We present a uniquely versatile and efficient mirror system capable of real-time fine-tuning in reflection and transmission properties across a broad wavelength range and at a high optical power. Leveraging the principles of the non-cyclic geometric phase (GP) acquired by the clockwise and counterclockwise beams of the Sagnac interferometer satisfying the anti-resonant condition on propagation through the quarter-wave plate, half-wave plate, and quarter-wave plate combination having fast axes oriented at 45° (fixed), θ (variable), and −45° (fixed) with respect to the vertical, respectively, our mirror system offers dynamic transmission control across 0–100% without the need for realignment. Notably, the GP-based mirror (GP-mirror) preserves the polarization state of the reflected beam, making it ideal for polarization-sensitive applications. The wavelength insensitivity of the GP enables seamless operation of the mirror across a wide wavelength range. As a proof-of-principle, we use the GP-mirror as the output coupler of a continuous-wave, green-pumped, doubly resonant optical parametric oscillator (DRO) based on a 30-mm-long MgO:sPPLT crystal and obtain stable operation at high powers over a wide wavelength tuning range. For a pump power of 5 W, the DRO provides an output power of 2.45 W at an extraction efficiency as high as 49% when operated at optimum output coupling. The DRO shows a maximum pump depletion of 89% and delivers an optimum output power across a tuning range ≥90 nm. The demonstrated concept offers a promising approach for advancing the capabilities and control of coherent optical sources tunable across different spectral regions and in all time scales from continuous-wave to ultrafast femtosecond domain.

Topics
Nonlinear optical processes, Geometrical optics, Interferometry, Optical parametric oscillators, Optical phase matching, Polarization, Gaussian beam, Coherent radiation sources, Geometric phases

At present, optical parametric oscillators (OPOs) based on second-order nonlinearities are established as highly versatile and practical sources of coherent radiation across vast spectral regions from visible to deep-infrared and are capable of operating in all time scales from continuous-wave to few-cycle femtosecond regime.1–4 In conventional designs delivering tunable radiations in the Gaussian spatial profile, OPOs have been used for a wide range of applications,5 while in modified architectures, they have been exploited for the generation of spatial structured beams,6,7 producing squeezed states,8–10 broadband frequency combs,11 and realizing coherent icing machine.12 Despite the tremendous advances over the past decades and the vast range of applications, the basic system architecture for an OPO, namely a nonlinear gain medium confined within an optical resonator, has remained unchanged, with typically one of the reflectors purposefully designed with partial transmission to act as an output coupler and extract usable output from the oscillator.

Using the coupled-wave equations,13 one can determine the transmission of the output coupler for the extraction of optimum output power/energy from an OPO.14,15 However, the optimum coupling is highly influenced by the nonlinear gain in an OPO. As the gain is dependent on the input pump power/energy and the wavelength of the interacting fields, the optimization of output coupling over a wide spectral range, while maintaining output stability and high extraction efficiency at an arbitrary pumping level,16 is a non-trivial task. To address this challenge, efforts have been made to replace the conventional output coupling mirror in an OPO with a Sagnac-based anti-resonant ring (ARR) interferometer17 providing variable output coupling by utilizing the angle-dependent transmission of the beam splitter (BS).18–20 While such an approach has proved promising for tunable output coupling over a wide wavelength range with optimum power/energy extraction, the change in the output coupling through the variation in the incidence angle on the beam splitter of the ARR interferometer results in OPO misalignment. As a result, ARR-based OPOs demand fine control of cavity alignment to enable optimum output coupling across the tuning range. At the same time, the small variation in transmission and reflection coefficients of the lossless beam splitter with the incidence angle restricts the percentage output coupling to a limited range.21 Therefore, it is imperative to devise new experimental schemes for optimum output coupling of OPOs and other optical oscillators, in general, including lasers, to extract maximum output power from such devices.

The geometric phase (GP)22 arising from the evolution of the polarization state of light on the Poincaré sphere plays a crucial role in understanding many physical concepts.23 The geometric phase acquired by the light beam due to the cyclic or non-cyclic evolution of its polarization state is proportional to the solid angle subtended on the Poincaré sphere with the trajectory representing the changes in the polarization state.22 Due to its easy manipulation and control, the geometric phase has found a wide range of applications, including guiding optical beams,24 frequency shift,25 structured beam generation,26 nonlinear frequency conversion,27 Steller interferometry,28 and weak-value measurement.29 Efforts have been made to introduce Berry’s topological phase30,31 inside a fiber loop to demonstrate tunable mirror transmission.32 As the cyclic evolution brings the system back to its original state while acquiring a phase without changing the overall system parameters, one can, in principle, use the concept of the geometric phase in the ARR interferometer and integrate it with optical oscillators (OPOs and lasers) for optimum output coupling across the tuning range in alignment-free configurations. However, the cyclic unitary (intensity-preserving) transformations of the polarization state of clockwise (CW) and counterclockwise (CCW) beams of the Sagnac loop obeying the SU(2) symmetry acquire a geometric phase of the same value and sign, resulting in constructive interference independent of the geometric phase. Therefore, it is imperative to devise an alternative unitary transformation trajectory for the effective use of the geometric phase to realize a tunable mirror.33,34

In this article, we report on a proof-of-principle demonstration, for the first time to our knowledge, of a stable, high-power, doubly resonant OPO (DRO) integrated with a non-cyclic GP-based Sagnac interferometer for alignment-free variable output coupling. The geometric phase is introduced through the rotation (θ) of a λ/2 plate (H) placed between two λ/4 plates (Q) in the QHQ configuration. The fast axes of the λ/4 plates are rotated at 45° and −45° with respect to the vertical. Keeping all parameters constant, a simple rotation (θ) of the λ/2 plate results in transmission variation in the GP-based Sagnac interferometer in the range 0.6%–98%. We have generated signal and idler radiations in the Gaussian beam profile with a maximum total output power as high as 2.4 W at 1.4% of output coupling. The maximum pump depletion of 89% is achieved near degeneracy, corresponding to an extraction efficiency of 49% at 5 W of input pump power.

To understand the origin of geometric phase in the non-cyclic evolution and the corresponding interference of the Sagnac interferometer designed in an ARR configuration, we use Jones matrix formalism.35 The system architecture used in the current study is shown in Fig. 1(a). The vertically polarized input beam (green arrow), after transformation into an anti-diagonal polarization by the λ/2 plate (H) with the fast axis rotated at +22.5° with the vertical, enters the two arms of the Sagnac interferometer comprised of a 50:50 beam splitter (BS) cube and two dielectric plane mirrors, M1 and M2. For the evolution of the polarization state of the beam over the Poincaré sphere, the λ/4 (Q), λ/2 (H), and λ/4 (Q) plate combination is oriented with fast axes making angles of +45° (fixed), θ (variable), and −45° (fixed) with the vertical, respectively. Using the Jones matrices for retarder plates, we can find the transfer matrix for the QHQ combination for the clockwise (gray arrow) and counterclockwise (red arrow) beams as
TCW=(−Q)HQ=−0e2iθ−e−2iθ0
(1)
and
TCCW=(−Q)(−H)Q=−0e−2iθ−e2iθ0.
(2)
Let the generalized state of polarization of the clockwise (CW) and counterclockwise (CCW) beams of the Sagnac loop be represented by the Jones vectors35 as
ECWI=cosαCWsinαCWeiθCW(0≤αCW≤π/2;−π≤θCW≤π)
(3)
and
ECCWI=eiβcosαCCWsinαCCWeiθCCW(0≤αCCW≤π/2;−π≤θCCW≤π),
(4)
where β is the initial relative phase between the beams, and the input intensity to the Sagnac loop is Iinput=Ein2=r2|ECWI|2+t2|ECCWI|2⁠. The polarization states of the CW and CCW beams after propagation through the QHQ combination can be represented, respectively, as
ECWF=−e2iθ⁡sinαCWeiθCW−e−2iθ⁡cosαCW
(5)
and
ECCWF=−eiβe−2iθ⁡sinαCCWeiθCCW−e2iθ⁡cosαCCW,
(6)
where θ is the fast axis angle of the λ/2 plate (H). In the present case, β = 0, αCW = αCCW, and θCW = θCCW, as the single input beam is divided into two beams of the Sagnac loop by the 50:50 beam splitter (BS). Considering the transmission and reflection coefficients as r and t, respectively, and the transfer matrix as MA=tr−rt⁠, we can calculate the amplitude of the reflected and transmitted electric fields of BS for the incident field, Ein, as
ECCWECW=tr−rt0Ein,ECCW=rEin,ECW=tEin.
(7)
Now, the transmitted (ECWF) and reflected (ECCWF) beams of the BS on propagation through QHQ acquire phase (Θ, same magnitude but opposite in sign) given by the transfer matrix of Eqs. (1) and (2), respectively. Subsequently, the incidence of these beams on the BS results in field amplitudes given by
ECWF(return)ECWF(out)=tr−rt0ECWF,ECWF(return)=rteiΘEin,ECWF(out)=t2eiΘEin.
(8)
Similarly, for the reflected beam, ECCW, the transmitted (return) and reflected (out) field amplitudes of the BS can be written as
ECCWF(out)ECCWF(return)=t−rrt0ECCW,ECCWF(out)=−r2e−iΘEin,ECCWF(return)=rte−iΘEin.
(9)
Therefore, the amplitude of the return and output beams can be represented as
Ereturn=ECWF(return)+ECCWF(return),Eoutput=ECWF(out)+ECCWF(out).
(10)
Now, according to Pancharatnam’s connections,22 the superposition of two fields yields a maximal intensity when they are in phase, that is, when
|ECWF+ECCWF|2=|ECWF|2+|ECCWF|2+2Re(ECWF⋅ECCWF*)
(11)
reaches its greatest value. Hence,
Im(ECWF⋅ECCWF*)=0,Re(ECWF⋅ECCWF*)>0.
(12)
Therefore, using Eqs. (8) and (9) in Eq. (10), we can find the intensities of the return beam and the output beam as
Ireturn=r2t2|eiΘ+e−iΘ|2Ein2=RT(2+2⁡cos⁡2Θ)Iin
(13)
and
Ioutput=|t2eiΘ−r2e−iΘ|2Ein2=(R2+T2−2RT⁡cos⁡2Θ)Iin.
(14)
Here, the angle, γ = 2Θ, is the geometric phase acquired by the superposed beams due to the QHQ combination. Using the transfer matrix of QHQ as given by Eqs. (1) and (2) and the polarization state of the input beam, one can find that Θ = 2θ. It is interesting to note from Eq. (13) that the maximal intensity of the return beam due to the interference in the Sagnac loop is independent of the polarization state of the input beam. At the same time, it does not carry any information about the polarization state of the return beam. However, to qualify the GP-based Sagnac loop as a mirror, the polarization state of the return beam needs to be the same as the input polarization state. Therefore, to confirm the polarization state of the return beam, we can calculate the scalar dot product between the initial and final fields of either CW or CCW beam using Eqs. (1)–(6) as follows:
ECWI⋅ECWF=−sinαCW⁡cosαCW(e2iθ⋅eiθCW−e−2iθ⋅e−iθCW).
(15)
It is evident from Eq. (15) that for both horizontal (α = 0) and vertical (α = π/2) polarization states of the input beam, the amplitude term (sin αCW cos αCW) vanishes, confirming that the polarization state of the return beam is orthogonal to that of the input beam. The value of (sin αCW cos αCW) varies with α, resulting in a maximum for α = π/4, the diagonal/anti-diagonal input polarization state, thus confirming that the return beam has the same polarization as the input beam. Therefore, in the present study, we have used the λ/2 plate before the BS to transform the vertically polarized beam into an anti-diagonal polarization state before entering the Sagnac loop.
FIG. 1.
(a) Schematic of the GP-mirror architecture. The evolution of the polarization state of the (b) clockwise and (c) counterclockwise beams of the Sagnac loop on the Poincare sphere reveals the origin of the non-cyclic geometric phase.
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(a) Schematic of the GP-mirror architecture. The evolution of the polarization state of the (b) clockwise and (c) counterclockwise beams of the Sagnac loop on the Poincare sphere reveals the origin of the non-cyclic geometric phase.

FIG. 1.
(a) Schematic of the GP-mirror architecture. The evolution of the polarization state of the (b) clockwise and (c) counterclockwise beams of the Sagnac loop on the Poincare sphere reveals the origin of the non-cyclic geometric phase.
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(a) Schematic of the GP-mirror architecture. The evolution of the polarization state of the (b) clockwise and (c) counterclockwise beams of the Sagnac loop on the Poincare sphere reveals the origin of the non-cyclic geometric phase.

Close modal
Now, to understand the origin of the geometric phase and the evolution over the Poincaré sphere, we have indicated different points in the Sagnac loop [see Fig. 1(a)] and evaluated the polarization state of those points and marked them on the Poincaré sphere, as shown in Figs. 1(b) and 1(c). For the CW beam (gray arrow), the points are P, O, O′ (after the λ/2 plate for θ = 0), N (after the λ/2 plate for θ ≠ 0), and M. Similarly, for the CCW beam (red arrow), the points are M, N, N′ (after the λ/2 plate for θ = 0), O (after the λ/2 plate for θ ≠ 0), and P. Using the Jones matrix for the polarization state (anti-diagonal) of CW and CCW beams,
A=−i21−1,
(16)
we calculate the polarization state of all points related to the evolution by the QHQ combination for the CW and CCW beams as
P=−i21−1,O=Q(+45°)⋅P=−ieiπ/421−1,O′=H(θ)|θ=0⋅O=−eiπ/42cos⁡2⁡θ−sin⁡2⁡θcos⁡2⁡θ+sin⁡2⁡θθ=0=−eiπ/4211,N=−eiπ/42cos⁡2⁡θ−sin⁡2⁡θcos⁡2⁡θ+sin⁡2⁡θ,M=Q(−45°)⋅N=−i2e2iθe−2iθ=−i2e2iθ1e−4iθ
(17)
and
M=−i21−1,N=Q(+45°)⋅M=−ieiπ/421−1,N′=H(−θ)|θ=0⋅N=−eiπ/42cos⁡2⁡θ+sin⁡2⁡θcos⁡2⁡θ−sin⁡2⁡θθ=0=−eiπ/4211,O=−eiπ/42cos⁡2⁡θ+sin⁡2⁡θcos⁡2⁡θ−sin⁡2⁡θ,P=Q(−45°).O=−i2e−2iθe2iθ=−i2e−2iθ1e4iθ.
(18)
Now, the geometric phase associated with the propagation of CW and CCW beams in the Sagnac loop can be determined using the geodesic rule: the half area on the sphere is bordered by the evolution curve and the shortest geodesic connecting the final and initial points,36,37 (M, P) for CW and (P, M) for CCW beams. The evolution and geodesic curves for the CW and CCW beams are CPOO′NM, GMP and CMNN′OP, GPM, respectively. Since CPOO′NM is in the southern hemisphere, GMP is below CPOO′NM passing through the south pole (left circular polarization).
Meanwhile, GPM is above CMNN′OP passing through the north pole (right circular polarization). Since the evolution of CW and CCW beams takes place on two hemispheres of the Poincaré sphere, the geometric phase acquired by the CW and CCW beams will be of the same magnitude but opposite in sign. The area corresponding to the geometric phase of CW beam covered by the evolution and geodesic curves, CPOO′NM and GMP, can be represented as the area covered by the curve, CPOO′MP (resulting from the λ/2 plate at angle θ = 0°), plus the blue shaded area enclosed by the curve, CO′NMO′. Similarly, for the CCW beam, it will be the area covered by CMNN′PM (resulting from the λ/2 plate at angle θ = 0°), plus the blue shaded area enclosed by the curve, CN′OPN′. However, the area enclosed by CPOO′MP and CMNN′PM is undefined, as the initial polarization states given by points P and M can take any path to reach O′ and N′ for the λ/2 plate at angle θ = 0°, respectively. Among different possibilities, we have pictorially shown one possibility as the curves, CPOO′MP and CMNN′PM. To address this issue, we use Pancharatnam’s geometric phase expression,22, |δ|=π−|E′|2⁠, based on spherical trigonometric results,38 where E′ is the spherical excess of the triangle enclosed by the sequence of polarization states, which is also equal to the solid angle formed at the center. If the direction of the sequence is reversed, then the sign flips. Meanwhile, since the evolution of CW and CCW beams takes place on two hemispheres of the Poincaré sphere, the geometric phase acquired by the CW and CCW beams will be of the same magnitude but opposite in sign. Therefore, we can calculate the resultant geometric phase difference between the superposed CW and CCW beams as (2π + X) or X, where X is the spherical excess area for the given evolution of states over O′MN and N′OP, that is, the columnar (a term for the relationship between two triangles of the same lune39) area of CPONM and CMNOP for the CW and CCW beams, respectively, as shown by the blue shaded area in Figs. 1(b) and 1(c). However, there is no direct mathematical function present to estimate the area, X. Meanwhile, the resultant (return) beam from the superposition of CW and CCW beams has the polarization state of (the amplitude is considered unity for simplicity)
S=ECWF+ECCWF=−irt2⁡cos⁡2⁡θ1211,
(19)
where the term cos 2θ represents the amplitude of the return beam. On propagation through the λ/2 plate (H), the polarization state of the return beam is transformed into the vertical, the same as that of the input beam, confirming that the GP-based Sagnac loop mirror does not change the polarization state of the reflected beam. As shown by Eqs. (13) and (14), the intensity of the return beam resulting from the interference of the CW and CCW beams is as follows:
Ireturn=RT(2+2⁡cos⁡4⁡θ)=4RTcos22θ,
(20)
Ioutput=R2+T2−2RT⁡cos(4θ),
(21)
where 4θ is the relative geometric phase acquired by the CW and CCW beams due to their evolution on the Poincaré sphere. Therefore, one can find the geometric phase associated with the blue-shaded area, X, to be 4θ, a function of the rotation angle of the λ/2 plate in the QHQ combination, controlling the reflectivity, RGP = Ireturn/Iinput, and transmissivity, TGP = Ioutput/Iinput, of the GP-mirror through the following equations:
RGP=RT(2+2⁡cos⁡4⁡θ)=4RTcos22θ
(22)
and
TGP=R2+T2−2RT⁡cos(4θ)=4R2sin22θ,
(23)
where R and T (R = T) are the reflectivity and transmissivity of the 50:50 beam splitter of the Sagnac loop.

The schematic of the experimental setup is shown in Fig. 2. A 10 W, continuous-wave, solid-state laser (Verdi V10) delivering a linearly polarized, single-frequency radiation at 532 nm in the TEM00 spatial profile with M2 < 1.1 and linewidth <5 MHz40 is used as the primary source in the experiment. A combination of λ/2 plate and polarizing beam splitter (PBS) cube is used to control the input pump power to the experiment. The second λ/2 plate is used to adjust the polarization state of the input beam for optimum phase-matching in the nonlinear crystal. A lens (L) of focal length, f = 150 mm, focuses the pump beam at the center of the nonlinear crystal to a waist radius of w0 = 42 μm, corresponding to a confocal parameter of b = 22.9 mm and a focusing parameter of ξ = 1.31. A 30-mm-long, 2-mm-wide, and 1-mm-thick periodically poled MgO-doped stoichiometrically grown lithium tantalate (MgO:sPPLT) crystal (C) with a single grating period of Λ = 7.97 μm is used as the nonlinear crystal for the parametric process. This crystal is housed in an oven, whose temperature can be varied up to 200 °C with the stability of 0.1 °C to satisfy the noncritical phase-matching condition. The crystal faces are antireflection-coated for pump, signal, and idler wavelengths. The OPO is configured in a standing-wave V-shaped cavity using two plano–concave mirrors, M3 and M4, with the radius of curvature, r = 100 mm, and the GP-based mirror. The GP-based mirror is designed in a Sagnac interferometer comprising a 50:50 beam splitter (BS) and two plane mirrors, M5 and M6. The BS is placed with its normal making an angle of ∼45° with the input beam. The reflectivity (RBS) to transmissivity (TBS) ratio of the BS is RBS:TBS ∼ 54:46. As a result, the maximum power reflected back to the OPO under the anti-resonant condition of the Sagnac loop is 4RBSTBS = ∼ 99.36%. All mirrors, M3–M6, have high transmission at the pump wavelength (T > 90%) and high reflectivity (R > 99%) for both signal and idler wavelengths over the range of 850–1200 nm, ensuring DRO operation. A λ/2 plate with the fast axis rotated by +22.5° with respect to the vertical is used before the BS to transform the polarization state (vertical) of resonant intracavity radiation into an anti-diagonal polarization.

FIG. 2.
Schematic of the experimental setup for the GP-mirror based out-coupled, continuous-wave, doubly resonant optical parametric oscillator designed in the V-cavity configuration. Laser: pump laser at 532 nm; λ/2: half wave-plate; PBS: polarizing beam splitter cube; M1–M6: mirrors; L1: lens; C: MgO:sPPLT crystal for nonlinear frequency conversion; BS: 50:50 non-polarizing beam splitter; and λ/4: quarter-wave plate.
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Schematic of the experimental setup for the GP-mirror based out-coupled, continuous-wave, doubly resonant optical parametric oscillator designed in the V-cavity configuration. Laser: pump laser at 532 nm; λ/2: half wave-plate; PBS: polarizing beam splitter cube; M1–M6: mirrors; L1: lens; C: MgO:sPPLT crystal for nonlinear frequency conversion; BS: 50:50 non-polarizing beam splitter; and λ/4: quarter-wave plate.

FIG. 2.
Schematic of the experimental setup for the GP-mirror based out-coupled, continuous-wave, doubly resonant optical parametric oscillator designed in the V-cavity configuration. Laser: pump laser at 532 nm; λ/2: half wave-plate; PBS: polarizing beam splitter cube; M1–M6: mirrors; L1: lens; C: MgO:sPPLT crystal for nonlinear frequency conversion; BS: 50:50 non-polarizing beam splitter; and λ/4: quarter-wave plate.
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Schematic of the experimental setup for the GP-mirror based out-coupled, continuous-wave, doubly resonant optical parametric oscillator designed in the V-cavity configuration. Laser: pump laser at 532 nm; λ/2: half wave-plate; PBS: polarizing beam splitter cube; M1–M6: mirrors; L1: lens; C: MgO:sPPLT crystal for nonlinear frequency conversion; BS: 50:50 non-polarizing beam splitter; and λ/4: quarter-wave plate.

Close modal

The combination of λ/4, λ/2, and λ/4 plates with the fast axes oriented at 45°, arbitrary angle, θ, and −45°, respectively, placed between the mirrors, M6 and M5, to introduce the same geometric phase but of opposite sign to the CW and CCW beams of the Sagnac interferometer. The interference of the CW and CCW beams on the BS results in the output beam and return beam, respectively, having a diagonal polarization. The intensity of these beams depends on the geometric phase acquired by the CW and CCW beams. The diagonal polarized return beam on propagation through the λ/2 plate is transformed into a vertical polarization and subsequently interacts with the vertically polarized pump beam in the nonlinear crystal for parametric application in each round-trip, ensuring OPO operation. All wave-plates used in the experiment are zero-order achromatic wave-plates with AR coating across 650–1200 nm.

First, we experimentally verify the concept of the GP-based mirror with controllable transmission using a continuous-wave Yb-fiber laser at 1064 nm. The laser beam of power, Pi = 1 W, to the GP-mirror is reflected by a PBS (not shown in Fig. 2) to ensure that the polarization state of the input beam to the GP-mirror is vertical. The same PBS also enables determining the orthogonal polarization component (through the power measurement in the transmission port) present in the return beam due to polarization rotation (if any) by the GP-based mirror. Measuring the output power, Poutput, of the GP-mirror and the transmitted power, Por, of the PBS corresponding to the orthogonal polarization state, we calculated the transmission (TGP = Po/Pi and polarization conversion factor, Cpol = Por/Pi, while changing the angle, θ, of the λ/2 plate of the QHQ combination). The results are shown in Fig. 3. As evident, the experimentally measured transmission of the GP-mirror (black dots) varies from 0.6% to 99% four times for the complete rotation (0–2π) of the λ/2 plate of the QHQ combination, in close agreement with Eq. (23).

FIG. 3.
Dynamic variation in the transmission (black dots) and the polarization conversion factor (red dots) of the GP-mirror with the rotation of the optic axis of the λ/2 plate. The solid line represents the theoretical data based on Eq. (23).
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Dynamic variation in the transmission (black dots) and the polarization conversion factor (red dots) of the GP-mirror with the rotation of the optic axis of the λ/2 plate. The solid line represents the theoretical data based on Eq. (23).

FIG. 3.
Dynamic variation in the transmission (black dots) and the polarization conversion factor (red dots) of the GP-mirror with the rotation of the optic axis of the λ/2 plate. The solid line represents the theoretical data based on Eq. (23).
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Dynamic variation in the transmission (black dots) and the polarization conversion factor (red dots) of the GP-mirror with the rotation of the optic axis of the λ/2 plate. The solid line represents the theoretical data based on Eq. (23).

Close modal

The base transmission of 0.6% for the λ/2 plate at 0° can be attributed to the imperfection of the reflectivity (R = 54%) and transmission (T = 46%) of the 50:50 beam splitter, BS. However, it is interesting to note that the polarization conversion factor, Cpol, as shown by red dots, remains close to zero irrespective of the rotation angle of the λ/2 plate, confirming the polarization of the return beam to be the same as the polarization of the input beam. Such observations confirm the successful development of the GP-mirror through the utilization of the non-cyclic geometric phase in an ARR interferometer having continuous variable transmission across 0%–100%, which preserves the polarization state of the input beam on reflection.

After complete characterization of the GP-mirror, we constructed the DRO with GP-mirror as the output coupler. Operating the DRO near degeneracy and pumping at a fixed power of 5 W, we varied the transmission of the GP-mirror from 0% to 40% by rotating the λ/2 plate from θ = 0° to 20° and recorded the output power and the operation threshold. The results are shown in Fig. 4. As evident from Fig. 4(a), at θ = 0°, the DRO has the output power of 2.1 W, corresponding to the output coupling of 0.6%. The output power of the DRO varies from 2.1 to 0.08 W (solid dots with dashed line) with the increase in transmission from 0.6% to 37% (solid line) by varying θ from 0° to 20°, with a maximum output power of 2.25 W for an optimum coupling of 1.4%.

FIG. 4.
Dependence of (a) output power and (b) threshold pump power of the GP-mirror based out-coupled DRO on the optic axis angle, θ, of the λ/2 plate in the QHQ combination. For output power measurement, the pump power to the DRO is maintained constant at 4.6 W. The solid line shows the variation in transmission of the GP-mirror. The dashed lines are guides to the eye.
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Dependence of (a) output power and (b) threshold pump power of the GP-mirror based out-coupled DRO on the optic axis angle, θ, of the λ/2 plate in the QHQ combination. For output power measurement, the pump power to the DRO is maintained constant at 4.6 W. The solid line shows the variation in transmission of the GP-mirror. The dashed lines are guides to the eye.

FIG. 4.
Dependence of (a) output power and (b) threshold pump power of the GP-mirror based out-coupled DRO on the optic axis angle, θ, of the λ/2 plate in the QHQ combination. For output power measurement, the pump power to the DRO is maintained constant at 4.6 W. The solid line shows the variation in transmission of the GP-mirror. The dashed lines are guides to the eye.
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Dependence of (a) output power and (b) threshold pump power of the GP-mirror based out-coupled DRO on the optic axis angle, θ, of the λ/2 plate in the QHQ combination. For output power measurement, the pump power to the DRO is maintained constant at 4.6 W. The solid line shows the variation in transmission of the GP-mirror. The dashed lines are guides to the eye.

Close modal

Due to the fourfold symmetry in the transmission curve, as shown in Fig. 3, we observe the same DRO output for eight different angles of the λ/2 plate. Since the parametric gain in the DRO is fixed for the given pump power, focusing condition, and crystal parameters, one can expect a rise in the operation threshold of the DRO with the increase in output coupling. We thus measured the variation in the DRO threshold with the output coupling, with the results shown in Fig. 4(b). The DRO has a pump power threshold of 100 mW when the GP-mirror is replaced by a high reflector (R > 99.5%). However, the incorporation of the GP-mirror into the cavity raises the DRO threshold to 400 mW, a fourfold increase due to the coating losses of the wave-plates and imperfections in the reflection and transmission of the 50:50 beam splitter of the GP-mirror. As evident from Fig. 4(b), the DRO threshold increases from 400 mW to 4.8 W due to the rotation of the λ/2 plate from θ = 0° to 23°, exactly following the transmission curve of the GP-mirror [see the solid line in Fig. 4(a)]. It is interesting to note that for a fixed pump power of 5 W, unlike singly resonant OPOs, the operation of the DRO can be achieved with a cavity transmission loss as high as 50%.

Furthermore, we studied the power scaling characteristics of the DRO under an optimum output coupling of 1.4%. Keeping the crystal temperature at 47 °C, corresponding to the signal and idler wavelengths of 1054 and 1074 nm, respectively, we recorded the output power and depletion of the DRO with the increase in pump power. The results are shown in Fig. 5. As evident, the signal and idler output powers from the DRO increase with input pump power after the threshold of 0.4 W is breached, reaching a maximum of 1.6 W of signal radiation at a slope efficiency of ηs = 37% and 0.8 W of idler radiation at a slope efficiency of ηi = 17%. Similarly, the pump depletion increases with input power from 7.5% at threshold to 74% at an input power of 2.5 W. A further increase in input power to 5 W leads to a slight variation in pump depletion, resulting in a maximum depletion of 87% for the DRO under optimum output coupling.

FIG. 5.
Variation in the extracted signal and idler powers and pump depletion as a function of pump power of the DRO for the optimum output coupling of 1.4%. The signal and idler wavelengths are 1054 and 1074 nm, respectively. The lines are guides to the eye.
View largeDownload slide

Variation in the extracted signal and idler powers and pump depletion as a function of pump power of the DRO for the optimum output coupling of 1.4%. The signal and idler wavelengths are 1054 and 1074 nm, respectively. The lines are guides to the eye.

FIG. 5.
Variation in the extracted signal and idler powers and pump depletion as a function of pump power of the DRO for the optimum output coupling of 1.4%. The signal and idler wavelengths are 1054 and 1074 nm, respectively. The lines are guides to the eye.
View largeDownload slide

Variation in the extracted signal and idler powers and pump depletion as a function of pump power of the DRO for the optimum output coupling of 1.4%. The signal and idler wavelengths are 1054 and 1074 nm, respectively. The lines are guides to the eye.

Close modal

We also studied the signal and idler output powers and spectral characteristics across the tuning range of the DRO under optimum output coupling. Keeping the pump power constant at 5 W and adjusting the angle of the λ/2 plate for an output coupling of 1.4%, we varied the crystal temperature from 47 to 53 °C and recorded the signal and idler powers together with the corresponding spectra using a dichroic separation mirror. The results are shown in Fig. 6. As can be seen, the signal (idler) power decreases from 1.67 W (0.78 W) at 1050 nm (1078 nm) to 1 W (0.5 W) at 1015 nm (1118 nm) as the DRO is tuned away from degeneracy. Measurements of signal and idler powers closer to the degeneracy (1064 nm) were restricted by the coating of the dichroic mirror, whereas wavelength tuning further away from the degeneracy was limited by the improper retardance of the achromatic wave-plates for wavelengths beyond 1100 nm. The use of super-achromatic wave-plates can further enhance the DRO tuning range. However, in this proof-of-principle experiment, we achieve a continuous wavelength tunability >100 nm across 1015–1118 nm, as shown in Fig. 6(b). The DRO produces total output power >1.5 W at a coupling efficiency of 30% over the entire tuning range.

FIG. 6.
Wavelength tuning. (a) Output power of the signal and idler radiations measured separately and the corresponding (b) spectrum across the tuning range of the GP-mirror based DRO operated at optimum output coupling. The pump power to the DRO is constant at 5 W.
View largeDownload slide

Wavelength tuning. (a) Output power of the signal and idler radiations measured separately and the corresponding (b) spectrum across the tuning range of the GP-mirror based DRO operated at optimum output coupling. The pump power to the DRO is constant at 5 W.

FIG. 6.
Wavelength tuning. (a) Output power of the signal and idler radiations measured separately and the corresponding (b) spectrum across the tuning range of the GP-mirror based DRO operated at optimum output coupling. The pump power to the DRO is constant at 5 W.
View largeDownload slide

Wavelength tuning. (a) Output power of the signal and idler radiations measured separately and the corresponding (b) spectrum across the tuning range of the GP-mirror based DRO operated at optimum output coupling. The pump power to the DRO is constant at 5 W.

Close modal

Finally, we measured the power stability and spatial beam quality of the output from the DRO, with the results shown in Fig. 7. Operating the DRO at signal and idler wavelengths of 1054 and 1074 nm, respectively, we recorded the total output power of the DRO for a fixed pump power of 5 W over 2 h. As evident from Fig. 7, the DRO exhibits a root-mean-square (rms) passive power stability better than 2.5% over 2 h. This is a remarkable level of power stability for a cw DRO under passive conditions and in the absence of active control. The passive power stability of the DRO can further be improved by isolating the system from the air currents and thermal fluctuations in the laboratory and by resonating the signal and idler wavelengths in two different paths by incorporating a dichroic mirror inside the cavity. Furthermore, we recorded the spatial profiles of the signal and idler wavelengths using a CCD camera. The results are presented in the inset of Fig. 7, where the TEM00 Gaussian beam profile with the ellipticity of 1.24 and 1.17 is confirmed for the signal and idler radiations, respectively. The ellipticity in the beams is due primarily to the tilt in the DRO concave mirror (M2), intentionally set higher than 10° to avoid experimental constraints imposed by mechanical restrictions.

FIG. 7.
Long-term power stability and spatial beam profiles of signal and idler radiations extracted from the optimum output coupling.
View largeDownload slide

Long-term power stability and spatial beam profiles of signal and idler radiations extracted from the optimum output coupling.

FIG. 7.
Long-term power stability and spatial beam profiles of signal and idler radiations extracted from the optimum output coupling.
View largeDownload slide

Long-term power stability and spatial beam profiles of signal and idler radiations extracted from the optimum output coupling.

Close modal

In conclusion, we have successfully demonstrated a novel, compact, and versatile mirror system based on a non-cyclic geometric phase, allowing dynamically and precisely variable reflection and transmission over a broad wavelength range. By integrating this mirror into a green-pumped continuous-wave DRO, we have shown the feasibility of using the concept of geometric phase to realize a finely tunable output coupler. The GP-mirror-based out-coupled DRO has been shown to exhibit an excellent output power stability and a wavelength coverage over 100 nm, from 1015 to 1118 nm, with a coupling efficiency of >30% over the entire tuning range. The DRO produces a maximum output power of 2.45 W at an extraction efficiency of 49% for an optimum output coupling of 1.4%, with the pump depletion reaching 89%. The generated signal and idler output beams exhibit Gaussian spatial profile. This proof-of-concept study provides a foundation for further exploration and application of the non-cyclic geometric phase-based mirror concept in different optical oscillators and various spectral regions. By selecting appropriate components, such as pump sources, broadband wave-plates, and suitable gain media, the alignment-free GP-based optimum output coupler can be exploited to extract maximum output power from any coherent source across the optical spectrum and in all time scales.

A.A. acknowledges the financial support through the DST-QuEST program, Government of India, to cover his research visit to PRL. M.E.-Z. acknowledges the support from Ministerio de Ciencia e Innovación (MCIN) and State Reserach Agency (AEI), Spain (Project No. PID2020-112700RB-I00); Project Ultrawave EUR2022-134051 funded by MCIN/AEI and the “European Union NextGenerationEU/PRTR”; “Severo Ochoa” Center of Excellence (Grant No. CEX2019-000910-S); Generalitat de Catalunya (CERCA); Fundación Cellex; and Fundació Mir-Puig.

The authors have no conflicts to disclose.

C.K., A.A., and A.G. developed the experimental setup and performed measurements, data analysis, and numerical simulation. A.A. led the experiment and wrote the original draft. C.K., S.D.G., and G.K.S. did the theoretical study, derived the analytical formulas, and performed the numerical simulation. R.P.S. and M.E.-Z. participated in the analysis and data interpretation. G.K.S. developed the ideas and led the project. All authors participated in the discussion and contributed to the manuscript writing.

Chahat Kaushik: Formal analysis (equal); Investigation (equal); Methodology (equal). A. Aadhi: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Anirban Ghosh: Formal analysis (equal); Investigation (equal). R. P. Singh: Resources (equal); Validation (equal). S. Dutta Gupta: Resources (equal); Validation (equal). M. Ebrahim-Zadeh: Resources (equal); Validation (equal); Writing – review & editing (equal). G. K. Samanta: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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