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Numerical modelling of continuous-wave holographic laser oscillators
1 March 2000
Optics Communications 175 Ž2000. 397–408 www.elsevier.comrlocateroptcom
Numerical modelling of continuous-wave holographic laser oscillators G.J. Crofts ) , M.J. Damzen Department of Physics, Laser Optics and Spectroscopy, Blackett Laboratory, Imperial College, London, SW7 2BW, UK Received 14 September 1999; received in revised form 8 December 1999; accepted 13 December 1999
Abstract The continuous-wave operation of a self-adaptive laser which oscillates by diffraction from a gain hologram is modelled numerically. We demonstrate the greater-than-unity diffraction efficiency of a single gain grating and show how the properties of an externally-written gain grating laser lead to the realization of an entirely self-starting system. The dependence of output power on parameters such as intracavity gain and output coupler reflectivity is investigated, thereby enabling the optimisation of power extraction and improving understanding of steady-state holographic laser action. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.55.Ah; 42.60.Da; 42.65.Hw Keywords: Laser theory; Laser resonators; Gain gratings; Phase conjugation
1. Introduction Experimental and theoretical studies have shown that three-dimensional gain gratings, optically written in a saturable laser amplifier, can act as very efficient Ž) 100%. diffractive optical elements w1–3x and, in a four-wave mixing ŽFWM. geometry, can produce extremely high phase-conjugate reflectivity w4–7x. A further development in the use of gain gratings for phase conjugation has been the employment of loop schemes to obtain self-pumped phase conjugation ŽSPPC. with only one input beam required w8–12x. In reality, these devices are actually a novel type of Žholographic. laser where the feedback ) Corresponding author. Tel.: q44-20-7594-7743; Fax: q4420-7594-7744; e-mail: [email protected]
is provided by diffraction from an externally-written gain-grating hologram w13x. Since the reflectivity of this type of SPPC device can be much greater than unity, it is possible to replace the input beam with a partial reflector and achieve self-oscillation by virtue of spontaneous gain-grating formation w12,14,15x. Such a dynamic holographic resonator has the ability to self-adapt and can thereby correct for intracavity distortions in real time. Self-adaptive laser architectures of this type are of particular interest because conventional solid-state laser oscillators are currently limited in beam quality at high output power by thermally-induced phase and polarisation distortions in the laser amplifiers w16x. Much of the recent experimental and theoretical work has concentrated on a transient regime where gratings are written with short Žnanosecond. pulses
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 7 - 0
398
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
and decay over an upper-state lifetime unless erased by a strong reading beam w10–15,17–21x. The lasing species are also usually pumped over a period similar to the upper-state lifetime. Early theoretical work did analyse the reading and writing of gainrabsorption gratings in a range of steady-state FWM interactions w22–24x and more recently the analysis has been extended to fully model an interaction with beams of arbitrary strength and where, in general, multiple gain gratings are simultaneously present w25x. As yet, however, there has been very little experimental or theoretical characterization of resonators which oscillate entirely via diffraction from a gain hologram in the fully steady-state regime of operation w26,27x. In this paper we present a numerical modelling of holographic lasers operating in a continuous-wave ŽCW. regime, as would be appropriate to a system based on a CW diode-pumped, solid-state laser medium, for example. We begin by developing a theory describing the core interaction and summarise the main characteristics of individual transmission and reflection gain gratings written by coherent CW beams in a continuously pumped laser amplifier. We then describe a self-intersecting loop geometry where the grating forms the basis of a diffractively coupled CW laser resonator which can also be viewed as a highly efficient self-pumped phase conjugator. Finally, we describe how this non-linear loop scheme can be used as the core element of an entirely self-starting, self-adaptive CW laser oscillator and characterize its performance. 2. Theory of CW gain grating interactions The physical mechanism for gain grating formation in a laser amplifier is the saturation of the gain in the presence of two or more interfering beams. The intense parts of the interference pattern result in local depletion of the population inversion leading to a spatially modulated gain or gain grating. In the steady-state regime, the saturation of the local amplitude gain coefficient a Žr. of a homogeneously broadened transition on resonance is described by the well-known equation w28x. a0 a Žr. s Ž 1. 1 q I T Ž r . rIS
where a 0 is the small-signal gain coefficient, I T Žr. is the total local beam intensity and IS is the characteristic saturation intensity of the medium Že.g. ; 1 kWrcm2 for Nd:YVO4 .. We will consider a four-wave mixing ŽFWM. interaction of the type shown in Fig. 1 where a complicated gain hologram is simultaneously written and read out by the optical beams present. Also shown in Fig. 1 are the two main gain gratings of interest, the transmission grating and the reflection grating, although other gratings can be formed w29x. The total electric field in the medium is taken to have the form ET Ž r,t . s
1 2
ET Ž r,t . exp Ž i v t . q c.c.
Ž 2.
where ET is the total, complex electric field amplitude in the medium which, for the four optical fields present, can be written as ET s A1exp Ž yik1 r . q A2 exp Ž yik2 r . qA3 exp Ž yik3 r . q A4 exp Ž yik4 r .
Ž 3.
where the complex slowly varying amplitude, A j , describes the amplitude, phase and polarisation state Žtaken to be transverse to the z-axis. of the jth electric field, and k j is the wave vector of the jth optical field with magnitude
ž ž
k3,4 s "kzˆY s "k yxsin ˆ
u 2
u qzcos ˆ
u 2
2
u qzcos ˆ
2
/ /
Ž 4a . Ž 4b .
where the q and y signs refer to indices 1, 3 and 2, 4 respectively and where xˆ and zˆ are the unit vectors along the two grating directions indicated in Fig. 1
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
Fig. 1. Schematic of a gain four-wave mixing geometry showing the interacting fields Ž A1 primary gratings of interest. X
Y
and zˆ and zˆ are the unit vectors along the beam propagation directions. The total, time-averaged intensity in the medium is given by I T s Ž1r2. n ´ 0 cET EUT , and when all the interacting fields are co-polarised, but the interference between A1 and A2 and between A3 and A4 is neglected, the resulting normalised intensity pattern of interest is given by I T Ž x , z ,t . IS
s s i q
1
Ž t eyi K 2 i
1 q
Ž r eyi K 2 i
t
rz
x
Ž 5.
si s
4
Ý A j Ž z ,t .
A2S
P A)j
Ž z ,t .
4
are the normalised mean intensity level, transmission pattern amplitude and reflection pattern amplitude, respectively, and A S is the saturation field strength defined such that IS s Ž1r2. n ´ 0 cA2S . As the angle between the beams is small, the distinction between z, zX and zY is neglected for the evolution of the slowly varying field amplitudes and they are assumed to vary along z. When the interference pattern of Eq. Ž5. is substituted for I T in Eq. Ž1., the gain hologram is found to have the form
s
where K t x s k Ž zX y zY . and K r z s k Ž zX q zY . are the fast spatially varying phase components and ft Ž z ., fr Ž z . are the slowly varying phase components of the transmission and reflection interference patterns, respectively, and where 1
™ A ., the coordinate system and the two
a Ž x, z.
q t i) eqi K t x .
q r i) eqi K r z .
399
Ž 6a .
a0 1q s i q
G0
s
1q Mt cos ct q Mr cos cr
Ž 7.
where ct Ž x, z . s K t x y ft Ž z . and cr Ž z . s K r z y fr Ž z . are the spatially varying phases of the transmission and reflection grating components, and where
G0 Ž z . s
a0 1 q si Ž z .
Ž 8a .
js1 qi ft
t i s
2 s
A2S
r i s < r i
2 A2S
A1 P A)3 q A)2 P A4 A1 P A4) q A2) P A3
Ž 6b .
Mt Ž z . s
Ž 6c .
Mr Ž z . s
Ž 8b . Ž 8c .
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
400
are the incoherently saturated gain coefficient, and a transmission and reflection grating modulation parameter, respectively. For simplicity we will consider the presence of only a single transmission Ž Mr s 0. or single reflection Ž Mt s 0. gain grating. This situation can easily be realised experimentally by judicious choice of beam polarization state w18x, but in any case the results will afford us significant insight into the underlying physics of CW gain gratings. In this regime, the gain grating of Eq. Ž7. is simplified and can be expanded in a Fourier cosine series as follows `
as
q
dz y
q
as
y
2
y
Ž z . exp Ž qi n cg . q c.c.
Ž 9.
G0
Ž 10a.
(
1 y M g2 2 G0
n
(
1 y M g2
ž
(
1 y 1 y M g2 Mg
n
/ Ž 10b.
and where the subscript g can refer to either the transmission grating Ž g s t . or the reflection grating Ž g s r .. It is noted that the odd harmonics, and the fundamental in particular, are negative and so are in antiphase to the interference pattern. The effect of the gain grating on the optical fields can be found by substituting the modulated gain of Eq. Ž7. into the non-linear Maxwell wave equation and applying the slowly varying envelope and plane-wave approximations w30x. Selection of the appropriate phase-matched components leads to the following sets of coupled equations, which agree with previous derivations w1,23x, describing the steady-state interaction with the transmission gain grating q
d A1 dz
y
d A2 dz
Ž 11c.
s gt A 4 q kt A 2
Ž 11d.
d A4 dz
s gr A1 q kr A 4
Ž 12a.
s gr A 2 q kr)A 3
Ž 12b.
s gr A 3 q kr A 2
Ž 12c.
s gr A 4 q kr)A1
Ž 12d.
with the coupling coefficients related to the lowest order grating harmonics by
with harmonic coefficients given by
a gŽ n. Ž z . s Ž y1 .
d A3 dz
ns0
a gŽ0. Ž z . s q
d A2 dz
`
Ý
d A1 dz
q
a gŽ n.
s gt A 3 q kt)A1
and with the reflection gain grating
ns0
1
d A4 dz
a gŽ n. Ž z . cos Ž n cg .
Ý
d A3
s gt A1 q kt A 3
Ž 11a.
s gt A 2 q kt)A 4
Ž 11b.
gg Ž z . s a gŽ0. Ž z .
Ž 13a .
1 k g Ž z . s a gŽ1. Ž z . eqi f g Ž z . . 2
Ž 13b.
As the diffractive coupling coefficient k g is directly proportional to the fundamental harmonic amplitude a gŽ1., which is negative, any field probing the grating will acquire a p-phase shift on diffraction. As we have noted in a previous publication w1x it is possible, from Eq. Ž11a. –Eq. Ž11d. and Eq. Ž12a. –Eq. Ž12d., to obtain the conserved quantities Ct s A1 A 2 q A 3 A 4 , for the transmission case, and Cr s A1 A 2 y A 3 A 4 , for the reflection case, which are independent of z and which can be used as a check on accuracy when solving the equations numerically. Using the theory developed in this section we can proceed to analyse the basic properties of transmission and reflection gain gratings.
3. The transmission and reflection gain grating The diffraction efficiency of a grating is a fundamental property that can be used to characterize its action on a optical beam. The diffraction efficiency of a gain grating written by two CW beams can be measured by a third, Bragg-matched beam which does not perturb the grating w1,3x. This requires that
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
the probing beam is weak enough not to saturate the gain, which would lead to a reduction in the average gain and partial erasure of the grating by the p-phase shifted, diffracted beam. A suitable interaction geometry, which can be modelled using the theory of the previous section, is shown in Fig. 1. The transmission grating is written by beams A1 and A 3 and probed by a weak beam A 2 generating a diffracted beam A 4 Žwith A 4Ž L. s 0.. The reflection grating is written by beams A1 and A 4 and probed by a weak beam A 2 generating a diffracted beam A 3 Žwith A 3 Ž0. s 0.. The intensity diffraction efficiency of the two gratings is then defined by
ht s
hr s
I4out I2in I3out I2in
s
s
< A 4 Ž 0. < 2
Ž 14a.
< A2 Ž L. < 2 < A3 Ž L. < 2 < A2 Ž L. < 2
.
401
intensity which maximises the diffraction efficiency. Above this level the writing beams themselves excessively extract energy from the amplifier, causing the gain and diffraction efficiency to fall. It is noteworthy that, for a 0 L G 2 Ži.e., G 0 G 55., the diffraction efficiency can exceed unity Že.g. h f 10 for a 0 L s 3.. This is possible due to extraction of the stored energy by the probe and diffracted beam. A further point to note is that, at higher values of gain, the reflection grating exhibits a larger peak diffraction efficiency Žhr s 11.7 for a 0 L s 3. than the transmission grating Žht s 8.1. and which occurs at a higher value of writing beam intensity, as previously noted w1x. This is a consequence of the different spatial distributions of the two gratings, and is the reverse of the behaviour observed and calculated for transient gain gratings w1,10x.
Ž 14b.
Fig. 2 shows the calculated diffraction efficiency of the transmission and reflection grating as a function of normalised writing beam intensity and for three values of small-signal gain-length product a 0 L Žrelated to small-signal intensity gain G 0 by G 0 s expw2 a 0 L x.. In each case, the two writing beams are of equal input intensity so as to write the deepest modulated grating and thereby optimise the grating diffraction efficiency. For each value of small-signal gain there is an optimum value of writing beam
Fig. 2. A plot of diffraction efficiency Žh . against normalised writing beam intensity for both a transmission and reflection gain-grating and for three values of small-signal gain-length product, a 0 L Žthe line h s1 is also shown..
4. The SPPC holographic loop resonator The greater-than-unity diffraction efficiencies demonstrated in Section 3 suggest that a CW-written gain grating could be used as the basis for a diffractively coupled laser resonator. Although such a system could be realised with a pair of external beams to write the gain hologram, an alternative Žand simpler. approach is to use a single external beam in a self-intersecting loop geometry, as shown in Fig. 3. Two possible configurations are shown which result in the writing of either a transmission Žt . grating ŽFig. 5Ža.. or a reflection Ž r . grating ŽFig. 5Žb.. in the FWM amplifier G FW M Žsmall-signal gain G 0 s expw2 a 0 L x.. Two additional elements included in the loop are: a non-reciprocal transmission element ŽNRTE. with clockwise intensity transmission factor Tq and counter-clockwise transmission Ty Žset to unity., and a loop amplifier GAm p Žsmall-signal gain GX0 s expw2 a 0X LX x. which can provide additional gain. The forward transmission factor Tq is set to be attenuating such that, despite the gain experienced by the input beam on passing around the loop, the two writing beams are of comparable strength for maximum grating modulation and diffraction efficiency. The self-consistent condition for CW oscillation in the counterclockwise direction is that the loop gain is unity, i.e. h Ty GX0 G 1. Thus the lasing flux initiates
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G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
whole device acts as a so-called self-pumped phase conjugator ŽSPPC. w12x. The loop boundary conditions for the transmission grating geometry ŽFig. 5Ža.. with no loop amplifier present are A 3 Ž 0 . s tq A1 Ž L .
Ž 15a.
A 2 Ž L . s ty A 4 Ž 0 .
Ž 15b.
A4 Ž L. s 0
Ž 15c.
and for the reflection grating geometry ŽFig. 5Žb.. are
Fig. 3. Schematic of the self-pumped phase-conjugate loop laser oscillating by diffraction from Ža. a transmission gain grating and Žb. a reflection gain grating written in the FWM amplifier G FW M . The loop incorporates a second amplifier GAm p and an NRTE comprising a Faraday rotator ŽFR., a half-wave retardation plate Ž l r2. and two polarisers ŽP1 , P2 ..
from noise which, via diffractive feedback, grows to a steady-state level determined by the writing beam strength and amplifier gains. Also, any phase distortions experienced by the input beam on passing around the loop are encoded in the gain hologram. Therefore, due to the nature of the FWM interaction, the optical fields propagating counterclockwise can form a self-consistent spatial mode if beams A 2 and A 4,3 Žfor the t , r-grating. are the phase conjugate of beams A1 and A 3,4 , respectively w11x. As a consequence, the system can thereby correct for phase distortions present in the loop Že.g. in amplifier GAm p . and in the FWM amplifier G FWM itself. In addition, as the output lasing beam A 2 Ž0. is the phase conjugate of the single input beam A1Ž0., the
A 4 Ž L . s tq A1 Ž L .
Ž 16a.
A 2 Ž L . s ty A 3 Ž L .
Ž 16b.
A 3 Ž 0. s 0
Ž 16c.
where t " are the forward Žq. and backward Žy. loop Žamplitude. transmission factors and are related to their intensity counterparts by T "s < t " < 2 . In order to compensate for the p-phase shift induced on diffraction, the transmission factor in the phase-conjugate lasing direction is chosen to have a value of U tys y1 such that the product tq ty- 0 w1x. This non-reciprocal p-phase shift can be achieved experimentally by appropriate setting of the NRTE waveplate and allows the lasing radiation in the phaseconjugate direction to be resonant with the input radiation w11x. It has, however, been shown experimentally that in the absence of the non-reciprocal phase shift, the system can still lase by slightly detuning in frequency to offset the p-phase shift w11x. Numerical solution of Eqs. Ž11a. and Ž11d. and Eq. Ž12a. –Eq. Ž12d., along with boundary conditions Ž15. and Ž16., leads to the normalised CW output I2outrIS s < A 2 Ž0.< 2rA2S and the characteristic SPPC reflectivity of the loop conjugator, defined as R SPPC s I2outrI1in s < A 2 Ž0.< 2r< A1Ž0.< 2 . Fig. 4Ža. and Fig. 4Žb., respectively, show the output intensity and loop reflectivity as a function of normalised input intensity I1in rIS for both a transmission and reflection geometry and for two values of forward intensity transmission factor Tq. In all cases, the gain parameters used were a 0 L s 3 Ž G 0 f 400. and a 0X LX s 0 Ž GX0 s 1., i.e. no loop amplifier. The graphs of Fig. 4 show that, for the parameters chosen, the transmission and reflection grating geometry are quite similar
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
Fig. 4. Ža. Normalised output intensity and Žb. SPPC reflectivity of the self-conjugating loop laser as a function of normalised input beam intensity for both a transmission and a reflection grating geometry. Curves for two values of forward intracavity transmisX X sion factor, Tq, are shown Žwith, a 0 Ls 3.0, a 0 L s 0.0 and Ty s1.0..
in their behaviour. Fig. 4Ža. shows that there is a threshold input intensity required to obtain an output Ž I1th rIS f 2.7 = 10y4 for Tqs 0.05.. As discussed in Section 3, the threshold condition is that the loop gain is unity Žh Tys 1. and, since Tys 1, this requires a sufficiently strong input for unity diffraction efficiency Žh s 1. to be achieved. In a similar manner to a conventional CW laser, the loop gain Žand hence h . is clamped to unity when operating above threshold. Once above threshold, the output of the loop conjugator grows approximately logarithmically with input, I2out A lnŽ I1in rI1th ., until the output starts to saturate. After reaching a peak Ž I2outrIS f 1.2 for
403
Tqs 0.05., the output falls as the input beam itself strongly saturates the gain. Fig. 4Žb. shows that just above threshold the corresponding SPPC reflectivity rises rapidly to a high value of f 400 and then falls due to the logarithmic growth of the output with input intensity. The maximum conjugate output that can be achieved is also dependent on the choice of Tq, as shown in Fig. 4Ža.. The value of Tqs 0.05 is a near optimum value whereas the lower value of Tqs 0.005 is clearly too attenuating. Values of Tq much greater than the optimum are too transmitting and allow the input beam to extract too much energy on passing around the loop. In fact, for the transmission case, the optimum value is Tqf 0.055 which yields a peak conjugate output when the combined input and lasing beams saturate the inversion such that effective gain seen Geff Žs I1outrI1in ., including diffractive losses, exactly offsets the forward loop losses, i.e. Geff s 1rTq. That is too say the optimum phase-conjugate lasing output occurs when the two writing beams Ž I1in and I3in . are precisely equal. This condition ensures that maximum modulation depth of grating is achieved and therefore maximum diffraction efficiency, as previously discussed. The parameters required to achieve this condition are not easily identified since, for a fixed gain, the output intensity is a non-linear function of both input intensity and loop transmission factor, I2out s f Ž I1in ,Tq .. Only by numerical integration of the interaction equations can this function be generated and the optimum values of I1in and Tq be deduced. The SPPC systems studied above contained only a FWM amplifier and so we now investigate the effect of including an amplifier inside the self-conjugating loop Ži.e. a 0X LX / 0.. This requires two more equations to describe the saturable amplification of the fields propagating clockwise Ž Aq . and counterclockwise Ž Ay . in the loop Žsee Fig. 3.. The formation of a standing-wave gain grating in the loop amplifier can also be included in the model, but for simplicity is neglected. It is also necessary to modify the boundary conditions Ž15. and Ž16. so as to relate the fields interacting in the FWM amplifier to the fields entering and leaving the loop amplifier in an appropriate manner. Although the addition of a further gain element can lead to a larger lasing output, a more important consideration is the dependence of
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G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
that lasing output on the gain distribution within the loop. In order to study this, we fix the total smallsignal gain in the loop, G 0 GX0 s expw2 a 0 L q 2 a 0X LX x, to be a constant. A total loop gain coefficient of a 0 L q a 0X LX s 3 is chosen and the other parameters given values of Tqs 0.05 and Tys 0.05 so as to correspond to the examples studied previously. Fig. 5Ža. and Fig. 5Žb. show the conjugate output and SPPC reflectivity of the loop system as a function of input intensity for both a transmission and reflection geometry and for two distributions of loop gain. Fig. 5Ža. shows that the conjugate output is suppressed and peaks at a larger input intensity when a portion of the loop gain is located in the loop amplifier. Fig. 5Žb. shows that, as a result, the peak SPPC reflectiv-
ity that can be achieved is significantly reduced. The clear result for both the transmission and reflection geometry is that maximum performance is obtained when all the available loop gain is concentrated in the FWM amplifier. This is a consequence of using diffractive coupling from a gain grating whose efficiency depends non-linearly on the gain coefficient of the FWM amplifier.
5. The self-starting holographic laser oscillator The self-conjugating loop geometry of the previous section was shown to be both a novel diffractively coupled laser oscillator and also a highly amplifying phase-conjugate mirror. The large SPPC reflectivities that can be achieved from such a system suggest that the external input beam can be replaced by a partial reflector and thereby an entirely self-starting, self-adaptive holographic laser oscillator ŽHLO. can be realised. The schematic diagram of Fig. 6 illustrates the concept for a transmission gain grating geometry. The core element is the self-conjugating loop, characterized by its reflectivity R SPPC , to which is added the partial reflector Žreflectivity R OC . to provide feedback to the loop and also to act as output coupling optic Žwith transmission TOC s 1 y R OC .. We identify four possible beams leaving the system with intensities as follows I PC s
1 2
I NPC s X I NPC s
X I PC s
Fig. 5. Ža. Normalised output intensity and Žb. SPPC reflectivity of the self-conjugating loop laser as a function of normalised input beam intensity for both a transmission and a reflection grating geometry. Curves for two different distributions of loop gain Ž a 0 Lq a 0X LX s 3.0. are shown Žwith Tq s 0.05 and Ty s1.0..
1 2
n ´ 0 c < A 2 Ž 0 . < 2 Ž 1 y R OC . 1
Ž 17a.
n´ 0 c < A3 Ž L. < 2
Ž 17b.
n ´ 0 c < A1 Ž L . < 2 Ž 1 y Tq .
Ž 17c.
n ´ 0 c < A 4 Ž 0 . < 2 Ž 1 y Ty . .
Ž 17d.
2 1 2
The unprimed quantities refer to the primary outputs whereas the primed quantities refer to the beams ejected from the NRTE. The subscript ‘PC’ refers to the phase-conjugate Žcounter-clockwise. lasing direction whereas the subscript ‘NPC’ refers to the nonphase-conjugate Žclockwise. lasing direction. The operation of the self-starting HLO is initiated by intracavity spontaneous emission which induces a
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
405
Fig. 6. Simplified schematic of the self-starting, self-adaptive holographic laser oscillator comprising the SPPC loop of Fig. 3Ža. with X X feedback from an ouput coupling optic of reflectivity R OC . Four possible ouput beams are identified: I PC , I NPC , I NPC and I PC .
weak gain grating in G FW M that in turn weakly diffracts radiation in the loop, causing an enhancement of the amplified spontaneous emission flux. The diffracted intracavity radiation that gives constructive interference to enhance the growth of the grating will be preferentially selected. This parametric feedback, involving the mutual growth of the gratings and the interacting fields, leads to the formation of a laser oscillator with spatial adaptability and spectral selectivity w15x. The CW operation of this device Žwith no loop amplifier, i.e. GX0 s 1. is still described by Eq. Ž12a. –Eq. Ž12d. and boundary conditions Ž15. Žwith tys y< ty <., but with the additional condition
(
A1 Ž 0 . s R OC A 2 Ž 0 . .
output of I PC rIS f 1.1 is obtained for R OC and Tq set to the low values of 2.7% and 4.2%, respectively. However, the output falls by only ; 10% if the values of R OC and Tq are doubled from their optimum. The minimum allowed values for R OC and Tq
Ž 18 .
The same condition would apply to a reflection grating based system along with Eq. Ž12a. –Eq. Ž12d. and boundary conditions Ž16., but for brevity we will confine our studies to the transmission grating geometry. Also, we will link Ty to the clockwise transmission factor by Tys 1 y Tq, in order to simulate the behaviour of a real experimental NRTE w11x. Fig. 7 shows a contour plot of the resulting phase-conjugate lasing output I PC as a function of R OC and Tq for a gain coefficient of a 0 L s 3 Žand a 0X LX s 0.. As can be seen, lasing action occurs over a wide range of the parameter space and is not overly sensitive to R OC and Tq. A peak conjugate
Fig. 7. A contour plot of the normalised conjugate output intensity, I PC r IS , as a function of both R OC and Tq for a self-starting HLO operating in a transmission grating geometry Žwith a 0 Ls X X 3.0, a 0 L s 0.0 and Ty s1.0..
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G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
are determined by the threshold condition for selfstarting CW laser action R OC R SPPC s 1.
Ž 19 .
For an optimum value of Tq Žwith a 0 L s 3. it was shown in Fig. 4Žb. that the peak value of R SPPC was in excess of 400, and therefore the minimum value of R OC required to reach threshold is less than 0.25%. As the maximum value for R OC is unity, the minimum value of R SPPC is also unity which places a lower limit on the small-signal gain required to reach threshold. Above threshold, R SPPC is clamped to a CW value of 1rR OC and the diffraction efficiency of the gain grating h is clamped to a CW value of 1rTy. A further relevant point to note from Fig. 4Žb. is that, except at the peak, there are two values of input intensity which yield the same SPPC reflectivity. This result suggests that for a given output coupler R OC , the self-starting system has two possible modes of operation, with either a low intracavity flux or a high intracavity flux. This property raises the possibility of bistable behaviour of the system. However, the preferred mode of operation will depend upon its perturbation stability and it is clear from Fig. 4Žb. that in the low power regime of operation, R SPPC is extremely sensitive to the input intensity, whereas it is much less sensitive in the higher power regime. It is in this higher power regime that the results of Fig. 7 were obtained, as will be all subsequent results. Although we have concentrated on the phase-conjugate output, it is informative to quantify the other beams leaving the system ŽEq. Ž17a. –Eq. Ž17d... Fig. 8 shows a diagonal cross-section Ž R OC s 2.5 = Tq . through the plot of Fig. 7 along with the corresponding data for the other output beams Žand their total.. When R OC is below optimum the low feedback leaves more energy in the amplifier as a weaker grating is formed resulting in lower levels of intracavity flux. At values of R OC above optimum, however, the large feedback leads to a considerable increase in the non-conjugate lasing output Žup to I NPC rIS f 2.9. at the expense of the conjugate output. Providing there are no significant phase distortions within the system, this beam can provide a useful alternative output with efficient energy extraction.
Fig. 8. A diagonal cross-section Ž R OC s 2.5=Tq . through the plot of Fig. 7, also showing the normalised intensity of the other output beams and their total.
In all the above simulations the small-signal FWM gain coefficient was fixed at a value of a 0 L s 3. We now investigate the performance of the holographic laser as a function of the FWM gain and determine the optimum values of R OC and Tq required to give maximum phase-conjugate output. Fig. 9Ža. shows the intensity of the three main output beams Žand their total. as a function of FWM gain for a selfstarting HLO that is optimised for phase-conjugate ŽPC. output. The results show a substantial increase X in I PC with increasing gain whilst I NPC and I NPC remain approximately equal and constant Žat f 0.7 = IS .. This indicates that in order to optimise I PC , the FWM gain is always saturated, independent of its small-signal value, to a level which gives a total output of f 1.4 = IS in the non-conjugate direction, with the additional energy extracted in the phaseconjugate direction. The trend is further emphasized if the absolute output intensities are replotted as a fraction of the available power density in the amplifier Ž IAvailrIS s 2 a 0 L., thereby giving an extraction efficiency Ž IrIAvail . as shown in Fig. 9Žb.. We see here that the PC output rises to a 39% extraction efficiency for a 0 L s 4.6, whereas the two non-PC beams extract less than 8% each. The near equality X of I NPC and I NPC corresponds to a near equality of FWM beams A1Ž0. and A 3 Ž0.. Thus, in a similar manner to the SPPC system, the maximum conjugate
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
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output occurs for near-equal writing beam intensity. The slight deviation from precisely equal writing beams stems from the presence of the output coupling optic which also links A1Ž0. and A 2 Ž0. ŽEq. Ž18... An important practical consideration is to know the values of R OC and Tq required to give this maximum phase-conjugate output. Fig. 10 shows the optimum values of R OC and Tq as a function of FWM gain for the simulations described above. As can be seen, the optimised self-starting system is not very sensitive to Tq, whose optimum value only Fig. 10. A plot of the optimum values of R OC and Tq as a function of small-signal FWM gain for the system modelled in Fig. 9.
varies between 6% and 2% over the range of gains shown. Over the same range the optimum value of R OC varies more rapidly, falling from 12% to 0.4% reflectivity and indeed, for most of the range shown has a lower reflectivity Ž- 4%. than a simple glass– air interface.
6. Conclusions
Fig. 9. A plot of Ža. normalised output intensity and Žb. extraction efficiency, as a function of small-signal FWM gain, for a selfstarting Žtransmission grating. HLO which is optimised for phaseX X conjugate output Žwith a 0 L s 0.0 and Ty s1yTq .. Data is shown for the three main ouput beams and their total.
In conclusion, we have demonstrated that a single steady-state gain grating written in a laser amplifier by CW monochromatic writing beams can exhibit a diffraction efficiency greatly in excess of unity and therefore be used as an efficient diffractive optical element. Furthermore, the use of a gain medium in a self-intersecting loop geometry leads to an efficient, self-pumped optical phase conjugator with very high Ž4 1. reflectivity. The device also acts as an externally controllable holographic laser oscillator with the ability to correct for intracavity phase distortions. The phase conjugate output can be maximised by adjusting a non-reciprocal loop transmission element to give equal writing beam intensities. Although higher phase-conjugate output can be achieved by the inclusion of an additional loop amplifier, we have shown that the most efficient extraction of the stored energy occurs when all the loop gain is concentrated within the FWM amplifier. Finally, we
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have shown that, by providing feedback to the loop system from an output coupling optic, an entirely self-starting, self-adaptive CW holographic laser oscillator can be realised. We have used our numerical model to determine the optimum output coupler and intracavity loss required to maximise phase-conjugate energy extraction, for a given level of gain within the system. An output of 3.7 = IS with an extraction efficiency approaching 40% has been demonstrated for a gain of 10 4 , and it is expected that this figure will be significantly higher in a real system due the formation of a combined transmission–reflection gain grating system which will give enhanced feedback w20x. The numerical modelling described in this paper is currently being used in the design and optimisation of practical Holographic Laser Oscillators based on solid-state gain materials Že.g. Nd:YVO4 . which are pumped by high-power CW diode lasers.
Acknowledgements The authors acknowledge support from the UK Engineering and Physical Science Research Council ŽEPSRC., grant number GRrL96455.
References w1x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 102 Ž1993. 288. w2x A. Brignon, J.-P. Huignard, Opt. Lett. 18 Ž1993. 1639. w3x M.J. Damzen, Y. Matsumoto, G.J. Crofts, R.P.M. Green, Opt. Commun. 123 Ž1996. 182. w4x J. Reintjes, B.L. Wexler, N. Djeu, J.L. Walsh, J. Phys. ŽParis. 44 Ž1983. C2–27. w5x S. Mailis, J. Hendricks, D.P. Shepherd, A.C. Tropper, N. Moore, R.W. Eason, G.J. Crofts, M. Trew, M.J. Damzen, Opt. Lett. 24 Ž1999. 972. w6x A. Brignon, J.-P. Huignard, Opt. Commun. 119 Ž1995. 171.
w7x P.J. Soan, A.D. Case, M.J. Damzen, M.H.R. Hutchinson, Opt. Lett. 17 Ž1992. 781. w8x I.M. Bel’dyugin, V.A. Berenberg, A.E. Vasil’ev, I.V. Mochalov, V.M. Petnikova, G.T. Petrovskii, M.A. Kharchenko, V.V. Shuvalov, Sov. J. Quantum Electron. 19 Ž1989. 740. w9x A.A. Ageichik, O.G. Kotyaev, Yu.A. Rezunkov, A.L. Safronov, V.V. Stepanov, Opt. Spectrosc. 76 Ž1994. 446. w10x K.S. Syed, R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 112 Ž1994. 175. w11x O. Wittler, D. Udaiyan, G.J. Crofts, K.S. Syed, M.J. Damzen, IEEE J. Quantum Electron. 35 Ž1999. 656. w12x P. Sillard, A. Brignon, J. Opt. Soc. Am. B 15 Ž1998. 2917. w13x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Lett. 19 Ž1994. 393. w14x M.J. Damzen, R.P.M. Green, K.S. Syed, Opt. Lett. 20 Ž1995. 1704. w15x A. Minassian, G.J. Crofts, M.J. Damzen, Opt. Commun. 161 Ž1999. 338. w16x R. Weber, B. Neuenschwander, M. MacDonald, M.B. Roos, H.P. Weber, IEEE J. Quantum Electron. 34 Ž1998. 1046. w17x A. Brignon, J.-P. Huignard, IEEE J. Quantum Electron. 30 Ž1994. 2203. w18x K.S. Syed, G.J. Crofts, R.P.M. Green, M.J. Damzen, J. Opt. Soc. Am. B 14 Ž1997. 2067. w19x P. Sillard, A. Brignon, J.-P. Huignard, J.-P. Pocholle, Opt. Lett. 23 Ž1998. 1093. w20x K.S. Syed, G.J. Crofts, M.J. Damzen, Opt. Commun. 146 Ž1998. 181. w21x P. Sillard, A. Brignon, J.-P. Huignard, IEEE J. Quantum Electron. 34 Ž1998. 465. w22x J. Reintjes, L.J. Palumbo, IEEE J. Quantum Electron. 18 Ž1982. 1934. w23x W.P. Brown, J. Opt. Soc. Am. 73 Ž1983. 629. w24x A.L. Gaeta, M.T. Gruneisen, R.W. Boyd, IEEE J. Quantum Electron. 22 Ž1986. 1095. w25x B. Ai, R.J. Knize, J. Opt. Soc. Am. B 13 Ž1996. 2408. w26x M.J. Damzen, R.P.M. Green, G.J. Crofts, Opt. Lett. 36 Ž1994. 34. w27x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 124 Ž1996. 488. w28x A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986, p. 207. w29x K.S. Syed, G.J. Crofts, M.J. Damzen, J. Opt. Soc. Am. B 13 Ž1996. 1892. w30x R.A. Fisher, Optical Phase Conjugation, Academic Press, New York, 1983, p. 7.
Optics Communications 175 Ž2000. 397–408 www.elsevier.comrlocateroptcom
Numerical modelling of continuous-wave holographic laser oscillators G.J. Crofts ) , M.J. Damzen Department of Physics, Laser Optics and Spectroscopy, Blackett Laboratory, Imperial College, London, SW7 2BW, UK Received 14 September 1999; received in revised form 8 December 1999; accepted 13 December 1999
Abstract The continuous-wave operation of a self-adaptive laser which oscillates by diffraction from a gain hologram is modelled numerically. We demonstrate the greater-than-unity diffraction efficiency of a single gain grating and show how the properties of an externally-written gain grating laser lead to the realization of an entirely self-starting system. The dependence of output power on parameters such as intracavity gain and output coupler reflectivity is investigated, thereby enabling the optimisation of power extraction and improving understanding of steady-state holographic laser action. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.55.Ah; 42.60.Da; 42.65.Hw Keywords: Laser theory; Laser resonators; Gain gratings; Phase conjugation
1. Introduction Experimental and theoretical studies have shown that three-dimensional gain gratings, optically written in a saturable laser amplifier, can act as very efficient Ž) 100%. diffractive optical elements w1–3x and, in a four-wave mixing ŽFWM. geometry, can produce extremely high phase-conjugate reflectivity w4–7x. A further development in the use of gain gratings for phase conjugation has been the employment of loop schemes to obtain self-pumped phase conjugation ŽSPPC. with only one input beam required w8–12x. In reality, these devices are actually a novel type of Žholographic. laser where the feedback ) Corresponding author. Tel.: q44-20-7594-7743; Fax: q4420-7594-7744; e-mail: [email protected]
is provided by diffraction from an externally-written gain-grating hologram w13x. Since the reflectivity of this type of SPPC device can be much greater than unity, it is possible to replace the input beam with a partial reflector and achieve self-oscillation by virtue of spontaneous gain-grating formation w12,14,15x. Such a dynamic holographic resonator has the ability to self-adapt and can thereby correct for intracavity distortions in real time. Self-adaptive laser architectures of this type are of particular interest because conventional solid-state laser oscillators are currently limited in beam quality at high output power by thermally-induced phase and polarisation distortions in the laser amplifiers w16x. Much of the recent experimental and theoretical work has concentrated on a transient regime where gratings are written with short Žnanosecond. pulses
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 7 - 0
398
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
and decay over an upper-state lifetime unless erased by a strong reading beam w10–15,17–21x. The lasing species are also usually pumped over a period similar to the upper-state lifetime. Early theoretical work did analyse the reading and writing of gainrabsorption gratings in a range of steady-state FWM interactions w22–24x and more recently the analysis has been extended to fully model an interaction with beams of arbitrary strength and where, in general, multiple gain gratings are simultaneously present w25x. As yet, however, there has been very little experimental or theoretical characterization of resonators which oscillate entirely via diffraction from a gain hologram in the fully steady-state regime of operation w26,27x. In this paper we present a numerical modelling of holographic lasers operating in a continuous-wave ŽCW. regime, as would be appropriate to a system based on a CW diode-pumped, solid-state laser medium, for example. We begin by developing a theory describing the core interaction and summarise the main characteristics of individual transmission and reflection gain gratings written by coherent CW beams in a continuously pumped laser amplifier. We then describe a self-intersecting loop geometry where the grating forms the basis of a diffractively coupled CW laser resonator which can also be viewed as a highly efficient self-pumped phase conjugator. Finally, we describe how this non-linear loop scheme can be used as the core element of an entirely self-starting, self-adaptive CW laser oscillator and characterize its performance. 2. Theory of CW gain grating interactions The physical mechanism for gain grating formation in a laser amplifier is the saturation of the gain in the presence of two or more interfering beams. The intense parts of the interference pattern result in local depletion of the population inversion leading to a spatially modulated gain or gain grating. In the steady-state regime, the saturation of the local amplitude gain coefficient a Žr. of a homogeneously broadened transition on resonance is described by the well-known equation w28x. a0 a Žr. s Ž 1. 1 q I T Ž r . rIS
where a 0 is the small-signal gain coefficient, I T Žr. is the total local beam intensity and IS is the characteristic saturation intensity of the medium Že.g. ; 1 kWrcm2 for Nd:YVO4 .. We will consider a four-wave mixing ŽFWM. interaction of the type shown in Fig. 1 where a complicated gain hologram is simultaneously written and read out by the optical beams present. Also shown in Fig. 1 are the two main gain gratings of interest, the transmission grating and the reflection grating, although other gratings can be formed w29x. The total electric field in the medium is taken to have the form ET Ž r,t . s
1 2
ET Ž r,t . exp Ž i v t . q c.c.
Ž 2.
where ET is the total, complex electric field amplitude in the medium which, for the four optical fields present, can be written as ET s A1exp Ž yik1 r . q A2 exp Ž yik2 r . qA3 exp Ž yik3 r . q A4 exp Ž yik4 r .
Ž 3.
where the complex slowly varying amplitude, A j , describes the amplitude, phase and polarisation state Žtaken to be transverse to the z-axis. of the jth electric field, and k j is the wave vector of the jth optical field with magnitude
ž ž
k3,4 s "kzˆY s "k yxsin ˆ
u 2
u qzcos ˆ
u 2
2
u qzcos ˆ
2
/ /
Ž 4a . Ž 4b .
where the q and y signs refer to indices 1, 3 and 2, 4 respectively and where xˆ and zˆ are the unit vectors along the two grating directions indicated in Fig. 1
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
Fig. 1. Schematic of a gain four-wave mixing geometry showing the interacting fields Ž A1 primary gratings of interest. X
Y
and zˆ and zˆ are the unit vectors along the beam propagation directions. The total, time-averaged intensity in the medium is given by I T s Ž1r2. n ´ 0 cET EUT , and when all the interacting fields are co-polarised, but the interference between A1 and A2 and between A3 and A4 is neglected, the resulting normalised intensity pattern of interest is given by I T Ž x , z ,t . IS
s s i q
1
Ž t eyi K 2 i
1 q
Ž r eyi K 2 i
t
rz
x
Ž 5.
si s
4
Ý A j Ž z ,t .
A2S
P A)j
Ž z ,t .
4
are the normalised mean intensity level, transmission pattern amplitude and reflection pattern amplitude, respectively, and A S is the saturation field strength defined such that IS s Ž1r2. n ´ 0 cA2S . As the angle between the beams is small, the distinction between z, zX and zY is neglected for the evolution of the slowly varying field amplitudes and they are assumed to vary along z. When the interference pattern of Eq. Ž5. is substituted for I T in Eq. Ž1., the gain hologram is found to have the form
s
where K t x s k Ž zX y zY . and K r z s k Ž zX q zY . are the fast spatially varying phase components and ft Ž z ., fr Ž z . are the slowly varying phase components of the transmission and reflection interference patterns, respectively, and where 1
™ A ., the coordinate system and the two
a Ž x, z.
q t i) eqi K t x .
q r i) eqi K r z .
399
Ž 6a .
a0 1q s i q
G0
s
1q Mt cos ct q Mr cos cr
Ž 7.
where ct Ž x, z . s K t x y ft Ž z . and cr Ž z . s K r z y fr Ž z . are the spatially varying phases of the transmission and reflection grating components, and where
G0 Ž z . s
a0 1 q si Ž z .
Ž 8a .
js1 qi ft
t i s
2 s
A2S
r i s < r i
2 A2S
A1 P A)3 q A)2 P A4 A1 P A4) q A2) P A3
Ž 6b .
Mt Ž z . s
Ž 6c .
Mr Ž z . s
Ž 8b . Ž 8c .
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
400
are the incoherently saturated gain coefficient, and a transmission and reflection grating modulation parameter, respectively. For simplicity we will consider the presence of only a single transmission Ž Mr s 0. or single reflection Ž Mt s 0. gain grating. This situation can easily be realised experimentally by judicious choice of beam polarization state w18x, but in any case the results will afford us significant insight into the underlying physics of CW gain gratings. In this regime, the gain grating of Eq. Ž7. is simplified and can be expanded in a Fourier cosine series as follows `
as
q
dz y
q
as
y
2
y
Ž z . exp Ž qi n cg . q c.c.
Ž 9.
G0
Ž 10a.
(
1 y M g2 2 G0
n
(
1 y M g2
ž
(
1 y 1 y M g2 Mg
n
/ Ž 10b.
and where the subscript g can refer to either the transmission grating Ž g s t . or the reflection grating Ž g s r .. It is noted that the odd harmonics, and the fundamental in particular, are negative and so are in antiphase to the interference pattern. The effect of the gain grating on the optical fields can be found by substituting the modulated gain of Eq. Ž7. into the non-linear Maxwell wave equation and applying the slowly varying envelope and plane-wave approximations w30x. Selection of the appropriate phase-matched components leads to the following sets of coupled equations, which agree with previous derivations w1,23x, describing the steady-state interaction with the transmission gain grating q
d A1 dz
y
d A2 dz
Ž 11c.
s gt A 4 q kt A 2
Ž 11d.
d A4 dz
s gr A1 q kr A 4
Ž 12a.
s gr A 2 q kr)A 3
Ž 12b.
s gr A 3 q kr A 2
Ž 12c.
s gr A 4 q kr)A1
Ž 12d.
with the coupling coefficients related to the lowest order grating harmonics by
with harmonic coefficients given by
a gŽ n. Ž z . s Ž y1 .
d A3 dz
ns0
a gŽ0. Ž z . s q
d A2 dz
`
Ý
d A1 dz
q
a gŽ n.
s gt A 3 q kt)A1
and with the reflection gain grating
ns0
1
d A4 dz
a gŽ n. Ž z . cos Ž n cg .
Ý
d A3
s gt A1 q kt A 3
Ž 11a.
s gt A 2 q kt)A 4
Ž 11b.
gg Ž z . s a gŽ0. Ž z .
Ž 13a .
1 k g Ž z . s a gŽ1. Ž z . eqi f g Ž z . . 2
Ž 13b.
As the diffractive coupling coefficient k g is directly proportional to the fundamental harmonic amplitude a gŽ1., which is negative, any field probing the grating will acquire a p-phase shift on diffraction. As we have noted in a previous publication w1x it is possible, from Eq. Ž11a. –Eq. Ž11d. and Eq. Ž12a. –Eq. Ž12d., to obtain the conserved quantities Ct s A1 A 2 q A 3 A 4 , for the transmission case, and Cr s A1 A 2 y A 3 A 4 , for the reflection case, which are independent of z and which can be used as a check on accuracy when solving the equations numerically. Using the theory developed in this section we can proceed to analyse the basic properties of transmission and reflection gain gratings.
3. The transmission and reflection gain grating The diffraction efficiency of a grating is a fundamental property that can be used to characterize its action on a optical beam. The diffraction efficiency of a gain grating written by two CW beams can be measured by a third, Bragg-matched beam which does not perturb the grating w1,3x. This requires that
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
the probing beam is weak enough not to saturate the gain, which would lead to a reduction in the average gain and partial erasure of the grating by the p-phase shifted, diffracted beam. A suitable interaction geometry, which can be modelled using the theory of the previous section, is shown in Fig. 1. The transmission grating is written by beams A1 and A 3 and probed by a weak beam A 2 generating a diffracted beam A 4 Žwith A 4Ž L. s 0.. The reflection grating is written by beams A1 and A 4 and probed by a weak beam A 2 generating a diffracted beam A 3 Žwith A 3 Ž0. s 0.. The intensity diffraction efficiency of the two gratings is then defined by
ht s
hr s
I4out I2in I3out I2in
s
s
< A 4 Ž 0. < 2
Ž 14a.
< A2 Ž L. < 2 < A3 Ž L. < 2 < A2 Ž L. < 2
.
401
intensity which maximises the diffraction efficiency. Above this level the writing beams themselves excessively extract energy from the amplifier, causing the gain and diffraction efficiency to fall. It is noteworthy that, for a 0 L G 2 Ži.e., G 0 G 55., the diffraction efficiency can exceed unity Že.g. h f 10 for a 0 L s 3.. This is possible due to extraction of the stored energy by the probe and diffracted beam. A further point to note is that, at higher values of gain, the reflection grating exhibits a larger peak diffraction efficiency Žhr s 11.7 for a 0 L s 3. than the transmission grating Žht s 8.1. and which occurs at a higher value of writing beam intensity, as previously noted w1x. This is a consequence of the different spatial distributions of the two gratings, and is the reverse of the behaviour observed and calculated for transient gain gratings w1,10x.
Ž 14b.
Fig. 2 shows the calculated diffraction efficiency of the transmission and reflection grating as a function of normalised writing beam intensity and for three values of small-signal gain-length product a 0 L Žrelated to small-signal intensity gain G 0 by G 0 s expw2 a 0 L x.. In each case, the two writing beams are of equal input intensity so as to write the deepest modulated grating and thereby optimise the grating diffraction efficiency. For each value of small-signal gain there is an optimum value of writing beam
Fig. 2. A plot of diffraction efficiency Žh . against normalised writing beam intensity for both a transmission and reflection gain-grating and for three values of small-signal gain-length product, a 0 L Žthe line h s1 is also shown..
4. The SPPC holographic loop resonator The greater-than-unity diffraction efficiencies demonstrated in Section 3 suggest that a CW-written gain grating could be used as the basis for a diffractively coupled laser resonator. Although such a system could be realised with a pair of external beams to write the gain hologram, an alternative Žand simpler. approach is to use a single external beam in a self-intersecting loop geometry, as shown in Fig. 3. Two possible configurations are shown which result in the writing of either a transmission Žt . grating ŽFig. 5Ža.. or a reflection Ž r . grating ŽFig. 5Žb.. in the FWM amplifier G FW M Žsmall-signal gain G 0 s expw2 a 0 L x.. Two additional elements included in the loop are: a non-reciprocal transmission element ŽNRTE. with clockwise intensity transmission factor Tq and counter-clockwise transmission Ty Žset to unity., and a loop amplifier GAm p Žsmall-signal gain GX0 s expw2 a 0X LX x. which can provide additional gain. The forward transmission factor Tq is set to be attenuating such that, despite the gain experienced by the input beam on passing around the loop, the two writing beams are of comparable strength for maximum grating modulation and diffraction efficiency. The self-consistent condition for CW oscillation in the counterclockwise direction is that the loop gain is unity, i.e. h Ty GX0 G 1. Thus the lasing flux initiates
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G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
whole device acts as a so-called self-pumped phase conjugator ŽSPPC. w12x. The loop boundary conditions for the transmission grating geometry ŽFig. 5Ža.. with no loop amplifier present are A 3 Ž 0 . s tq A1 Ž L .
Ž 15a.
A 2 Ž L . s ty A 4 Ž 0 .
Ž 15b.
A4 Ž L. s 0
Ž 15c.
and for the reflection grating geometry ŽFig. 5Žb.. are
Fig. 3. Schematic of the self-pumped phase-conjugate loop laser oscillating by diffraction from Ža. a transmission gain grating and Žb. a reflection gain grating written in the FWM amplifier G FW M . The loop incorporates a second amplifier GAm p and an NRTE comprising a Faraday rotator ŽFR., a half-wave retardation plate Ž l r2. and two polarisers ŽP1 , P2 ..
from noise which, via diffractive feedback, grows to a steady-state level determined by the writing beam strength and amplifier gains. Also, any phase distortions experienced by the input beam on passing around the loop are encoded in the gain hologram. Therefore, due to the nature of the FWM interaction, the optical fields propagating counterclockwise can form a self-consistent spatial mode if beams A 2 and A 4,3 Žfor the t , r-grating. are the phase conjugate of beams A1 and A 3,4 , respectively w11x. As a consequence, the system can thereby correct for phase distortions present in the loop Že.g. in amplifier GAm p . and in the FWM amplifier G FWM itself. In addition, as the output lasing beam A 2 Ž0. is the phase conjugate of the single input beam A1Ž0., the
A 4 Ž L . s tq A1 Ž L .
Ž 16a.
A 2 Ž L . s ty A 3 Ž L .
Ž 16b.
A 3 Ž 0. s 0
Ž 16c.
where t " are the forward Žq. and backward Žy. loop Žamplitude. transmission factors and are related to their intensity counterparts by T "s < t " < 2 . In order to compensate for the p-phase shift induced on diffraction, the transmission factor in the phase-conjugate lasing direction is chosen to have a value of U tys y1 such that the product tq ty- 0 w1x. This non-reciprocal p-phase shift can be achieved experimentally by appropriate setting of the NRTE waveplate and allows the lasing radiation in the phaseconjugate direction to be resonant with the input radiation w11x. It has, however, been shown experimentally that in the absence of the non-reciprocal phase shift, the system can still lase by slightly detuning in frequency to offset the p-phase shift w11x. Numerical solution of Eqs. Ž11a. and Ž11d. and Eq. Ž12a. –Eq. Ž12d., along with boundary conditions Ž15. and Ž16., leads to the normalised CW output I2outrIS s < A 2 Ž0.< 2rA2S and the characteristic SPPC reflectivity of the loop conjugator, defined as R SPPC s I2outrI1in s < A 2 Ž0.< 2r< A1Ž0.< 2 . Fig. 4Ža. and Fig. 4Žb., respectively, show the output intensity and loop reflectivity as a function of normalised input intensity I1in rIS for both a transmission and reflection geometry and for two values of forward intensity transmission factor Tq. In all cases, the gain parameters used were a 0 L s 3 Ž G 0 f 400. and a 0X LX s 0 Ž GX0 s 1., i.e. no loop amplifier. The graphs of Fig. 4 show that, for the parameters chosen, the transmission and reflection grating geometry are quite similar
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
Fig. 4. Ža. Normalised output intensity and Žb. SPPC reflectivity of the self-conjugating loop laser as a function of normalised input beam intensity for both a transmission and a reflection grating geometry. Curves for two values of forward intracavity transmisX X sion factor, Tq, are shown Žwith, a 0 Ls 3.0, a 0 L s 0.0 and Ty s1.0..
in their behaviour. Fig. 4Ža. shows that there is a threshold input intensity required to obtain an output Ž I1th rIS f 2.7 = 10y4 for Tqs 0.05.. As discussed in Section 3, the threshold condition is that the loop gain is unity Žh Tys 1. and, since Tys 1, this requires a sufficiently strong input for unity diffraction efficiency Žh s 1. to be achieved. In a similar manner to a conventional CW laser, the loop gain Žand hence h . is clamped to unity when operating above threshold. Once above threshold, the output of the loop conjugator grows approximately logarithmically with input, I2out A lnŽ I1in rI1th ., until the output starts to saturate. After reaching a peak Ž I2outrIS f 1.2 for
403
Tqs 0.05., the output falls as the input beam itself strongly saturates the gain. Fig. 4Žb. shows that just above threshold the corresponding SPPC reflectivity rises rapidly to a high value of f 400 and then falls due to the logarithmic growth of the output with input intensity. The maximum conjugate output that can be achieved is also dependent on the choice of Tq, as shown in Fig. 4Ža.. The value of Tqs 0.05 is a near optimum value whereas the lower value of Tqs 0.005 is clearly too attenuating. Values of Tq much greater than the optimum are too transmitting and allow the input beam to extract too much energy on passing around the loop. In fact, for the transmission case, the optimum value is Tqf 0.055 which yields a peak conjugate output when the combined input and lasing beams saturate the inversion such that effective gain seen Geff Žs I1outrI1in ., including diffractive losses, exactly offsets the forward loop losses, i.e. Geff s 1rTq. That is too say the optimum phase-conjugate lasing output occurs when the two writing beams Ž I1in and I3in . are precisely equal. This condition ensures that maximum modulation depth of grating is achieved and therefore maximum diffraction efficiency, as previously discussed. The parameters required to achieve this condition are not easily identified since, for a fixed gain, the output intensity is a non-linear function of both input intensity and loop transmission factor, I2out s f Ž I1in ,Tq .. Only by numerical integration of the interaction equations can this function be generated and the optimum values of I1in and Tq be deduced. The SPPC systems studied above contained only a FWM amplifier and so we now investigate the effect of including an amplifier inside the self-conjugating loop Ži.e. a 0X LX / 0.. This requires two more equations to describe the saturable amplification of the fields propagating clockwise Ž Aq . and counterclockwise Ž Ay . in the loop Žsee Fig. 3.. The formation of a standing-wave gain grating in the loop amplifier can also be included in the model, but for simplicity is neglected. It is also necessary to modify the boundary conditions Ž15. and Ž16. so as to relate the fields interacting in the FWM amplifier to the fields entering and leaving the loop amplifier in an appropriate manner. Although the addition of a further gain element can lead to a larger lasing output, a more important consideration is the dependence of
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that lasing output on the gain distribution within the loop. In order to study this, we fix the total smallsignal gain in the loop, G 0 GX0 s expw2 a 0 L q 2 a 0X LX x, to be a constant. A total loop gain coefficient of a 0 L q a 0X LX s 3 is chosen and the other parameters given values of Tqs 0.05 and Tys 0.05 so as to correspond to the examples studied previously. Fig. 5Ža. and Fig. 5Žb. show the conjugate output and SPPC reflectivity of the loop system as a function of input intensity for both a transmission and reflection geometry and for two distributions of loop gain. Fig. 5Ža. shows that the conjugate output is suppressed and peaks at a larger input intensity when a portion of the loop gain is located in the loop amplifier. Fig. 5Žb. shows that, as a result, the peak SPPC reflectiv-
ity that can be achieved is significantly reduced. The clear result for both the transmission and reflection geometry is that maximum performance is obtained when all the available loop gain is concentrated in the FWM amplifier. This is a consequence of using diffractive coupling from a gain grating whose efficiency depends non-linearly on the gain coefficient of the FWM amplifier.
5. The self-starting holographic laser oscillator The self-conjugating loop geometry of the previous section was shown to be both a novel diffractively coupled laser oscillator and also a highly amplifying phase-conjugate mirror. The large SPPC reflectivities that can be achieved from such a system suggest that the external input beam can be replaced by a partial reflector and thereby an entirely self-starting, self-adaptive holographic laser oscillator ŽHLO. can be realised. The schematic diagram of Fig. 6 illustrates the concept for a transmission gain grating geometry. The core element is the self-conjugating loop, characterized by its reflectivity R SPPC , to which is added the partial reflector Žreflectivity R OC . to provide feedback to the loop and also to act as output coupling optic Žwith transmission TOC s 1 y R OC .. We identify four possible beams leaving the system with intensities as follows I PC s
1 2
I NPC s X I NPC s
X I PC s
Fig. 5. Ža. Normalised output intensity and Žb. SPPC reflectivity of the self-conjugating loop laser as a function of normalised input beam intensity for both a transmission and a reflection grating geometry. Curves for two different distributions of loop gain Ž a 0 Lq a 0X LX s 3.0. are shown Žwith Tq s 0.05 and Ty s1.0..
1 2
n ´ 0 c < A 2 Ž 0 . < 2 Ž 1 y R OC . 1
Ž 17a.
n´ 0 c < A3 Ž L. < 2
Ž 17b.
n ´ 0 c < A1 Ž L . < 2 Ž 1 y Tq .
Ž 17c.
n ´ 0 c < A 4 Ž 0 . < 2 Ž 1 y Ty . .
Ž 17d.
2 1 2
The unprimed quantities refer to the primary outputs whereas the primed quantities refer to the beams ejected from the NRTE. The subscript ‘PC’ refers to the phase-conjugate Žcounter-clockwise. lasing direction whereas the subscript ‘NPC’ refers to the nonphase-conjugate Žclockwise. lasing direction. The operation of the self-starting HLO is initiated by intracavity spontaneous emission which induces a
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Fig. 6. Simplified schematic of the self-starting, self-adaptive holographic laser oscillator comprising the SPPC loop of Fig. 3Ža. with X X feedback from an ouput coupling optic of reflectivity R OC . Four possible ouput beams are identified: I PC , I NPC , I NPC and I PC .
weak gain grating in G FW M that in turn weakly diffracts radiation in the loop, causing an enhancement of the amplified spontaneous emission flux. The diffracted intracavity radiation that gives constructive interference to enhance the growth of the grating will be preferentially selected. This parametric feedback, involving the mutual growth of the gratings and the interacting fields, leads to the formation of a laser oscillator with spatial adaptability and spectral selectivity w15x. The CW operation of this device Žwith no loop amplifier, i.e. GX0 s 1. is still described by Eq. Ž12a. –Eq. Ž12d. and boundary conditions Ž15. Žwith tys y< ty <., but with the additional condition
(
A1 Ž 0 . s R OC A 2 Ž 0 . .
output of I PC rIS f 1.1 is obtained for R OC and Tq set to the low values of 2.7% and 4.2%, respectively. However, the output falls by only ; 10% if the values of R OC and Tq are doubled from their optimum. The minimum allowed values for R OC and Tq
Ž 18 .
The same condition would apply to a reflection grating based system along with Eq. Ž12a. –Eq. Ž12d. and boundary conditions Ž16., but for brevity we will confine our studies to the transmission grating geometry. Also, we will link Ty to the clockwise transmission factor by Tys 1 y Tq, in order to simulate the behaviour of a real experimental NRTE w11x. Fig. 7 shows a contour plot of the resulting phase-conjugate lasing output I PC as a function of R OC and Tq for a gain coefficient of a 0 L s 3 Žand a 0X LX s 0.. As can be seen, lasing action occurs over a wide range of the parameter space and is not overly sensitive to R OC and Tq. A peak conjugate
Fig. 7. A contour plot of the normalised conjugate output intensity, I PC r IS , as a function of both R OC and Tq for a self-starting HLO operating in a transmission grating geometry Žwith a 0 Ls X X 3.0, a 0 L s 0.0 and Ty s1.0..
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are determined by the threshold condition for selfstarting CW laser action R OC R SPPC s 1.
Ž 19 .
For an optimum value of Tq Žwith a 0 L s 3. it was shown in Fig. 4Žb. that the peak value of R SPPC was in excess of 400, and therefore the minimum value of R OC required to reach threshold is less than 0.25%. As the maximum value for R OC is unity, the minimum value of R SPPC is also unity which places a lower limit on the small-signal gain required to reach threshold. Above threshold, R SPPC is clamped to a CW value of 1rR OC and the diffraction efficiency of the gain grating h is clamped to a CW value of 1rTy. A further relevant point to note from Fig. 4Žb. is that, except at the peak, there are two values of input intensity which yield the same SPPC reflectivity. This result suggests that for a given output coupler R OC , the self-starting system has two possible modes of operation, with either a low intracavity flux or a high intracavity flux. This property raises the possibility of bistable behaviour of the system. However, the preferred mode of operation will depend upon its perturbation stability and it is clear from Fig. 4Žb. that in the low power regime of operation, R SPPC is extremely sensitive to the input intensity, whereas it is much less sensitive in the higher power regime. It is in this higher power regime that the results of Fig. 7 were obtained, as will be all subsequent results. Although we have concentrated on the phase-conjugate output, it is informative to quantify the other beams leaving the system ŽEq. Ž17a. –Eq. Ž17d... Fig. 8 shows a diagonal cross-section Ž R OC s 2.5 = Tq . through the plot of Fig. 7 along with the corresponding data for the other output beams Žand their total.. When R OC is below optimum the low feedback leaves more energy in the amplifier as a weaker grating is formed resulting in lower levels of intracavity flux. At values of R OC above optimum, however, the large feedback leads to a considerable increase in the non-conjugate lasing output Žup to I NPC rIS f 2.9. at the expense of the conjugate output. Providing there are no significant phase distortions within the system, this beam can provide a useful alternative output with efficient energy extraction.
Fig. 8. A diagonal cross-section Ž R OC s 2.5=Tq . through the plot of Fig. 7, also showing the normalised intensity of the other output beams and their total.
In all the above simulations the small-signal FWM gain coefficient was fixed at a value of a 0 L s 3. We now investigate the performance of the holographic laser as a function of the FWM gain and determine the optimum values of R OC and Tq required to give maximum phase-conjugate output. Fig. 9Ža. shows the intensity of the three main output beams Žand their total. as a function of FWM gain for a selfstarting HLO that is optimised for phase-conjugate ŽPC. output. The results show a substantial increase X in I PC with increasing gain whilst I NPC and I NPC remain approximately equal and constant Žat f 0.7 = IS .. This indicates that in order to optimise I PC , the FWM gain is always saturated, independent of its small-signal value, to a level which gives a total output of f 1.4 = IS in the non-conjugate direction, with the additional energy extracted in the phaseconjugate direction. The trend is further emphasized if the absolute output intensities are replotted as a fraction of the available power density in the amplifier Ž IAvailrIS s 2 a 0 L., thereby giving an extraction efficiency Ž IrIAvail . as shown in Fig. 9Žb.. We see here that the PC output rises to a 39% extraction efficiency for a 0 L s 4.6, whereas the two non-PC beams extract less than 8% each. The near equality X of I NPC and I NPC corresponds to a near equality of FWM beams A1Ž0. and A 3 Ž0.. Thus, in a similar manner to the SPPC system, the maximum conjugate
G.J. Crofts, M.J. Damzenr Optics Communications 175 (2000) 397–408
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output occurs for near-equal writing beam intensity. The slight deviation from precisely equal writing beams stems from the presence of the output coupling optic which also links A1Ž0. and A 2 Ž0. ŽEq. Ž18... An important practical consideration is to know the values of R OC and Tq required to give this maximum phase-conjugate output. Fig. 10 shows the optimum values of R OC and Tq as a function of FWM gain for the simulations described above. As can be seen, the optimised self-starting system is not very sensitive to Tq, whose optimum value only Fig. 10. A plot of the optimum values of R OC and Tq as a function of small-signal FWM gain for the system modelled in Fig. 9.
varies between 6% and 2% over the range of gains shown. Over the same range the optimum value of R OC varies more rapidly, falling from 12% to 0.4% reflectivity and indeed, for most of the range shown has a lower reflectivity Ž- 4%. than a simple glass– air interface.
6. Conclusions
Fig. 9. A plot of Ža. normalised output intensity and Žb. extraction efficiency, as a function of small-signal FWM gain, for a selfstarting Žtransmission grating. HLO which is optimised for phaseX X conjugate output Žwith a 0 L s 0.0 and Ty s1yTq .. Data is shown for the three main ouput beams and their total.
In conclusion, we have demonstrated that a single steady-state gain grating written in a laser amplifier by CW monochromatic writing beams can exhibit a diffraction efficiency greatly in excess of unity and therefore be used as an efficient diffractive optical element. Furthermore, the use of a gain medium in a self-intersecting loop geometry leads to an efficient, self-pumped optical phase conjugator with very high Ž4 1. reflectivity. The device also acts as an externally controllable holographic laser oscillator with the ability to correct for intracavity phase distortions. The phase conjugate output can be maximised by adjusting a non-reciprocal loop transmission element to give equal writing beam intensities. Although higher phase-conjugate output can be achieved by the inclusion of an additional loop amplifier, we have shown that the most efficient extraction of the stored energy occurs when all the loop gain is concentrated within the FWM amplifier. Finally, we
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have shown that, by providing feedback to the loop system from an output coupling optic, an entirely self-starting, self-adaptive CW holographic laser oscillator can be realised. We have used our numerical model to determine the optimum output coupler and intracavity loss required to maximise phase-conjugate energy extraction, for a given level of gain within the system. An output of 3.7 = IS with an extraction efficiency approaching 40% has been demonstrated for a gain of 10 4 , and it is expected that this figure will be significantly higher in a real system due the formation of a combined transmission–reflection gain grating system which will give enhanced feedback w20x. The numerical modelling described in this paper is currently being used in the design and optimisation of practical Holographic Laser Oscillators based on solid-state gain materials Že.g. Nd:YVO4 . which are pumped by high-power CW diode lasers.
Acknowledgements The authors acknowledge support from the UK Engineering and Physical Science Research Council ŽEPSRC., grant number GRrL96455.
References w1x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 102 Ž1993. 288. w2x A. Brignon, J.-P. Huignard, Opt. Lett. 18 Ž1993. 1639. w3x M.J. Damzen, Y. Matsumoto, G.J. Crofts, R.P.M. Green, Opt. Commun. 123 Ž1996. 182. w4x J. Reintjes, B.L. Wexler, N. Djeu, J.L. Walsh, J. Phys. ŽParis. 44 Ž1983. C2–27. w5x S. Mailis, J. Hendricks, D.P. Shepherd, A.C. Tropper, N. Moore, R.W. Eason, G.J. Crofts, M. Trew, M.J. Damzen, Opt. Lett. 24 Ž1999. 972. w6x A. Brignon, J.-P. Huignard, Opt. Commun. 119 Ž1995. 171.
w7x P.J. Soan, A.D. Case, M.J. Damzen, M.H.R. Hutchinson, Opt. Lett. 17 Ž1992. 781. w8x I.M. Bel’dyugin, V.A. Berenberg, A.E. Vasil’ev, I.V. Mochalov, V.M. Petnikova, G.T. Petrovskii, M.A. Kharchenko, V.V. Shuvalov, Sov. J. Quantum Electron. 19 Ž1989. 740. w9x A.A. Ageichik, O.G. Kotyaev, Yu.A. Rezunkov, A.L. Safronov, V.V. Stepanov, Opt. Spectrosc. 76 Ž1994. 446. w10x K.S. Syed, R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 112 Ž1994. 175. w11x O. Wittler, D. Udaiyan, G.J. Crofts, K.S. Syed, M.J. Damzen, IEEE J. Quantum Electron. 35 Ž1999. 656. w12x P. Sillard, A. Brignon, J. Opt. Soc. Am. B 15 Ž1998. 2917. w13x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Lett. 19 Ž1994. 393. w14x M.J. Damzen, R.P.M. Green, K.S. Syed, Opt. Lett. 20 Ž1995. 1704. w15x A. Minassian, G.J. Crofts, M.J. Damzen, Opt. Commun. 161 Ž1999. 338. w16x R. Weber, B. Neuenschwander, M. MacDonald, M.B. Roos, H.P. Weber, IEEE J. Quantum Electron. 34 Ž1998. 1046. w17x A. Brignon, J.-P. Huignard, IEEE J. Quantum Electron. 30 Ž1994. 2203. w18x K.S. Syed, G.J. Crofts, R.P.M. Green, M.J. Damzen, J. Opt. Soc. Am. B 14 Ž1997. 2067. w19x P. Sillard, A. Brignon, J.-P. Huignard, J.-P. Pocholle, Opt. Lett. 23 Ž1998. 1093. w20x K.S. Syed, G.J. Crofts, M.J. Damzen, Opt. Commun. 146 Ž1998. 181. w21x P. Sillard, A. Brignon, J.-P. Huignard, IEEE J. Quantum Electron. 34 Ž1998. 465. w22x J. Reintjes, L.J. Palumbo, IEEE J. Quantum Electron. 18 Ž1982. 1934. w23x W.P. Brown, J. Opt. Soc. Am. 73 Ž1983. 629. w24x A.L. Gaeta, M.T. Gruneisen, R.W. Boyd, IEEE J. Quantum Electron. 22 Ž1986. 1095. w25x B. Ai, R.J. Knize, J. Opt. Soc. Am. B 13 Ž1996. 2408. w26x M.J. Damzen, R.P.M. Green, G.J. Crofts, Opt. Lett. 36 Ž1994. 34. w27x R.P.M. Green, G.J. Crofts, M.J. Damzen, Opt. Commun. 124 Ž1996. 488. w28x A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986, p. 207. w29x K.S. Syed, G.J. Crofts, M.J. Damzen, J. Opt. Soc. Am. B 13 Ž1996. 1892. w30x R.A. Fisher, Optical Phase Conjugation, Academic Press, New York, 1983, p. 7.