Spatial hole burning in thin-disk lasers and twisted-mode operation
Karsten Schuhmann Klaus Kirch Francois Nez Gunther Wichmann Randolf Pohl Aldo Antognini
physics.optics




Controlling heat transport and flow structures in thermal turbulence using ratchet surfaces
    Chao Sun
    December 30, 2023
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Spatial hole burning prevents single-frequency operation of thin-disk lasers when the thin disk is used as a folding mirror.

We present an evaluation of the saturation effects in the disk for disks acting as end-mirrors and as folding-mirrors explaining one of the main obstacles towards single-frequency operation.

It is shown that a twisted-mode scheme based on a multi-order quarter-wave plate combined with a polarizer provides an almost complete suppression of spatial hole burning and creates an additional wavelength selectivity that enforces efficient single-frequency operation.




§ INTRODUCTION


The various longitudinal modes of a laser cavity undergoing optical
amplification in the same active medium experience the phenomenon of
gain competition which leads to the prevailing of one mode
over all the others and thus to single-frequency operation <cit.>.

A tiny difference of the effective gain  seen by a mode
is sufficient to initiate this prevailing process.

However, the domination of one mode over all the others can be
sustained only when the stimulated emission induced by this mode
causes gain saturation not only for itself (self saturation), but with
equal strength also for the other modes (cross saturation).

This cross saturation occurs when the amplification of the various
modes addresses the same regions of the active medium, so that the
amplification of one mode depletes also the population inversion
experienced by the other modes.

If the various modes address the distribution of the population inversion in different ways, as in the case of “spatial hole burning” (SHB)  <cit.>, the amplification process of one mode only partially affects the population inversion experienced by the other modes.

Several modes can thus coexist in steady-state conditions in a linear laser cavity preventing single-frequency operation.



The principle of SHB and its relation to single
frequency operation is illustrated in
Fig. <ref>.

In the center of a linear cavity, the standing wave pattern of the 14^th longitudinal mode (black continuous curves) has maxima where the 13^th longitudinal mode (black dashed curves) has minima.

Therefore, the depletion of the population inversion caused by the 14^th mode reduces the gain of the 14^th mode (self saturation) but does not significantly decrease the gain of the 13^th mode (negligible cross saturation).

As a consequence, the gain competition between these two adjacent
longitudinal modes is strongly suppressed and the laser operates
simultaneously in several longitudinal modes.







Spatial hole burning is often eliminated by the use of unidirectional ring-laser cavities. 

However, this is not an option for high-power thin-disk lasers 
as Faraday rotators, required to enforce the unidirectional operation, lack the power handling capability <cit.>. 

A well established method to eliminate SHB in a linear cavity, is given by the so called twisted-mode scheme <cit.> where the counter-propagating beams in the active medium have orthogonal polarizations so that no interference occurs.

To our knowledge, this method has so far not been applied to thin-disk lasers, except in our previous work <cit.>.

By reducing the length of a laser cavity (in the micrometer range
for Yb:YAG as active medium), the effect of spatial hole burning in preventing
single-frequency operation can be eliminated.

Indeed for such a short cavity only one longitudinal mode falls into the spectral region where the active medium has a gain.

Yet, these short cavities are not apt for high-power lasers as they can not sustain modes with large transverse size.

Wavelength selective elements as gratings or Fabry-Perot etalons
could be used to favor one of the modes, but also in these cases an
efficient selectivity can be achieved only for relatively short
cavity lengths.

Another method to suppress the disrupting effects of SHB and to achieve single-frequency operation is obtained by using a short active medium placed close to a cavity end-mirror.

This is often called a “gain-at-the-end” configuration.

In fact, as can be deduced from Fig. <ref> in this region the intensity maxima of neighboring modes are located at similar positions so that these modes sample the active medium with nearly identical patterns.

Thus a strong cross saturation between neighboring modes exists which leads to the prevailing of the dominant mode over the others.

Because of this, a thin-disk laser with the disk acting as end-mirror M1 is suitable for high-power lasers in single-frequency operation given the mitigation of the effects of SHB and the power scalability of the thin-disk <cit.>.



Improper understanding of the “gain-at-the-end” configuration has sometimes lead to erroneously neglecting SHB when the thin-disk is used as a folding mirror of the resonator.



In such a case, the SHB pattern is more complex than
presented in Fig. <ref> because it results from
the interference of four circulating beams as recognized by Vorholt and Wittrock in reference <cit.>.

However, their model does not include the dependence on the distance between the active medium and the end-mirror (see D_3 in Fig. <ref>).

Spatial hole burning for disks acting as folding-mirrors has also been
discussed for mode-locked lasers <cit.>, though this is not applicable for cw and ns pulses.

This study aims to clarify the SHB effects taking place in thin-disk lasers and their relation to single-frequency operation.

In Sec. <ref> we introduce the fundamentals of SHB for the simple situation presented in Fig. <ref> similar to the studies in references <cit.>.

The obtained results are then applied in Sec. <ref> to the case of thin-disk lasers with the
thin-disk acting as cavity end-mirror. 

In this design SHB favors longitudinal modes with a large frequency spacing from the lasing (saturating) mode, that can be easily suppressed using standard frequency selective elements. 



Section <ref> presents SHB when the thin disk acts as a folding-mirror. 

The resulting interference of four beams in the active medium increases the susceptibility for SHB. 

In Sec. <ref> we apply the twisted-mode scheme <cit.> to this cavity layout to reduce the four-beam to a two-beam interference nearly eliminating the SHB.

It is concluded with Sec. <ref> presenting an implementation of the twisted-mode scheme in a Q-switched thin-disk laser <cit.>.








§ SPATIAL HOLE BURNING FOR A LINEAR CAVITY
 


In this section, the well known principle of SHB is derived for the linear cavity in Fig. <ref> by using the generic building block of an amplifier given in Fig. <ref>. In this amplifier a saturating Gaussian beam at a wavelength λ_0 passes an active medium, travels to a mirror M1 that is reflecting the beam back onto itself. 

Thus the beam crosses the medium a second time. 

The origin of the coordinate system used to compute SHB is located in the center of the active medium with z the propagation and x,y the transverse directions.

The position- and time-dependent amplitude of the forward propagating beam inside the active medium  (z∈[-L/2, +L/2] with the length of the active medium L) can be written as:


    E_1(x,y,z,t, λ_0) = E_0 cos(2π (-n z/λ_0- ν_0 t))·exp(-x^2 + y^2/2 w^2),


with w the width of the transverse profile of the beam, n the refractive index of the active medium, t the time, E_0 the amplitude of the beam on its symmetry axis, ν_0 and λ_0 = c/ν_0 the frequency and wavelength of the beam, respectively.

For simplicity we neglected the increase of amplitude occurring in the active medium. This simplification allows for analytical results and is well justified for low-gain media as the thin-disk.  

Furthermore it provides qualitatively valid results also for high-gain media.

Note that the laser gain will be accounted for separately (see e.g. Eq. (<ref>)).

The corresponding backward propagating beam is given by


    E_2(x,y,z,t, λ_0) = E_0 cos(2π (2d+n z/λ_0 - ν_0 t))·exp(-x^2 + y^2/2 w^2),
 

where d = D_1 + n L/2 is the optical distance between the center of the active medium and M1 with D_1 the distance between the active medium and M1 (cf. Figs. <ref> and <ref>).

The total amplitude in the active medium resulting from the superposition of the two beams is thus


    E(x,y,z,t, λ_0) = E_1 + E_2=2 E_0 cos(2π ( d/λ_0-ν_0 t))cos(2πd + n z/λ_0 )·exp(-x^2 +y^2/2 w^2).


The position dependency of the time-averaged intensity distribution resulting from the two counter-propagating beams is given by


    I(x,y,z,λ_0 ) = 2 I_0 (1+cos(4π n z+d/λ_0 )) exp(-x^2 + y^2/w^2),


where I_0 = c nε_0|E_0^2|/2 is the intensity of a single laser beam on its symmetry axis.

From this intensity distribution I(x,y,z,λ_0), it is possible to obtain the position-dependent density of the population inversion in the active material N(x,y,z, λ_0) by using the known rate equation for a four-level laser <cit.>


    dN (x,y,z,t, λ_0)/dt = R_p - N(x,y,z,t, λ_0)I(x,y,z,t,λ_0 )/F_sat - N(x,y,z,t, λ_0 )/τ,


where R_p is the pump rate (homogeneous within
the active medium), F_sat the saturation fluence
and τ the lifetime of the upper state.

The decrease of the upper state population by the stimulated emission is proportional to the local intensity of the circulating light I(x,y,z,λ_0 ).

The steady-state solution of this rate equation (dN/dt = 0) reads


    N(x,y,z, λ_0)    =   R_p τ/1+ τI(x,y,z,λ_0 )/F_sat
       ≈    R_p τ - R_p τ^2 I(x,y,z,λ_0 )/F_sat + 𝒪(I^2).


In Eq. (<ref>) we assumed that τI(x,y,z,λ_0 )/F_sat≪ 1, which is valid for a weakly saturating beam intensity

[For efficient laser operation high saturation of the active medium is required. In this case this assumption does not hold. Nevertheless, the general behaviour presented in this study remains valid.], e.g. for a laser cavity with an out-coupling mirror of low reflectivity.

The position dependent gain g(x,y,z, λ_0) is
proportional to the local density of the population inversion and can be expressed as


    g(x,y,z, λ_0)≈ g_0-g_0τI(x,y,z,λ_0 )/F_sat,


where g_0 is the unsaturated gain.

So far we described the SHB effect of the dominant laser beam with a wavelength λ_0 amplified in a dual-pass configuration as depicted in Fig. <ref>.

As a next step we evaluate the saturation spectrum of the active medium experienced by a second (probing) beam at a wavelength λ.

This beam forms a similar interference pattern I(x,y,z,λ) in the active medium slightly shifted relative to I(x,y,z,λ_0). 

The cross-saturation gain G experienced by this beam is given by the integral 


    G(λ) = ∫∫∫ I(x,y,z,λ ) g(x,y,z,λ_0) dx dy dz/P_laser.

where P_laser is the incident laser power at wavelength λ_0.

The gain G can be described as the product of the unsaturated laser gain G_0 and the saturation spectrum S_SHB(λ) so that

    P_out =e^G(λ) P_in = e^S_SHB(λ) G_0(λ) P_in .


The saturation spectrum S_SHB(λ) for the dual-pass configuration of Fig. <ref> is thus given by


    [ S_SHB(λ) = G/G_0 =∫_-∞^∞∫_-∞^∞∫_-L/2^L/2 I(x,y,z,λ) g(x,y,z,λ_0) dz dx dy/∫_-∞^∞∫_-∞^∞∫_-L/2^L/2I(x,y,z,λ) g_0 dz dx dy. ]


Combining Eq. (<ref>) with Eq. (<ref>) we find


    [                                                                  S_SHB(λ)=; 1-τ/I_0  F_sat L∫_-∞^∞∫_-∞^∞∫_-L/2^L/2 I(x,y,z,λ _0) I(x,y,z,λ) dz dx dy . ]




Inserting the time-averaged intensity of Eq. (<ref>) into Eq. (<ref>), and assuming that d and L are much larger than λ and λ_0, and also that Δ≡ (λ_0 - λ )/( λ_0λ )≪ 1 /λ_0 we find

[To obtain this result we have made use of the approximation


    ∫_-L/2^L/2(1+ cos(2πn z+ d/λ_0))(1+ cos(2πn z+d/λ))dz ≈ L(1+1/2sinc(π n L Δ) cos(2π d Δ) ).

] 


    S_SHB(λ ) ≈ 1-2 τI_0/F_sat(1+1/2sinc(π n L Δ) cos (2π d Δ)).


A plot of the saturation spectrum S_SHB(λ ) for realistic values of d and L is shown in Fig. <ref> as a function of the wavelength λ of the probe beam assuming a saturating beam at λ_0 = 1030 nm.

When applying this results to a cavity, it is necessary to take into account that the wavelength λ and λ_0 can only have discrete values as a consequence of the resonance conditions.

For the simplified cavity layout in
Fig. <ref> with D_1 = D_2, the longitudinal modes neighboring the saturating mode are located exactly at the two maxima of S_SHB(λ ) as indicated by the blue hollow circles in Fig. <ref>.

These two modes experience thus a gain decrease due to saturation
effects significantly smaller compared to the gain decrease at the
saturating wavelength λ_0.

Therefore amplification of these neighboring modes occurs leading
to disruption of single-frequency operation.

Because the spectral distance between modes for realistic layouts of rod lasers with typical cavity round-trip length of about 0.1 m is 1.5 GHz (5 pm at 1030 nm), the suppression of the neighboring modes using selective elements would require extremely narrow spectral filters with active regulation to prevent significant cavity losses at the operating wavelength λ_0.



§ SPATIAL HOLE BURNING FOR A THIN DISK ACTING AS CAVITY END-MIRROR








For a thin disk acting as a cavity end-mirror M1 (see  Fig. <ref>), the effects related with SHB are mitigated.

Indeed  for this “gain-at-the-end” configuration the intensity pattern in the active medium (see Fig. <ref>) change so slowly with the wavelength λ that the  maxima and minimima of the intensity pattern of two neighboring axial cavity modes are located at similar positions (cf. Fig. <ref>).

This implies that the cross saturation between neighboring modes is nearly identical to the self saturation. 

In other words, the saturation spectrum S_SHB(λ) is a slowly varying function of λ.



The reduction of gain caused by the saturation spectrum S_SHB(λ) for thin-disk lasers with disks acting as cavity end-mirrors can be quantified using Eq. (<ref>) and setting D_1 = 0 (cf. Figs. <ref> with <ref>)


    S_SHB(λ ) ≈  1 - 2 τ I_0/F_sat(1+1/2sinc (π n L Δ )) .





S_SHB is plotted in Fig. <ref> as a function of the wavelength λ for a realistic thin-disk laser (D_2=2 m) having a saturating mode at λ_0 = 1030 nm.

A long cavity was assumed in this case to provide an intra-cavity beam with large w required for the high-power operation.

Because of this, the wavelength difference between neighboring modes in the 1030 nm region is only about 0.25 pm (75 MHz) apart, as indicated by the vicinity of the blue hollow circles and the black hollow square in Fig. <ref>.

Thus, competition between the various modes located around the broad minimum centered at 1030 nm exists, which can be used to achieve single-frequency operation.

In Fig. <ref> the red squares indicate the maxima of the saturation spectrum S_SHB at a spectral distance of 2.11 nm (615 GHz) from λ_0.

These modes could grow in the laser cavity and disrupt single-frequency operation.

However, due to the large spectral distance of these maxima, the laser operation at these wavelengths can be easily suppressed by introducing a standard frequency selective element (e.g. a birefringent filter or a grating <cit.>) without active stabilization.










§ SPATIAL HOLE BURNING FOR A THIN-DISK AS A FOLDING-MIRROR



A more stable output beam for variations of the thin-disk thermal lens can be obtained by using the thin disk as a folding mirror in a V-shaped cavity (see Fig. <ref>). 

The stability properties of a laser containing a thermal lens have been studied in  <cit.>. 

The thermal lens splits the cavity in two parts.

The part which is longer (effective length given by the B-element of the corresponding ABCD-matrix) has a complex beam parameter q which exhibits a linear (first-order) dependence on variations of the thermal lens.

By contrast, the q-parameter in the shorter part exhibits a second-order dependence on variations of the thermal lens around the center of stability. 

Thus, to minimize variations of size and divergence of the output beam, 
the out-coupling mirror has to be placed in the short part of the cavity.

When the thin disk is used as an end-mirror of the cavity, it would be necessary to use the rear side of the disk as an out-coupling mirror, which is not feasible.

As a result the beam out-coupled  from a thin-disk laser with the disk acting as end-mirror suffers from a large sensitivity to variations of the thermal lens of the active medium.

On the contrary, in a V-shaped cavity the disk is used as folding mirror, and the out-coupling mirror can be placed in the short (stable) part of the cavity leading to an out-coupled beam insensitive to variations of the thermal lens <cit.>.



In contrast to the simple linear cavity of Fig. <ref>, in the V-shaped cavity with disk acting as folding-mirror M1, the consideration of the SHB requires the evaluation of the interference of 4 beams similar to what was presented in <cit.>.

Following the same methodology of Sec. <ref> to compute the interference pattern we proceed first by omitting M2 and computing the interference of the four beams for the amplifier configuration shown in Fig. <ref>.

In a second step, the other end-mirror (M2) is considered: it provides the resonance condition for the cavity.



 




The amplitude of the first beam E_1(x,y,z,t, λ_0) traveling within the
active medium with refractive index n from the disk front side
towards the disk rear side can be described as


    E_1(x,y,z,t, λ_0)= E_0 ·cos(2 π-n z cos(α ) + n y sin(α )/λ_0 - 2πν_0 t)·exp(-x^2 + ( (z+L/2) sin(α ) - y cos(α))^2/2 w^2),


where α is the propagation angle relative to the disk normal
within the active medium.

Again, we assume that the origin of the reference system is located at the
center of the disk, thus at a distance L/2 from the disk's rear and front
sides.

The 2^nd beam originates from the reflection on the high-reflection (HR) at the rear side of the disk (M1).

Within the active medium the amplitude of this second beam E_2
(x,y,z,t, λ_0) is given by


    E_2 (x,y,z,t, λ_0) = E_0    ·cos(2π ( L + z) n cos(α ) + n y sin(α)/λ_0 - 2πν_0 t )
       ·exp(-x^2 + ( (z + L/2) sin(α) + y cos(α))^2/2 w^2).


A phase delay and an inversion of the propagation in z-direction has been used to obtain this second beam from the first one, while size and direction in x- and
y-directions are unchanged.

This beam travels to M3 (given in green in Fig. <ref>) that reflects the beam in itself (inversion of all three propagation directions) back towards the active medium.

The amplitude of this back-reflected beam traveling in the active medium from the front-side to the rear-side takes the form


    E_3 (x,y,z,t, λ_0) = E_0    ·cos( 2π 2 d + n (L-z) cos(α ) - n y sin(α )/λ_0 - 2πν_0 t )
       ·exp(- x^2 +( (z + L/2) sin (α) - y cos(α))^2/2 w^2),


where d=D_3+nL/2 is the optical distance between the active medium center and M3, with D_3 and L as defined in Fig. <ref>.

After a second reflection on the HR-coating of the disk, a fourth beam moving from the disk rear-side to the front-side is ensued whose amplitude takes the form


    E_4(x,y,z,t,λ_0) = E_0    ·cos(2π2d + (2 L + z)n cos(α) - n  y sin(α)/λ_0 - 2πν_0 t )
       ·exp(-x^2 + ( (z + L/2)  sin(α) + y cos(α))^2/2 w^2) .




The time-averaged intensity distribution resulting from the interference of the four traveling waves takes the form



    I(x,y,z, λ_0 ) = 4 I_0 (1 + cos(4 πn z cos(α) + L/2/λ_0))  ·(1 + cos(4πn y sin(α)+d/λ_0)) exp(-x^2 + y^2/w^2)


when walk-off effects are neglected. This approximation is justified given the small thickness of the disk (L<0.5 mm) relative to the typical size of the cavity eigenmodes (w>1 mm) and the small angle (α<5^∘).

An example of such  an intensity pattern is shown in Fig. <ref>.

By inserting Eq. (<ref>) into Eqs. (<ref>) and (<ref>) the gain decrease related with the saturation effects is obtained:


    S_SHB(λ) ≈ 1 - 4 τ I_0/F_sat(1 + 1/2sinc(π n L Δ cos (α))) ·(1 + 1/2cos( 4 π Δ d ) exp(-2 Δ^2 π^2 n^2 w^2 sin^2 ( α)  ) ).




The saturation spectrum S_SHB(λ) for D_3=1 m and α=4.5^∘ is shown in Fig. <ref> as a function of the probe beam  wavelength λ.

The self saturation is depicted by the the black square.

The gray shaded region in the spectral plot of S_SHB represents rapid oscillations caused by the large optical distance D_3 between the active medium and the end-mirror M3.

These rapid oscillations are visible in Fig. <ref> where the horizontal axis is magnified by a factor of 2× 10^4  in the wavelength region around the saturating wavelength λ_0 = 1030 nm.

In the corresponding cavity (D_3=D_2=1 m) the modes (indicated by the blue empty circles) nearest to the saturating mode  experience the maximal values of S_SHB.

As the two neighboring modes (and others) have a larger small-signal gain, they will grow until they significantly contribute to the saturation of the gain medium.

Hence, single-frequency operation is disrupted.

Also in this case a suppression of these modes would require an extremely narrow-band filter with active stabilization to prevent significant losses at the wavelength of the saturating mode.




§ TWISTED-MODE OPERATION OF A V-SHAPED THIN-DISK LASER





The twisted-mode scheme <cit.> represents a well-known possibility to eliminate SHB.

Its principle is to insert two quarter-wave plates, one before and one after the active medium, as presented in Fig. <ref> such that there is no interference in the active medium, because the two counter-propagating beams have orthogonal polarizations.

The absence of the interference between the two circulating beams leads to the complete elimination of the SHB paving the way for single-frequency operation <cit.>.


The twisted-mode scheme could be implemented also for thin-disk lasers with V-shaped cavities as shown for example in Fig. <ref>.

Yet, contrary to the cavity of Fig. <ref>, in this case the twisted-mode scheme does not completely eliminate the interference.

The interference between the beams traveling to, and being back-reflected from the mirror M1 (rear side of the disk) can not be eliminated.



The resulting interference pattern which is shown in Fig. <ref> reduces thus from a four-beam to a two-beam interference similar to Fig. <ref>.




This reduction of the four-beam interference to a two-beam interference can be seen in the mathematical expression of the saturation spectrum S_SHG(λ) which takes the form:


    S_SHB(λ)≈ 1 - 4τ I_0/F_sat(1 + 1/2sinc(π n L cos(α) Δ))


similar to Eq. (<ref>).

This equation also represents the saturation spectrum for a unidirectional ring cavity provided 4I_0 is substituted by 2I_0.

Practically, this twisted-mode scheme can be implemented using only a single quarter-wave plate as shown in Fig. <ref>.

This design is possible given the small incident angle α that can be realized and the availability of commercial waveplates with retardation showing only small position and incident angle dependences (the retardation scales quadratically with the incident angle) <cit.>.


The usage of a multi-order waveplate is advisable as it provides higher damage threshold compared to a single-order waveplate.



Beside enabling the implementation of the twisted-mode scheme the multi-order waveplate combined with a polarizer also acts as a birefringent filter <cit.>.

The wavelength-dependent retardation in a round-trip that occurs in the four passes in the  quarter-wave plate leads to a wavelength-dependent round-trip transmission B(λ)


    B(λ) = ln( 1-cos^2(2π K  λ -λ_0/λ_0)),


where  2π K is the round-trip retardation experienced in the quarter-wave plate at the λ_0 wavelength (K = 58 for four passes in a 1.5 mm thick quartz plate at 1030 nm).

The logarithm is used to present the transmission as coefficient of the exponential in Eq. (<ref>). 

The improved frequency selectivity achieved with the multi-order waveplate is demonstrated in Fig. <ref>.

The black curve represents a schematic representation of the unsaturated roundtrip gain G_0 of the cavity depicted in Fig. <ref> using a thin disk made of Yb:YAG (G_0(1030 nm) = 0.26 ).



The red curve 

represents the round-trip gain when accounting for the saturation effects caused by the dominant mode at λ_0=1030 nm.

It is computed by multiplying S_SHB from Eq. (<ref>) with the unsaturated gain (black curve).

Because the modes located at the two maxima of this curve see a larger gain than the dominant laser mode, they will grow until their saturation effects become relevant.

Single-frequency operation is thus disrupted.

The blue curve is the total round-trip gain that also includes the wavelength-dependent transmission B(λ) of Eq. (<ref>) for a quarter-wave plate based on a 1.5 mm thick quartz plate.

This curve shows maximal gain for λ_0. 

Hence, the twisted-mode scheme can also be used for V-shaped cavities to obtain single-frequency operation. 







§ DEMONSTRATION OF A Q-SWITCHED THIN-DISK LASER OPERATED IN TWISTED-MODE



For the measurement of the 2S-2P splitting in muonic helium
ions <cit.> we realized a
Q-switched thin-disk oscillator as sketched in Fig. <ref>.



The disk was used as a folding mirror and a multi-order quarter-wave plate was inserted in front of the disk to implement the twisted-mode scheme and to provide the additional polarization dependent losses B(λ).

The waveplate was acting also as window for the pump module (see
Fig. <ref>) which shielded the hot front surface of the disk
from air flows that would lead to optical distortions of the laser
beam.











The other optical elements of the cavity, namely the Pockels-cell (PC), a second quarter-wave plate and a thin film polarizer (TFP) were used to control the pre-lasing operation and the Q-switch dynamics as explained for a similar system in reference <cit.>.

Pulses up to an energy of 110 mJ at a repetition rate of 800 Hz were delivered by the oscillator.

This laser was operated flawlessly for several months  at the high-intensity proton accelerator of the Paul Scherrer Institute, Villigen, Switzerland.


The effectiveness of the twisted-mode scheme in reducing SHB manifests itself also in the measured temporal profile of the emitted laser pulses.

Without the quarter-wave plate, or with its axis misaligned, the SHB causes simultaneous lasing at a few longitudinal modes, resulting in mode beating with a frequency given by the round-trip time <cit.> (see Fig. <ref>). 

This beating (fast oscillations) is suppressed when the quarter-wave plate is correctly aligned as demonstrated in Fig. <ref>.



The measurement of the pulse trace is thus a sensitive method to show
the suppression of multi-mode operation <cit.>.

In principle also a spectrometer could be used but a resolution
better than 50 MHz is needed.





§ CONCLUSIONS


In this paper we presented the wavelength-dependent decrease of the
gain in an active medium caused by saturation effects.

This decrease has been expressed as S_SHB (λ) for various linear-cavity layouts: simple cavity (Eq. (<ref>) and Fig. <ref>), thin-disk laser with the thin disk acting as end-mirror (Eq. (<ref>) and Fig. <ref>), and thin-disk laser with the disk as folding-mirror (Eq. (<ref>) and Fig. <ref>).

We found that the spectral distance between adjacent minima and maxima in S_SHB around the saturating wavelength is large when the disk is used as  end-mirror of the cavity. 

By contrast, it is small when the disk is used as a folding-mirror. 

Hence, the “gain-at-the-end” configuration allows for easy suppression of interference effects using standard selective elements as Fabry-Perot etalons. 

For the more elaborate geometry where the disks acts as folding-mirror, suppression of these maxima is more challenging given the small spectral distance (in the order of 100 MHz).

We have demonstrated that the implementation of the twisted-mode scheme
in thin-disk lasers with V-shaped layouts suppresses only partially the
SHB in the disk.

Using the twisted-mode scheme with a multi-order waveplate an additional wavelength-dependent loss arises that is spectrally sufficiently narrow to suppress the residual SHB effects that prevent single-frequency operation.

The simplicity, efficiency and robustness (in term of optical damage) of this scheme pave the way for achieving single-frequency operation of thin-disk lasers in the multi kW range.



§ FUNDING INFORMATION


We acknowledge the support from the Swiss National Science Foundation Project SNF 200021_165854, the European Research Council ERC CoG. #725039 and ERC StG. # 279765. 
The work was supported by ETH Research Grant ETH-35 14-1.
The study has been also supported by the ETH Femtosecond and Attosecond Science and Technology (ETH-FAST) initiative as part of the NCCR MUST program.


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