the ability of the laser to be mode-locked. Finally, the pulse lengthening effect due to inhomogeneous broadening predicted by Kafri et al. (.1972) was not observed. LIST OF REFERENCES Bean, Brent L., and Jerald R. Izatt, "Verification of the Kramers-Kronig relations in optically pumped ruby," JOSA, 63_, .832 (.1973) . Born, Max, and Emil Wolf, Principles of Optics, Pergamon Press, New York, 1959. Bradley, D. J., T. Morrow, and M. S, Petty, "Intensity dependent quenching of two-photon fluorescence displays of a mode-locked ruby laser," Opt. Commun., 2, 75 (1970). Cutler, C. C., "The Regenerative Pulse Generator," IRE, 43, 140 (1955). D'Haenens, I, J., and C. K„ Asawa, "Stimulated and fluores cent optical emission in ruby from 4.2° to 300°K: zero-field splitting and mode structure," J. Appl, Rhys., 33_, 3201 (19 62). Duguay, M. A . , S. L. Shapiro, and P, M. Rentzepis, "Spontaneous appearance of picosecond pulses in ruby and Nd:glass lasers," Phys. Rev. Letter, 19, 1014 (1967). Garmire, Elsa, and Amnon Yariv, "Laser mode^locking with saturable absorbers," IEEE J. Quantum Electronics, QE-3, 222 (1967), Grutter, A. A,, H, P. Weber, and R. Dandliker, "Imperfectly mode-blocked laser emission and its effect on non linear optics," Phys. Rev., 185, 629 (1969). Hausherr, B., Ernst Mathieu, and Horst Weber, "Influence of saturable absorber transmissions and optical pumping on the reproducibility of passive mode-locking," IEEE J, Quantum Electronics, QE^g, 445 (19 73). Kafri, O., S. Kimel, and Joseph Shamir, "Description of laser pulses using Fourier analysis,"-' IEEE J . Quantum Electronics, QE^-8 , 295 (1972), 75 76 Kuznetsova, T . I ., "Statistics of appearance of ultrashort light pulses in a laser with a bleachable filter," Sov. Phys., JETP, 3iQ, 904 (1970). Letokhov, V. S., "Concerning the investigation of the statistical properties of incoherent light," Sov. Phys. JETP, 2 3 _ , 506 (1966), Letokhov, V. S., "Generation of ultrashort light pulses in a laser with a nonlinear absorber," Sov. Phys. JETP, 28_, 562 (1969) . Mack, M. E ., "Mode-Locking the ruby laser," IEEE J. Quantum Electronics, QE-4, 1016 (1968). Maitland, A., and M. H. Dunn, Laser Physics, American Elsevier Publishing Company, Inc., New York, 1969. Mandel, L., and E. W. Wolf, "Coherence properties of optica fields," Rev. Mod, Phys., 3_7, 231 (1965), McAbel, Walter E ., Probe Measurements, Tektronix, Inc., Beaverton, Oregon, 19 69. " Mocker, H. W,, and R. J. Collins, "Mode competition and self ^-locking effects in a Q-switched laser," Appl, Phys, Letters, 1_, 270 (1965), Papoulis, Athanasias, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Company, New York, 1965. Sacchi, C. A., G. Sancini, and O. Svelto, "Self-locking of modes in a passive Q-switched laser," Nuovo Cimento, 4_8, 58 (1967) . Schawlow, A. L,, "Fine structure and properties of chromium fluorescence," in Advances in Quantum Electronics, J, R, Singer, Ed,, Columbia University Press, New York, 1961. She, C, Y,, and Ang-Tiek Tan, "The effect of absorber concentration on a pulsed laser system," Appl. Phys. Letters, 9_, 198 (1966). Slegman, A. E ,, An Introduction to Lasers and Masers, McGraw-Hill Book Company, New York, 1971, Smith, P. W., "Mode-locking of lasers," Proc. IEEE, 58, 1342 (1970). Sooy, W. R., "The natural selection of modes in a passive Q-switched laser," Appl. Phys. Letter, 1 _, 36 (1965). Tolansky, S., High Resolution Spectroscopy, Pitman Publishing Corporation, New York, 1947.