Oscillation characteristics of a coupled unidirectional ring resonators with photorefractive crystals

Optics & Laser Technology 43 (2011) 1041–1053

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Oscillation characteristics of a coupled unidirectional ring resonators with photorefractive crystals M.K. Maurya a, T.K. Yadav a, Dheerendra Yadav b, R.A. Yadav a,n a b

Lasers and Spectroscopy Laboratory, Department of Physics, Banaras Hindu University, Varanasi-221005, India Department of Physics, Indian Institute of Technology, Kanpur-208016, India

a r t i c l e i n f o

abstract

Article history: Received 22 July 2010 Received in revised form 1 January 2011 Accepted 2 January 2011 Available online 26 March 2011

For a coupled unidirectional photorefractive ring resonator (UPRR), the oscillation characteristics have been studied in details in terms of the photoconductive and dielectric constant of the photorefractive (PR) crystals under the assumption of the plane-wave approximation based on non-degenerate twowave mixing in the photorefractive materials. It has been found that the steady oscillations are possible when the two resonators oscillate independently. Using the plane-wave approximation and steady state oscillation conditions, the effect of the frequency detuning, photoconductivity and dielectric constant of the PR crystals on the relative intensity and frequency of oscillation of the secondary resonator in the coupled UPRR have been studied. It has been found that the relative oscillation frequency of the secondary resonator could be enhanced by selecting PR crystal A of higher absorption strength relative to PR crystal B and the higher photoconductivity of the crystals B as compared to that of the crystal A. Due to the non-reciprocal energy transfer between the oscillating beams and the additional PR phase-shift in the PR crystals A and B, the magnitude of the relative oscillation frequency of the secondary resonator could be controlled by the absorption strength, dielectric constant and photoconductivity of the two crystals. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Coupled unidirectional photorefractive ring resonators Oscillation characteristics Photoconductivity and dielectric constant of the coupled photorefractive crystals

1. Introduction Photorefractive (PR) materials have long been known to be suitable recording media for holography [1–6] and, hence for data storage. For these optical processing applications, many experimental and theoretical analyses have been performed using multi-wave mixing in such crystals [7–11]. The two-wave mixing can also be used to provide parametric gain for oscillation in ring resonators [12–22]. When these devices operate by means of a four-wave mixing (FWM) process, optical bistability has also been observed [23–25]. Recently, this phenomenon was theoretically predicted in PR two wave-mixing [23–25]. Unidirectional photorefractive ring resonators (UPRR) have been a subject of recent studies because of their potential uses in optical computing systems, e.g., associative memory systems [26,27], and in various other dynamic devices involving optical bistability [28,29] and beam competition [30]. The steady-state properties of UPRR have been extensively studied [14–19,31–35]; however, many applications require a thorough understanding of their dynamic properties. In a UPRR, oscillation develops when light from a laser is used to pump a PR crystal within an optical cavity [14–19]. The resonator

n

Corresponding author. Tel.: +91 8874228680; fax: + 91 5422368390. E-mail addresses: [email protected], [email protected] (R.A. Yadav).

0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.01.001

beam interferes with the pump beam inside the crystal and, when the crystal is properly oriented, the resulting interference pattern creates refractive index variations that refract pump beam into the resonator [17]. Consequently, the resonator field grows until saturation effects balance the amplification. As in the laser, the spatial pattern of the oscillation depends on the geometry of the optical cavity and the characteristics of the active medium (PR crystal) [36]. This PR crystal can be considered as an optical cavity which supports a large number of modes. Many of these modes can be excited by the incidence of a laser beam. All these oscillations may contribute to FWM processes that generate phase conjugate beam [37–43]. Since these modes overlap inside the crystal, they are strongly coupled via PR FWM. The coupling leads to energy transfer between the modes of these internal oscillations. The energy transfer between them may affect the stability of self-pumped phase conjugation [37–43]. The aim of this work is to present the photoconductive and dielectric constant dependence oscillation characteristics of a coupled UPRR configuration based on the non-degenerate two-wave mixing in the coupled PR crystals. In this paper, we have analyzed the problem of photoconductive and dielectric constant dependence of two-beam coupling in the coupled PR crystals pumped by an external pump beam for a coupled UPRR in the case of nondegenerate two wave mixing. We have also derived the expressions for the relative intensity and frequency of oscillation of the secondary resonator with respect to the primary resonator in a coupled

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UPRR under the appropriate boundary conditions and discuss the influences of frequency detuning of the oscillation beams relative to the pump beams, photoconductivity and dielectric constant of the coupled PR crystals on its performances.

2p) [17]. Additional phase- shift induced by the PR coupling can compensate for the resonator detuning and hence, it can allow oscillations to occur at frequencies different from the frequency of the pumping beam [17]. 2.2. Basic equations

2. Theoretical model 2.1. Coupled UPRR The schematic diagram of a coupled UPRR under consideration is shown in Fig. 1. It consists of two unidirectional ring resonators, which are coupled via PR two-wave mixing. The first (primary) resonator contains the two PR crystals A and B and three plane mirrors M1, M2 and M3 and second (secondary) resonator contains only the PR crystal B and three mirrors M4, M5 and M6. The PR crystal A is pumped by an external laser beam inserted into the primary resonator. The PR crystal B is pumped by an internal oscillating beam of the primary resonator and is employed to provide the gain for the secondary resonator. In such a case, the crystal A acts as a gain element for the primary resonator while crystal B as loss element. For the secondary resonator, the crystal B acts as a gain element. The primary and the secondary resonators are interlinked to each other to form a coupled system [37–39]. In order to analyze the oscillation characteristic properties of the coupled UPRR, the non-reciprocal energy transfer and PR phase-shift induced by the coupling between the beams in the PR crystals A and B of a coupled UPRR system is considered. The UPRR with a single resonator has been investigated in details by a number of workers [14,17,19]. In such a resonator amplification due to PR two-wave mixing in the PR crystal is responsible for the sustained oscillations, which occur when the two-wave mixing gain dominates cavity losses and the round-trip optical phase reproduces itself (to within an integral multiple of

In order to solve the problems of the photoconductive dependence of PR two-beam coupling coefficient in a coupled UPRR, the expressions for the intensities of the two beam and PR phaseshifts induced by the PR two-beam coupling are required. Expressions for the intensities and phase shifts of the pump beam A A (IpA ,Dcp ) and the oscillating beam (IoA ,Dco ) in the crystal A are given by [19] ! ð1 þmA ÞexpðaA zÞ A A Ip ðzÞ ¼ Ip ð0Þ ð1Þ mA þexpðg0A s2A z=ðs2A þ O2A e2A ÞÞ

DcAp  cAp ðlA ÞcAp ð0Þ ¼

OA eA 2sA

log

!

1 þmA

mA þ expðg0A s2A lA =ðs2A þ O2A e2A ÞÞ ð2Þ

IoA ðzÞ ¼ IoA ð0Þ

ð1 þmA Þ expðaA zÞ

¼

2sA

ð3Þ

1 þ mA expððg0A s2A z=ðs2A þ O2A e2A ÞÞÞ

DcAo  cAo ðlA ÞcAo ð0Þ OA eA

!

log

1 þ mA

!

1þ mA expððg0A s2A lA =ðs2A þ O2A e2A ÞÞÞ

ð4Þ

where IpA is the intensity of the pump beam; IoA is the intensity of the A A oscillating beam in the crystal A, cp and co are their phases and DcAp , DcAo are the corresponding PR phase-shifts. Here, z is the distance measured from the front face of the crystal A, aA is the bulk material absorption coefficient of the crystal A, lA is the thickness of

Fig. 1. Schematic diagram of a coupled UPRR.

M.K. Maurya et al. / Optics & Laser Technology 43 (2011) 1041–1053

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the crystal A, g0A is the intensity coupling constant [19] for the case of degenerate two-wave mixing (i.e. OA ¼0), and mA is the beam intensity ratio at zA ¼0. The degenerate intensity coupling constant g0A is a function of the frequency detuning (OA) between the pump beam and the oscillating beam of the primary resonator defined by

where g0B is the intensity coupling constant for the case of degenerate two-wave mixing (i.e. OB ¼0). It is to be noted that oAo ¼ oBp , as the oscillating beam inside the crystal A becomes the pump beam inside the crystal B and therefore, Eq. (5) becomes

OA  oAp oAo

OA  oAp oBp

ð5Þ

A p

A o

where o and o are the frequencies of the pump and oscillating beams in the crystal A. In PR media like BaTiO3, which operate by diffusion only (i.e., no external static field), using Eq. (6), the intensity coupling constant (gA) for the case of non-degenerate two-wave mixing (i.e., OA a0) depends on sA, eA and OA through the relation [19] 2 0A A 2 þ O2 2 A A A

g s

gA ¼

s

ð6Þ

e

where eA is the dielectric constant of the crystal A [45,46], sA is the photoconductivity of the crystal A [44], g0A is the intensity coupling constant for the case of degenerate two-wave mixing (i.e. OA ¼0) defined elsewhere [17,19]. The beam intensity ratio mA for the crystal A is defined as the ratio of the pump beam intensity to the oscillation beam intensity inside the crystal A and is given by mA 

IpA ð0Þ

ð7Þ

IoA ð0Þ

Similarly, the intensities and phase shifts of the two beams in the crystal B are given by ! ð1 þ mB Þ expðaB zÞ IpB ðzÞ ¼ IpB ð0Þ ð8Þ mB þ expðg0B s2B z=ðs2B þ O2B e2B ÞÞ

DcBp  cBp ðlB ÞcBp ð0Þ ¼

OB eB 2sB

log

!

1 þmB

!

ð1 þ mB ÞexpðaB zÞ

IoB ðzÞ ¼ IoB ð0Þ

2 zÞ=ð 2 þ O2 2 ÞÞ B B B B

1 þ mB expððg0B s

DcBo  cBo ðlB ÞcBo ð0Þ ¼ 

OB e B 2sB

log

s

e

1 þ mB

ð10Þ !

1 þ mB expððg0B s2B lB Þ=ðs2B þ O2B e2B ÞÞ

ð11Þ IpB

IoB

is intensity of the pump beam; is intensity of the where B B oscillating beam in the crystal B; and cp and co are their phases B B and Dcp , Dco are the corresponding PR phase shifts. Here, z is the distance measured from the front face of the crystal B, aB is the bulk material absorption coefficient, and lB is thickness of the crystal B; g0B is the intensity coupling constant for the case of degenerate twowave mixing (i.e. OB ¼ 0), and mB is the beam intensity ratio at z¼0. The beam intensity ratio mB is defined as the ratio of the pump beam intensity to the oscillation beam intensity mB 

IpB ð0Þ

ð12Þ

IoB ð0Þ

The expressions for the frequency detuning (OB) between the pump beam and the oscillating beam of the secondary resonator and the intensity coupling constant (gB) for the case of nondegenerate two-wave mixing (i.e., OB a0) are similar to those for the crystal A and are given as

OB  oBp oBo

gB ¼

2 0B B 2 þ O2 2 B B B

g s

s

e

When both the cavities oscillate simultaneously, there is a possibility of oscillation in which the steady state oscillation is established in the primary resonator only but not in the secondary cavity. In such a case, no two-beam coupling occurs within the crystal B (if one ignores the beam-fanning effect) and the crystal B behaves as a linear loss element in the primary resonator. Under these conditions, the expressions (Eqs. ((8)–(11))) need to be replaced by the relations IpB ðzÞ ¼ IpB ðz ¼ 0ÞexpðaB zÞ

ð16Þ

DcBp ¼ 0

ð17Þ

and the expressions for the non-degenerate two beam intensity coupling constant (gB) and beam intensity ratio mB of the secondary resonator lose their meanings. 2.3. PR two-beam coupling gain The amplification of the signal beam inside the coupled UPRR is responsible for the steady state oscillations for the coupled system. The PR gains for the primary (gA) and secondary (gB) resonators, when both the resonators oscillate in the steady state are defined by [17] "( ) # IpB ðlB Þ IoA ðlA Þ ð18Þ gA  IoA ð0Þ IPB ð0Þ

mB þ expðg0B s2B lB =ðs2B þ O2B e2B ÞÞ ð9Þ

ð13Þ

ð14Þ

ð15Þ

gB 

IpB ðlB Þ

ð19Þ

IPB ð0Þ

Substituting the expressions for the intensities IoA ðlA Þ, IpB ðlB Þ and IoB ðlB Þ respectively from Eqs. (3), (8) and (10)) into Eqs. (18) and (19) one gets the expressions for gA and gB as ( ) ð1 þ mB ÞexpðaB lB Þ gA ¼ 1 þ mB expððg0B s2B lB Þ=ðs2B þ O2B e2B ÞÞ ( ) ð1 þ mA Þ f expðaA lA Þ ð20Þ  1 þ mA expððg0A s2A lA Þ=ðs2A þ O2A e2A ÞÞ ( gB ¼

ð1 þ mB ÞexpðaB lB Þ

)

1 þ mB expððg0B s2B lB Þ=ðs2B þ O2B e2B ÞÞ

ð21Þ

Eq. (20) describes the net gain due to the two crystals in the primary resonator, including the gain in the crystal A and the loss in the crystal B whereas Eq. (21) describes the net gain due to the crystal B in the secondary resonator. For the oscillation, when only the primary cavity resonator oscillates at steady state while the secondary resonator is not active, the crystal B does not provide gain for the secondary resonator [37–39]. 2.4. PR phase-shifts The PR phase-shift for each resonator is one of the most important factors responsible for the oscillation characteristics of the coupled UPRR. If the values of IoA ,IpA ,IpB and IoB are known, the PR phases for the primary and secondary resonators of the coupled UPRR can be obtained by Eqs. (2), (4), (9) and (11). The phase shift in traversing through the PR crystals A and B for

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the primary resonator is given as

Dc ¼ Dc0 þðDcÞPC

ð22Þ

where Dc0 represents the phase-shift in the absence of PR coupling resulting due to passage of the signal beam through the PR crystals A and B only and it is given by,

Dc0 ¼

2p

l

ðn0A lA þ n0B lB Þ

ð23Þ

where the constant term n0A and n0B appearing in Eq. (23) are the refractive indices of crystal A and B in the absence of light, respectively. The second term (Dc)PC on the RHS of Eq. (22) represents the additional phase-shift in the presence of PR twobeam coupling for the primary resonator and it is given by A B ðDcÞPC ¼ Dco þ Dcp

and MB are integers. Without loss of generality, the discussion may be confined to a single mode oscillation for the two resonators and take MA ¼MB ¼0, provided that the intensity (energy) coupling constants g0A and g0B are not too large [14–19,37–39]. 2.5.2. Analysis of the threshold conditions The threshold conditions for a single UPRR has been studied in [14,17,19] which suggest that a steady oscillation inside the resonator would be observable only when the two-beam coupling constant of the PR crystals is greater than or equal to its threshold value. Eqs. (20) and (27) yield the value of mA as ( mA ¼

1Req expðaA lA Þ

)

Req expððg0A s2A lA =ðs2A þ O2A e2A ÞÞÞexpððg0A s2A lA Þ=ðs2A þ O2A e2A ÞÞ

ð24Þ

ð31Þ

and from Eqs. (4) and (9) into Substituting the values of Eq. (24) one gets the PR phase-shifts for the primary resonator as ! O e 1þ mA ðDcÞPC ¼  A A loge 2sA 1 þ mA expððg0A s2A lA Þ=ðs2A þ O2A e2A ÞÞ ! OB eB 1 þ mB ð25Þ þ loge 2sB mB þ expðg s2 lB =ðs2 þ O2B e2 ÞÞ

where Req represents an equivalent cavity loss parameter for the primary resonator and is defined as ! g0B s2B lB RA Req ¼ exp  ð32Þ RB s2 þ O2B e2

DcAo

DcBp

0B

B

B

B

Eq. (25) represents the phase-shift for the primary resonator due to traversal through the crystals A and B in the presence of PR two-beam coupling of the coupled UPRR. Similarly, the phaseshift for the secondary resonator can be obtained from Eq. (11) as ! OB eB 1 þmB B ðDcÞSC ¼ Dco ¼  loge 2sB 1þ mB expððg s2 lB =ðs2 þ O2B e2 ÞÞ 0B

B

B

B

ð26Þ The above Eq. (26) gives the phase-shift for secondary resonator due to traversal through the crystal B only in the presence of PR twobeam coupling. These additional PR phase-shifts (Eqs. (25) and (26)) of the primary and secondary resonators in the coupled crystals A and B are responsible for the oscillation characteristics of the coupled UPRR. From Eqs. ((25) and (26)), it can be seen that the PR phase-shift for each resonator of the coupled UPRR is independent of the material absorption coefficients aA and aB. 2.5. Output characteristics 2.5.1. Oscillation conditions During steady oscillation, the electric field must reproduce itself, both in the phase and intensity, after each round-trip, which leads to the following oscillation conditions for the two resonators gA RA ¼ 1

ð27Þ

DcAo þ DcBp ¼ DGA þ 2MA p

ð28Þ

gB RB ¼ 1

ð29Þ

DcBo ¼ DGB þ 2MB p

ð30Þ

In the case where both the primary and secondary resonators oscillate simultaneously, Eqs. (27)–(30)) represent the necessary oscillation conditions to ensure the steady oscillations of the two resonators. In the case where only the primary resonator oscillates in the coupled system, the approximate oscillation conditions are B given by Eqs. (27)–(28)) with Dcp ¼ 0, of Eqs. (16) and (17). Here, A B B Dco , Dcp and Dco are the additional phase-shifts due to the nondegenerate PR two-beam coupling, and Dcj( p r DGj r p,j¼A,B) and Rj(j¼A,B) are the cavity detuning parameters and products of the mirror reflectivities of the two resonators, respectively, and MA

B

B

Likewise Eqs. (21) and (29) one gets the value of mB in the secondary resonator as ( ) 1RB expðaB lB Þ mB ¼ ð33Þ RB expðaB lB Þexpððg0B s2B lB =ðs2B þ O2B e2B ÞÞÞ In Eq. (31) mA must be positive and Req expðaA lA Þ o 1 (since Req o1 and aB 40), therefore, the numerator of Eq. (31) is positive. Thus, for positive mA the denominator of Eq. (31) must be positive, which yields the threshold oscillation condition for the primary resonator as [14,17]

gA lA Z aA lA þ aB lB log RA  ðgA lA Þth

ð34Þ

Similarly, using Eq. (33), threshold condition for the secondary resonator to attain the steady oscillation is given as

gB lB Z aB lB log RB  ðgB lB Þth

ð35Þ

where (gAlA)th and (gBlB)th are the threshold parametric energy coupling strengths in the primary and secondary resonators, respectively. As the two-beam intensity (energy) coupling constant gA is a function of the frequency detuning parameter OA, it is obvious from Eq. (34) that the parametric gain is above the threshold only in a finite spectral regime. Using Eq. (6), Eq. (34) becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     g0A lA OA eA  o sA 1 ð36Þ aA lA þ aB lB loge RA The PR ring resonator has sustained oscillations only when the oscillation frequency shift (OA) of the primary resonator falls within the spectral regime given by Eqs. (34)/(36). Hence, using Eqs. (29) and (36) one also obtains the oscillation condition for the primary resonator in terms of the cavity parameters (cavity detuning, reflectivity of mirrors , linear loss and degenerate two-beam intensity coupling constant of the PR medium) and it can be written as   aA lA þ aB lB loge RA 9DGA þ 2MA p9 r 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   # g0A lA 1 ð37Þ aA lA þ aB lB loge RA Similarly, using Eqs. (14), (30) and (35) one obtains the oscillation condition for the secondary resonator as 9DGB þ 2MB p9r

"

aB lB loge RB 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  ffi# g0B lB 1 aB lB loge RB

ð38Þ

M.K. Maurya et al. / Optics & Laser Technology 43 (2011) 1041–1053

The resonator parameters of the two cavities must satisfy the conditions given by Eqs. (37) and (38) in order to sustain steady oscillations inside them without which no oscillation can occur in the two resonators even if the pumping beam exists. The coupled resonator is formed by combining the two independent oscillating resonators. Since the oscillating modes of the two resonators are overlapped inside the crystal B, these are strongly coupled via PR twowave mixing. This coupling leads to energy transfer between the two cavity modes and induces a PR phase shift [37–39]. Hence, the oscillation behavior of the two resonators in the coupled system would be different from that observed when these oscillate independently. 2.5.3. Steady state oscillation characteristics There are two possible oscillation configurations of the coupled UPRR one in which the two resonators oscillate simultaneously and the other in which only the primary resonator oscillates and the secondary resonator does not oscillate. 2.5.3.1. Behavior of the steady state oscillations with the primary resonator oscillating. In this case, the parameters of the secondary resonator are such that the two-beam coupling constant of the PR crystal B is smaller than its threshold value (Eqs. (35) and (38)) and oscillation inside the resonator does not remain sustained. Under this condition, only the primary resonator oscillates in the coupled system and the crystal A provides the two-beam coupling gain by the external pumping beam and the crystal B behaves only as a linear loss element (i.e., no two-wave mixing occurs) [37–39]. Using Eqs. (7) and (27), the oscillating intensity inside the primary resonator can be written as ! RA expðaA lA aB lB Þexpððg0A s2A lA =ðs2A þ O2A e2A ÞÞÞ A IoA ð0Þ ¼ Ip ð0Þ 1RA expðaA lA aB lB Þ

1045

Putting the values of mA and mB from Eqs. (31) and (33) into Eqs.(4) and (8) and using Eqs. (13) and (28), one gets the oscillation frequency shift (OA) inside the primary resonator as

OA ¼ oAp oBp ¼ sA

! 2ðDGA þ DGB þ 2MA p þ 2MB pÞ þ ðg0B s2B lB OB eB =ðsB ðs2B þ O2B e2B ÞÞÞ eA ðloge Req aA lA Þ

ð44Þ To determine the relative oscillation intensity and frequency of the secondary resonator of the coupled UPRR, the following boundary condition has been used IpB ð0Þ ¼ IoA ðlA Þ

ð45Þ

Applying the above boundary condition and using Eqs. (3) and (27–30), one gets the relative oscillation intensity of the secondary resonator as ( ) 2 1 RB expðaB lB Þexpððg0B s2B lB =ðs2B þ OB e2B ÞÞÞ B Io ð0Þ ¼ Req 1RB expðaB lB Þ ( ) Req expðaA lA Þexpððg0A s2A lA =ðs2A þ O2A e2A ÞÞÞ A  Ip ð0Þ 1Req expðaA lA Þ ð46Þ Similarly, using Eqs. (44) and (46), the expression for the relative oscillation frequency of the secondary resonator becomes   2sB ðDGB þ 2MB pÞ OB ¼ oBo oAp ¼ eB ðloge RB aB lB Þ ! 2ðDGA þ DGB þ 2MA p þ 2MB pÞ þ ðg0B s2B lB OB eB =ðsB ðs2B þ O2B e2B ÞÞÞ þ sA eA ðaA lA loge Req Þ ð47Þ

ð39Þ Using Eqs. (5), (28) and (34), the oscillation frequency inside the primary resonator can be written as

oBp oAp ¼

2sA ðDGA þ 2MA pÞ eA ðaA lA þ aB lB loge RA Þ

ð40Þ

2.5.3.2. Behaviors of the steady state oscillations with both the resonators oscillating. In this case, the parameters of the two resonators satisfy the threshold conditions given by Eqs. (37) and (38) i.e., steady oscillations must be sustained inside each resonator when the two resonators oscillate independently. On combing the two resonators to form the coupled system two-beam coupling occurs in the crystals A and B [37–39]. Eqs. (27)–(30) may be used to solve for the unknown quantities,mA, mBand OA, OB. Using Eqs. (12) and (33), the oscillation intensity for the secondary resonator can be written as ! 1RB expðaB lB Þ B IP ð0Þ ¼ ð41Þ IoB ð0Þ RB expðaB lB Þexpððg0B s2B lB Þ=ðs2B þ O2B e2B ÞÞ Substituting Eqs. (13) and (33) into Eq. (11) and using Eq. (30), the oscillation frequency shift (OB) inside the secondary resonator of the coupled UPRR can be written as

OB ¼ oBp oBo ¼

2sB ðDGB þ 2MB pÞ eB ðloge RB aB lB Þ

! Req expðaA lA Þexpððg0A s2A lA Þ=ðs2A þ O2A e2A ÞÞ B Ip ð0Þ 1Req expðaA lA Þ

!

RA exp

!

g0A s2A lA g0B s2B lB aA lA aB lB 4RB exp aB lB 41 2 2 2 s A þ OA eA s2B þ O2B e2B ð48Þ

Eq. (48) dictates that the coupled resonators oscillate only in a finite spectral regime. With the help of Eqs. (6), (48) can be written as ! ! g0A s2A lA g0B s2B lB RA o aA lA þ aB lB log RA o aA lA log þ 2 RB s þ O2B e2 s2 þ O2A e2 B

ð42Þ

Eqs. (7) and (31) lead to the expression for the oscillation intensity inside the primary resonator of the coupled UPRR given by

IoA ð0Þ ¼

It is important to note that since the pumping beam of the secondary resonator is same as the oscillating beam of the primary resonator, the steady state oscillation of the secondary resonator can be maintained only after the steady oscillation of the primary resonator has been established [37–39]. As the oscillating beam intensity inside the coupled UPRR must be positive, one can determine the regimes where the resonator can sustain the oscillation and for this purpose mA and mB both should be greater than zero. Using Eqs. (31) and (33) the steady state threshold condition of the coupled UPRR is obtained as

ð43Þ

B

A

A

ð49Þ The above equation represents the spectral regime where the two cavities in the coupled UPRR system would oscillate simultaneously. On comparing Eq. (49) with Eqs. (34) and (35), one can see that when the two independent oscillating resonators are combined to form a coupled system, owing to the coupling of the two cavity modes due to the crystal B, only some ranges of the parameters can keep the two resonators oscillating simultaneously.

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For the other range i.e. !

RB exp

2 0A A lA 2 þ O2 2 A A A

!

g s g0B s2B lB aB lB 4 RA exp aA lA aB lB 4 1: s2B þ O2B e2B s e ð50Þ

The parameters of the two resonators satisfy simultaneously Eqs. (34) and (35), i.e., the steady oscillation can be sustained when the two resonators oscillate independently; however, no steady oscillations can be maintained when the two resonators form a single coupled system [37–39]. Under this condition, the process for establishing oscillation in the coupled UPRR is unstable in time. The main reasons for such instability of oscillation are the nonreciprocal energy transfer in the coupled crystal B and the competition for energy between the two cavity modes [37–39].

3. Results and discussions The relative oscillation intensity ðIoB ð0Þ=IpA ð0ÞÞ of the secondary resonator (Eq. (46)) with respect to the primary resonator depend on the photoconductivity (sA and sB), dielectric constants (eA and eB), frequency detuning (OA and OB), degenerate intensity (energy) coupling strengths (g0AlA and g0BlB), cavity loss parameter Req for the primary cavity and absorption strengths (aAlA and aBlB) of the PR crystals A and B. For the calculations of the relative intensity and frequency of oscillation of the secondary resonator with respect to the primary resonator we have used the following parameters RA ¼80%, RB ¼80%, g0BlB ¼4.0 and aBlB ¼1.0. Variations of the relative oscillation intensity ðIoB ð0Þ=IpA ð0ÞÞ of the secondary resonator with the frequency detuning (OB) of the secondary resonator for different values of OA (fixed sB ¼8 S/cm, eB ¼5.0, sA ¼10 S/cm, eA ¼2.0, aAlA ¼ 1.0 and g0AlA ¼6.0), g0AlA (fixed sB ¼8 S/cm, eB ¼ 5.0, sA ¼ 10 S/cm, eA ¼2.0, aAlA ¼1.0 and OA ¼1.0 Hz), sA (fixed sB ¼8 S/cm, eB ¼5.0, g0AlA ¼6.0, eA ¼2.0, aAlA ¼1.0 and OA ¼1.0 Hz), eA (fixed sB ¼ 8 S/cm, eB ¼5.0, g0AlA ¼6.0, sA ¼10 S/cm, aAlA ¼1.0 and OA ¼1.0 Hz), and aAlA (fixed sB ¼8 S/cm, eB ¼5.0, g0AlA ¼6.0, sA ¼10 S/cm and OA ¼1.0 Hz) are shown in Fig. 2(a)–(e), respectively. From Fig. 2(a) it is obvious that the relative oscillation intensity of the secondary resonator with respect to the primary resonator decreases with the increase of frequency detuning (OB) of the secondary cavity; reaches to its minimum value; afterwards increases and reaches to its initial value. For a given value of the frequency detuning of the secondary resonator, higher value of the intensity of oscillation can be achieved at much lower value of frequency detuning (OA 50.1 Hz) of the primary resonator. Converse of this effect is also found in the case of photoconductivity (sA) of the PR crystal A, which can be seen from Fig. 2(c). Similar variations of the relative oscillation intensity of the secondary resonator with the degenerate energy coupling strength (g0AlA), dielectric constant (eA) and absorption strength (aAlA) of the crystal A can be seen from Fig. 2(b), (d) and (e). From Fig. 2(a)–(e), it can be seen that the all the curves are symmetrical about OB ¼ 0.0 Hz. However, it is interesting to note that for a given value of OB, the higher value of the relative oscillation intensity of the secondary resonator with respect to the primary resonator can be achieved if one selects a PR crystal A, which has lower value of the absorption strength (aAlA 50.1) as compared to the absorption strength of the crystal B in the coupled system. This greatly improves the performance of the coupled UPRR systems used in the selfpumped phase conjugators, spatial-temporal dynamics, temporal instability in self-pumped phase conjugators, nonlinear dynamical characteristics, the optical flip–flop, feature extractor, and stability analysis [47–53].

Fig. 3(a)–(e), respectively, shows the variations of the relative oscillation intensity IoB ð0Þ=IpA ð0Þ of the secondary resonator with the photoconductivity (sB) of the PR crystal B for different values of OA (fixed OB ¼1.0, eB ¼5.0, sA ¼ 10 S/cm, eA ¼2.0, aAlA ¼1.0 and g0AlA ¼6.0), g0AlA (fixed OB ¼1.0, eB ¼ 5.0, sA ¼10 S/cm, eA ¼2.0, aAlA ¼ 1.0, and OA ¼1.0), sA (fixed OB ¼1.0, eB ¼5.0, g0AlA ¼6.0, eA ¼2.0, aAlA ¼ 1.0, and OA ¼ 1.0),eA (fixed OB ¼1.0, eB ¼5.0, g0AlA ¼6.0, sA ¼10 S/cm, aAlA ¼1.0, and OA ¼1.0), and aAlA (fixed OB ¼ 1.0, eB ¼5.0, g0AlA ¼6.0, sA ¼ 10 S/cm, and OA ¼1.0). From Fig. 3(a) it is evident that the relative oscillation intensity of the secondary resonator increases with the increase in photoconductivity (sB) of the PR crystal B; reaches to a maximum value and afterwards slowly decreases. However, for a given value of the photoconductivity (sB) of the PR crystal B, the higher value of the relative oscillation intensity of the secondary resonator can be obtained at much lower value of frequency detuning (OA) of the primary cavity. Reverse of this is found in case of the energy coupling strength (g0AlA) and photoconductivity (sA) of the PR crystal A, which could be seen from the Fig. 3(b) and (c). Similar variations of the relative oscillation intensity of the secondary resonator have been observed with degenerate energy coupling strength (g0AlA), dielectric constant (eA) and absorption strength (aAlA) of the crystal A, which could be seen from Fig. 3(d) and (e). From Fig. 3(e), it is obvious that the relative oscillation intensity of the secondary resonator with respect to the primary resonator increases exponentially with the increasing photoconductivity (sB) of the PR crystal B and finally attains a saturation value. It could be noted here that the saturation value of relative oscillation intensity decreases with the increase of absorption strength (aAlA) of the crystal A. Variations of the relative oscillation intensity ðIoB ð0Þ=IpA ð0ÞÞ of the secondary resonator with the dielectric constant (eB) of the PR crystal B for different values of OA (fixed OB ¼ 1.0, sB ¼8 S/cm, sA ¼10 S/cm, eA ¼2.0, aAlA ¼1.0 and g0AlA ¼6.0), g0AlA (fixed OB ¼1.0, sB ¼ 8 S/cm, sA ¼10 S/cm, eA ¼2.0, aAlA ¼1.0 and OA ¼1.0), sA (fixed OB ¼1.0, sB ¼8 S/cm, g0AlA ¼6.0, eA ¼2.0, aAlA ¼1.0 and OA ¼1.0), eA (fixed OB ¼1.0, sB ¼8 S/cm, g0AlA ¼6.0, sA ¼10 S/cm, aAlA ¼ 1.0 and OA ¼1.0) and aAlA (fixed OB ¼1.0, sB ¼8 S/cm, g0AlA ¼6.0, sA ¼10 S/cm and OA ¼1.0) are shown in Fig. 4(a)–(e). From Fig. 4(a), it is clear that the relative oscillation intensity of the secondary resonator increases with the increasing dielectric constant (eB) of the crystal B; reaches to a maximum value; afterwards decreases witheB. However, for a given value ofeB, the higher value of the relative oscillation intensity of the secondary resonator could be obtained at much lower value of frequency detuning (OA) of the primary cavity. Opposite behavior is found for the cases of the energy coupling strength (g0AlA) and photoconductivity (sA) of the crystal A, which could be seen in the Fig. 4(b) and (c). Similar variations have been observed for the relative oscillation intensity of the secondary resonator with the dielectric constant (eA), and absorption strength (aAlA) of the crystal A, which could be seen in Fig. 4(d) and (e). From the Fig. 4(d), one may also conclude that for a given value of OB, the higher value of the relative oscillation intensity of the secondary resonator in the coupled UPRR system can be achieved by choosing a PR crystal A having much lower value of the dielectric constant as compared to the PR crystal B. The oscillation frequency of the secondary resonator (oBo oAp ) (Eq. (47)) with respect to the primary resonator depends on the cavity-length detuning (DGA/p and DGB/p), cavity loss parameter Req of the primary and secondary resonators and photoconductivity (sA and sB), dielectric constant (eA and eB), intensity (energy) coupling strength (g0BlB) and absorption strength (aAlA and aBlB) of the PR crystals A and B. Fig. 5(a)–(d), respectively, depicts the variations of the relative oscillation frequency (oBo oAp ) of the secondary resonator with

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Fig. 2. The variations of the relative oscillation intensity ‘IoB ð0Þ=IpA ð0Þ’ of the secondary cavity with the frequency detuning ‘OB’ of the secondary resonator for the different values of (a) OA; (b) g0AlA ; (c) sA; (d) eA and (e) aAlA.

the cavity-length detuning (DGB/p) of the secondary resonator for different values of DGA/p (fixed eB ¼5.0, sB ¼ 8 S/cm, sA ¼10 S/cm, eA ¼2.0 and aAlA ¼1.0), aAlA (fixed eB ¼5.0, sB ¼8 S/cm, sA ¼10 S/cm, eA ¼2.0 and DGA/p ¼0.1), sA (fixed, eB ¼5.0, sB ¼8 S/cm, eA ¼2.0, aAlA ¼1.0 and DGA/p ¼0.1) and eA (fixed eB ¼5.0, sB ¼8 S/cm, sA ¼10

S/cm, aAlA ¼1.0 and DGA/p ¼0.1). From Fig. 5(a), it might be seen that the oscillation frequency of the secondary resonator with respect to the primary resonator increases with the increasing value of cavitylength detuning (DGB/p)of the secondary resonator. However, magnitude of the relative oscillation frequency is higher at much higher

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Fig. 3. The variations of the relative oscillation intensity ‘IoB ð0Þ=IpA ð0Þ’ of the secondary cavity with the photoconductivity ‘sB’ of the PR crystal B for the different values of (a) OA; (b) g0AlA ; (c) sA; (d) eA and (e) aAlA.

value of the cavity-length detuning (DGA/p b0.1) of the primary cavity. The reverse variation is found in the case of photoconductivity (sA) of the PR crystal A, which can be seen in Fig. 5(c). Fig. 5(b) and

(d) shows similar variations of the relative oscillation frequency of the secondary resonator with the absorption strength (aAlA) and dielectric constant (eA) of the PR crystal A. From Fig. 5(b), one could conclude

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Fig. 4. The variations of the relative oscillation intensity ‘IoB ð0Þ=IpA ð0Þ’ of the secondary cavity with the dielectric constant ‘eB’ of the PR crystal B for the different values of (a) OA; (b) g0AlA ; (c) sA; (d) eA and (e) aAlA.

that the oscillation frequency of the secondary resonator with respect to the primary resonator could be increased to the higher value by selecting a PR crystal A of much lower value of the absorption strength, which could be existed at a higher value of cavity-length

detuning of the secondary resonator. It is interesting to note that for a given value of DGB/p, the relative oscillation frequency of the secondary resonator is found to be higher at much lower absorption strength of the PR crystal A (Fig. 5b) as compared to the other

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Fig. 5. The variations of the relative oscillation frequency ‘oBo oAp ’ of the secondary cavity with the cavity-length detuning ‘DGB/p’ of the secondary resonator for the different values of (a) DGA/p; (b) aAlA; (c) sA and (d) eA.

parameters such as cavity-length detuning (DGA/p), photoconductivity (sA), and dielectric constant (eA; Fig. 5(a)–(d)). Variations of the relative oscillation frequency (oBo oAp ) of the secondary resonator with the photoconductivity (sB) of the PR crystal B for different values of DGA/p (fixed eB ¼5.0, DGB/p ¼0.1, sA ¼10 S/cm, eA ¼ 2.0 and aAlA ¼1.0), aAlA (fixed eB ¼5.0, DGB/ p ¼0.1, sA ¼10 S/cm, eA ¼2.0, and DGA/p ¼0.1), sA (fixed, eB ¼5.0, DGB/p ¼ 0.1, eA ¼2.0, aAlA ¼1.0 and DGA/p ¼ 0.1) and eA (fixed, eB ¼5.0, DGB/p ¼0.1, sA ¼10 S/cm, aAlA ¼1.0, and DGA/ p ¼0.1) are shown in Fig. 6(a–d), respectively. From Fig. 6(a), it is obvious that for a given value of the photoconductivity (sB) of the PR crystal B the magnitude of the relative oscillation frequency is found to be higher at higher value of the cavitylength detuning of the primary resonator. This is because of the gains (i.e., gain due to the two crystals in the primary resonator and gain due to the crystal B in the secondary resonator of the coupled UPRR system), which are above the round-trip cavity losses and the frequencies fall within oscillation spectral regions (Eq. (50)). For a given value of photoconductivity (sB) of the PR crystal B, the magnitude of the relative oscillation frequency is lower at higher value of photoconductivity (sA) of the crystal A

(Fig. 6(c)). Fig. 6(b) and (d) shows similar variations of the relative oscillation frequency of the secondary resonator with the absorption strength (aAlA) and dielectric constant (eA) of the PR crystal A. It is obvious from the Fig. 6(b) that the relative oscillation frequency of the secondary resonator is found to be higher for higher value of the absorption strength of the crystal A and lower value of the photoconductivity of the crystal B. From Fig. 6(b) and (c), one may also conclude that the relative oscillation frequency of the secondary resonator can be enhanced by selecting PR crystal A of higher absorption strength relative to PR crystal B provided the photoconductivity of the crystal B is more than that of the crystal A. The non-reciprocal energy transfer between the oscillating beams and the additional PR phase shift in the coupled crystals, the magnitude of the oscillation frequency of the secondary resonator with respect to the primary resonator could be controlled by changing the absorption strength and photoconductivity of the PR crystal A. Fig. 7(a)–(d), respectively, present the variations of the relative oscillation frequency of the secondary resonator (oBo oAp ) in a coupled UPRR system with the dielectric constant (eB) of the PR crystal B for different values of DGA/p (fixed sB ¼8 S/cm, DGB/p ¼0.1,

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Fig. 6. The variations of the relative oscillation frequency ‘oBo oAp ’ of the secondary cavity with the photoconductivity ‘sB’ of the PR crystal B for the different values of (a) DGA/p; (b) aAlA; (c) sA and (d) eA.

sA ¼10 S/cm, eA ¼2.0 and aAlA ¼ 1.0 ), aAlA (fixed sB ¼ 8 S/cm, DGB/ p ¼0.1, sA ¼10 S/cm, eA ¼2.0 and DGA/p ¼0.1), sA (fixed sB ¼8 S/cm, DGB/p ¼0.1, eA ¼2.0, aAlA ¼1.0 and DGA/p ¼0.1) and eA (fixed sB ¼8 S/cm, DGB/p ¼0.1, sA ¼10 S/cm, aAlA ¼1.0, and DGA/p ¼0.1). With the increase in value of the dielectric constant of the PR crystal B, the magnitude of the relative oscillation frequency of the secondary resonator decreases very rapidly (Fig. 7(a)). For a given value of the dielectric constant of the PR crystal B, there is a lowering in the relative oscillation frequency of the secondary resonator, which is higher for lower value of the cavity-length detuning of the primary resonator. Similar variations of the relative oscillation frequency of the secondary resonator with the absorption strength (aAlA), photoconductivity (sA) and dielectric constant (eA) of the PR crystal A, could be observed in Fig. 7(b)–(d). It is interesting to note that the relative oscillation frequency of the secondary resonator could be enhanced by inserting the PR crystal B of small dielectric constant in the coupled UPRR system. Fig. 7(b)–(d) shows the plots of the relative oscillation frequency of the secondary resonator (oBo oAp ) in the coupled UPRR system with dielectric constant (eB) of the PR crystal B. It is evident that these curves coincide for all values of the absorption strength, photoconductivity and dielectric constant of the PR crystal

A, which means that the oscillation frequency of the secondary resonator with respect to the primary resonator in the coupled UPRR is independent of the absorption strength, photoconductivity and dielectric constant of the PR crystal A.

4. Conclusions In the present paper, dependence of the oscillation characteristics of the coupled UPRR, on the photoconductivity and dielectric constant of the PR crystals under the assumption of the planewave approximation based on the non-degenerate two-wave mixing in the coupled PR materials has been studied in details. The relative oscillation intensity of the secondary resonator with respect to the primary resonator could be increased by inserting the PR crystal A of higher photoconductivity and lower dielectric constant as compared to the PR crystal B. It is also found to be higher at the higher value of the frequency detuning of the secondary resonator (Figs. 2 and 3). The higher relative oscillation intensity of the secondary resonator greatly improves the performance of the coupled UPRR characteristics. The relative

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Fig. 7. The variations of the relative oscillation frequency ‘oBo oAp ’ of the secondary cavity with the dielectric constant ‘eB’ of the PR crystal B for the different values of (a) DGA/p; (b) aAlA; (c) sA and (d) eA.

oscillation intensity of the secondary resonator is higher for the PR crystal A having lower value of the absorption strength (aAlA 50.1) as compared to the absorption strength of the crystal B and could be increased by decrease in the frequency detuning of the primary resonator. This means that the relative oscillation intensity of the secondary resonator can be enhanced by choosing the frequency detuning of the primary resonator as low as possible. It is also found that the relative oscillation frequency of the secondary resonator could be enhanced by selecting PR crystal A of higher absorption strength relative to PR crystal B and the higher photoconductivity of the crystals B compared to that of the crystal A. Due to the non-reciprocal energy transfer between the oscillating beams and the additional PR phase shift in the PR crystals A and B, the magnitude of the relative oscillation frequency of the secondary resonator could be controlled by the absorption strength, dielectric constant and photoconductivity of the two crystals. In comparison with a single PR unidirectional ring cavity [19], the results show that the oscillation intensity and mode frequency for one resonator will change even if the resonator is kept fixed when the other is detuned. The theory developed in the present paper is applicable to the self-pumped phase conjugators, spatial-temporal dynamics, temporal instability,

the optical flip–flop, feature extractor and stability analysis of the coupled UPRR systems [47–53]. The relative oscillation intensity of the secondary resonator with respect to the primary resonator can be enhanced by selecting the PR crystal A of higher energy beam coupling strength g0A lA ð b10:0Þ, higher photoconductivity sA(425 S/cm), lower absorption strength aAlA(o0.1) and lower dielectric constant eA(o2.0) and the PR crystal B of lower photoconductivity sB(¼8.0 S/cm) and higher dielectric constant eB(¼4.0) provided the frequency detuning parameters of the primary and secondary resonators are kept as lower as possible. For obtaining the higher value of the relative frequency of the secondary resonator the required parameters for the PR crystals A and B are reverse of those required for the enhanced oscillation intensity of the secondary resonator.

Acknowledgments M.K. Maurya gratefully acknowledges the financial support from the CSIR, New Delhi in the form of a Senior Research Fellowship and Mr. T.K. Yadav is thankful to the UGC, New Delhi, India for the financial assistance.

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