A theory of the mode-locked, internally frequency-doubled laser

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-11, NO. 1, JANUARY 1975 21

output voltage was analyzed up to 30 MHz with a Heathkit RG52 communications receiver and up to 870 MHz with an Eddystone 990s UHF receiver. The spectrum was found to consist of lines having a transform-limited halfwidth of 2.5 MHz with subsidiary satellites, separated by the axial mode spacing of 125 MHz.

CONCLUSIONS It has been shown that the shape of a ruby laser pulse can be

controlled by the use of a Pockels cell whose transmission is governed by the intensity of light in the laser cavity. The latter is related to the potential difference across the Pockels cell by the use of a photodiode and a combination of negative and positive feedback.

Using the Wagner and Lengyel’s giant-pulse equations coupled with the equations of the electric circuit, pulse shapes have been computed which bear a close resemblence to those actually measured in the laboratory.

A correct choice of feedback parameters permits the generation of approximately rectangular laser pulses whose length can be varied at will between 120 and 600 ns, and which, more- over, are free from high-frequency oscillations. 15-MW rectangular pulses have been amplified without distortion.

ACKNOWLEDGMENT The authors wish to thank Dr. H. A. B. Bodin for inviting

them to carry out this work as part of the Toroidal Pinch program at Culham, and are grateful to the UKAEA for the facilities provided. They also wish to thank P. Tripathi of the

University of California for assistance in setting up the compu- tational model, and R. Hamilton for experimental assistance.

REFERENCES E. R. Wooding and M. L. Yeoman, “A laser scattering homodyne technique for studying plasma drift waves,” in Proc. 10th Znt. Con$ Phenomena inzonized Gases (1971), p. 417. W. R. Callen, R. H. Pantell, and J. Warszawski, “Phase stretching of Q-switched lasers,” Opto-Electron., vol. 1, pp. 123-127, Aug. 1969. R. H. Pantell and J. Warszawski, “Laser power stabilisation by means of non-linear absorption,” Appl. Phys. Lett., vol. 11,

A. Hordvik, “Pulse stretching utilizing two-photon-induced light absorption,” IEEEJ. Quantum Electron.,vol. QE-6, pp. 199-203, Apr. 1970. J. F. Young, J. E. Murrey, R. B. Miles, and S. E. Harris, “Q- switched laser with controllable pulse length,” Appl. Phys. Lett., vol. 18, pp. 129-130, Feb. 1971. F. R. Marshall and D. L. Roberts, “Use of electro-optical shutters to stabilize ruby laser operation,” Proc. IRE (Corresp.), vol. 50, p. 2108, Oct. 1962. C. H. Thomas and E. V. Price, “Feedback control of a Q-switched ruby laser,” ZEEE J. Quantum Electron. (1966 International Quantum Electronics Conference), vol. QE-2, pp. 617-623, Sept. 1966. V. I. Mishin, “Ruby laser for generating microsecond light pulses having a narrow spectrum,”Pri. Tekh. Eksp., no. 4, pp. 181-182, July/Aug. 1971. R. Burlisch and J. Stoer, “Numerical treatment of ordinary dif- ferential equations by extrapolation methods,” J. Num. Maths,

W. G . Wagner and P. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys., vol. 34, pp. 2040-2046, July 1963. L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys., vol. 34, pp. 2346-2349, Aug. 1963.

pp. 213-215, Oct. 1967.

V O ~ . 8, pp. 1-13,1966.

A Theory of .the Mode-Locked, Internally Frequency= Doubled Laser

Ahrroct-An AM mode-locked laser with a frequency doubler internal to the laser cavity is analyzed. Mathematical expressions are derived for laser pulsewidth, saturated gain, and secondharmonic output power. It is shown that the maximum second-harmonic power output available from the mode-locked laser is a factor of &less than that available from an identical free-running laser.

Numerical examples are given for an Nd :YAG laser with a Ba2NaNb5 OI5 internal frequency doubler.

Manuscript received June 26, 1974; revised September 17, 1974. This work was partially supported by the Advanced Development Program 405B, Air Force Avionics Laboratory, Wright-Patterson AFB, Ohio, under a subcontract from the McDonnell Douglas Astronautics Com- pany East.

The author is with the Electro-Optics Organization, GTE Sylvania Inc., Mountain View, Calif. 94042.

A I. INTRODUCTION

MODE-LOCKED LASER provides an attractive means of producing a stream of Kigh repetition-rate pulses use-

ful for optical communication. Frequency doubling of a mode-locked laser is often useful for shifting the output of a laser source from a wavelength of limited utility to a more useful region of the optical spectrum. Limited atmospheric transmission, poor detector, and modulator response at the fundamental wavelength are three possible reasons for considering frequency doubling of a mode-locked laser designed for optical communications applications. Frequency doubling may be accomplished internally or externally to the laser’s optical cavity. The internally frequency-doubled laser, be-

22 IEEE JOURNAL OF QUANTUM ELECTRONICS, JANUARY 1975

_. - - - 0 - - ,e A -

/ \

L-.---.-- 0

0 - - L -

- MI MODULATOR - AMPLITUDE OSECOND-HARMONIC LASER ROD

GENERATOR M2

Fig. 1. The optical cavity.

cause of the higher fundamental fields available within the laser cavity, accomplishes the frequency doubling in a more efficient manner than if it is performed externally to the laser cavity. On the other hand, because the second-harmonic process provides a power-dependent loss to the mode-locked laser pulse, the laser’s temporal width is increased by the doubling process [ 11. This increase in pulsewidth at the fundamental wavelength is an important determinant of the pulse shape for the internally doubled laser.

Ths paper theoretically analyzes the mode-locked, frequency- doubled, homogeneously broadened laser. Mode locking produced by amplitude modulation is treated. The self- consistent approach introduced by Kuizenga and Siegman for describing a mode-locked homogeneously broadened laser is employed [2]. In the analysis that follows the propagation of a temporally Gaussian pulse is traced around the optical cavity shown schematically in Fig. 1. The effects of each component on the Gaussian pulse shape are considered, and it is required that the pulse envelope repeat itself after a round-trip traversal of the optical cavity. The principal extension of the Kuizenga- Siegman theory is the inclusion and analysis of a frequency doubler within the laser cavity.

Other authors have attempted to treat the mode-locked doubled laser analytically but analyses previously published P I , ~31:

1) have neglected the effects of laser fluorescent linewidth and hence present results which have only limited validity;

2) have not given an explicit expression for laser pulsewidth; 3) have not treated the effects of modulator detuning.

The analysis presented here leads to expressions for pulsewidth and laser power, and a prediction of laser performance when the modulator is detuned from its ideal driving frequency.

This paper fipt considers the component transforms of each of the elements in the cavity shown in Fig. 1. Self-consistent equations are then developed and simplified. Numerical examples relevant to the Nd : YAG laser are given.

11. COMPONENT TRANSFORMS The effect of each of the intracavity components on a tem-

porally Gaussian pulse is considered. The pulse is assumed to have a Gaussian spatial distribution with a I /e beam size of wo . The generalized electric-field pulse shape, including both time and radial variations, is

E, ( t , r ) = A exp ( - r2 /w$) E, ( t ) exp jw, t t C.C. (1)

where the function E,(t) = exp - [ y ( t t t o ) 2 ] describes the temporal variation of the envelope of E, ( t , r ) . The parameters y and to determine the laser’s pulsewidth and the time of the peak pulse amplitude. In the analysis that follows y is assumed

to be real, and any frequency chirping on the pulse is ignored. The Fourier transform of E,(t , r) , or a description of the

propagating pulse in the frequency domain is

F[E,( t ,r)] = E , ( o , r ) = A f i e x p ( - r 2 / w i )

. exp [ito (O - 0,) - (w - ~ , ) ~ / 4 y ] . (2)

Thus the pulse’s optical spectrum is centered at w = w,.

A . The Laser Rod The transfer function of the laser rod is best described in the

frequency domain. For a homogeneously broadened medium, the amplifier single-pass field transfer characteristic TA (a) has a Lorentzian line shape [2]

TA (0) = exp +j2(w - ~, ) /Aw,) l (3)

or

TA (w) exp [g - (2jg(u - q J A d

. - 4g(w - w,)2/Aw:] (4)

where g is the saturated single-pass field gain at line center (0 = o,), and Aw, is the homogeneous linewidth of the laser transition. The three-term Taylor-series expansion given in (4) for TA (0) is valid for I O, - w, I << Am,.

The temporal output field of the amplifier rod is given by

( t , r> = F [TA ( 0 ) E u (a, r)]

where F -’ indicates an inverse Fourier transform. Thus

Eb(t, r) =A’exp ( - r 2 / w $ ) exp [-y‘(t t t ; ) ’ ]

exp jw, t + C.C. (5)

and

A’ = A exp (g)/@, y’ -y/G, t ; to - 2g/Awo

where

G E 1 + 16yg/Awi.

Hence the output pulse from the laser amplifier remains Gaussian. The effects of the laser rod are to delay the pulse by 2g/Ao, to lengthen it by decreasing y to y/G, and to increase the pulse amplitude. These effects have been previously reported by Kuizenga and Siegman and are repeated here only

rfor completeness [2]. In writing (1)-(4) it was tacitly assumed that the center of

the envelope of the optical spectrum of the Gaussian pulse lies exactly at the center of the amplifier’s passband. In practice the centermost axial mode may be as far as one-half the longitudinal-mode spacing removed from line center. This small frequency displacement has no apparent practical consequence and is neglected in the remainder of this paper.

B. The Loss Modulator The single-pass field transfer function of the loss modulator

is [2]

aL ( t ) = exp [-6L sin2 (w, t /2 ) ] ( 6 4

or

FALK: MODE-LOCKED, INTERNALLY FREQUENCY-DOUBLED LASER 2 3

aL ( t ) = exp [- 6L (sin2 (w, to / 2 )

w, (t f t o ) sin (0, t o ) 2

[

where exp ( - 6 ~ / 2 ) is the time-average modulator amplitude transmission and w, /2 is. the modulator drive frequency. The loss seen by the laser varies with frequency w, , which is twice the modulator drive frequency. It is important to realize that the exact form of the modulator transfer function is dependent upon the physical means employed to loss modulate. Equation (6a) is only approximately correct for the two most common means of loss modulation, namely electrooptic or acoustooptic [4] , [ 5 ] . However, (6b) is valid in the temporal region of interest, where 6 L sin2 (w,t/2) << 1, i.e., when the optical pulse passes through the loss modulator with compara- tively little loss.

C. The Second-Harmonic Generator The frequency-doubling process has pronounced effects on

both the spatial and the temporal shape of the Gaussian pulse. For this analysis the effects on the spatial variation can be treated in a conventional manner where coupli.ng to the fundamental-field cavity mode is considered. Changes in the Gaussian temporal shape of the pulse, however, are not so easily treated. The investigation of the effect of second- harmonic generation on the temporal shape is facilitated by a physically reasonable approximation.

The analysis given is strictly valid only for small conversion efficiency to the second harmonic. As a practical matter this limitation is of little consequence because most internally frequency-doubled lasers produce the maximum output power with only a small conversion of the fundamental-wavelength circulating power to second-harmonic output power.

I ) Spatial-Mode Considerations: In this section the effect of the second-harmonic generation process on the spatial variation of the generalized Gaussian pulse given by (1) is considered. For notational simplicity the time variation of every electric field is rewritten as a product of an optically varying phase and of an envelope that varies both in time and in space

E, (t, r ) = I E,(t, r ) I exp -jw,t + C.C. (7)

where

IE,(t, r)l =AE,(t) exp (-r2/wg). (8)

If second harmonic is generated with 90" phase matching, i.e., without Poynting-vector walkoff, the envelope of the second- harmonic field produced in the doubler crystal is

(9)

where K is dependent solely upon crystal parameters [6]. Similarly, the envelope of the fundamental frequency field emerging from the crystal is

IEb(t,r)I = IEu(t,r)I sech [KIE,(t,r)I]. (10)

If the conversion efficiency is small, i.e., if KI E, (t, r)l << 1, (10) becomes

lEb(t,I)I EE,(t)A exp (-p2/wg) [ l - (K2A2/2)E,2(t)

* exp (-2r2/wg)]. (1 1)

For a laser operating in the TEMoo mode the only part of I E, ( t , r) I which couples to the cavity mode is that which over- laps a Gaussian beam of radius wo . In general, I Eb (t , r ) I can be expanded as a sum of products of Gaussian and higher order Hermite or Laguerre polynomials. The envelope of the TEMoo mode exiting from the crystal can then be calculated using the orthogonality properties of the Laguerre functions. Hence, if I EL (t , r ) I is the TEMoo component of I Eb (t , r ) I ,

IEb( t , r ) IeXp(-r2 /Wg)2~rdF

IEb(f,r)I = exp (-r2/wg)

[exp (-r2/wg)] 2 ~ r dr

(12)

I EA ( t , r) I = I Eu (t , r) I [1 - (KA/2I2 E," ( t) l . (13)

Thus the time-varying loss seen by the Gaussian envelope I E, (t, r) I is (KA/2)2 E; (t).

2) Temporal Considerations: An exact analysis of the temporal effects of the second-harmonic process within the frame- work of Gaussian pulses is, impossible because the second- harmonic process does not maintain the envelope of Gaussian input pulse. That is, treatment of the second-harmonic generation process by use of the transfer function described in (13) destroys the Gaussian temporal pulse shape.

However, it has been noted that the experimentally observed pulse shape and optical spectrum observed from a mode-locked frequency-doubled laser are very nearly Gaussian [7]. The effect of many passes through the doubling crystal is to alter the Gaussian-pulse parameters y and to while essentially maintaining a Gaussian envelope.

Therefore, the temporal effect of the second-harmonic generation is treated as if it maintains the Gaussian envelope of an input pulse, and produces an output whose peak amplitude and curvature at the time of the peak pulse amplitude are the same as the actual TEMoo output pulse described by (13). Thus the output of the doubler crystal is approximated by

I EL (t , r) I = A exp ( - r2 /wi ) (exp -a) exp [-yb(t + to)2] (14)

where a and b are the curve-fitting parameters that can be adjusted so that (13) and (14) yield identical 1) peak amplitudes,

IEL(t=-to,r)I

and 2) curvatures

a2 at2 - lEL(t=-tO,r)l.

We note that both expressions for I EL ( t , r) I yield equal (zero) slopes at t = - to . These three time derivatives, which are


identical if calculated from either (13) or (14), are the first three coefficients of a Taylor expansion for IEf,(t , r)l . Thus, with parameters a and b appropriately chosen, (14) describes the nonlinear transmission of the doubler crystal correctly to second order in t f to . This is the same degree of accuracy used for the description of both the amplitude modulator (6b) and the amplifier rod (4).

Equating the three time derivatives, as outlined above, it is found that the parameters a and b are given by

a = -1n [ 1 - (KA/2)' 1 , a (KA/2)' (1 5)

b = [ l - 3(KA/2)'] [ l - (KA/2)']-',

b 1 - (KA)'/2 (1 6)

where the approximate values hold for (KA)' << 1. Equations (1 5) and (1 6) relate curve-fitting variables a and b

to physical parameters K and A . Although, in principle, K and A are either measurable or calculable precise evaluation of either is difficult and furthermore is unnecessary because only the product KA appears in (13)-(16). It is more convenient and physically meaningful to relate K A , hence a and b , to the single-pass power loss associated with the second-harmonic process. This time-varying loss a( t ) E [Pz ( t ) ] / [ P , ( t ) ] = (second-harmonic power)/(fundamental power) can be calculated from (9), (14)-(16). The peak value of a(t) , i.e., the peak conversion efficiency or peak single-pass power loss to the second harmonic, is the conversion efficiency at the peak of the fundamental pulse and is given by

Similarly, it can be shown that the average single-pass power loss to the second harmonic,

is related to KA via

Hence the peak loss to the second harmonic is f i t imes the average power loss. It will later become apparent that the difference between peak and average second-harmonic losses leads to a unique power saturation behavior for the mode-locked frequency-doubled laser.

With values of CYSHG, the average single-pass power loss or second-harmonic conversion efficiency, determined by (18), the parameters a and b may be rewritten

111. SELF-CONSISTENCY It is required that after one round-trip traversal of the optical

cavity, the pulse envelope repeat itself. With the required component transforms developed in the preceding sections of this paper the mathematical formalism necessary to follow a pulse around the optical cavity now exists.

Consider a pulse with electric-field envelope at point 1 (input to the doubler crystal) in Fig. 1 given by the generalized Gaussian

I El ( t , r ) I = A exp (-r'/lw$) exp [ -y( t + to)' 1. (20)

The TEMoo output pulse resulting from a single pass through the second-harmonic crystal and thus the pulse into the amplifier rod is determined from (14). The output resulting from two passes through the amplifier rod is found by two applications of (5). Hence the rod output is

I E ~ (t ,r)l =A" exp (-~'/Iw$) exp [-y"(t + tb')']

where

and

G' = 1 t 32 ybg /Ao; .

Similarly, I E3 ( t , r ) I is the input for the second pass through the doubler crystal, so that the output-field envelope I E4 ( t , r ) I is

I E ~ ( t , r ) I = (A /@) exp (-r'/w$) exp (-2a' + 2g)

- exp [(-yb'/G')(t t to - 4 g / A ~ , ) ~ l (24)

where

b' = 1 - a ( a S H G 1 +aSHG2) = 1 - 2fiaSHG. (26)

Note that QSHG~, the efficiency of conversion to the second- harmonic for the second pass through the doubler crystal, is not necessarily equal to (YSHG~ (the first pass conversion efficiency) 181. The parameter, aSHG E (QSHG~ t asHG2)/2, defines a measure of the effective single-pass losses to the second harmonic. Continuing the tracing of the optical pulse with two passes through the loss modulator

IE5(t,r)I=IE4(t,r)I'aZ(t> (27)

where aL(t), the loss modulator's amplitude transmission is given by (6b).

Finally, the round trip for the pulse is completed by including an additional time delay 2L0/c, and an effective loss a. The time delay allows for the time of round-trip propagation through the cavity of optical length Lo and the effective loss a accounts for all linear cavity losses.

FALK: MODE-LOCKED, INTERNALLY FREQUENCY-DOUBLED LASER 25

For self-consistency the envelope of the pulse at any point in the cavity must repeat itself after one period of the modulation cycle. Thus

El (t - T,, r ) = E6 ( t , r ) exp (-io) (28)

where TM = 2n/o, and 8 allows for an optical phase pre- cession [2].

Equation (28) may be rewritten by using (24)-(27) and by equating powers of t. This leads to three coupled complex equations which completely describe the behavior of the mode-locked frequency-doubled laser. These equations are

y = yb‘/G’ + (6L0;/2) COS O, to (29)

T, - ~ L o / c = 4b’g/(G’A~,)

+ [ ~ L W , / ( ~ Y ) I sin o m to (3 0)

and

exp [-?(TIM - 2LO/c)2]

= (c’>-’I2 eXp [2g - 2fZ’ - 01 t j e i-jW,(TM - 2Lo/C)

- 16g2 yb’ / (A~iG’ ) - 26, sin2 (o,to/2)]. (31)

In the absence of second-harmonic generation these equations, with minor notational changes, reduce to those given in [2].

Although (29)-(31) provide a complete description of the laser’s performance, their complexity precludes any meaningful physical understanding without further simplification.

Fortunately, simplification is possible if a low-gain laser (g << l), such as CW-pumped Nd : YAG is considered, and if note is taken of the usual magnitudes of the variables in (29)-(31). First, for internal frequency doubling, the conversion efficiency QSHG must be much less thatl unity if over- coupling [9] of the laser is to be avoided. Further, the pulse spectral extent Ao,, defined at its 10-percent power points, must be less than the laser fluorescent linewidth. Subject only to these two assumptions, (29) becomes

(3 2)

This equation essentially describes laser pulsewidth as a function of modulation depth (6,), second-harmonic conversion efficiency (CYSHG), and detuning. P,ulsewidth (7,) is defined as the full pulsewidth of the second-harmonic output as measured as its 10-percent power points. This definition of pulsewidth is useful when dealing with optical communication systems because it represents the time which an optical shutter or modulator must be open in order to pass almost all (97 percent) of the energy in the second-harmonic output pulse [ 101. Pulsewidth and y are related by r, = [In lO/y] 1/2. Note that maximum y, or minimum pulsewidth (T,), occurs with w, to = 0, i.e., when there is no phase shift between the optical pulse train and the loss modulation. Further note that minimum pulsewidth occurs at minimum value of QSHG, i.e., where the effects of second-harmonic generation are least. It should

be pointed out that although DLSHG is treated as an independent variable in (32) it is in fact an implicit function of pulsewidth (e.g., CVSHG increases for shorter pulsewidths) and hence cannot simply be varied at will.

It is interesting to evaluate the expression for pulsewidth in the region of large second-harmonic generation, i.e., where

(Y’$HG >> 8 g 6 l ( ~ & / A o f ) COS O, to. (33)

In this case a Taylor expansion of (32) yields

Y = (a& 6, / 4 a ) (COS a m t o )/asHG (34)

and

In this regime y and 7p are independent of g and Am,. That is, for high second-harmonic conversion efficiency, pulsewidth is .independent of the characteristics of the laser amplifier medium and dependent only upon a competition between the doubler which favors long pulses and the modulator which prefers short pulses [ l ] . The simple explicit dependence of pulsewidth on w, and 6~ , given in (35), is markedly different from that which occurs for the mode-locked, nondoubled laser [2] where 7 - (I/u,)’/~ (l/6,)‘l4. It is interesting to note that the dependence of r on QSHG, w,, and 6, given implicitly by Bernecker [3] (who neglects the effects of laser fluorescent linewidth) is identical to that predicted by (35).

The phase shift, described by (30), between the optical pulse train and modulator drive, may also be simplified using the small gain, small CYSHG approximation. Then

Af = u&/(4n2) [4g/A0Q + (6,5/2~) 0, sin w m to] (36)

where A f = c/2L0 - w,/2n is the deviation of twice the modulator drive frequency from the cavity axial mode spacing. In a practical laser system this phase-shift-detuning characteristic may be used as a discriminant in order to maintain mode locking with wto = 0 over long periods of time [ 111.

Similarly, (31), which determines laser saturated gain and hence laser output power, may be written

2g = f l a S H G t a t 2 6 L sin2 (a, to/2)

- 2 (In 10) sin2 om to

- 4g6L(w,/Aw,) sin om to t (2glln 10)(Aw,/Aw,)2

(37)

where Ao, is the pulse spectral width measured at its 10- percent power points. The terms on the right-hand side of (37) can be physically identified. The first three terms are due to the second-harmonic conversion, linear loss, and loss produced by the modulator if the pulse center does not pass through the modulator at a time of zero loss. The next three terms are a consequence of the finite amplifier linewidth and finite pulse spectral width. Typical values for an Nd :YAG mode-locked frequency-doubled laser are w, = 2n(250 MHz), Ao, = 2n(120GHz),and Ao, = 2n(lOGHz). Hence o,/AwQ << 1,

26

(Awp/Ao,)2 << 1, and the last two terms are negligible. The fourth term, -2 (In lO)62 . ( ~ , / A w , ) ~ sin2 w, to is maximum when Ao, is minimum. This occurs when the coupling to the second harmonic QISHG is large. In that case Aw, can be determined from (35) and therefore an upper bound for the fourth term is given by @,SL CYSHG sin2 w, to /cos o, to. In the usual frequency-doubled Nd : YAG laser, CYSHG < 0.0 1, and therefore the fourth term of (37) is negligible compared to t 2 6 ~ sin2 (o,t0/2) over nearly the entire detuning range.

Hence the saturated round-trip field gain is determined approximately by

2g = @cYSHG t a t 26L sin2 (w, to/2). (38)

The form of this equation determines the power output from the mode-locked frequency-doubled laser discussed in the next section.

IV. LASER SATURATION AND OUTPUT POWER It has been noted previously [I21 that the gain of a homo-

geneously broadened laser saturates according to

g = Go/U t PPC) (39)

where Go is the unsaturated or small-signal single-pass electric- field amplitude gain, P is a saturation parameter which depends only upon laser-material parameters and the cross-sectional area of the oscillating mode, and PC is the laser’s average circulating power. Equations (38) and (39) thus serve to determine the circulating power that can be expected internal to a mode-locked frequency-doubled laser. Harmonic output power is given by Pout = 2aSHGPc, and

Maximum power output occurs with om to = 0 and

and is given by

These values of optimum output coupling and optimum output power are a factor of @less than those expected from either a frequency-doubled non-mode-locked laser or a mode- locked laser operating at its fundamental wavelength.

For either of these lasers, power gain saturates at a value equal to single-pass power loss. For the frequency-doubled free-running laser, this means that saturated gain is found from 2g = a + QISHG. Similarly, gain saturation in the mode-locked fundamental-wavelength laser reduces gain until 2g = CY + T/2 , where T is the power transmission of the output coupler. The maximum powe; available from either of these lasers is 2/P [(2Go)l/’ - l 2 and occurs with CYSHG = (2Go)l” . a1l2 - a for the doubled laser and with T/2 = (2 Go)’/’ CY’/’ - CY for the mode-locked fundamental-wavelength laser [ 9 ] .

The basic mathematical difference that distinguishes the mode-locked frequency-doubled laser from these lasers is that

IEEE JOURNAL OF QUANTUM ELECTRONICS, JANUARY 1975

fected by ~ C Y S H G , while only CYSHG contributes to the single- direction output power. For either a frequency-doubled free- running laser or for a mode-locked fundamental-wavelength laser, all loss terms which contribute to output power appear in an identical form in both output-power and laser-saturation equations.

There is one basic physically distinctive feature of the mode- locked frequency-doubled laser that accounts for a reduction in its available power output as compared to the free-running or fundamental-wavelength laser. In contrast to the latter two lasers, the loss encountered by the Gaussian pulse in passing through the amplitude modulator of a frequency-doubled laser is not negligible and is in fact proportional to CYSHG.

For the special case of a long pulsewidth, without detuning, a simple physical explanation exists which clarifies this result. In this case, the Gaussian pulse occupies a relatively narrow spectral width and the amplifier saturated gain may be considered a temporal constant g. In the steady state the sum of all single-pass cavity power-losses must be equal to 2g and thus must be time independent. Although second-harmonic loss and modulator losses each vary with time, their sum must be a constant if the total cavity loss is to be time invariant. At t = 0, i.e., at the peak of the Gaussian pulse, modulator loss is zero and second-harmonic loss is maximum and given by f i a s ~ ~ . As time increases from t = 0 , doubling losses are reduced and modulator losses increase. Their sum nevertheless remains constant and may be calculated by its value at t = 0 and is thus given by ~ C Y S H G .

In the case of no detuning and where the line shape of the amplifying medium is unimportant, the dependence of the sum of modulator and second-harmonic loss on CYSHG is mathe- matically easily demonstrated. Consider the field-amplitude transmission of a modulator as given by (6b). For a laser operating without detuning, i.e., for modulator-drive frequency adjusted such that to = 0 , this transmission may be written aL( t ) = exp [ - 6 ~ ( w , t)’/4]. The amplitude transmission through the second-harmonic generator is found from (7) and (14) and is given by

T ~ H G ( t ) = (exp - a) exp [ - y t 2 (b - I ) ] (43)

where a = ( f i j 2 ) C Y ~ H G and b = 1 - @CYSHG. Thus a single pass through the modulator, frequency-doubler combination results in a power-transmission function

T = [ T s ~ ~ ( t ) a ~ ( t ) ] ’ =exp (-2a)

exp [-2t2 [y(b - 1) t 6,0&/4]]. (44)

The pulse propagation constant y is determined from (34) and hence the single-pass modulator, frequency-doubler power loss is

1 - T = I - eXp- 2a fiCYsHG. (45)

Consequently, @CYSHG accounts for the effect of both the modulator and the doubler in saturating the laser gain. On the other hand, only C Y ~ H G , the time average losses to the second harmonic, is available as single-direction output coupling. The remaining loss is caused by the modulator and does not con-

in the mode-locked frequency-doubled laser saturation is af- tribute to output power.


V. SOME NUMERICAL EXAMPLES A. The Significance of C Y ~ H G : Laser Perfomance as a Function of CYSHG

Before considering the analytical implications of the theory developed in this paper it is useful to examine further the physical significance of CYSHG, the second-harmonic conversion efficiency. Although in the theory herein CYSHG is treated as an independent variable and as a determinant of circulating power and pulsewidth, CYSHG actually changes both with circulating power and with pulsewidth.

The treatment of QSHG as an independent, rather than dependent variable, has the advantage that it decouples the analysis of effects of laser saturation from the analysis of the effects of mode locking and frequency doubling. Hence it allows the characterization of the output mode-locked pulsewidth given in Section I11 without direct reference to laser saturation. This treatment permits the generation of generalized curves which describe the pulsewidth rp and phase shift w, to of the mode-locked pulse as a function of second- harmonic conversion efficiency CYSHG, and detuning parameter Af. Figs. 2-4 present these curves for a internally frequency- doubled Nd : YAG laser. The laser parameters used to calculate the curves are Aw, = 2n(120 GHz), 0, = 2n(250 MHz), and

Fig. 2 shows, for various ,modulation indices, 6~ the variation of pulsewidth rp with increasing second-harmonic conversion efficiency CYSHG. This curve is plotted from (32) for zero phase shift between the amplitude-modulation and the mode- locked pulse train (w, t o = 0). Note that even for small CYSHG rather substantial broadening of the laser pulsewidth from its second-harmonic-free value occurs. It is interesting to note

2g = 0.05 + f i a s ~ ~ .

‘ s H G - - %

Fig. 2. Second-harmonic pulsewidth rp (10-percent power points) as a function of the second-harmonic conversion efficiency OLSHG.

that for CYSHG greater than approximately 0.3 percent the pulsewidth follows a square-root dependency on CYSHG, as described by (35), and is consequently independent of the assumed value of saturated laser gain. That is, for CYSHG > 0.3- percent pulsewidth is determined solely by the modulator and second-harmonic process and does not depend on the amplifier line shape.

Fig. 3 shows the variation of pulsewidth T~ with detuning A f. The detuning characteristic is approximately symmetric about a detuning Af = ( ~ , / 2 n ) ~ (4g/Aw,). A positive Af corresponds to a modulation frequency lower than the axial mode spacing, c/2L0. The occurrence of a minimum pulsewidth at a modulation period different from the round-trip cavity time is due to the time delay seen by the pulse passing through the Nd :YAG amplifier rod. Note also that as CYSHG is increased the effect of detuning is decreased.

The change in the mode-locked pulse phase shift w,to with detuning A f is shown in Fig. 4. A positive value of 0, to denotes that the laser pulse train leads the amplitude modulation. Here again note that detuning effects are less pronounced at high CYSHG. The reduced sensitivity of the phase shift w, to to detuning may be viewed as the cause of the insensitivity of pulsewidth to changes in modulation frequency when CYSHG is large. That is, from (32) it is seen that the primary way detuning affects pulsewidth is via a change in pulse position urn to. Hence, if pulse position is insensitive to detuning, pulsewidth will also vary slowly with changes in modulation frequency.

Physically, the lessened sensitivity of pulse position to detuning at higher second-harmonic conversion efficiencies is due to the relatively long pulse durations associated with large values of CYSHG. This can be understood by examining the physical implications of modulator detuning. For any modulation period, the requirement that the pulse envelope repeat itself after a cavity round trip determines om t o , the phase shift between the modulation and the output pulse train.

A f -- IkHd

Fig. 3. Second-harmonic pulsewidth rp (10-percent power points) as a function of detuning Af. The modulation depth 6~ is 0.1.


Af -- kHz

Fig. 4. Pulse position wmto relative to the amplitude modulation, as a function of detuning Af. The modulation depth 6~ is 0.1.

If the drive frequency is chosen optimally, i.e., for minimum pulsewidth, then the pulse train will leave the modulator in phase with the modulation, complete a traversal through the optical cavity, and return in phase with the modulation wave- form. A pulse thus propagating will suffer minimum loss in the modulator. In that case the modulation period T, is the sum of the linear cavity propagation time and the two-pass amplifier delay time, i.e.,

T, (optimum) = 2Lo/c t 4g/Aw,. (46)

Now what happens if the modulation period is not chosen according to (46)? A pulse leaving the modulator in phase with the amplitude modulation will return to the modulator after a cavity round trip with a nonzero phase shift (I$) with respect to .the amplitude modulation. First consider the case shown in Fig. 5 where the modulation period is slightly less than T, (optimum) so that when the pulse reenters the modulator, I$ > 0. At 0 < qj< ?T the modulator loss is increasing with increasing time and hence the pulse’s trailing portion is at- tenuated more strongly than its leading edge. As seen in Fig. 5, this nonsymmetrical loss results in a negative delay of the pulse centroid. Steady-state operation of the detuned laser is achieved when the pulse leaving the modulator with phase $I returns to it with the same phase I$. Thus in the steady state the phase shift between the modulation and pulse train determines a delay in the modulator. This delay exactly compen- sates for the deviation of modulation period from T, (optimum). The delay in the modulator is determined from the generalized Gaussian pulse shape given by (1) and the amplitude modulator transfer function given by (6b). From these equations it can be shown that the delay t o , due to the amplitude modulator, is given by

MODULATOR

PULSE INPUT

Fig. 5. Pulse delay in the amplitude modulator. If the pulse enters the modulator with a phase @J relative to the modulation it experiences an effective delay to.

t o = ( & L a r n sin urnto)7; / (2 In IO). (48)

In Fig. 5 steaay-state operation occurs with urn to < 0 so that to is negative.

On the other hand, if the modulation period is greater than T, the pulse’s leading edge sees more loss than its trailing portion and in the steady state the pulse position urn to > 0 so that a positive pulse delay results.

The delay t o , required by the modulator, is determined solely by the deviation of the modulation period from T, (optimum). From (48) it is seen that the phase shift urn to required for a particular delay to is least when the laser produces long pulses. Longest pulsewidth occurs when the conversion to the second harmonic is greatest and hence the sensitivity of pulse phase shift to detuning is least at highest conversion to the second harmonic.

3. Laser Performance as a Function of Beam Size in the Frequency-Doubler Crystal

Thus far this paper has analyzed the performance of mode- locked internally frequency-doubled laser in terms of a single parameter, ~ S H G , the second-harmonic conversion efficiency. Because of the dependent nature of ~ S H G , a parametric analysis based only on (YSHG is necessarily incomplete and the variation of CYSHG with circulating power, laser beam size, and pulsewidth must be unraveled before a prediction of laser performance as a function of independent variables can be made. That is, a prediction of laser performance as a function of cavity mode sizes or pump-lamp input power requires an evaluation of CYSHG in terms of more basic laser parameters.

Recall that ~ S H G is defined as the ratio of average single- direction second-harmonic power generated to the average circulating fundamental power, i.e.,

(49)

where the overbar denotes a time average. The average second-harmonic power produced can be found

from the single-pulse second-harmonic shape, (9), by integrat- ing over both the spatial and temporal pulse extent. Assuming the maximum instantaneous conversion efficiency is small the second-harmonic power may be written

or in terms of the pulsewidth


where z2w is the crystalline impedance at the second harmonic, wo is the l/e2 fundamental wavelength beam radius, and c/2L0 is the pulse repetition frequency. Other quantities in (50) are defined in (1) and (9).

The average fundamental power, calculated from (1) in a similar manner, is given by

PC = [A2 ~ ~ ~ / ( 4 ~ J ( c / 2 L 0 ) [ 7 1 / ( 2 ~ ) 1 ~ ’ ~ (5 1)

where z, is the crystalline impedance of the fundamental wavelength. Thus the second-harmonic conversion efficiency can be described in terms of incavity beam sizes, fundamental circulating power, and second-harmonic pulsewidth as

~ S H G = [ 2 K 2 / ( 7 d ) I [ z L / ~ 2 ~ 1

‘ Pc(2LO/C) r n / ( f i T p ) . (52)

In performing the temporal integrations necessary to evaluate PC and Pz( t ) it was implicitly assumed that rp << 2Lo/c, so that temporal overlap of consecutive pulses is negligible.

It is instructive to compare the evaluation of O~SHG for the mode-locked laser to that of a identical single axial mode [6 , p. 1331 free-running laser. A similar calculation of the second-harmonic conversion efficiency for the free-running laser and a comparison with (52) shows that the factor

(2Lo/c)[-)/fiI ( 1 / T p ) ( ~ L O / C ) ( O - ~ ~ / T ~ )

-

is the enhancement due to the mode locking. Now with a mechanism established to relate (YSHG to basic

cavity parameters, the results of this paper can be well illus- trated by a numerical example. An Nd : YAG laser, AM mode locked and internally frequency doubled by a Ba2NaNbsOls crystal is considered. Where appropriate, parameters are taken equal to those analyzed in [2]. This choice of numerical values facilitates comparison with the mode-locked fundamental- wavelength laser.

A laser pumped by a single krypton lamp is considered, and the unsaturated power gain 2Go is assumed to vary linearly wlrn lamp input power and to be described by [ 131

GO = k P 1 a m p - PO 1. (53)

This linear approximation to unsaturated gain is approximately valid for Plamp > 2Po. The parameter k depends on the elec- trical efficiency of the pump lamp, on the efficiency of coupling into the Nd : YAG rod, and upon Nd :YAG material parameters. The parameter Po is the value of power input to the pump lamp at which the gain Go equals zero in the linear approximation. P b p is the input power to the pump lamp.

Assumed values for these parameters and other parameters are given in Table I. The values given are typical values for a laser consisting of a 3-mm-diamY 50-mm-long Nd:YAG laser crystal pumped by a 4-mm-bore krypton lamp.

Fig. 6 shows the predicted variation in second-harmonic output power as a function of lamp power for various beam sizes in the frequency-doubler crystal. For reference, the mode-locked second-harmonic power P,,, available from the laser is also shown (42). The curve for a beam size of wo = 100 p is slightly under coupled over much of the range of pumping powers. Hence while decreasing the beam size in the

TABLE I LASER PARAMETERS: FIGS. 6-10

k pumping-efficiency constant = 2.2 X 10-4 W)-l PO lamp-threshold parameter = 385 W 01 single-pass linear power loss = 0.05

2LO - laser axial mode spacing = 250 MHz

LO lasercavity optical length = 60 cm. Awa laser fluorescent linewidth (full width at half maximum) =

277(120 GHz) w, AM frequency = 2x

modulation drive frequency = 277 (250 MHz) 8p saturation parameter = 0.1 (W)-1 K second-harmonic nonlinearitya = 1.96 X 10-7 m/V

aThe second-harmonic nonlinearity K (9) is calculated assuming a 90Ophase-matched BazNaNb5015 of 5-mm length. The crystal parameters used in evaluation of K are from S . Singh, D. A. Draegert, and J. E. Geusic, “Optical and ferroelectric properties of barium sodium niobate,” Phys. Rev. B , vol. 2, pp. 2709-2724, Oct. 1970.

XI

Fig. 6. Second-harmonic output power PSHG as a functlon of the krypton-lamp input power Plamp. The modulation depth 6~ is 0.1.

Fig. 7. Second-harmonic pulsewidth (10-percent power points) T as a function of the krypton-lamp input power Plamp. The modu%tion depth 6~ is 0.1.

doubler crystal to slightly below 100 1.1 would provide a small increase in output power, decreasing the beam size well below 100 p would actually lead to a reduction of output power. A curve of second-harmonic output power for wo = 50 p falls nearly coincident with the 100-p curve.

Fig. 7 shows predicted second-harmonic pulsewidths over the same range of pump powers and spot sizes. As lamp input


power increases, pulses broaden due to the increase of second- harmonic conversion efficiency. Because pulsewidths increase monotonically with decreasing beam sizes (increasing (YSHG) a decrease in the beam size in the doubler crystal below 100 p would result in a large increase in pulsewidth with only a small increase in power output.

Figs. 8-10 show the response of the frequency-doubled laser to detuning of the modulation drive frequency. As expected from Fig. 4, detuning effects are less pronounced for smaller beam sizes in the frequency doubler, i.e., at higher values of CYSHG. Figs. 8-10 show, respectively, the change in pulse position, output power, and pulsewidth with detuning. This large

Af -- kHz

Fig. 10. Second-harmonic pulsewidth (10-percent power points) as a function of detuning Af. Plamp = 1000 w, 6L = 0.1.

A f -- kHz

Fig. 8. Pulse position w,to relative to the amplitude modulation as a function of detuning Af. The lamp input power Plamp is 1000 W and the modulation depth 6~ is 0.1.

350 1

m

250

150 " L " L

150 I

Fig. 9. Second-harmonic power PSHG as a function of detuning Af. Plamp = 1000 W, 6~ = 0.1.

theoretical detuning range is not apt to be aclueved in practice because the most common types of amplitude modulators, acoustooptic or electrooptic, do not usually present constant modulation depths over frequency ranges as large as the 700-kHz range shown in Figs. 8-10 [14].

The reduction in pulsewidth with detuning shown in Fig. 10 is caused by the reduction in circulating power due to modulator loss as the modulator is detuned from its optimal drive frequency (or as the cavity length is changed from optimum). As the detuning is increased the pulse position changes and the modulator presents an internal cavity loss that reduces PC. This decrease in circulating power results in a decrease in CYSHG and hence reduction in the laser's pulsewidth.

VI. EXPERIMENTAL RESULTS A preliminary confirmation of this theory was made with a

frequency-doubled Nd: YAG laser pumped by a K-Rb lamp. The frequency doubler was a barium sodium niobate crystal and relevant laser parameters were g E 0.0 1, a L = 0.03, c /2L = 5 * lo8 Hz, and ~ S H G ~0.005. The theory developed here predicts a pulsewidth of 460 ps in rough agreement with the measured pulsewidth (corrected for detector response) of 390 ps.

VII. SUMMARY

This paper has analyzed the performance of an internally frequency-doubled mode-locked laser. The results reported herein have been derived by considering the effects of each intracavity component upon a generalized Gaussian pulse shape. This technique was introduced by Kuizenga and Siegman [2] for the analysis of a mode-locked fundamental- wavelength laser. The effects of the internal frequency doubler are accounted for by a single parameter (YSHG, defined as the average power conversion efficiency to the second-harmonic wavelength. The second-harmonic process provides a power-


dependent loss to the intracavity propagating pulse, presenting more loss at its peak than at its extremities. This nonlinear loss leads to substantial broadening of the laser pulsewidth. The variation of pulsewidth, pulse position, and laser saturated gain with CYSHG has been predicted. The equations derived differ. markedly from those describing the fundamental- wavelength laser. The pulse broadening caused by second- harmonic generation has been shown to be substantial even at small CYSHG.

In contrast to the fundamental-wavelength mode-locked laser, losses due to the amplitude modulator have been shown to be an important determinant of total laser losses and hence greatly affect the available second-harmonic output power. In fact, unavoidable modulator losses limit the output power theoretically extractable from the mode-locked frequency- doubled laser to a factor of f i l e s than that available from an identical free-running laser. It has been further shown that, if the value of (ILSHG necessary to extract the maximum available laser power is exceeded, the pulsewidth increases while the output power decreases with increasing CVSHG.

The response of the mode-locked frequency-doubled laser to detuning has been predicted. It has been shown that at higher values of C Y ~ H G the laser is less affected by detuning of the modulator from its optimal frequency.

Numerical examples have been given for predicted laser performance for an Nd:YAG laser, internally frequency doubled by a Baz NaNb5 OI5 crystal.

ACKNOWLEDGMENT The author wishes to thank A . E. Siegman for discussions

regarding the treatment of second-harmonic generation used in Section I1 of this paper. He also wishes to thank J. D. Barry, C. B. Hitz, C. J. Kennedy, D. J. Kuizenga, N. Blackman, R. S. Reynolds, W.A. Specht, and P. J. Titterton for suggestions and

comments on the manuscript. He especially wants to thank this JOURNAL’S reviewers for valuable criticisms of the manuscript.

REFERENCES

[ 11 C. B. Hitz and L. M. Osterink, “Simultaneous intracavity frequency doubling and mode locking in a Nd :YAG laser,” Appl. Phys. Lett., vol. 18, pp. 378-380, May 1, 1971.

[2] D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser-Part I: Theory,” IEEE J. Quantum Electron., vol. QE-6, pp. 694-708, Nov. 1970.

[3] 0. Bernecker, “Limitations for mode-locking enhancement of internal SHG in a laser,” IEEE J. Quantum Electron., vol. QE-9, pp. 897-900, Sept. 1973.

[4] A. E. Siegman and D. J. Kuizenga, “Active mode-coupling phenomena in pulsed and continuous lasers,” Opto-Electron., vol. 6, pp. 43-66, 1974.

[SI A. Yariv, Quantum Electronics. New York: Wiley, 1967, pp.

[6] N. Bloembergen, Nonlinear Optics. New York: W. A. Benjamin,

171 C. B. Hitz, GTE Sylvania, Mountain View, Calif., private communication.

[8] J. M. Yarborough, J. Falk, and C. B. Hitz, “Enhancement of optical second harmonic generation by utilizing the dispersion of air,”Appl. Phys. Lett., vol. 18, pp. 70-72, Feb. 1, 1971.

[9] R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron., vol. QE-6, pp. 215-223, Apr. 1970.

[ 101 P. J. Titterton, “Effect of system inaccuracies on the probability of bit error for pulsed communication systems,” Appl. Opt., vol. 13, pp. 1034-1040, May 1974.

[ 111 T. S . Kinsel, “A stabilized mode-locked Nd :YAlG laser using electronic feedback,” IEEE J. Quantum Electron. (Part I of Two Parts), vol. QE-9, pp. 3-8, Jan. 1973.

[12] A. Yariv, Introduction to Optical Electronics. New York: Holt, Rinehart and Winston, 1971, pp. 109-112.

[ 131 G. A. Massey, “Measurement of device parameters for Nd: YA103 lasers,” IEEE J. Quantum Electron., vol. QE-8, pp. 669-674, July 1972.

[ 141 I. P. Kaminow and E. H. Turner, “Electrooptic light modulators,” Proc. IEEE (A Special Joint Issue on Optical Electronics with Applied Optics), vol. 54, pp. 1374-1390, Oct. 1966.

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A theory of the mode-locked, internally frequency-doubled laser

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