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Membrane deformable mirror for adaptive optics: performance limits in visual optics

Enrique J. Fernández and Pablo Artal

Open Access Open Access

Abstract

The performance of a membrane deformable mirror with 37 electrodes (OKO Technologies) is studied in order to characterize its utility as an adaptive optics element. The control procedure is based on knowledge of the membrane’s response under the action of each isolate electrode, i.e., the influence functions. The analysis of the mathematical techniques to obtain the control matrix gives useful information about the surfaces that are within the device’s range of production, thus predicting the best performance of the mirror. We used a straightforward iterative algorithm to control the deformable membrane that permits the induction of surfaces in approximately four iterations, with an acceptable level of stability. The mirror and the control procedure are tested by means of generating Zernike polynomials and other surfaces. The mirror was incorporated in an adaptive optics prototype to compensate the eye’s aberration in real time and in a closed loop. Double-pass retinal images with and without aberration correction were directly recorded in a real eye in order to evaluate the actual performance of the adaptive optics prototype.

©2003 Optical Society of America

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Supplementary Material (5)

Media 1: AVI (1175 KB)     
Media 2: AVI (1186 KB)     
Media 3: AVI (1027 KB)     
Media 4: AVI (1602 KB)     
Media 5: AVI (1450 KB)     

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Figures (10)
Fig. 1.
Fig. 1. (a) Experimental apparatus to test the deformable mirror. A reference beam is introduced in the system that measures the induced aberrations by means of a Hartmann-Shack wave-front sensor. (b) AO prototype for the human eye. Some modifications of the system permit the measurement and compensation of the ocular aberrations. Simultaneous recording of retinal images or visual task can be performed during the retrieval and correction of the wave.

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Fig. 2.
Fig. 2. Influence functions produced by the action of each isolated electrode over the membrane. The electrode structure underneath the deformable mirror and the 9.2 mm of diameter pupil on the membrane are presented. The deformation of the membrane is shown with a color-coded representation together with the corresponding electrode in dark blue.

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Fig. 3.
Fig. 3. Representative spatial modes of the deformable mirror. Any linear combination of them can theoretically be induced on the membrane.

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Fig. 4.
Fig. 4. Range of production of the Zernike coefficients for the 21st polynomials in the 9.2-mmdiameter pupil on the deformable membrane.

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Fig. 5.
Fig. 5. Examples of the membrane’s capability to reproduce some arbitrary surfaces. The required surface, at the top left side, is described by Zernike polynomials as Z 4=0.3 µm, Z 7=0.4 µm, and the rest of the coefficients equal to zero. The corresponding residual, at top center, is 0.16 µm. The voltage distributions on the electrodes underneath to produce the desired shape on the mirror are also shown. At the bottom the required surface is described by Z 5=0.2 µm, Z 8=0.3 µm, and the other coefficients equal to zero. The obtained residual was 0.8 µm in this case.

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Fig. 6.
Fig. 6. Iterative generation of some Zernike polynomials as a function of time performed by the deformable mirror. The membrane’s pupil is 9.2 mm in diameter. Points in the curves are 0.04 ms apart. The dotted horizontal curves are the programmed level of aberration in micrometers.

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Fig. 7.
Fig. 7. (a) Surfaces obtained by the deformable mirror. Some individual Zernike polynomials are replicated with programmed values of Z 8=0.4 µm, Z 9=-0.6 µm, and Z 15=-0.4 µm. The pupil was 9.2 mm in diameter for the three cases. The final measured values were Z 8=0.32 µm, Z 9=-0.56 µm, and Z 15=-0.34 µm, keeping the other coefficients less than 0.1 µm respectively. (1.14, 1.15, and 1 MB, respectively) Videos of the wave fronts during the generations are also shown in the next column[[?]]. (b) Final surfaces obtained in the preceding generations in terms of Zernike’s polynomials in micrometers.

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Fig. 8.
Fig. 8. Closed-loop aberration correction in the living eye by use of the deformable mirror. The evolution for the rms of the wave front through 6 s during the real-time compensation is shown. The red line corresponds to the best estimated compensation (0.098 µm). The eye’s pupil studied was 5.52 mm in diameter under natural viewing conditions. Both the initial and the predicted best wave aberration correction are also shown in a color-coded representation.

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Fig. 9.
Fig. 9. (1.56 MB, 1.46 MB) Video of the measured wave aberration along the compensation together with the evolution of its associated PSF.

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Fig. 10.
Fig. 10. Retinal images with and without AO. The eye pupil studied was 5.52 mm in diameter under natural viewing conditions. At the top, estimated PSF from the Hartmann-Shack measurements are presented. The retinal images recorded with the auxiliary camera are shown below. To compensate the speckle effects, six frames were added to obtain these figures. At the bottom, the 3D representation of the recorded images in arbitrary units (A.U.) is shown. The increment in intensity at peak maximum is 2.86 times during AO aberration correction.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

IFM * V → = S → ,
S → = [ a 1 a 2 … … a m ] ; S = ∑ j = 1 m a j Z j .
CM * S → = V → .
r = ∣ IFM · v ′ → − s → ∣ .
r = ∣ ∑ i a ′ i Z i − ∑ j a j Z j ∣ = ∣ ∑ j ( a ′ j − a j ) · Z j ∣ = ∑ j ( a ′ j − a j ) 2 = rms ,
∇ 2 S ( x , y ) = − P ( x , y ) T = − ε 0 [ V ( x , y ) ] 2 T d 2 ,
V → ( t k + 1 ) = β · CM · ( S → req . − S → measur . ( t k ) ) + V → ( t k ) ,
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