1. INTRODUCTION

Wavefront sensing is fundamental in a number of applications, such as quality testing of laser beams and optical components, atmospheric turbulence correction for telescopes with adaptive optics, and aberration measurements of eyes. A common wavefront sensor used today is the Hartmann–Shack wavefront sensor (HSWS), which uses a lenslet array to focus an incoming wavefront into discrete spots on a camera sensor. The local tilt of the wavefront at each lenslet translates to a lateral shift in the position of the spot on the sensor. Thus, by measuring the shifts of all spots on the sensor, the wavefront profile on the lenslet array can be reconstructed [1,2].

The HSWS measures the wavefront at the lenslet array. However, often it is neither practical nor possible to place the HSWS in the position of the wavefront to be measured. In these cases, it is necessary to use an optical system to relay the wavefront of interest from its origin to the lenslet array. Such a relay system may also be used to magnify the wavefront to better match the size of the camera sensor. The relay system has to be phase-conserving so that both lateral and angular magnification are well-defined. Conventionally, a 4f system is used, where two positive lenses (back focal lengths ${f^\prime _1}$ and ${f^\prime _2}$, respectively) with coinciding focal points create an afocal system (i.e., a telescope). A wavefront of interest (the “object”) is placed in the front focal plane of the first lens of the telescope. This creates an image of the wavefront at the back focal plane of the second lens of the telescope [1]. Consequently, the length of the relay system from object to image is $2{f^\prime _1} + 2{f^\prime _2}$, hence the name “4f system.”

The choice of focal lengths for the telescope lenses depends on the required magnification. Additionally, in some applications, further constraints to the focal lengths are imposed by necessary distances between the system components. This is exemplified in the area of optometry and ophthalmology, when an ocular wavefront is measured with an open field of view wavefront sensor. In the open-field configuration, a large hot mirror is placed at an angle between the eye and the telescope, thereby allowing the telescope to be placed outside the field of view. This, however, results in a relatively long distance (tens of centimeters) between the eye and telescope. For a 4f system, this long distance, together with the magnification, determines the focal lengths of the telescope and results in a very long system length from eye to HSWS. An alternative to the 4f system is necessary if the system is to be made smaller while still maintaining the same distance to the eye: this would enable lighter and more portable instruments, which in turn would enable wider clinical use of the instruments.

A non-4f relay system solves this problem by decoupling the focal lengths of the relay telescope from the distance to the object or image plane [3]. With the non-4f relay telescope design, the object and image planes are displaced from the focal planes of the telescope lenses. This allows us to choose the focal lengths relatively independently from the distances to the relay telescope, which in turn allows for more varied designs in terms of size, components, and cost. However, it has not yet been investigated how a non-4f relay telescope performs compared to a 4f relay in a real setting with non-paraxial lenses. There is also a need for a new alignment approach for conjugating the object with the lenslet array since both are displaced from the focal planes in the non-4f setup. The aims of this paper are:

 

Fig. 1. Ray diagram of non-4f system. The focal points of the two telescope lenses are marked in green on the optical axis. The extents of the object and image are marked in black for the non-4f system and in gray for the corresponding 4f system. By following the red and blue rays, we see that the angular magnification is the same for both systems. The green ray, along with the red and blue, shows that the transverse magnification is also the same for both systems.

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2. METHODS

We will first demonstrate theoretically that a non-4f system has the same wavefront relaying properties as its corresponding 4f system. We will then describe a method for aligning non-4f HSWS systems. Finally, we will describe how the new alignment method was used to align an experimental setup, in which we compared the wavefront relaying properties of a 4f and non-4f system.

A. Theoretical Calculations

Let us first consider a conventional 4f system used for relaying a wavefront. A system like this has a telescope consisting of two lenses (with back focal lengths ${f^\prime _1}$ and ${f^\prime _2}$, respectively), where the first lens’s back focal plane coincides with the front focal plane of the second lens, thus creating an afocal system. The wavefront of interest is in the front focal plane of the first lens (Badal lens); thus, it is imaged at the back focal plane of the second lens, i.e., at the conjugate plane. This image is a scaled version of the wavefront at the object plane, with transverse magnification $m = - {f^\prime _2}/{f^\prime _1}$ and angular magnification $M = - {f^\prime _1}/{f^\prime _2}$ [1]. In other words, the system is phase-conserving; a ray with height $h$ and angle $\alpha$ at the object plane will have height $mh$ and angle $M\alpha$ at the image plane.

1. Paraxial Optics

We will now consider the corresponding non-4f system. It only differs from the 4f system in that the object and image planes are displaced from the focal planes by ${x_1}$ and ${x^\prime _2}$, respectively (see Fig. 1). We use the Newtonian lens formula, $xx^\prime = ff^\prime $, to calculate how ${x^\prime _2}$ depends on ${x_1}$:

The transverse magnification of the non-4f system is the magnification of the first lens times the magnification of the second lens:

2. Fourier Optics

To be certain that the wavefront (i.e., field) is imaged without additional phase factors, we use Fourier optics. Though this has been shown before by Freimann et al. [3], the calculations here are made simpler through the use of the operator notation described by Goodman [4]. The operators used here are:

We will use the following relations between the operators:

We begin by calculating the operator sequence for a 4f system. From Goodman [4], we know that the operator sequence for a field transmitted from the front to the back focal plane of a single lens is ${\cal V}[{\frac{1}{{\lambda f^\prime}}}]{\cal F}$. This lets us describe the 4f system by

Using the operator relations in Eqs. (3), (5), and (6), we can simplify Eq. (8):

Thus, for a 4f system, the field at the back focal plane of the second lens is simply a scaled version of the field at the front focal plane of the first lens, which confirms what we showed paraxially.

For a non-4f system, we split the propagation into three parts: free-space propagation from the object plane to the front focal plane of the first lens (${\cal R}[- {x_1}]$); propagation from the front focal plane of the first lens to the back focal plane of the second lens, as in a 4f system (${\cal V}[M]$); and free-space propagation from the back focal plane of the second lens to the image plane (${\cal R}[{x^\prime _2}]$). Thus, the whole operator sequence is ${\cal R}[{x^\prime _2}]{\cal V}[M]{\cal R}[- {x_1}]$. This expression can be simplified with Eqs. (1), (4), and (7):

As can be seen, a non-4f system has the very same propagation as its corresponding 4f system, i.e., a simple scaling of the field both angularly and transversely.

B. Vergence-Independent Diameter Method for Finding the Conjugate Plane

The non-4f design allows us to more freely choose the focal lengths and distances used for the relay telescope, thus enabling more varied sizes and constructions. However, since the object and image are no longer in the focal planes of the telescope lenses, the alignment of such a wavefront relaying system is not straightforward. Even for 4f systems, it is not trivial to align a HSWS with the telescope so that the wavefront is imaged on the lenslet array of the HSWS, rather than on the HSWS’s sensor. Here we propose a method (called the vergence-independent diameter method) for conjugating the object and HSWS, which works for both 4f and non-4f systems.

Our method utilizes the fact that the image diameter is only dependent on the size of the object, and not on the vergence of the light. This means that, when the lenslet array is in the image plane, the reconstructed image diameter (calculated from the number of illuminated lenslets on the HSWS’s lenslet array) will not change if we change the vergence of the object. Thus, the image plane can be found by fixing the object size and then altering the vergence of the object and adjusting the HSWS’s position (or the object’s position) until the reconstructed diameter does not change when the vergence is altered. We will here assume that the reconstruction has a resolution limit of one lenslet diameter, though this may vary between systems. In this case, the precision of the vergence-independent diameter method is limited by the induced vergence $L$ of the object and by the pitch $2{r_{{\rm lenslet}}}$ of the lenslets in the HSWS. At worst, the error (i.e., the distance between the image plane and the lenslet array) will be ${e_{{\rm max}}} = {r_{{\rm lenslet}}}/|M{r_o}L|$, where $M$ is the angular magnification of the relay telescope and ${r_o}$ the radius of the object. For details, see Supplement 1. If we induce both positive and negative vergences, the maximum error is reduced to

Using Eq. (1), we see that this corresponds to a maximum error of ${M^2}{e_{{\rm max}}}$ for the displacement of the object.

C. Experimental Validation

The theoretical calculations demonstrated that 4f and non-4f systems have the same wavefront relaying properties. However, the calculations assume paraxial conditions, but in a real setup we might also have aberrations from the telescope lenses that could affect the imaged wavefront differently depending on the relay design. For instance, aberrations such as lens-induced coma and spherical aberration depend on the distances to the object and image planes, which change when we go from a 4f to a non-4f setup. Therefore, we built an optical setup to compare the wavefront imaging properties of a real 4f system with those of a corresponding non-4f system. That is, we wanted to see whether any changes induced by the relay design were discernible from natural fluctuations in the wavefront in a typical experimental setup. The optical setup was chosen to match a conventional relay telescope for visual optics applications in terms of magnification, lens types, and focal lengths. The setup is shown in Fig. 2, and its key aspects are listed below.

 

Fig. 2. Top: 4f setup. Bottom: non-4f setup. Components from left to right: collimation lens; progressive lens; compensating negative ophthalmic lens; holder for trial lens for the vergence-independent diameter alignment method; aperture; lens 1 of telescope; lens 2 of telescope; HSWS.

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To create the 4f system, the aperture of the test object unit had to be placed in the front focal plane of the first telescope lens. This position was found before aligning the telescope by looking through the first lens at the test aperture with a handheld telescope subjectively focused on infinity. The distance between the test object unit and first lens was then measured with a caliper. This ensured that the test object unit could be removed and replaced in the 4f position by simply measuring the distance with a caliper. The second lens of the telescope was then added and aligned. The position of the HSWS was found with the vergence-independent diameter method, using two ophthalmic trial lenses (${-}20\,{\rm D}$ and ${+}8\,{\rm D}$) and an aperture radius of approximately 2.5 mm. Using Eq. (11), the maximum error for the HSWS position was 0.41 mm. Finally, the progressive lens and compensating negative trial lens were added to the test object unit.

To create the non-4f setup, the test object unit was simply moved 50 mm further from the telescope, and the position of the HSWS was adjusted with the vergence-independent diameter method. This ensured that the telescope was the same in both systems and that the only things that changed were the positions of the test object unit and HSWS relative to the telescope.

1. Measurements

The test object unit was moved to the 4f position, and the position of the HSWS was found with the vergence-independent diameter method. The Zernike coefficients and diameter were then measured for 5 s. The test object unit was then moved to the non-4f position, and the position of the HSWS was found with the vergence-independent diameter method. The Zernike coefficients and diameter were again measured for 5 s. The measurements were repeated 10 times for each system type, with realignment of the test object unit and HSWS between each measurement. This was to account for the small changes in the wavefront caused by realigning and refastening of the test object unit to the optical rail. By repeating the measurements and realigning between each measurement, these variations were present for both system types and could, therefore, not be misinterpreted as a difference between them.

2. Post-Processing and Statistical Analysis

The Zernike coefficients were calculated for an aperture radius of 2.4 mm, and the mean coefficients for each measurement were calculated. To test if these 10 sets of mean coefficients differed significantly between the systems, a two-sample $t$-test was performed for each Zernike coefficient. To correct for the multiple comparisons, Bonferroni correction was used to adjust the significance level (${\alpha _{{\rm adjusted}}} = 0.05/30$). The mean and 95% confidence interval for each Zernike coefficient and system type were also calculated.

3. RESULTS

The theoretical calculations showed that non-4f and 4f systems have the same wavefront relaying properties. This was demonstrated first with paraxial calculations, then with Fourier optics using the operation notation described by Goodman [4]. The results confirm those by Freimann et al. [3].

In the experimental part, non-paraxial wavefront relaying properties of a typical setup for visual optics were analyzed. Figure 3 shows the 95% confidence intervals for the measured Zernike coefficients of the highly aberrated wavefront, as relayed by the two relay systems. The variation seen for each coefficient is caused by two factors: differences due to realignment between measurements and random fluctuations within each measurement. As can be seen in Fig. 3, the Zernike coefficients of the two systems match each other, with overlapping confidence intervals for all coefficients. Moreover, the statistical analysis did not show any significant differences between the Zernike coefficients of the two systems. This demonstrates that, even with a highly aberrated wavefront and non-paraxial lenses, this non-4f relay telescope had equal performance to that of the 4f relay telescope. In other words, any difference between the relay designs was not discernible within the natural variation of the wavefront seen in the real-life experimental setup.

 

Fig. 3. Measured Zernike coefficients (mean and 95% confidence intervals, aperture radius 2.4 mm) for the test object unit when relayed by a 4f system (shown in blue) and a non-4f system (shown in orange). Note that the scale varies between the different graphs.

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The vergence-independent diameter method was successfully employed to align the systems in the experimental part. The maximum possible alignment error was calculated to be 0.41 mm for the position of the HSWS.

4. DISCUSSION

In this study, we show that a HSWS with a non-4f relay system has the same wavefront imaging properties as with a 4f system. The phase-conserving properties of non-4f systems have been theoretically described by [3] with the same conclusions as in this study. Although not stated explicitly, this has also been assumed in the analysis of off-axis mirror telescopes by [5]. In our study, we have taken the theory to the optical table, showing how to implement the non-4f design with a HSWS, and verifying that it yields the same measurements as with a 4f system. We hope that this lowers the barrier for scientists to try out the non-4f design in their systems.

Compared to a conventional 4f system, the main advantage of a non-4f system is that it gives us a larger flexibility when choosing the distances and focal lengths for the telescope. This in turn allows for more varied designs in terms of size, components, and cost. As previously mentioned, for open-field ocular wavefront sensors it is necessary to have a large distance between the eye and the telescope, which in turn results in a bulky instrument when using a 4f system since the focal lengths also have to be long. However, by using a non-4f system, it is possible to keep this long distance to the eye while at the same time having shorter focal lengths for the telescope. At most, the distance from eye to HSWS can be reduced by 50% (see Appendix B in [6]). In other applications, it might be desirable to have a short distance between the object and telescope, and/or to have longer or shorter focal lengths for the telescope, all of which can be more easily achieved with the non-4f system type.

It should be noted that in our experiment the lenses were chosen to match a typical instrument for visual optics, and they were the same for the 4f and non-4f setup. When redesigning a 4f system to a non-4f system, one might want to change the focal lengths and, thus, also the lenses. In that case, any aberrations from the telescope lenses present in the original 4f system might be different in the new design as both chromatic and monochromatic aberrations vary between lenses. The displacement from the focal planes might also be more or less than what we tested in our experimental setup, depending on the needs for each specific application.

Though the non-4f design has many advantages, it also lacks some aspects of the 4f design that might be desired in some applications. The first lens of a 4f relay system will be a Badal lens, which images the wavefront at infinity. This makes it possible to utilize a Badal system (a system of mirrors that can be used to alter the optical path distance between the lenses of the telescope in order to compensate for defocus) in the telescope. This is not possible for a non-4f system since the image of the wavefront is no longer at infinity in between the telescope lenses. For ocular wavefront sensors, the implementation of a pupil camera to find the correct position of the patient also needs more careful planning when using a non-4f system since that normally also utilizes the pupil being imaged at infinity between the telescope lenses.

When aligning a HSWS, the main difficulty is getting the image of the wavefront onto the lenslet array rather than onto the sensor, especially when using an encased HSWS where the lenslet array and sensor are not directly accessible. An encased HSWS restricts the angle of incoming and reflected light to/from the lenslet array, with the advantage that stray light is reduced. However, this also limits our alignment method choices. A tempting option is to find the conjugate plane by looking at the image of the object aperture on the sensor, or equivalently, to image a point source. The assumption would be that the sharpest image seen is when the image is on the lenslet array. However, this is unfortunately not true. When a point source is imaged properly onto the lenslet array, the rays will diverge after the lenslet array, resulting in a large spot on the sensor. A semi-sharp image will instead occur when the image plane roughly coincides with the sensor, when spots from multiple lenslets overlap on the sensor. As the image never gets fully sharp, but rather has a range where it is approximately semi-sharp, this method is not only inaccurate but also imprecise.

The vergence-independent diameter method ensures that the wavefront of interest is imaged on the lenslet array. However, there are some practical limits to this alignment method, apart from the theoretical limits described previously. First, the diameters of the telescope lenses limit the strength of the trial lenses we can use to induce vergence. Second, with too strong positive vergences, the spots from the lenslets start to overlap on the sensor. Therefore, the trial lenses for this alignment method need to be carefully considered so that too high powers are avoided while still ensuring a small enough error for the alignment.

5. CONCLUSION

Non-4f systems have the same wavefront relaying properties as 4f systems and allow for more varied instrument design choices. The vergence-independent diameter method can aid in the implementation of lighter and more space-efficient instruments with non-4f designs.