Counting Near Infrared Photons with Microwave Kinetic Inductance Detectors

W. Guo1, X. Liu1, Y. Wang1,2∗, Q. Wei1, L. F. Wei1,3†, J. Hubmayr2,

J. Fowler2, J. Ullom2, L. Vale2, M. R. Vissers2, and J. Gao2

1) Quantum Optoelectronics Laboratory, School of Physical Science and Technology,Southwest Jiaotong University, Chengdu, 610031, China

2) National Institute of Standards and Technology, Boulder, CO 80305, USA‡

3) State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

(Dated: May 10, 2017)

We demonstrate photon counting at 1550 nm wavelength using microwave kinetic inductancedetectors (MKIDs) made from TiN/Ti/TiN trilayer films with superconducting transition temper-ature Tc ≈ 1.4 K. The detectors have a lumped-element design with a large interdigitated capacitorcovered by aluminum and inductive photon absorbers whose volume ranges from 0.4 µm3 to 20 µm3.The energy resolution improves as the absorber volume is reduced. We achieved an energy resolutionof 0.22 eV and resolved up to 7 photons per optical pulse, both greatly improved from previouslyreported results at 1550 nm wavelength using MKIDs. Further improvements are possible by opti-mizing the optical coupling to maximize photon absorption into the inductive absorber.

Photon-number-resolving (PNR) detectors at near in-frared wavelengths have important applications in a num-ber of frontier fields, such as quantum secure commu-nications [1], linear optical quantum computing [2] andoptical quantum metrology [3]. Compared to more con-ventional detectors at this wavelength, such as silicon-based detectors [4], superconducting detectors have lowerdark-count rate, higher sensitivity, and broadband re-sponse. They show great promise in serving as the basicbuilding blocks for efficient PNR devices. For example,by spatial or temporal multiplexing of superconductingnanowire single-photon detectors (SNSPDs) [5–8], pho-tons can be counted at high speed. But the single-element nanowire has no intrinsic PNR and energy-resolving capabilities. Alternatively, single-element tran-sition edge sensors (TESs) [9] have demonstrated highquantum efficiency and multi-photon discrimination attelecommunication wavelengths [10–12]. Recently, count-ing up to 29 photons and intrinsic energy resolution≈ 0.11 eV at 1550 nm wavelength have been achievedin Ti/Au TESs [13–15].

Another type of superconducting detector possessingintrinsic photon-number-resolving and energy-resolvingpower is the microwave kinetic inductance detector(MKID) [16]. MKIDs are cooper pair breaking detectorsbased on high-quality factor (high-Q) superconductingresonators [17, 18]. The absorption of a photon with en-ergy higher than twice the gap energy (hν > 2∆) canbreak Cooper pairs into quasiparticles, changing the sur-face impedance of the resonator and resulting in a lowerresonance frequency fr and higher internal dissipation (orlower quality factor Qi). When applying a short opticalpulse to the detector and probing the resonator with a

∗Electronic mail: qubit@home.swjtu.edu.cn†Electronic mail: weilianfu@gmail.com‡Contribution of the U.S. government, not subject to copyright

microwave tone near the resonance frequency, one can ob-tain a pulse response in the complex forward transmissionS21, as shown in Fig. 1(a). This photon response can bemeasured using a homodyne detection scheme (Fig. 1(d))and the signal can be decomposed into frequency and dis-sipation responses (Fig. 1(a),(b)) for pulse analysis.

Compared to TESs, MKIDs are easy to fabricate andmultiplex into large arrays. A large array of MKIDs canbe measured using a pair of coaxial cables, which greatlyreduces the complexity of the instrument design. Pre-viously, MKIDs with PNR capability have mostly beenconsidered for astronomy applications at the visible wave-length [19]. Single-photon counting at telecommunica-tion wavelengths (near infrared) with titanium-nitride(TiN) MKIDs was first demonstrated in Ref. [20], where afull-width-at-half-maximum (FWHM) energy resolution∆E ≈ 0.4 eV was achieved and up to 2-photon eventswere resolved. In this letter, we present an optimizedMKID design based on TiN/Ti/TiN trilayer films andimproved photon counting performance at 1550 nm wave-length: energy resolution ∆E ≈ 0.22 eV is obtained andup to 7-photon events can be resolved.

Our detectors are made from a 20 nm thickTiN/Ti/TiN trilayer film [21] (Tc ≈ 1.4 K) deposited on ahigh-resistivity Si substrate. Such TiN trilayer films wereinitially developed for feedhorn-coupled MKIDs whichhave recently demonstrated photon-noise limited sensi-tivity at submillimeter wavelengths [22]. As shown inFig. 1(c), our detectors comprise a large IDC shuntedby a meandered inductive strip. The latter serves as asensitive photon absorber. The IDC area is ≈ 0.7 mm× 0.7 mm, with 5 µm finger/gap width. This large areaIDC is used to suppress the two-level system (TLS) noisein the substrate [23]. The IDC is covered with a 100 nm-thick layer of aluminum (Al). Because of the low currentdensity in the IDC and the much lower kinetic inductanceof Al than TiN, the response from a photon hitting theIDC area is negligible. We designed 13 resonators on a10 mm by 5 mm chip, with inductor strip width rang-

arX

iv:1

702.

0799

3v3

[ph

ysic

s.in

s-de

t] 9

May

201

7

2

dissipation

IDC covered by Aluminum

TiN absorber(a) (c)

0 200 4000

0.5

1

Res

pons

e [a

.u.]

(b)

Synthesizer

SplitterMKID

(40 mK)HEMT(4 K)

Amplifier(300K)

Mixer

I Q

A/D

(d)

μs]Time [

frequencydissipation

frequency

FIG. 1: [Color online] (a) Pulse response in the complex S21

plane. The blue circle represents the resonance loop from afrequency sweep. The red line is the averaged response pulseafter photon absorption and the red arrow shows the rising-edge of the pulse. This response can be projected to frequencyand dissipation responses, with directions tangent and normalto the resonance loop. (b) The averaged frequency and dis-sipation pulse responses in time domain. (c) A schematic ofthe MKID design. The resonator has a lumped-element de-sign, with a small volume of meandered inductive strip (red)in parallel with a large interdigitated capacitor (IDC), whichis capped by a layer of aluminum (blue). (d) Homodyne de-tection scheme used to read out MKIDs.

ing from 1 µm to 20 µm, length from 10 µm to 100 µmand volume from 0.4 µm3 to 20 µm3, to systematicallystudy the dependence of the detector performance on theabsorber geometry. All the resonance frequencies are de-signed to be around 6 GHz and all the resonators arecoupled to a common microstrip feedline with couplingquality factor Qc ≈ 1.5× 104.

The detectors are cooled in a dilution refrigerator toa base temperature of 40 mK. At this temperature, theinternal quality factors of the resonators are measuredto be around 105. A 1550 nm laser diode driven by afunction generator at room temperature is used to gen-erate optical pulses with a width of 200 ns at a repeti-tion frequency of 120 Hz. The incident photons are thenattenuated and guided into the device box mounted atthe mixing chamber stage through a bare optical fiber.In this demonstration experiment, we did not optimizethe optical coupling to the absorber and the light ex-iting the fiber flood illuminates the entire chip insteadof being focused only onto the absorber area. As a re-sult, the optical efficiency is rather low, which we planto improve in future experiments. As shown in Fig. 1(d),the standard homodyne scheme is used to read out the

resonators. We probe the resonators at a microwave fre-quency that maximizes the frequency response δS21/δfrand the microwave power is chosen to be 2 dB below bi-furcation power to avoid the strong non-linear effects [17]in the resonator. For each optical pulse, the correspond-ing response of the detector is digitized at a samplingrate of 2.5 Ms/s. The raw data are converted to the fre-quency and dissipation responses. Only the frequencyresponse data are further analyzed, because the dissi-pation response is smaller compared to the frequency re-sponse and the dissipation pulse decay time is much faster(see Fig1. (b))due to the anomalous electrodynamic effectfound previously in TiN films [20, 22, 24]. Note that wehave used a rigorous non-linear fitting procedure to di-rectly convert the pulse trajectory in the IQ plane to thefractional frequency shift, because the response in frac-tional frequency shift unit is always linearly proportionalto the change in the quasiparticle density, even when thepulse response is large (approaching the resonator line-width) and the phase shift becomes nonlinear. We an-alyze the pulse data by using standard Weiner optimalfilter procedures and the filtered pulse height data areused to generate photon-counting statistics.

Fig. 2(a) shows a histogram of the optimally filteredpulse height data for 2×104 pulse events measured fromthe resonator with absorber width of 2 µm and volumeof 1.92 µm3. The first 3 peaks, which correspond to theevents of 0, 1, and 2 photons being absorbed in the de-tector, are clearly observed. We fit the histogram to amodel of a superposition of 4 Gaussian peaks with inde-pendent heights and widths, as shown by the red profilein Fig. 2(a). The FWHM energy resolution ∆En of then-photon peak is related to the standard deviation σn ofthe n-th Gaussian peak by:

∆En = 2√

2ln(2)σn

An −An−1hν, n = 1, 2, ... (1)

where hν = 0.80 eV is the energy of a single 1550 nmphoton and An is the pulse height of the n-photon peak.The obtained FWHM energy resolutions for the 1-photonand 2-photon peaks are ∆E1 = 0.34 eV and ∆E2 =0.42 eV respectively. Here we claim a peak is resolvedif ∆E/hν < 1. According to this criterion, this detec-tor has the sensitivity to resolve the first 3 peaks (0-,1- and 2-photon). According to the stochastic nature ofthe photon detection process, the n-photon events shouldobey Poisson statistics. Indeed, as shown in the inset ofFig. 2(a), the counts in the n-photon peak (proportionalto the area of each Gaussian) normalized by the totalcounts match a Poisson distribution with λ = 0.61. λis the mean photon number absorbed by the detector,suggesting that our detector detects an average of 0.61photons per pulse event.

Fig. 2(b) shows the photon counting histogram at ahigher input optical power, corresponding to a mean pho-ton number λ = 1.95. The first 6 (0- to 5-photon) peaksare resolved with the energy resolutions of ∆E1 = 0.36 eVand ∆E2 = 0.45 eV for the 1-, and 2-photon peak re-

3

0 2 4 6 0

1

2

3

Cou

nts

x 103

0 1 2 30

0.2

0.4

0.6

0.8

n−photon event

0 1 2 3 4 0

0.4

0.8

1.2

Cou

nts

x 103

0 1 2 3 4 50

0.1

0.2

0.3

0.4

n−photon event

dataP−dist

0 1 2 3 4 0

1

2

3

4

O.F. pulse height

Cou

nts

x 102

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

n−photon event

Prob

.

(a) (b)

(c) (d)

0 10 20 300

1

2

3

Number of incident photons

λ

datafitting

Prob

.

Prob

.

O.F. pulse height O.F. pulse height

dataP−dist

dataP−dist

FIG. 2: [Color online] (a) A histogram of the optimally filtered(O.F.) pulse height (normalized by the template pulse) usingfrequency readout. A 4-peak Gaussian fit to the data is shownby the red line. Inset: the probability of the n-photon event(calculated by the area in each Gaussian peak normalized bythe total area) fit to a Poisson distribution with λ = 0.61. (b)Photon counting histogram (λ = 1.95), fit by a superpositionof 6 Gaussian peaks. (c) Photon counting histogram (λ =3.78) where 7-photon events are resolved. (d) The detectedmean photon number per pulse (red dots) vs. the estimatedtotal number of incident photons onto the absorber area. Theslope of the linear fitting (blue curve) suggests the photon-device coupling efficiency is ≈ 10%.

spectively, which both slightly increase from Fig. 2(a).Fig. 2(c) shows the histogram at an even higher opticalpower with a mean photon number of λ = 3.78, wherethe first 8 (0- to 7-photon) peaks are resolved.

In the 3 histograms shown in Fig. 2, we see that the 1-and 2-photon peaks are clearly broadened as compared tothe 0-photon peak, indicating that additional noise ariseswhen photons are absorbed and the energy resolution forthe n-photon peak (n ≥ 1) is not dominated by the back-ground noise of the detector in the dark environment.We speculate the broadening might be related to severalfactors, including position-dependent response of the ab-sorber, parasitic response from the photons hitting thenon-absorber area (e.g., IDC, substrate, feedline), someunknown sources of photon-induced noise. We have sim-ulated the current distribution using Sonnet (an electro-magnetic simulation software) and the results show thatthe current is very uniform throughout the inductor stripto be within 0.4%. This is expected because the dimen-sions of the inductors (< 100 µm) are much smaller than

the microwave wavelength (> 1 cm around 6 GHz). Sincethe resonator frequency response is proportional to thelocal kinetic inductance change weighted by the squareof the current distribution [25], broadening of the photonpeak due should not be dominated by the non-uniformcurrent distribution in the inductive absorber.

In Fig. 2(d), we plot the detected mean photon numberas a function of the estimated total number of photonsincident onto the absorber area, which is perfectly linearas expected. The incident photon number is estimatedfrom the total optical power measured by a power meterand the solid angle covered by the absorber area at thedistance from the absorber to the fiber tip. Due to thelow photon absorption efficiency, our detector can absorband detect only 1 photon for approximately 10 incomingphotons hitting the absorber.

In this work, we have 13 resonators with different ab-sorber volumes, which allows us to compare the photoncounting statistics. The main results are summarized inFig. 3. Fig. 3(a) shows the 1-photon responsivity (frac-tional frequency shift δfr/fr induced by absorbing 1 pho-ton) as a function of the absorber volume V . The mea-sured responsivity is fitted well by a linear relation with1/V . This is expected because δfr/fr ∝ δnqp ∝ 1/V ,where nqp is the quasiparticle density.

Fig. 3(b) shows the widths (i.e., the standard devi-ations σ0 and σ1 converted to δfr/fr which is a mea-sure of the frequency noise) of the 0-photon peak (blackdots) and 1-photon peak (green dots) as a function of V .Both widths roughly fit onto a power-law of V −0.7 andthe 1-photon peak is about ∼ 4.5 times wider than the0-photon peak. Combining the responsivity data fromFig. 3(a) and the noise data from Fig. 3(b), we derivethe 1-photon energy resolution ∆E1 from Eqn. (1) asa function of V , which is plotted in Fig. 3(c). We seethat ∆E1 increases with V and scales as ≈ V 0.3. Our re-sults suggest that the energy resolution improves as theabsorber volume is reduced. The best ∆E1 we obtainedis 0.22 eV, corresponding to an energy-resolving powerof R = hν/∆E1 = 3.7 at 1550 nm, which is achievedin the resonator with the smallest absorber volume of0.4 µm3 and also the narrowest inductor width of 1 µm.In Fig. 3(d), we plot the maximum number of photonsthat can be resolved by each detector Nr as a functionof its absorber volume V . We see that Nr drops at bothsmallest and largest V . Nr drops at large V because theenergy resolution degrades as V is increased (Fig. 3(c)).Nr also drops at small V because the large responsivityand high photon number lead to “saturation” of the de-tector, where the frequency shift of the pulse exceeds theresonator bandwidth and the signal-to-noise ratio is de-graded. To increase the bandwidth for operation, we candesign resonator with lower Qc and/or higher resonancefrequency fr.

The best theoretical energy-resolving power that canbe achieved by a MKID as a pair-breaking detector is

given by R = 12.355

√ηhνF∆ , where η ≈ 0.57 is the con-

4

1-photon response linear fitting

Res

pons

e (in

δf/f r

)1-photon peak0-photon peak

0.2

0.5

11-photon

0

2

4

6

8R

esol

ved

phot

on n

umbe

r Resolved photons

1 10V [ m3]μ

10−6

10−5

∆E [

eV]

∆E1

1 10V [ m3]μ

0 10 20V [ m3]μ

Wid

th σ

(in δf r/f

r)

10−7

10−6

r1

1 10V [ m3]μ

(a) (b)

(c) (d)

FIG. 3: [Color online] Log-log plot of 1-photon responsivityvs. absorber volume V . The data fit onto a straight line witha slope of -1 (blue line), indicating the measured responsivityis proportional to 1/V . (b) Log-log plot of the widths of0-photon and 1-photon peak vs. V . The data points fallinto two groups and both can be fitted by straight lines (thetwo blue lines) with the same slope of -0.7, suggesting bothwidths roughly scale as V −0.7. (c) Log-log plot of 1-photonenergy resolution ∆E1 vs. V and the fitted red line indicatesa V 0.3 scaling of ∆E1. (d) The maximum number of resolvedphotons Nr vs. V .

version efficiency from photons to quasiparticles [26], hνis the energy of the incident photons, ∆ = 1.72kBTc isthe superconducting gap energy of the absorber material,and F is the Fano factor [27]. This predicts a theoret-ical R = 45 at 1550 nm (a typical value of F = 0.2 isassumed), which is an order of magnitude higher thanthe R = 3.7 achieved by our best detector. Coinciden-tally, the optical lumped-element MKIDs [28, 29] madefrom 20-60 nm substoichiometric TiN films have a typ-ical energy-resolving power R = 16 at 254 nm, whichis also an order of magnitude below their Fano limitR = 150. While this suggests that TiN-based photoncounting detectors have large room to improve, it is im-portant to understand why they “underperform” theirtheoretical prediction. In fact, η ≈ 0.57 is the ideal con-version efficiency when photons are absorbed in a bulksuperconductor. Our film is only 20 nm thick and thehigh energy phonons may quickly escape the film into thesubstrate before breaking more quasiparticles, leading toa efficiency η smaller than 0.57 and a smaller response.

This phonon loss process may also fluctuate and causeadditional noise, as observed in the thin film supercon-ducting tunnel junction photon detectors [26]. In futureexperiments, we plan to futher explore this phonon losseffect, as well as the V 0.3 energy resolution scaling, bytesting different thickness of TiN films and by makingthe absorber on a suspended membrane.

Many aspects in our design and experimental setupcan be improved. If the responsivity and noise trendsstill hold below 0.4 µm3, we expect that better energyresolution can be achieved by using an absorber volumeeven smaller than 0.1 µm3. Instead of using Tc ≈ 1.4 Ktrilayer, a lower Tc TiN film with a lower gap energy mayfurther boost the responsivity. Suspending the absorberon a thin silicon membrane may increase the quasiparti-cle recombination time and the conversion efficiency, assuggested by the “phonon recycling” scheme [30, 31]. Ac-cording to the optical measurement on thin TiN films byVolkonen[32], we estimate that the reflectance and trans-mittance for our 20 nm TiN film are about 60% and 10%respectively, indicating approximately only 30% photonsare absorbed. The photon absorption efficiency can begreatly enhanced by adding anti-reflection coating andembedding the absorber in an optical structure [33]. Toefficiently collect every photon, the input light should beprecisely confined onto the absorber active area, whichcan be realized using advanced alignment and couplingtechniques, such as direct fiber coupling to the detector[34] or through a fusion-spliced microlens [35].

In conclusion, we have demonstrated photon count-ing at 1550 nm using TiN/Ti/TiN trilayer MKIDs. En-ergy resolution as low as ∆E ≈ 0.22 eV is obtained andup to 7-photon events can be resolved. By studying de-vices with a variety of geometries, we have systematicallyinvestigated the dependence of photon counting perfor-mance on the absorber volume. The energy resolutionimproves as the absorber volume is reduced. Further im-provements in these detectors are possible by improvingthe detector design and optimizing the optical couplingto maximize the photon absorption into the absorber.With the energy resolution of our MKID photon count-ing detectors approaching the performance of TESs (cur-rently a factor of two better), the multiplexing advantageof MKIDs may stand out in applications where a largearray of detectors with high photon-resolving power isneeded.

The MKID devices were fabricated in the NIST-Boulder microfabrication facility. This work was sup-ported in part by the National Natural Science Founda-tion of China (Grant Nos. 61301031, U1330201). L. F.Wei thanks Profs. C. D. Xie and K. C. Peng for theirencouragements and useful discussions.

5

[1] P. Hiskett, D. Rosenberg, C. Peterson, R. Hughes, S.Nam, A. Lita, A. Miller, and J. Nordholt, New J. Phys.8, 193 (2006).

[2] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409,46 (2012).

[3] J. Zwinkels, E. Ikonen, N. Fox, G. Ulm, and M. Rastello,Metrologia 47, R15 (2010).

[4] G. Finger, I. Baker, D. Alvarez, D. Ives, L. Mehrgan,M. Meyer, J. Stegmeier and H. J. Weller, Proc. of SPIE9148, 914817 (2014).

[5] G. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A.Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C.Williams and R. Sobolewski, Appl. Phys. Lett. 79, 705(2001).

[6] A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni,F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O.Minaeva, G. Gol’tsman, K. Lagoudakis, M. Benkhaoul,F. Levy, and A. Fiore, Nature Photonics 2, 302 (2008).

[7] E. Dauler, A. Kerman, B. Robinson, J. Yang, B. Voronov,G. Goltsman, S. Hamilton, and K. Berggren, Journal ofModern Optics 56, 364 (2009).

[8] F. Mattioli, Z. Zhou, A. Gaggero, R. Gaudio, R. Leoni,and A. Fiore, Optics Express 24, 9067 (2016).

[9] K. D. Irwin, Appl. Phys. Lett. 66, 1998 (1995).[10] A. J. Miller, S. W. Nam, J. M. Martinis, and A. V.

Sergienko, Appl. Phys. Lett. 83, 791 (2003).[11] A. Lita, A. Miller and S. Nam, Optics Express 16, 3032

(2008).[12] B. Calkins, P. Mennea, A. Lita, B. Metcalf, W. Koltham-

mer, A. Linares, J. Spring, P. Humphreys, R. Mirin, J.Gates, et al., Optics Express 21, 22657 (2013).

[13] L. Lolli, E. Taralli, and M. Rajteri, J Low Temp Phys167, 803 (2012).

[14] G. Brida, L. Ciavarella, I. Degiovanni, M. Genovese, L.Lolli, M. Mingolla, F. Piacentini, M. Rajteri, E. Taralli,and M. Paris, New. J. Phys. 14, 085001 (2012).

[15] L. Lolli, E. Taralli, C. Portesi, E. Monticone, and M.Rajteri, Appl. Phys. Lett. 103, 041107 (2013).

[16] P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, andJ. Zmuidzinas, Nature 425, 817 (2003).

[17] J. Zmuidzinas, Annual Review of Condensed MatterPhysics 3, 169 (2012).

[18] Y. Wang, P. Zhou, L. Wei, H. Li, B. Zhang, M. Zhang,Q. Wei, Y. Fang, and Chunhai Cao, J. Appl. Phys. 114,153109 (2013).

[19] B. A. Mazin, B. Bumble, S. R. Meeker, K. OBrien,S. McHugh, and E. Langman, Opt. Express 20, 1503(2012).

[20] J. Gao, M. Vissers, M. Sandberg, F. Silva, S. Nam, D.Pappas, D. Wisbey, E. Langman, S. Meeker, B. MazinH. Leduc, J. Zmuidzinas, and K. Irwin, Appl. Phys. Lett.101, 142602 (2012).

[21] M.R. Vissers, J. Gao, M. Sandberg, S. M. Duff, D. S.Wisbey, K.D. Irwin, and D. P. Pappas. Appl. Phys. Lett.102, 232603 (2013).

[22] J. Hubmayr, J. Beall, D. Becker, H.-M. Cho, M. Devlin,B. Dober, C. Groppi, G. C. Hilton, K. D. Irwin, D. Li,P. Mauskopf, D. P. Pappas, J. Van Lanen, M. Vissers,Y. Wang, L. F. Wei, and J. Gao, Appl. Phys. Lett. 106,073505 (2015).

[23] J. Gao, M. Daal, J. Martinis, A. Vayonakis, J. Zmuidzi-

nas, B. Sadoulet, B. Mazin, P. Day, and H. Leduc, Appl.Phys. Lett. 92, 212504 (2008).

[24] J. Gao, M. R. Vissers, M. Sandberg, D. Li, H. M. Cho,C. Bockstiegel, B. A. Mazin, H. G. Leduc, S. Chaudhuri,D. P. Pappas,and K. D. Irwin, J Low Temp Phys, 176,136 (2014).

[25] J. Gao, Ph.D. thesis, Caltech, 2008.[26] D. D. E. Martin, P. Verhoeve, A. Peacock, A. G. Ko-

zorezov, J. K. Wigmore, H. Rogalla, and R. Venn, Appl.Phys. Lett. 88, 123510 (2006).

[27] U. Fano, Phys. Rev. 72, 26(1947).[28] D. Marsden, B. A. Mazin, B. Bumble, Seth Meeker, K.

O’Brien, S. McHugh, M. Strader, and E. Langman, Proc.SPIE 8453, 84530B (2012).

[29] B. A. Mazin, S. R. Meeker, M. J. Strader, P. Szypryt, D.Marsden, J. C. van Eyken, G. E. Duggan, A. B. Walter,G. Ulbricht, M. Johnson, B. Bumble, K. O’Brien, and C.Stoughton, Publications of the Astronomical Society ofthe Pacific, 125, 1348 (2013).

[30] A. Fyhrie, C. McKenney, J. Glenn, H. G. LeDuc, J. Gao,P. Day, and Jonas Zmuidzinas, Proc. SPIE 9914, 99142B(2012).

[31] G. Ulbricht, B. A. Mazin, P. Szypryt, A. B. Walter,C. Bockstiegel, and B. Bumble, Appl. Phys. Lett. 106,251103 (2015).

[32] E. Valkonen, C. Ribbing, and J. Sundgren, Applied Op-tics 25, 3624 (1986).

[33] A. Lita, B. Calkins, L. Pellouchoud, A. Miller, and S.Nam, Proc. of SPIE 7681, 76810D-1 (2010).

[34] A. Miller, A. Lita, B. Calkins, I. Vayshenker, S. Gruber,and S. Nam, Optics Express 19, 9102 (2011).

[35] E. A. Dauler, M. E. Grein, A. J. Kerman, F. Marsili, S.Miki, S. W. Nam, M. D. Shaw, H. Terai, V. B. Verma,and T. Yamashita, Opt. Eng. 53, 081907 (2014).