Optimization of adaptive optics (AO) wavefront con-trollers have historically concentrated on attemptingto minimize the residual root mean square (RMS)wavefront error contributions from atmospheric tur-bulence and sensor noise [1,2]. However, increases inreal-time computer processing speed and improveddetectors with faster integration times have resultedin AO wavefront controllers with sufficient distur-bance rejection bandwidths to significantly reducethe atmospheric component of the wavefront resi-duals, under most conditions, to levels where thedominant residual component is due to structural vi-brations or resonances of the telescope rather thanseeing. This is particularly true for many of the largeaperture telescopes currently in operation [3–5].Figure 1 shows the closed-loop power spectral den-sity of Zernike tilt, collected during aMultiple MirrorTelescope (MMT) Natural Guide Star (NGS) AO sys-tem run in November 2009. The seeing conditionswere from 1–1:2 arcsec at 0:5 μm, which is somewhatworse than the historical median seeing at the MMTof 0:77 arcsec. The dominant residual wavefront
error seen at 18:75Hz is due to a local structural vi-bration of the telescope secondary hub assembly[4,6]. Analysis of image data obtained from theMMT NGS-AO system has shown that diffraction-limited images (Fig. 2) can be obtained in H-band(1:65 μm) and above when vibrational tip/tilt canbe removed from the system by applying postproces-sing techniques [7,8].
A major exception to this assumption is when fast-moving high-altitude atmospheric layers are pres-ent, as with the jet stream, where wind velocitiesare high enough that system delays in measuring theatmospheric component remain an important contri-buting factor to the residual wavefront. Neverthe-less, even in these cases, structural vibration is stilla significant contributor to the total residual wave-front error, the effects of which must be accountedfor in the design and optimization process of theAO wavefront controller if optimal performance isto be obtained [9].
For the next generation of very large aperture tele-scopes currently being developed, assessing the ef-fects of structural vibration on closed-loop AOsystem performance should be an important part ofthe overall telescope and AO system design. It istherefore necessary to have a better understandingof these effects and how they impact the performance
of the closed-loop AO system. The theoretical ap-proach begins by defining the angular resolution Rfor a telescope with aperture D (meters) at an obser-vation wavelength of λ (micrometers):
R ðradÞ ¼ 1:22λ ðmÞD ðmÞ
¼ 1:22
�206265 ðarcsecÞ
rad
�� μm106 m
�
¼ R ðarcsecÞ ¼ 0:2516λ ðμmÞD ðmÞ ; ð1Þ
where the angular resolution is proportional to wave-length and inversely proportional to telescope diam-eter, thus the principle reason for building largeaperture telescopes with AO systems. However, tele-scope structural vibrations can cause significant de-gradation of image quality, resulting in observed spotfull width at half-maximum (FWHM) and angular re-
solutions far worse than the theoretical limit. Strehland peak intensity can also be significantly degradedby structural vibration as energy is dispersed over amuch larger area of the detector. Note that structuralvibration is completely different in character than at-mospheric jitter. Structural vibration is seen as alightly damped, sinusoidal oscillation, often at highfrequencies relative to the AO wavefront controllerbandwidth, while atmospheric jitter results in ran-dom image motion observed at much lower frequen-cies, resulting in a Gaussian broadening of the pointspread function (PSF). Structural vibration frequen-cies are often high enough to be outside the distur-bance rejection bandwidth of the AO system, sothere is often little attenuation or possibly evenamplification of the mode. This is in stark contrastto the atmospheric tip/tilt component of the residualwavefront, where power is typically at or below 5Hz(Fig. 1), which can be greatly attenuated by thewavefront controller.
We can determine the effect of structural vibrationon the closed-loop image motion through the use ofthe disturbance rejection transfer function [10,11].The disturbance rejection transfer function acts as ahigh-pass filter, attenuating low frequency responsebut passing through, or amplifying, high-frequencyresponse. This transfer function can be used to deter-mine the relationship between open-loop and closed-loop image motion at the detector modified by the AOwavefront controller. Figure 3 shows the theoreticaldisturbance rejection transfer function for the MMTNGS-AO integral wavefront controller under typicaloperating conditions. The wavefront controller sam-ple rate is 550Hz and the integral gain is set to anominal value of 0.3, which minimizes the residualwavefront under average seeing conditions. We seethat the system disturbance rejection bandwidthfrequency, defined as the 0dB crossover point, is16:5Hz. Thus, in the case of the local structural
Fig. 2. Logarithmically scaled H-band image of a star from theMMT-AO system showing four Airy rings when tip/tilt is removedthrough postprocessing.
Fig. 3. Theoretical disturbance rejection frequency response forthe MMT-AO integral wavefront controller. Resonances affectingthe MMT NGS-AO system are located at 18.75, 61, and 66Hz.The 0dB crossover is at approximately 16:5Hz, allowing thestructural vibration at 18:75Hz to be slightly amplified.
Fig. 1. Power spectral density of residual Zernike tilt obtainedduring an MMT NGS-AO run in November 2009. The log scale il-lustrates the dominant residual effect of the 18:75Hz structuralvibration compared with the atmospheric component. Resonancesare also seen at 61 and 66Hz.
vibrational mode in the secondary hub at 18:75Hz,the AO wavefront controller is actually amplifyingimage motion at the science camera by 1:5dB or ap-proximately 20%. Additional resonances are presentat 61 and 66Hz that are also amplified slightly by thewavefront controller. From Fig. 3, we see that, at5Hz, the disturbance rejection transfer function at-tenuates the signal −12dB, so the atmospheric com-ponent of tip/tilt at or below 5Hz is reduced by atleast a factor of 4.
The amplitude of image motion at the detector dueto local structural vibrations can be quite significant.Analysis of images and telemetry data from the de-formable mirror has shown typical levels of imagemotion at the detector of 0.02 to 0:04 arcsec ampli-tude from zero to peak, and vibration levels of0:05 arcsec are not uncommon. Data taken for verifi-cation of the analytical equations showed amplitudesas high as 0:09 arcsec zero to peak. However, this is arather extreme case where the telescope was pointeddirectly into a 25–30mph wind specifically to inducevibration.
With the f =15 secondary mounted on theMMT, thesecondary is the stop, giving an entrance pupildiameter of D ¼ 6:3m. Therefore, at H-band(1:65 μm), the theoretical angular resolution isR ¼ 0:065 arcsec. Obviously, image motion of 0.02 to0:04 arcsec zero to peak will significantly degrade im-age quality both in terms of Strehl and FWHM. Forfuture extremely large telescopes, this problem willbe exacerbated; as the effective telescope diameterincreases and the theoretical resolution limit be-comes smaller, image motion due to structural vibra-tion must be suppressed to even higher levels toobtain diffraction-limited images and maximum sys-tem performance. There has also been significantinterest in AO system observations at shorter wave-lengths, from the near-infrared down to visible wave-lengths at or near 0:7 μm [12], making assessmentand control of structural vibrations even more criti-cal for obtaining good AO system performance in thisregime. In addition to AO system applications, theanalytical techniques developed in this paper are ap-plicable to any imaging system subject to vibration,such as point-to-point free-space communications oroptical alignment.
2. Derivation of the Normalized Intensity Distributiondue to Structural Vibration
A. Derivation of Exact, Time-Varying Solution forTransient Responses and Short Exposure Times
To understand the effects of structural vibration onimage quality, we first derive the theoretical inten-sity distribution for a normalized Gaussian spot withstandard deviation σ, beam radius w ¼ 2σ, andFWHM ¼ 2
ffiffiffiffiffiffiffiffiffiffiffiffi2 ln 2
pσ, moving sinusoidally with fre-
quency ω and amplitude A zero to peak. The primaryassumption in this derivation is that the closed-loopAO system has removed the majority of the atmo-spheric contribution and that the dominant residual
effect is due only to the structural vibration. Theassumption that the spot is Gaussian rather thanan Airy pattern is reasonable if we set the standarddeviation to σA ¼ 0:44λ=D, which gives the Gaussianprofile the same FWHM as an Airy function. In ad-dition, a Gaussian function simplifies the math with-out significantly changing the conclusions. However,it should be noted that the same analysis can be em-ployed regardless of whether the PSF is Gaussian,composite of core and halo, or a diffraction-limitedAiry pattern.
The time-varying normalized intensity distribu-tion is defined as
INorm_Shortðx; y; tÞ≡R tfinal0 Iaðx; y; tÞdt
max j R tfinal0 Iuðx; yÞdtj
; ð2Þ
where Ia and Iu are the aberrated and unaberratedintensity distribution functions respectively, andtfinal is the total integration time. The generalizedtwo-dimensional intensity distribution for a Gaus-sian spot moving sinusoidally due to multiple struc-tural modes is given by
Iaðx; y; tÞ ¼P
2πσxσyexp
�−
�½x − μxðtÞ�22σ2x
þ ½y − μyðtÞ�22σ2y
��;
ð3Þ
where P is power in watts, Ia is given in W=m2, and
μxðtÞ ¼Xnk¼1
Axk sinðωxktþ ϕxkÞ;
μyðtÞ ¼Xnk¼1
Ayk sinðωyktþ ϕykÞ: ð4Þ
Here, Axk , Ayk are the x- and y-axis motion ampli-tudes of the kth structural mode; ωxk , ωyk are thex- and y-axis components of the vibrational frequen-cies; ϕxk , ϕyk are initial phase angles; and σx, σy arethe one sigma Gaussian PSF standard deviations.The unaberrated intensity distribution function(W=m2) is given by
Iuðx; yÞ ¼P
2πσxσyexp
�−
�x2
2σ2xþ y2
2σ2y
��: ð5Þ
Thus, for the general time-varying normalizedintensity distribution, we have
when the image motion is assumed to be dominatedby a single vibrational mode aligned with the x axis;thus μyðtÞ ¼ 0. The time-varying normalized inten-sity distribution can then be determined by numeri-cally integrating Eq. (7). As an example, image datataken from the ARIES imager and spectrograph [13]is compared with data derived from the theoreticalmodel. The ARIES image seen in Fig. 4(a) is a 2 s ex-posure in K-band (2:2 μm). Seeing was 0.7–0.8 in K-band and the telescope was pointed directly into a25–30mph wind. Using slope data from the deform-able mirror telemetry data, the amplitude of thevibration was determined to be approximately0:09 arcsec zero to peak at a frequency of 18:75Hz.Numerically evaluating the time-varying solutionfor an amplitude A ¼ 0:09, ω ¼ 18:75ð2πÞ rad=s, andtfinal ¼ 2 s, gives the two-dimensional intensity distri-bution shown in Fig. 4(b).
The “dumbbell” shaped spot pattern seen in theARIES image clearly shows the dominance of thestructural vibration over the atmospheric compo-nent. The residual atmospheric effect can be seenin the ARIES image as an elongated halo surround-ing the two cores, which is not modeled under the as-sumption of a Gaussian PSF. However, the estimatesfrom the theoretical model of peak intensity, Strehl,and spot FWHM in the axis of vibration are still quitereasonable compared with the actual ARIES image.Normalized peak intensity for the theoretical modelwas 0.348 compared with 0.333 for the ARIES image,a difference of less than 5%. The Strehl and FWHMfor the theoretical model and ARIES image werevirtually identical at 0.27 and 0.265, respectively.FWHM for the theoretical model was 0:304 arcsecand was 0:31 arcsec for the ARIES image.
The separation between peaks in the PSF is deter-mined solely by the vibrational amplitude A, which is0:09 arcsec in this case, resulting in peak-to-peak se-paration for the theoretical model and ARIES image
of 0.180 and 0:178arcsec, respectively. The ratio ofpeak intensity to Strehl is also an important metricfor model accuracy, which is 1.28 and 1.23 for the the-oretical model and the ARIES image, respectively.This rather severe case, where the telescope waspointed directly into a 25–30mph wind, providesan excellent case to verify the analytical modelperformance.
B. Derivation of a Time-Invariant Analytical Approximationto the Intensity Distribution for Long Exposures
Interestingly, the long-term normalized intensitydistribution is not a function of the vibration fre-quency ω, and for long exposure times, as tfinalbecomes large, integration of Eq. (7) results in a nor-malized PSF that converges to a steady-state solu-tion. This leads to an interesting approximationfor the normalized intensity distribution where theunaberrated intensity distribution Iu is convolvedwith the average spot distribution over time, which,for a sinusoidally moving object, is given by thearcsine distribution.
Letting tfinal → ∞ and using the fact that taking theinverse Fourier transform of the Fourier transform ofa function returns the original function, we get
INorm_Longðx; y; tÞ
¼ 1tfinal
F−1x;y
�Fξ;η
�lim
tfinal→∞
Ztfinal
0Iaðx; y; tÞdt
��: ð8Þ
Employing the Fourier transform shift theoremgives
INorm_Longðx; y; tÞ ¼1
tfinalF−1
x;y
�Fξ;η½Iuðx; yÞ�
× limtfinal→∞
Ztfinal
0expf−iξA sinωtgdt
�:
ð9ÞAfter some additional manipulation and noting thatthe sine term is periodic overΔT ≡ 2π
ω, we can approx-imate Eq. (9) as
INorm_Longðx; yÞ ≅ F−1x;y½Fξ;η½Iuðx; yÞ��
⊗ F−1x ½J0ðAξÞ�; δðyÞ; ð10Þ
where the term J0ðAξÞ is the zero-order Bessel func-tion of the first kind. The inverse Fourier transformof J0ðAξÞ is given by the generalized arcsine distribu-tion [14,15] as
dμA ¼(dμA ¼ 1
jAjπ�1 − x2
A2
−12χAðxÞdx
dμA ¼ δ0ðxÞ
); ð11Þ
χAðxÞ ¼�1 if x ∈ A0 if x∉A
�; ð12Þ
Fig. 4. (a) 2 s exposure from the ARIES imaging camera showingthe effects of the 18:75Hz mode on the PSF. Data were obtained atthe MMT in November 2009. (b) Theoretical PSF of a 2 s exposureobtained by numerically integrating the time-varying solution andassuming the atmospheric component is completely removed.
where χAðxÞ is the indicator function and δ0ðxÞ is theDirac measure concentrated on the point 0. WhenA ¼ 0, we have the unaberrated intensity distribu-tion convolved with a unit amplitude delta function,which simply returns the original function. For thetime-invariant approximation, we therefore have
INorm_Longðx; yÞ ¼ Iuðx; yÞ ⊗ dμA; δ0ðyÞ: ð13Þ
As an example, for a Gaussian PSF with a vibra-tional mode aligned with the x axis, we have
INorm_Longðx; yÞ ¼1
2πσxσyexp
�−
�x2
2σ2xþ y2
2σ2y
��
⊗1
jAjπ�1 −
x2
A2
�−12χAðxÞ; δ0ðyÞ: ð14Þ
This equation has a simple intuitive interpreta-tion. We can think of it as the unaberrated intensitydistribution weighted by the amount of time, onaverage, spent at each location x as the spot movesacross the detector. Figure 5 shows a comparisonof the exact time-varying solution with parametersof A ¼ 0:08 arcsec, ω ¼ 117:8 rad=s (18:75Hz), andtfinal ¼ 2 s (solid curve) with the analytical time-invariant expression (dashed curve) in Eq. (14).Although there is still some oscillation present inthe time-varying solution, the values of FWHM andStrehl are very close to those given by the analyticalapproximation. As tfinal approaches infinity, the exactsolution will converge to the time-invariant solution.
The analytical equation for long exposures allowsfor rapid evaluation of the intensity distributionscompared with the somewhat more lengthy processof numerically integrating the exact solution, thus al-lowing for rapid computation of Strehl, FWHM, andother parameters as functions of telescope diameter,observation wavelength, and vibration amplitude.The above expressions can be useful when determin-ing specifications on the allowable image motion atthe detector, or when determining the required dis-
turbance rejection bandwidth necessary to obtainsufficient attenuation of the image motion.
3. Image Quality Predictions Using the Time-InvariantAnalytical Approximation
Equation (14) can readily be used to assess the effectsof vibration amplitude and observing wavelength onimage quality for the next generation of large optical/infrared telescopes. For example, applying Eq. (14) tothe Giant Magellan Telescope with an effective dia-meter of D ¼ 25m, at a wavelength of 1:65 μm (H-band), and assuming the atmospheric componentis completely removed, we see that a vibrational am-plitude of 0:02 arcsec zero to peak results in a Strehlratio of approximately 0.24 and a vibrational ampli-tude of 0:040 arcsec will give a Strehl of only 0.11[Fig. 6(a)]. For the same telescope diameter and ob-servation wavelength, a vibration amplitude of only0:01 arcsec zero to peak is required to double theFWHM in the direction of the vibration [Fig. 6(b)].Any additional uncorrected atmospheric componentwill, of course, degrade the image quality further.
The effects will be significantly worse when tryingto observe at or near visible wavelengths. Again, for atelescope diameter of D ¼ 25m and an observationwavelength of 0:7 μm, we see that, for vibration am-plitudes of 0.01 and 0:02 arcsec, the resulting Strehlratios are 0.19 and 0.09, respectively [Fig. 6(c)].Figure 6(d) shows that a vibration amplitude of only0:005 arcsec is required to double the FWHM alongthe axis of vibration. Qualitatively, as the spot sizebecomes smaller, there are two effects that reducethe Strehl ratio. First, energy is distributed overthe length of vibration amplitude 2A, which reducesthe peak intensity, and second, the sinusoidal motionof the PSF means that the spot will spend more timeat the inflection points, x ¼ �A, where the velocity ofthe spot is at or near zero, rather than at the center ofthe detector. In addition, as the spot size decreasesrelative to the amplitude of the vibration, the FWHMapproaches 2A and it becomes completely dominatedby the vibration amplitude. Again, these figuresshow the best possible image quality given the speci-fied level of closed-loop vibration.
At J-band and below, where the atmosphericcomponent of the residual wavefront becomes in-creasingly important, we can use the analytical ap-proximation as an additional structural vibrationcorrection factor. In this case, an estimate of the coreand halo height can be computed given the atmo-spheric conditions and observation wavelength.Equation (13) can then be used on the turbulence-degraded PSF to estimate the final PSF given thelevel of vibration specified.
As an example, the core and halo heights and dia-meters were computed using methods given in [11].The model of the turbulence-degraded PSF was thenconvolved with the generalized arcsine distribution,as in Eq. (13), resulting in a structural vibration cor-rection factor to the intensity distribution. Theresulting PSF is compared with the actual ARIES
Fig. 5. Comparison of exact time-varying solution for K-band(2:2 μm), t ¼ 2 s, frequency of 18:75Hz, and the time-invariantanalytical approximation for an amplitude of 0:08arcsec zero topeak, D ¼ 6:5m.
image intensity profile (Fig. 7). As expected, the haloin the ARIES image is matched significantly betterby the theoretical PSF when modeled as a compositeof core and halo. In addition, the peak intensity in thetheoretical model is reduced, as energy is dispersedfrom the core to the halo, which results in a bettermatch in peak intensity of the composite model overthe Gaussian approximation.
4. Conclusions
The significance of understanding and accounting forthe effects of structural vibration were discussedwith an emphasis on how the closed-loop vibrationamplitude affects the image Strehl and FWHM. Ageneral equation was then derived to compute thetime-varying normalized intensity distribution fora Gaussian spot moving sinusoidally due to structur-
al vibration. The integral has no analytical solutionbut can be solved numerically for transient responsesor for short exposure times to determine Strehl or im-age FWHM. An example using the time-varying ap-proximation was given that modeled a K-band PSFfrom the MMT and compared this with an actualPSF obtained from the ARIES imager and spectro-graph. The resulting theoretical PSF matched theARIES image well in terms of peak intensity, Strehl,and FWHM. However, the assumption of a GaussianPSF does not account for the elongated halo seen inthe actual image.
An analytical approximation for the time-invariant normalized intensity distribution was thenderived for long exposure times and compared withthe exact time-varying solution. The analytical ex-pression was used to determine the effects of vibra-tion on Strehl and spot FWHM for different telescopediameters and observation wavelengths. The resultsshow that both Strehl and spot FWHM are signifi-cantly affected by structural vibration as the tele-scope diameter is increased or the observationwavelength is decreased. Using the time-invariantapproximation, an example was shown of how amod-el of a partially corrected image can be constructedby estimation of the core and halo, which is then ap-plied to Eq. (13) to give a structural vibration correc-tion factor that more accurately predicts the actualPSF peak intensity and halo.
The equations derived for both the time-varyingand time-invariant intensity profiles can be quiteuseful in determining the effects of structural vibra-tion on AO system performance as well as predictionof the final image PSF. The time-invariantexpressions allow for rapid evaluation of specifica-tions on allowable image motion at the detector, orwhen determining the required disturbance rejection
Fig. 7. Comparison of ARIES image intensity profile (solid curve)with the theoretical prediction (dashed curve) using a composite ofestimated core and halo. Vibration amplitude was 0:09arcsec onaverage over the 2 s exposure.
Fig. 6. Strehl ratio and FWHM as a function of closed-loop vibration and three representative telescope diameters, D ¼ 6:5m (dashedcurves), D ¼ 10m (dashed–dotted curves), D ¼ 25m (solid curves): (a) H-band Strehl ratio, (b) H-band spot FWHM measured along thevibration axis, (c) Strehl at an observational wavelength of 0:7 μm, and (d) FWHM at an observational wavelength of 0:7 μm.
bandwidth necessary to obtain the desired imagequality.
These results illustrate the importance of account-ing for structural vibration in the design and analy-sis of future AO systems. It also illustrates the needto develop techniques that will significantly attenu-ate the effect on image motion due to structural vi-bration, particularly as observation wavelengthsmove toward the visible and for the next generationextremely large aperture telescopes currently underdevelopment.
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