**3.2. Optimization of low spatial frequencies**

The sharpness of an optical imaging system depends strongly on the wavefront quality. Re‐ cently [28] demonstrated that the low spatial frequency in an image can be used as a metric to perform the optimization. The process takes place by the acquisition of a series of images with the application of a predetermined aberration. The images, then, contain the informa‐ tion about the corrections which have to be applied to cancel the aberrations. This technique is very powerful, especially if coupled to the Lukosz modes aberration expansion. The Lu‐ kosz modes are similar to the Zernike polynomials: the difference is that the Zernike polyno‐ mials are normalized such that a coefficient of value 1 generates a wavefront with a variance of 1 rad2 , while the Lukosz functions are normalized such that a value 1 coefficient corre‐ sponds to a rms spot radius of λ/(2πNA), where λ is the wavelength and NA is the numeri‐ cal aperture of the focusing lens.

aging of the mirror membrane that has been used. This stability is inherent in the adopted approach to the problem, which is less ambitious than correcting the wavefront aberration. The described technique has been verified by means of the rather simple optical setup shown in Fig. 10. The radiation emitted by a LED diode source (SOU) is condensed by a microscope objective lens (Lcond) on a pinhole (PH). The radiation emerging from the pinhole is collected by a zoom collimating lens (Lcoll). The collimated beam passes through a diaphragm (DIA) and a beam splitter (BS) and impinges normally onto a deformable mirror (Mdef). After reflec‐ tions from Mdef and BS, the beam is compressed by an a-focal Newtonian system (Lcomp1

) and can, then, follow two different paths: either a) a focusing two-lens system Lfoc that makes the image of the pinhole on a CMOS digital camera (DET), or b) a flip mirror (Mflip) which deviates the beam on a wavefront analyzer (WFA). The latter has been used to measure the wavefront aberrations before and after the correction performed by the DM. With this sys‐

trolled amounts of aberrations on the nominal pinhole image. Then, by the suitable image analysis and consequent estimate of the aberrations, the parameters needed to drive the de‐

**Figure 10.** Schematic representation of the optical setup used for testing the capability of correcting the system aber‐ rations with a sensorless technique. SOU: source LED diode; Mdef: deformable mirror; DET: CMOS camera for detection;

Even if the apparatus performance was constrained by the limited unidirectional sag of the deformable mirror, the obtained results proof the principle of the adopted methodology. This is clearly demonstrated in Fig. 11, which shows the wavefront error measured with the WFA before and after the deformable mirror correction for three different cases. From these graphs, and more quantitatively from the detailed analysis described in [29], it can be seen

tem, both by varying the focal length of Lcoll and tilting Lcomp1

WFA: wavefront analyzer. See text for a complete description.

formable mirror to improve the image quality have been derived.

Lcomp2

and

55

, it was possible to introduce con‐

Devices and Techniques for Sensorless Adaptive Optics

http://dx.doi.org/10.5772/53550

The peculiarity of this expansion is that the effect of the Lukosz polynomials coefficients {*ai*}, on the image sharpness *I(ai)* is quadratic:

$$I\left(a\_{i}\right) = \sum\_{i} \left|a\_{i}\right|^{2}.$$

This implies that the optimization of each mode can be performed independently and re‐ quires just the acquisition of three images. Then, the best point for each aberration can be found by interpolating the result with a quadratic function.

### **3.3. Point Spread Function (PSF) optimization**

Another example of application of wavefront sensorless AO [29] consists in projecting a known point-like source through the optical system under test and then analyzing its image by means of a suitable software [36]. With the information obtained by the analysis of the point source image, the shape of a deformable mirror inserted along the optical path is modified. This process is iteratively repeated through a defined hierarchy, to gradually re‐ move the optical aberrations.

With this technique, the point source image to be analyzed is not directly available and has to be somehow created. As an example, in the case of a fundus camera dedicated to the ob‐ servation of the human retina, an illuminated pinhole can be projected on the retina itself through a dedicated optical path; this is a standard technique for this type of applications and is not going to introduce a significant complexity in the system. The light that is backdiffused by the retinal fundus acts as a point source, and its wavefront can be analyzed to estimate the aberrations present along the optical path from the retina to the detector.

#### *3.3.1. Application of the PSF optimization in a visual optics setup*

The closed loop method of correcting the aberrations of an optical system has been verified to be very stable, at least with respect to possible misalignments of the deformable mirror or aging of the mirror membrane that has been used. This stability is inherent in the adopted approach to the problem, which is less ambitious than correcting the wavefront aberration.

**3.2. Optimization of low spatial frequencies**

of 1 rad2

54 Adaptive Optics Progress

cal aperture of the focusing lens.

move the optical aberrations.

on the image sharpness *I(ai)* is quadratic:

found by interpolating the result with a quadratic function.

*3.3.1. Application of the PSF optimization in a visual optics setup*

**3.3. Point Spread Function (PSF) optimization**

The sharpness of an optical imaging system depends strongly on the wavefront quality. Re‐ cently [28] demonstrated that the low spatial frequency in an image can be used as a metric to perform the optimization. The process takes place by the acquisition of a series of images with the application of a predetermined aberration. The images, then, contain the informa‐ tion about the corrections which have to be applied to cancel the aberrations. This technique is very powerful, especially if coupled to the Lukosz modes aberration expansion. The Lu‐ kosz modes are similar to the Zernike polynomials: the difference is that the Zernike polyno‐ mials are normalized such that a coefficient of value 1 generates a wavefront with a variance

, while the Lukosz functions are normalized such that a value 1 coefficient corre‐

sponds to a rms spot radius of λ/(2πNA), where λ is the wavelength and NA is the numeri‐

The peculiarity of this expansion is that the effect of the Lukosz polynomials coefficients {*ai*},

This implies that the optimization of each mode can be performed independently and re‐ quires just the acquisition of three images. Then, the best point for each aberration can be

Another example of application of wavefront sensorless AO [29] consists in projecting a known point-like source through the optical system under test and then analyzing its image by means of a suitable software [36]. With the information obtained by the analysis of the point source image, the shape of a deformable mirror inserted along the optical path is modified. This process is iteratively repeated through a defined hierarchy, to gradually re‐

With this technique, the point source image to be analyzed is not directly available and has to be somehow created. As an example, in the case of a fundus camera dedicated to the ob‐ servation of the human retina, an illuminated pinhole can be projected on the retina itself through a dedicated optical path; this is a standard technique for this type of applications and is not going to introduce a significant complexity in the system. The light that is backdiffused by the retinal fundus acts as a point source, and its wavefront can be analyzed to

estimate the aberrations present along the optical path from the retina to the detector.

The closed loop method of correcting the aberrations of an optical system has been verified to be very stable, at least with respect to possible misalignments of the deformable mirror or

*I*(*ai* )≈∑ *i ai* 2 . The described technique has been verified by means of the rather simple optical setup shown in Fig. 10. The radiation emitted by a LED diode source (SOU) is condensed by a microscope objective lens (Lcond) on a pinhole (PH). The radiation emerging from the pinhole is collected by a zoom collimating lens (Lcoll). The collimated beam passes through a diaphragm (DIA) and a beam splitter (BS) and impinges normally onto a deformable mirror (Mdef). After reflec‐ tions from Mdef and BS, the beam is compressed by an a-focal Newtonian system (Lcomp1 and Lcomp2 ) and can, then, follow two different paths: either a) a focusing two-lens system Lfoc that makes the image of the pinhole on a CMOS digital camera (DET), or b) a flip mirror (Mflip) which deviates the beam on a wavefront analyzer (WFA). The latter has been used to measure the wavefront aberrations before and after the correction performed by the DM. With this sys‐ tem, both by varying the focal length of Lcoll and tilting Lcomp1 , it was possible to introduce con‐ trolled amounts of aberrations on the nominal pinhole image. Then, by the suitable image analysis and consequent estimate of the aberrations, the parameters needed to drive the de‐ formable mirror to improve the image quality have been derived.

**Figure 10.** Schematic representation of the optical setup used for testing the capability of correcting the system aber‐ rations with a sensorless technique. SOU: source LED diode; Mdef: deformable mirror; DET: CMOS camera for detection; WFA: wavefront analyzer. See text for a complete description.

Even if the apparatus performance was constrained by the limited unidirectional sag of the deformable mirror, the obtained results proof the principle of the adopted methodology. This is clearly demonstrated in Fig. 11, which shows the wavefront error measured with the WFA before and after the deformable mirror correction for three different cases. From these graphs, and more quantitatively from the detailed analysis described in [29], it can be seen that a RMS wavefront error as low as *λ*/10 (@527.5 nm) can be obtained, which is a signifi‐ cant result for a sensorless AO system. The correction was not particularly effective only in those cases in which the unidirectionality of the mirror deformation did not allow aberration compensation, as in case of astigmatism. However, with a different choice of AO system, the system is very effective in identifying and correcting aberrations.

It has to be mentioned that these tests verified that the image analysis algorithm takes less than 100 cycles to reach the optimal condition; since one cycles takes approximately 1/20 - 1/25 s on a standard computer, the whole optimization takes just 4-5 s. Comparing this time with the typical times necessary to optimize other sensorless AO systems, it is evident the significant advantage of this technique, once implemented. The only limitation of this tech‐ nique is that starting PSF image should give enough signal. In fact, if the point source image is too spread out, the signal to noise ratio can be very poor, substantially inhibiting the sys‐

Devices and Techniques for Sensorless Adaptive Optics

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57

Optical coherence tomography, OCT, is an imaging modality allowing acquisition of micro‐ meter-resolution three-dimensional images from the inside of optical scattering media (e.g. biological tissue). OCT is analogous to ultrasound imaging, except that it makes use of light instead of sound. It relies on detecting interferometric signal created by the light back scat‐ tered from the sample and from a reference arm in a Michelson or Mach-Zehnder interfer‐ ometer. OCT has many applications in biology and medicine and can be treated as a sort of

optical biopsy without requirement of tissue processing for microscopic examination.

*<sup>Δ</sup><sup>z</sup>* <sup>=</sup> 2ln2 *π*

The lateral resolution (∆x) in OCT is defined similarly to the confocal scanning laser oph‐ thalmoscopy (cSLO), since OCT is based on a confocal imaging scheme. In many imaging systems, however, the confocal aperture exceeds the size of the Airy disc, which degrades

*<sup>Δ</sup><sup>x</sup>* =1.22*<sup>λ</sup> <sup>f</sup>*

Therefore, as for standard microscopy, AO enhanced devices might be necessary to achieve diffraction limited transverse resolution. As a result, only a combination of OCT with AO has the potential to achieve high and isotropic volumetric resolution. The use of broadband light sources that are necessary for OCT and the complexity of both the AO and the OCT technique, make the combination very challenging [39]. In general, any AO-OCT instrument can be divided into two subsystems: an adaptive optics subsystem, with wavefront sensing and wavefront correction, and an interferometric OCT subsystem. In every implementation of AO-OCT all the elements of the AO subsystem are located in the sample arm of the OCT

*D* .

*λ*0 2 *Δλ* .

length (λ0) and the bandwidth (∆λ) of the light source as [37]:

the resolution to the value known from microscopy, i.e. [38]:

One of the interesting features of OCT is that, unlike in most optical imaging techniques, the axial and lateral resolutions are decoupled, thus allowing for an improved axial resolution, which is independent of transverse resolution. The axial resolution Δz is determined by the roundtrip coherence length of the light source and can be calculated from the central wave‐

tem to make a correct image analysis.

**3.4. Optical Coherence Tomography (OCT)**

**Figure 11.** Wavefront plots (obtained with the wavefront analysis, WFA) before and after the correction applied by the deformable mirror for three considered cases. Top: the main aberration is defocus; Middle: the main aberrations are astigmatism and coma; Bottom: the main aberrations are defocus and astigmatism. The blue dashed lines over plotted to the Z axis represent the total wavefront excursion.

It has to be mentioned that these tests verified that the image analysis algorithm takes less than 100 cycles to reach the optimal condition; since one cycles takes approximately 1/20 - 1/25 s on a standard computer, the whole optimization takes just 4-5 s. Comparing this time with the typical times necessary to optimize other sensorless AO systems, it is evident the significant advantage of this technique, once implemented. The only limitation of this tech‐ nique is that starting PSF image should give enough signal. In fact, if the point source image is too spread out, the signal to noise ratio can be very poor, substantially inhibiting the sys‐ tem to make a correct image analysis.
