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“Achromatic Limits” of Pancharatnam Phase lenses

COMRUN YOUSEFZADEH1, AFSOON JAMALI1, COLIN MCGINTY1 AND PHILIP J. BOS 1,*

1Liquid Crystal Institute, Kent State University, 1425 University Esplanade, Kent, OH.

*Corresponding author: pbos@kent.edu

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

Lenses based on Pancharatnam phase have the advantages of being thin and inexpensive. Unfortunately their

optical effect is strongly wavelength dependent, and in general their applications are limited by the requirement of

a monochromatic source. However low power lenses based on Pancharatnam phase can be considered for

applications over the visible range. In this paper, we provide intuitive “limits” for the lens power below which

these devices can be considered for use with the eye and visible light imaging applications. © 2017 Optical Society

of America

OCIS codes: (230.3720) Liquid-crystal devices; (130.5440) Polarization-selective devices; (350.1370) Berry's phase; (090.1970)

Diffractive optics; (080.3630) Lenses.

http://

1. INTRODUCTION

Pancharatnam phase refers to the phase shift acquired by an

electromagnetic wave that undergoes a continuous sequence of

polarization state transformations following a closed path in the space

of polarization states [1]. Unlike conventional phase or amplitude

gratings, Pancharatnam phase devices (PPDs) operate by locally

modifying the polarization state of light waves passing through them.

Their unique optical properties have been utilized recently for

wide-angle non-mechanical beam steering applications

[2,3,4,5,6,7,8,9,10,11]. A particularly interesting application for PPDs is

an optical lens made by modulating the Pancharatnam phase to

provide an appropriate profile across an aperture. [12,13,14,15]

A PPD lens is accomplished by fabricating a half-wave retarder that

has its optics axis in the plane of the film while the azimuthal angle

(

),is spatially varied along the radial direction forming concentric

rings of constant [16]. The phase gained by circularly polarized light

exiting a PPD is related to the angle

between the optic axis of the half-

wave retarder with respect to a fixed lab axis by

.A

Pancharatnam lens can be fabricated by meeting the condition

r

f, which defines a parabolic phase profile for a lens with

radius r (meter) and focal length f (meter) at a designed wavelength

(meter). Combing these relations, the focal length of a PDD lens is

given by

(1)

Where D is the power (diopter) of the lens and

(r) is the azimuthal

angle (radian) defined above, along a lens radius of length r. [17,18,19]

A lens fabricated in this way can changes its sign (e.g. from a

‘positive’ lens to a ‘negative’ lens) when the handedness of the input

circularly polarized light is changed. [16,20]

Lens made with this method have many positive characteristics.

They require a device thickness of only a few microns and this

thickness is independent of aperture size. Additionally, they employ

simple fabrication processes which can be easily extended to mass

production. [21]

However, Pancharatnam Phase devices are very chromatic. While

of exceptionally high efficiency when used with monochromtic light,

there are two issues with polychromatic light. One is that a PPD device

is designed to be a half-wave retarder for maximum efficiency at a

particular wavelength, and the other is that the focal length of a PPD

lens is wavelength dependent. (Eqn 1)

The first issue, with the device not being a half-wave retarder for

wavelengths other than that of the the design wavelength, increases

the intensity of light “leakage” that is not steered or focused by the

PPD device and that exits with the same handedness of circular

polarization as the input value. This problem has been addressed by

Escuti[16,22], who showed how to fix this problem by utilizing a dual

twist design. Also, the non-deflected light for lenses with f/# > f/2, the

percentage of leakage light is small and can be blocked with a circular

polarizer that has the opposite handedness as the input light. [23]

While viable solutions exist to solve or mitigate the chromaticity

effects from the issue described above , the second issue of a

wavelength-dependent focal length as given by Eqn 1 is more

problematic and is the subject of this paper.

If a PPD lens is designed to have a focal length fg for green light,

then the focal length when illuminated with blue or red light is given

by:

(2a)

(2b)

One way to conceptualize the effect of wavelength is to consider the

spot size on the focal plane for the design wavelength (showing by line

in Fig 1). For this simple example, the effect of diffraction is neglected.

Consider that the focal distance of a PPD lens for green light is designed

to be fg, and that we would like to know the spot size of light obtained

for different wavelengths on the plane that is at a distance fg from the

lens.

If a diameter (L) is chosen for the lens, then the spot size on the

plane at a distance fg from the lens can be calculated.

Fig. 1. Schematic representation of the wavelength dependence of the

focal length. a) green and red light focal lengths b) green and blue

light focal lengths.

Figure 1 shows that red and blue light will focus at different points

than the green light that the lens is designed for. It can be seen that the

diameter of the wave-front at the distance fg from the lens is Sb for blue

light and Sr for red light:

(3a)

(3b)

Using equation 2, equation 3 can be rewritten as:

(4a)

(4b)

These spot sizes are interestingly independent of the value of fg, but

very dependent on the diameter of the lens as well as the difference

between the design wavelength and the measurement wavelength

While Equations 2a-b as well as 4a-b demonstrate the basic

problem when considering use of a PPD lens with a polychromatic

source, it is expected that there are applications, possibly where the

power and aperture of the PPD lens is small, that the chromatic effect

may be acceptable.

The goal of this paper is to provide insight into what the limits of the

PDD lens power and aperture are for applications involving the human

eye, and for camera type applications.

2. MODELING

A. Modeling of the effect of polychromatic light on a PPD

lens for use with the eye

Related to human perception, the optical system was modeled as

shown in figure 2.

In the test system the retina is modeled as a flat plane 20mm behind

the eye. The eye is modeled as a 50 diopters lens with a parabolic

phase profile and a 3mm aperture that will provide a diffraction

limited spot on the retina for an object located at infinity.

Fig. 2. Schematic representation of the modeled system

The PPD device is assumed to meet the condition provided in eqn. 1

which yields a parabolic phase profile. The spatial variation of the

azimuthal angle of the optic axis ( is chosen to provide a power

(Dg) of 0, 0.25, 0.5, 1.5, 2 , 4 and 10 diopters for λg which corresponds

to green light. It is noted that the power will change for red and blue

wavelengths such that

for blue light and

for

red light.

In the case of green light, a point source on the image plane is

expected to yield a diffraction limited spot on the image plane. Red and

blue wavelengths are expected to yield a larger spot size since the focal

length is wavelength dependent.

The spot size calculation utilizes scalar diffraction theory [24,25,26]

which is expressed by equation (5).

(5)

In equation 5, λ represents the wavelength of incident light, z is the

distance between the source plane and observation plane, r is the

distance between a point (ξ , η) on the source plane and a point (x , y)

on observation plane (

2 2 2

( ) ( )r z x y

) and k is the

wave number. In eqn. 5, is the complex electric field amplitude exits

component “n” and

is the field from component “n” projection onto

the entrance plane of component “n+1” . The output field () from

component “n+1”, determined by (

), is the value of

the input field () to component “n+1” with the added spatially

dependent phase term This derivation is repeated until

the scalar amplitude of the field on the image plane is known. Finally,

the light intensity at the image plane (point spread function PSF) is

determined (Fig. 3). The Modulation Transfer Function (MTF) can be

then calculated by taking the Fourier transform of PSF (Fig. 4). The

calculation of eqn 5 is done using a discrete Fourier transform method

[26].

Figure 3 shows a representative result of the PSF and MTF for the

case of Dg=2 diopters for the cases of λg = 540 nm; λb=460 nm, and

λr=620nm . The results are shown in normalized curves due to the fact

that the output of the calculation depends on the user defined input

that can be adjusted arbitrarily.

a

b

Fig. 3. Normalized PSF and MTF for a 2D PPD devices at wavelenths of

460nm, 540nm and 610 nm. In this case the distance of the image

plane is fixed at the focal length for 540 nm light.

Figure 4 summarizes the results of the calculations for other values of

Dg, by plotting the cycle/degree value corresponding to an MTF value

of 0.8.

Fig. 4. Graph of cycles/degree corresponding to an MTF value of 0.8 vs

PPD lens power. Note that a break has been placed between 6 and 48 D

on the x-axis for visualization purposes.

It is reported that the resolution limit of the human eye is between 6

and 10 cycles/degree [27], it can be inferred from figure 4 that PPD

lens powers of less than 2 diopters have the potential for providing a

level of image degradation that is not observable by the human eye.

For a more intuitive interpretation of these numerical results, it is

helpful to consider the case of white light comprised of three

wavelengths across the visible spectrum: λb, λg and λr. For a

particular PPD lens, the wavelengths will be focused at distances: fb, fg,

and fr, correspond to lens powers of

(=Db) ;

(=Dg); and

(=Dr).

For lenses used in a human visual system, it is generally assumed that

changes in lens power below value of DDmax=0.25 diopter are

insignificant. Therefore, if the power of the PPD lens changes by less

than DDmax, it will not be observable by the human eye. Explicitly, if:

(6a)

and if :

(6b)

Then the spectral components of the white light will appear in focus

to the eye. Plugging Eqn 2 into Eqn 6 and solving for Dg yields:

(7a)

Or

(7b)

Therefore, for given values of wavelength ratios and DDmax = 0.25,

the limit on Dg for achromatic perception can be determined by using

the smaller value from eqn 7a-b.

As a specific example consider that the wavelength for λb, λg and λr

are as used in figures 3 and 4, and say that DDMax is 0.25 diopter.

In this case

= 0.85 and

=1.15, and when both are plugged into

eqn 7a, gives a maximum power of the lens for the design wavelength

(λg) of 1.68 diopters.

To implement this lens, the spatial variation of the azimuthal angle

of the optic axis is given by equation 1 in which the power of

the lens can be related to the focal length as a function of wavelength,

azimuthal angle and radius (r).

As another example, the spot size for is considered when is in

focus and compared to the model explained above. For this numerical

evaluation, a glass lens paired with the PPD lens is considered.

Therefore, when considering eqn 3 as a predictor of the spot size, the

focal length considered must be the effective focal length of the PPD

lens and the glass lens (

) where is the effective focal

length, is the PPD lens focal length and is the focal length of the

glass lens considered that could be the eye lens or camera lens depend

on the system. By plugging the calculated fg into eqn 2, the spot size for

light of a wavelength different than the design wavelength can be

obtained.

For the case above, It is found that the spot size for the red

wavelength considered, is 17.3 . When this spot size and effective

focal length are considered the predicted 50% MTF is 10.3

The value predicted from the simple formulation, only differs slightly

from the prediction of the model presented in fig. 3.

This result shows that the wavelength dependence of the focal

length of a PPD lens is not significant as long as the power of the lens is

less than 1.5-2 diopters. It has been shown both by analytic and

numerical calculations that the performance of PPD lens is suitable for

use in applications with the human eye. Of course, this is only a

guideline, with the actual value being dependent on user perception.

B. The effect of polychromatic light on a PPD lens for use

in an imaging application

The use of a PPD lens with a camera is similar to use with the human

eye but introduces variables such as adjustable aperture, varying pixel

size, and varying lens quality. To keep the analysis general and

applicable to a wide variety of camera systems, the approach taken

here is to consider the effect on the depth of field (DOF) of the camera

system.

If a camera is focused at infinity and a PPD lens with a focal length of

fg is placed in front of it, a green object at a distance fg from the PPD

lens will appear in focus. However, for blue or red light the object

distance to achieve sharp focus would be fb or fr respectively due to the

wavelength dependence of the PPD lens focal length. If a white object

was placed at distance fg, It would appear acceptably in focus if the

DOF of the camera system is greater than the difference between fr and

fb. This condition can be written explicitly as:

(8)

3. DATA RELATED TO IMAGING APPLICATION

As an example, If the power of the PPD lens is considered to be 1.4

diopters, then fb and fr are determined to be 0.84 m and 0.62 m

respectively in order to allow all wavelengths to be in focus. This

means that the DOF of the camera system at fg should be greater than

22 cm. To test this prediction experimentally, a 1.4D PPD lens was

fabricated.

For the fabrication of the lens, a polarization holography exposure

on a thin photo-alignment layer on a glass substrate was used followed

by a photo-polymerization of a reactive mesogen (RM). The

holographic setup is used to record the wavefront pattern of a

“template” lens (TL) as a physical element is based on the modified

Mach-Zehnder interferometer [Fig5. a]. This employs the idea of

interference of a phase modified beam using a physical element (i.e.

template lens in our case) with a reference beam [16]. Through the

quarter wave plate (QWP), the two beams exit the second beam

splitter (BS2) are orthogonally circular polarized. Both beams

combine again at the location of the PPD and results in an interference

pattern that is linearly polarized with its polarization axis being a

function of the radial distance from the center of the pattern [28]. The

azodye Brilliant Yellow, coated on a glass substrate is placed in the

location of PPD in Fig 5.a, as close as possible to the beam combiner

(BS2). Figure 5.c shows the interference pattern from the setup

captured by a CCD placed at the location of PPD. Figure 5.d is the image

of the actual 1.4 diopters lens while using a green color filter at front of

it.

Fig. 6. Shows an experimental set up to observe the effect of the

DOF of the camera system to the detectability of chromatic

problems with a 1.4 diopters PPD lens (a), and spectrum of the

white light used in background (b).

Fig. 5. a) Shows the schematic of the interference hologram

setup used to record the desired pattern on the substrate (PPD)

where BE is beam expander, LP is linear polarizer, BS is beam

splitter, QWP is quarter wave plate and TL is template lens.

b) shows the direction of polarization inside each circular ring

and its radial dependence. C) Interference pattern of the setup

d) Picture of the actual 1.4 D lens, using the design wavelength

color filter e) Schematic view of the template and the PPD lenses

and their respective focal lengths.

a

b

As indicated in the Fig5. e, the designed focal length of the PPD will

be shorter than the focal length of the template lens based on the

distance of the two components so that FPPD=FTL – D where FTL is the

focal length of the template lens, FPPD is the focal lens of the PPD lens

and D is the distance between the focal point of the template lens, and

the PPD lens. During fabrication special care was taken to minimize

this distance and keep the focal length of the PPD lens within 10% of

the value of the template lens.. The sample was then exposed for 10

minutes with a blue and the total power of the

combined beam was 7 mW. It should be noted that the blue laser light

used, would define the focal length of the PPD for blue wavelength of

475nm. Using equations 2.a & 2.b, the green and the red wavelength

focal lengths can be found respectively.

The exposure of the azodye is then followed by coating a reactive

mesogen (RM257) solution layer on it, followed by exposure by UV

(365 nm) light at an intensity of 3.5

for 10 minutes. The coating of

the RM layer, is repeated until the retardation of the RM film is a half

wavelength of the design wavelength (in this case, green wavelength).

The setup used for evaluating the performance of the PPD lenses is

shown in Fig. 6. The lens evaluated was placed in front of a camera

focused at infinity and the image quality was determined for varying

object distance and F/#. Previously, it was determined that the DOF of

the camera system should be at least 22 cm to overcome the

wavelength dependence of the PPD lens focal length. Figure 7 shows a

series of images taken with a 1.3D glass lens placed in front of the

camera. The f-number was adjusted to be either f/4 or f/22 and the

object distance that provided best focus was determined to be 82.5 cm.

The object distance was changed from 74 to 96 cm to correspond to a

DOF of 22 cm. Fig. 7 shows that the DOF for f/22 appears to be at least

22 cm since the image remains focused as the object distance is

changed. On the other hand images taken at f/4 are noticeably blurred

over a 22 cm DOF.

Figure 8 &9 show pictures taken at best focus for green light with

the 1.3 D glass lens removed and the 1.4 D PPD lens put in place.

Figure 8 shows PPD lens performance when the target was illuminated

with white light. It can be seen in figure 8d that some chromatic effects

are seen when considering the f/22 image in the form of a blue shadow

image. This is related to the first issue with the wavelength

dependence of PPD lens that was discussed in the introduction, the

effect of the device not being a half-wave retarder for blue light. This

“leakage” light can be blocked by adding a second circular polarizer

that has the opposite handedness of the input circular polarizer.

Figures 8b and 8.e show that including the 2nd polarizer noticeably

reduce the chromatic effects. However, due to the smaller DOF the

pictures images taken at f/4 still show some chromatic effects. For a

more detailed analysis, pictures 8c and 8f show the zoomed and

cropped versions of the middle region from 8b and 8e respectively. In

order to examine this further, figure 9 shows a circular part of the

resolution target with 2cm diameter that includes lines of group (-1)

with elements 3-6 using blue(459nm), green(545nm) and

red(610nm) color filters attached to the object which is placed at the

best focus location for green light.

a

b

c

f

e

d

Fig. 7. Pictures of images with a camera focused at infinity, and with a 1.3D glass lens. Figures (a) ,(b) and (c) are for the camera with f/4

setting, while figures (d), (e) and (f) are with f/22. (b)&(e) are with the object at the “best focus” distance of 82.5 cm. (a)&(d) are taken

with the object at 74cm, while (c)&(f) are taken with the object at 96cm.

a

b

c

f

e

d

a

b

c

f

e

d

Fig. 8. Pictures of image with PPD lens at the “best focus” distance of 71cm. (a)&(b) are taken with the camera setting of f/4,

while (d)&(e) are taken with f/22, (a)&(d) are taken with a single circular polarizer, while (b)&(e) are taken with a circular

polarizer on both sides of the PPD lens as explained in the text. (c) &(f) are just the middle region of the pictures (b)&(e), zoomed

and cropped.

Fig. 9. Pictures of the group (-1) elements using PPD lens with blue ( left )(459nm), green (center)(545nm) and red ( right)

(610nm) color filters attached to the object and circular polarizer on both sides at the “best focus” distance of 71cm .(a),(b)&(c) are

taken with f/4 and ¼ second exposure time while (d),(e)&(f) are taken with f/22 and 3 seconds exposure time.

As expected, the green light is fairly at focus for both f/4 and f/22

setting while the blue and red are not at focus for the case of f/4. These

last two colors show large amount of haze when depth of field is small

(f/4). However, in the case of illumination by blue and red light, the

camera set at f/22 shows improved performance thanks to the large

DOF. This confirms that the DOF limit of Eqn 8 can be used as a

guideline to determine the limits of the PPD lens power and useable

optical bandwidth.

Knowing the pixel size of our camera and the USAF 1951 target

resolution dimensions data, the magnification and corresponding

spatial frequency value for each element of interest shown in the Fig.9

was determined.

Using the modulation definition (

where

are maximum and minimu m intensity near the bars

respectively), the contrast from peak to valley difference was found as

shown in Fig 10. Then, the intensity is normalized to the absolute black

and white values in the large areas nearby the group elements of

interest. Next, the MTF values were determined for each element

shown in Fig.9. Fig 10c shows a sample of the intensity modulation of

the pixels from element 3 in the Fig. 9c from which the MTF value was

determined. In this particular case, the peak to valley amount was

found 114-45=69 which resulted in contrast ratio of

Then the contrast ratio was normalized with respect

to the absolute values of the white and black (which was 116 & 9

respectively) pixels; the resulting MTF value for this example was

determined to be 0.50. Using different color filters, this analysis was

performed to the same element number for green (Fig.9b) and blue

light (Fig 9a) and the resulted MTF values of 0.56 and 0.14 was found

respectively (Fig 10b &10a).

Furthermore, the results of the MTF of element 3 of Fig.9c was

compared with expectations from the “simple model” prediction of

Eqn.3 and from numerical calculations introduced in section 2A.

For the prediction of the “simple model”, the value of L in Eqn 3 was

determined to be 12.5 mm from the f/# of the camera setting (f/4)

along with the 50 mm camera lens focal length. Using thin lens

combination formula and the PPD lens design power of 1.4D, the spot

siz e for the red light found from equations 2,3 to be 0.097mm. This

leads to an expected value of

for a MTF value of 50%.

For the prediction from a numerical calculation, the method

described in section 2.A was used but with the “eye lens” replaced by

the camera lens that has an aperture of 12.5 mm and a focal length of

50mm, and with t he PPD lens having a power of 1.4D. In this way, the

spatial frequency was determined to be 5.5

for a corresponding

MTF of 50% that agrees very closely with the “simple model” and

validates the proposed design criteria.

The predictions from both approaches can be compared with the

data shown in Fig.10, for the element group that has an MTF of 0.5 and

a spatial frequency of 8.7

. This comparison demonstrates that the

proposed design criteria is quite useful.

CONCLUSION

This paper has provided a guideline for the limits on PDD lens

power and aperture for applications involving the human eye or

imaging with a camera [5,15]. While PPD lenses have a strong

chromatic effect, acceptable performance can be obtained, for

applications involving the eye, providing that the lens power is around

1.6 D [27,29]. Finally, for imaging applications [29], a criteria for the

DOF of the camera that allows acceptable performance has been

established for a given PPD lens power and an optical bandwidth

range.

Funding Information. MIT/Lincoln Laboratory (Dr. Harrold Payson,

technical contact) and the ARO agreement number (W911NF-14-1-

0650), (Dr. Michael Gerhold , program manager)

Fig. 10. Intensity modulation of the element 3 of group (-1)

in the fig.9.a for blue (a), green in the fig.9.b (b) and red in the

fig.9.c (c)

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29. E.J. Galvez, “Applications of Geometric Phase in Optics” Department of

Physics and Astronomy, Colgate University (2017).