1. Introduction
Recent experiments in which broadband femtosecond combs are coupled to passive optical
cavities have demonstrated great potential for new optical devices and measurement techniques. While some experiments utilize the increased frequency sensitivity of a cavity to more precisely measure optical parameters such as absorption [
1
1
.
M.J.
Thorpe
,
K.D.
Moll
,
R.J.
Jones
,
B.
Safdi
, and
J.
Ye
, “
Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection
,”
Science
311
,
1595
(
2006
).
[CrossRef] [PubMed]
] and group-delay dispersion [
2
2
.
M.J.
Thorpe
,
R.J.
Jones
,
K.D.
Moll
,
J.
Ye
, and
R.
Lalezari
, “
Precise measurements of optical cavity dispersionand mirror coating properties via femtosecond combs
,”
Opt. Express
13
,
882
(
2005
).
[CrossRef] [PubMed]
] over a broad bandwidth, other experiments make use of the pulse-energy enhancement inherent to the cavity to generate more intense sources [
3
3
.
R.J.
Jones
and
J.
Ye
, “
High-repetition-rate coherent femtosecond pulse amplification with an external passive optical cavity
,”
Opt. Lett.
29
,
2812
(
2004
).
[CrossRef] [PubMed]
] and enhance nonlinear interactions [
4
4
.
V.P.
Yanovsky
and
F.W.
Wise
, “
Frequency doubling of 100-fs pulses with 50% efficiency by use of a resonant enhancement cavity
,”
Opt. Lett.
19
,
1952
(
1994
).
[CrossRef] [PubMed]
,
5
5
.
F.O.
Ilday
and
F.X.
Kartner
, “
Cavity-enhanced optical parametric chirped-pulse amplification
,”
Opt. Lett.
31
,
637
(
2006
).
[CrossRef] [PubMed]
]. One particularly interesting experiment is the recent demonstration of generating a high repetition rate vacuum-and extreme-ultraviolet (VUV/XUV) source by coupling the light from a Ti:sapphire oscillator to a passive cavity [
6
6
.
R.J.
Jones
,
K.D.
Moll
,
M.J.
Thorpe
, and
J.
Ye
, “
Phase-coherent frequency combs in the EUV via high-harmonic generation inside a femtosecond enhancement cavity
,”
Phys. Rev. Lett.
94
,
193201
(
2005
).
[CrossRef] [PubMed]
,
7
7
.
C.
Gohle
, et al., “
A frequency comb in the extreme ultraviolet
,”
Nature
436
,
234
(
2005
).
[CrossRef] [PubMed]
]. However, coupling the generated radiation out of the cavity presents many challenges. In this paper, we explore possible output-coupling methods to efficiently extract the high-harmonic radiation from the cavity.
The recent experimental demonstrations of cavity-based high-harmonic generation (HHG)
utilized an intracavity sapphire plate oriented at Brewster’s angle to couple the HHG from the cavity. A small Fresnel reflection occurs for the HHG because the sapphire plate is dispersive. However, this method has may limitations. For instance, the sapphire plate can limit the cavity energy enhancement. In order to reach the necessary intensities to achieve HHG with a cavity-enhanced technique, several conditions must be satisfied. First, the cavity must be low loss over the spectral bandwidth of the femtosecond laser. Because the sapphire plate must be placed near the intracavity focus, the entire spatial profile of the diffracting beam cannot be aligned at Brewster’s angle. This additional loss limits the finesse that can be achieved. A second condition is that each component of the femtosecond comb must overlap a corresponding cavity mode. This requires that the cavity dispersion must be minimized such that the free spectral range of the cavity is uniform as a function of wavelength. In order to compensate the dispersion of the sapphire plate, a negative group-delay dispersion (NGDD) mirror must be utilized. However, NGDD mirrors are not ideal for low-loss/broadband because of the increased number of coating layers. Additionally, compensating for higher-order dispersive terms of the sapphire can be
problematic.
A second limitation of the intracavity plate based method of output coupling arises from the fact that the sapphire plate must be located near the intracavity focus in order to couple out the HHG before it hits another cavity mirror. For high-finesse cavities, the high intensities inside the plate lead to significant spectral distortions due to nonlinear processes in the plate [
8
8
.
K.D.
Moll
,
R.J.
Jones
, and
J.
Ye
, “
Nonlinear dynamics inside femtosecond enhancement cavities
,”
Opt. Express
13
,
1672
(
2005
).
[CrossRef] [PubMed]
] which reduce the ability to couple the entire comb to the cavity. A final limitation of the sapphire plate is that the output coupling efficiency is low. At low-order harmonics, the index of refraction of the sapphire has not changed sufficiently to lead to a large Fresnel reflection. At higher-order harmonics, the sapphire becomes absorbing, and even though the light is reflected from the front surface, absorption of the evanescent wave penetrating the sapphire leads to a maximum reflectivity of approximately 20 percent.
An alternative output coupling method, which will be the focus of this article, is to couple the HHG through a small hole in the concave mirror after the intracavity focus. By coupling a higher-order spatial beam to the cavity, the pump beam will avoid the hole, maintaining the high finesse. However, the HHG generated at the focus will diffract at a smaller angle due to its decreased wavelength and will partially couple through the hole. This technique eliminates many of the problems associated with the sapphire plate. An additional technique that we will briefly discuss towards the end of this paper is a noncollinear geometry in which the HHG is generated by two intracavity pulses intersecting at a slight angle. This technique bears similarity to the first approach except that the two beams are separate instead of being derived from the two lobes of the higher-order spatial mode.
Fig. 1. Schematic of the ring cavity under investigation. A hole of radius a is drilled in one of the curved mirrors to allow the high-harmonic light to escape from the cavity. The curved mirrors have radius of curvature R
_{1} and R
_{2} and the distances of separation between the mirrors are denoted as d_{i}
.
2. Computational method
where (
r,
ϕ) represent the radial and angular coordinates of the field amplitude
u_{ν}
,
ν is the angular mode order number, and
p is the radial mode order number. Physically, the weighting coefficient
c_{p}
represents the amount of the unapertured cavity mode with radial mode-order
p that is contained in the physical cavity mode of the apertured cavity.
R and
w represent the position dependent radius of curvature and width of the unapertured modes inside the cavity and can be calculated with ABCD matrix techniques,
k = 2
π/
λ where
λ is the laser wavelength,
Lpν
is the associated Laguerre polynomial [
10
10
.
G.B.
Arfken
and
H.J.
Weber
,
Mathematical methods for physicists, 4th ed
., (
Academic, San Diego, California
,
1995
).
], and
Ψ
_{p,ν} is the position dependent geometrical phase.
Using this representation, the weighting coefficients of the cavity mode can be solved for using matrix eigenvalue methods. The associated eigenvalue β, which encodes the round-trip cavity loss and longitudinal frequency, can be calculated by solving the matrix equation:
where
c⃗ is a vector with elements
c_{p}
, and
A is a matrix which is responsible for modeling the
aperture by decomposing the field
u(
r) before the mirror into a field which is zero for
r<
a after the mirror. Diagonal matrices
D_{i}
account for the fact that different transverse modes accumulate different geometrical phase by traveling distance
d_{i}
in the cavity where (as shown in
Fig. 1) we use
d
_{1} to represent the overall path outside the two curved mirrors and
d
_{2} for the distance between the two curved mirrors. Fortunately, the elements of the aperture matrix
Fig. 2. (a) Comparison of expected cavity loss for a gaussian (ν = 0) and donut-mode (ν = 1) beam coupled to a cavity with a hole drilled in one of the mirrors. The losses were also approximated by integrating the power of the unperturbed beam over the hole area which are shown as solid and dashed lines for the Gaussian and donut mode, respectively. (b) Percentage of light that can be coupled to cavity mode using an ideal gaussian or donut-mode.
where
can be easily calculated by making use of the recurrence relationship
where y
_{0} = 2a
^{2}/w02 and w
_{0} is the field 1/e half-width of the unapertured gaussian mode on the apertured mirror. The coefficient vector c⃗ is truncated at a finite number of terms, and β is solved by inverse iteration as the size of the aperture adiabatically increases.
3. Results
As a specific example to study, we have chosen cavity parameters similar to those used experimentally in Refs. [
6
6
.
R.J.
Jones
,
K.D.
Moll
,
M.J.
Thorpe
, and
J.
Ye
, “
Phase-coherent frequency combs in the EUV via high-harmonic generation inside a femtosecond enhancement cavity
,”
Phys. Rev. Lett.
94
,
193201
(
2005
).
[CrossRef] [PubMed]
,
7
7
.
C.
Gohle
, et al., “
A frequency comb in the extreme ultraviolet
,”
Nature
436
,
234
(
2005
).
[CrossRef] [PubMed]
]. The curved mirrors have a radius of curvature
R
_{1} =
R
_{2} = 10 cm, the long arm of the cavity has length
d
_{1} = 2.9 m and the short path of the cavity (
d
_{2} ≈ 10 cm) will be varied to probe the output-coupling scheme for different levels of cavity stability. In order to calculate the beam width of the cavity mode, a laser wavelength of
λ = 800 nm was used.
Fig. 3. (a) Comparison of loss for the donut mode at varying levels of cavity stability. (b)
Fortunately, even after renormalizing the hole size by the mode diameter, for larger hole
sizes the cavity loss is less when the cavity is operated near the edge of stability where the intracavity focus is tighter (△ and ☐).
Another issue that arises when comparing performance between the gaussian and donut mode is the input coupling efficiency. The reorganization of the gaussian beam that allows relatively low loss to be maintained leads to a dramatic reduction in the input-coupling efficiency (see
Fig. 2(b)) since the cavity mode looks less like a gaussian. Large diffraction fringes also develop because the hard-edged aperture acts on the high-intensity portion of the gaussian beam. In contrast, high coupling efficiency can be maintained for the donut-mode at moderately large hole diameter.
While the donut mode offers better performance compared to the gaussian in terms of minimizing loss and improved input-coupling efficiency, there are several drawbacks. First, it is very difficult to achieve a donut mode directly from a mode locked laser. Second, the peak intensity at the focus for a donut mode is only 1/e≈37% the peak intensity of the gaussian if the two beams have the same power. Considering the highly nonlinear nature of HHG, this reduction in peak intensity can lead to a significant reduction in the HHG yield. Third, slight astigmatism in the ring cavity breaks the longitudinal-mode degeneracy between the TEM_{01} and TEM_{10} modes. These difficulties can be partially addressed by inserting a phase-mask before the cavity which produces a beam with a gaussian intensity distribution but introduces a π phase shift between the two halves of the spatial profile. Computing the overlap integral of the beam with the TEM_{01} reveals that a conversion efficiency of 93% can be achieved. While a phase plate that increases ν by one could have been used, manufacturing a phase plate that creates a TEM_{01} is far simpler. In addition, the peak intensity of the TEM_{01} is twice that of the donut yielding only a 26% reduction in peak intensity relative to the gaussian. Since the TEM_{01} mode can be expressed as a superposition of the ν = ±1 donut modes, the expected coupling efficiency and loss experienced by the TEM_{01} will be the same as that already calculated for the donut mode.
To experimentally verify the numerical results presented here, a phase plate was fabricated as well as mirrors with holes drilled in them. A 725-μm step was reactive ion etched onto a sapphire window which was inserted into the laser beam (with the step line cutting through the middle of the spatial profile) at Brewster’s angle between the laser and enhancement cavity. This phase plate worked sufficiently well over the spectral bandwidth of ~20 nm that could be coupled well into the cavity due to intracavity dispersion. With unapertured mirrors in the cavity, the coupling efficiency of the TEM_{01} mode was 85% of what could be achieved with the gaussian mode, compared to a theoretical limit of 93%. By filling the entire vacuum chamber with 1-Torr of Xenon, a strong plasma was observed indicating that sufficient peak intensity
buildup in excess of 10^{13}W/cm^{2} at the intracavity focus was still maintained with the TEM_{01} mode. An apertured mirror was then fabricated by mechanically counterboring a 3-mm hole into a glass substrate to within 1-mm of the reflecting surface. The remaining 1-mm was then laser machined to produce holes from 100 to 300 μm in diameter. The surface was then polished to the desired radius of curvature and coated with a high-reflector coating. When inserted into the cavity to which a TEM_{01} was coupled, a 100-μm hole changed the cavity finesse from 3300 to 2650 (as measured by cavity ringdown) corresponding to an additional 0.05% loss. This is comparable to that predicted by the simulations presented above. After coupling all aspects of the experiment together (input coupling efficiency, mode conversion, spectral filtering, hole loss), an intracavity pulse-energy enhancement of 275 was still experimentally achieved leading to visible ionization of Xenon at the intracavity focus.
One solution that addresses these issues is to integrate
λ /2 phase masks onto both concave mirrors of the cavity. In the long arm of the cavity, a TEM
_{01} beam with low on-axis intensity is coupled to the cavity. Upon reflection from the concave mirror, the two lobes of the beam become in phase but retains a low-intensity on-axis. However, as the beam propagates to the focus, a strong on-axis intensity develops leading to strong axially-generated HHG [
11
11
.
J.
Peatross
,
J.L.
Chaloupka
, and
D.D.
Meyerhofer
, “
High-order harmonic-generation with an annular laser-beam
,”
Opt. Lett.
19
,
942
(
1994
).
[CrossRef] [PubMed]
]. Upon propagating to the apertured concave mirror, the fundamental beam redevelops the low-intensity node on-axis such that low loss is maintained. Finally, reflection from the apertured mirror with the integrated phase mask reproduces a TEM
_{01} character to the beam.
Fig. 4. Ninth-harmonic intensity profiles at the apertured mirror for gaussian, donut (ν =1), and TEM_{01} input beams. The high-harmonic donut has a ν = 9 character. The white circle represent the 225-μm diameter aperture.
Even though a TEM_{01} mode would be used in an actual experiment, it is easier to do the calculations in the basis of the donut mode. The action of a phase mask that maintains the intensity distribution but increases the azimuthal symmetry parameter, ν, by one can be incorporated into the model as an additional transfer matrix, N, with components,
where
_{3}
F
_{2} is the generalized hypergeometric function [
12]. While calculating the matrix elements directly from Eq.
6 can be cumbersome for large
p and
q, a recurrence relationship can be derived to allow efficient computation of the matrix elements:
The loss resulting from the integration of a pair of phase masks on the two curved mirrors can therefore be computed by solving the matrix eigenvalue problem
where the ν = 1 beam is propagating in the d
_{1} arm of the cavity.
We have calculated the expected output coupling efficiency when a TEM
_{01} mode is coupled to the cavity with and without the integrated phase masks. As shown in
Fig. 6, the integrated phase plates lead to a dramatic increase in the output coupling, especially at lower order harmonics. Even for a relatively large hole of 250
μm, the loss can be maintained below 0.1% and an output coupling efficiency in excess of 40% at the fifteenth harmonic can be achieved. In addition, as stated above, HHG will be more efficiently generated because the peak intensity at the intracavity focus will be higher for the hybrid mode.
Fig. 5. (a) Intracavity loss near the inner edge of stability for a cavity that has phase masks
integrated on the curved mirrors. A hybrid mode which has ν = 0 and ν = 1 character in different parts of the cavity is supported. For comparison, the round-trip loss without the integrated masks is also computed for an input gaussian and donut mode. (b) The hybrid mode can also be excited with similar coupling efficiency as the donut mode.
Fig. 6. Output coupling efficiency of the HHG for a cavity (a) without and (b) with phase
masks integrated onto the concave mirrors. Note the vertical scale difference between (a)
and (b)
An additional output coupling method is to utilize a cavity that is twice as long as the laser cavity. As such, two pulses will be inside the buildup cavity. By using two sets of concave mirrors that share a common focus, HHG can be achieved in a noncollinear geometry [
13
13
.
S.V.
Fomichev
,
P.
Breger
,
B.
Carre
,
P.
Agostini
, and
D.F.
Zaretsky,
“
Non-collinear high-harmonic generation
,”
Laser Phys.
12
,
383
(
2002
).
,
14
14
.
S.V.
Fomichev
,
P.
Breger
, and
P.
Agostini
, “
Far-field distribution of third-harmonic generation by two crossed beams
,”
Appl. Phys. B
76
,
621
(
2003
).
[CrossRef]
] and the high-harmonic light can escape through a small gap between the curved mirrors. (See
Fig. 7). This method presents some advantages compared to the previously discussed schemes. First, separation of the high-harmonic from the fundamental light is simple because they are propagating in different directions. Second, no complicated fabrication is necessary. Finally, there is an inherent increase in the intensity at the focus by a factor of two since there are two separate pulses colliding. However, several disadvantages may lead to experimental challenges. First, since the cavity is twice as long, more mirrors would have to be used to fold the cavity into a reasonable size. This will increase the losses and dispersion and could limit the energy enhancement achieved. Second, the noncollinear geometry will limit the effective interaction length of the two pulses. Lastly, the cavity geometry must be arranged such that the two ultrashort pulses overlap in time at the focus. In addition, the two halves of the cavity need to be stabilized such that the two pulses will maintain constructive interference on axis.
Fig. 7. Output coupling method using noncollinear geometry. The length of the optical cavity is twice the laser-cavity length such that two pulses inside the cavity can simultaneously focus into a gas sample. The noncollinearly generated HHG is output coupled through a gap between the two focusing mirrors.