WO2019178822A1

WO2019178822A1 - Methods and systems for measuring optical shear of birefringent devices beyond diffraction limit

Info

Publication number: WO2019178822A1
Application number: PCT/CN2018/080104
Authority: WO
Inventors: Xian Chen; Shengwang Du; Teng ZHAO; Luwei ZHAO; Zhuohui ZENG; Hoi Chun CHIU
Original assignee: Light Innovation Technology Limited
Priority date: 2018-03-23
Filing date: 2018-03-23
Publication date: 2019-09-26
Also published as: CN112219096A

Abstract

The methods and systems directly measure the optical shear angle and lateral displacement of a light beam passing through a birefringent device (212) with resolution beyond the diffraction limit. The system for measuring the shear angle comprises an illumination module (310), a polarization control unit (209) or polarizer (309), the said birefringent device (212), a lens module (215), and a data acquisition module (218) for recording the light intensity distribution. When the polarization control unit (209) is used, the polarization of the input light beam (201) from the illumination module (310) is controlled such that we can record the two light spots (206, 207) with orthogonal polarizations separately at different frames. When the polarizer (309) is used, with mixed polarizations of the input beam (201) from the illumination module (310), the polarizer (309) is placed before the data acquisition module (218) to record the two spots (206, 207) with orthogonal polarizations separately at different frames. Then it applys localization analysis to determine the central positions of the two spots (206, 207) with perpendicular polarizations and calculate the value of the shear angle/displacement between laterally sheared beams. This method can resolve the shear angle/displacement beyond optical diffraction limit.

Description

Methods and systems for measuring optical shear of birefringent devices beyond diffraction limit

FIELD

The present disclosure relates to optical characterization of the optical shear of birefringent devices.

BACKGROUND

A birefringent crystal or prism can be used to split the incident light into two orthogonally polarized beams either along different directions or shifting by some lateral displacement. Figure 1 illustrates schematics of such devices. In Fig. 1A, an input light ray 101 is incident to a birefringent device 102 and is split into two (104 and 105) at the shear plane 103, which have orthogonal polarizations (P ₊ and P _-) with a shear angle ε. In Fig. 1B, an input light ray 106 is incident to a birefringent device 107 and is split into two parallel rays (109 and 110) at the shear plane 108 which have orthogonal polarizations (P ₊ and P _-) with a lateral shear displacement S. An example of such a device is the Nomarski prism, consisting of two birefringent crystal wedges aligned with different optical axes. It is the key component of a differential interference contrast (DIC) microscope. For the prisms used in a DIC microscope, the shear angles are typically about 10 ^-5 rad or even smaller. While this shear angle is critical for a DIC microscope's spatial resolution, contrast, and depth, commercial manufacturers usually do not provide any such information. The increasing interests in label-free biological imaging and surface topography push the development of the quantitative DIC microscopy, which demands an accurate determination of the beam shear parameters of the prism. However, by far only indirect methods have been proposed and demonstrated, such as use of calibrated specimens or standard optical wedges, dual-focus fluorescence correlation spectroscopy, spatial interference, and retardance derivative. These measurements require complicated setups and data analysis.

Consider a collimated monochromatic light beam (e.g. laser) of wavelength λ and diameter D passing through a birefringent prism. The output is two orthogonally polarized light beams propagating along directions separated by a small angle ε. Restricted by the diffraction effect of light, the shear angle has to be larger than the diffraction angle of the beam, i.e.,

so that the spatially sheared beams can be resolved. This condition suggests that a sufficiently large incident beam (i.e.

) is required if we want to directly measure the separation of the scattered beams with a small shear angle. For example, in a typical configuration with λ =400 nm and a shear angle ε=10 μrad (μrad=10 ^-6rad) , the required beam size should be greater or equal than 5 cm. However, the sizes of most birefringent devices are much smaller than that, which hinders the development of the direct method in this field.

For measuring the lateral displacement, it is also not feasible using the direct method if the displacement S is less and equal than the diffraction limit

where n is the refractive index.

In this invention, we revisit the direct spatial measurement of the optical shear of birefringent devices with the application of the localization analysis, which can precisely determine the each of the centroids of the two light waves beyond the diffraction limit. The novelty is that the momentum and polarization are not overlap in the

joint space.

SUMMARY

We describe the methods and systems to directly measure the optical shear angle and lateral displacement of a light beam passing through a birefringent device with resolution beyond the diffraction limit. The system for measuring the shear angle comprises an illumination module, a polarization control unit or polarizer, the said birefringent device, a lens module, and a data acquisition module for recording the light intensity distribution. The system for measuring the shear displacement comprises an illumination module, a polarization control unit or polarizer, the said birefringent device, an imaging module, and a data acquisition module for recording the light intensity distribution. When the polarization control unit is used, the polarization of the input light beam from the illumination module is controlled such that we can record the two light spots with orthogonal polarizations separately at different frames. When the polarizer is used, with mixed polarizations of the input beam from the illumination module, the polarizer is placed before the data acquisition module to record the two spots with orthogonal polarizations separately at different frames. Then we apply localization analysis to determine the central positions of the two spots with perpendicular polarizations and calculate the value of the shear angle/displacement between laterally sheared beams. This method can resolve the shear angle/displacement beyond optical diffraction limit.

DRAWINGS

FIG. 1 shows the schematic drawings of the optical shear effects of two different birefringent devices. FIG. 1A illustrates the shear angle effect when a light ray passes through a type of birefringent device. FIG. 1B illustrates the shear displacement effect when a light ray passes through another type of birefringent device.

FIG. 2A is the schematic drawing of the optical setup for measuring the shear angle of a birefringent device using the polarization control unit (PCU) . FIG. 2B shows the block diagram of the corresponding system for measuring the shear angle of a birefringent device using the polarization control unit (PCU) .

FIG. 3A is the schematic drawing of the optical setup for measuring the shear angle of a birefringent device using the polarizer. FIG. 3B shows the block diagram of the corresponding system for measuring the shear angle of a birefringent device using the polarizer.

FIG. 4A is the schematic drawing of the optical setup for measuring the shear displacement of a birefringent device using the polarization control unit (PCU) . FIG. 4B shows the block diagram of the corresponding system for measuring the shear displacement of a birefringent device using the polarization control unit (PCU) .

FIG. 5A is the schematic drawing of the optical setup for measuring the shear displacement of a birefringent device using the polarizer. FIG. 5B shows the block diagram of the corresponding system for measuring the shear displacement of a birefringent device using the polarizer.

FIG. 6 shows the intensity distributions collected by the data acquisition module and the results of the localization analysis. FIG. 6A shows the intensity distribution of two overlapped spots with mixed polarizations P ₊ and P _-. FIG. 6B shows the intensity distribution of the spot with polarization P ₊. FIG. 6C shows the intensity distribution of the spot with polarization P _-.

FIG. 7 is the block diagram showing the procedures of the methods in measuring the shear angle and displacement of a birefringent device.

DETAILED DESCRIPTION

The first optical setup for directly measuring the shear angle of a birefringent device is illustrated in Fig. 2A. A light beam 201 with a diameter D is incident into the birefringent device, which spatially shears the output light beam 203 of polarization P ₊ relative to the output light beam 204 of polarization P _- by an angle ε after the shear plane 202. The polarizations P ₊ and P _- are orthogonal to each other. The polarization control unit (PCU) 209 is used to control the polarization state of the input light beam 201 and alter the intensity contributions to P ₊ and P _-components of the output sheared beams. A lens 205 with focal length f is used to focus the two sheared

beams

203 and 204 to two

spots

206 and 207 whose centers are separated by Δ on the focal plane, which is the Fourier transform of the two beams in the momentum space. The intensity distribution I (x, y) of the spots is collected by the record device 208 on the focal plane. The centroid separation between the two

spots

206 and 207 is estimated as

The spot size of each of the beams on the focal plane is estimated as

Using conventional direct measurement, the necessary condition for resolving the two

spots

206 and 207 in an image is Δ>d, i.e.

In our system, we can control the input beam polarization using PCU 209 so that we can individually collect the intensity distribution from each of the spots at different camera frames to avoid spot overlapping. It is implemented by means of: 1) at P=P ₊, we take an image showing P ₊-polarized spot I ₊ (x, y) ; 2) at P=P _-, we take an image with P _--polarized spot I _- (x, y) .

Depending on the polarization state, the PCU comprises different optical components. 1) In the case of nonpolarized incident light, the PCU is made up of a switchable polarization filter for selecting P ₊ or P _-. 2) In the case of polarized incident light, the PCU can be a half-wave plate or a combination of a half-wave plate and quarter-wave plate for selecting P ₊ or P _-.. The polarization components P ₊ and P _- are either linearly orthogonal or circular orthogonal to each other, depending on the type of the birefringent crystal used.

FIG. 2B shows the block diagram of the corresponding system for measuring the shear angle of a birefringent device in the first optical setup. Such a system, to implement the function of Fig. 1A, is made up of an illumination module 210 that delivers a light beam, a PCU 209, a birefringent device 212, a lens module 215, and a data acquisition module 218.

Fig. 3A illustrate the second optical setup for directly measuring the shear angle of a birefringent device. Different from the first one in Fig. 2A, the polarizer 309 is placed in front of the light intensity distribution record device 208. In this setup, the incident beam consists of a mixed polarizations of both P ₊ and P _-, of which the polarizer 309 only allows one passing through and being collected at 208. By switching the bypass of the polarizer 309, we can individually collect the intensity I ₊ (x, y) and I _- (x, y) at different data frames without overlapping.

FIG. 3B shows the block diagram of the corresponding system for measuring the shear angle of a birefringent device in the second optical setup. Such a system, to implement the function of Fig. 3A, is made up of an illumination module 310 that delivers a light beam, the birefringent device 212, a lens module 215, a switchable polarizer 309, and a data acquisition module 218.

Fig. 4A illustrate the first optical setup for directly measuring the shear displacement of a birefringent device. A light beam 401 is focused on the shear plane 402 where the two output

orthogonal polarization components

403 and 404 are laterally shifted by displacement S. Then the shear plane is imaged onto the light intensity distribution record device 208 through an imaging system 405 with a magnification M. Finally two

spots

406 and 407 are collected 208. The PCU 209 is used to control the polarization state of the incident light beam such that I ₊ (x, y) and I _- (x, y) can be taken separately at different data frames.

FIG. 4B shows the block diagram of the corresponding system for measuring the shear displacement of a birefringent device in the first optical setup. Such a system, to implement the function of Fig. 4A, is made up of an illumination module 410 that delivers a focused light beam, a PCU 209, the birefringent device 412, an imaging module 415, and a data acquisition module 218.

Fig. 5A illustrate the second optical setup for directly measuring the shear displacement of a birefringent device. The switchable polarizer 309 is placed in front of the light intensity distribution record device 208. In this setup, the incident light beam comprises mixed polarization states, i.e. P ₊ and P _-, of which the polarizer 309 allows only one passing through and being collected at 208 so that the intensity distributions I ₊ (x, y) and I _- (x, y) are separately recorded at 208 at different data frames.

FIG. 5B shows the block diagram of the corresponding system for measuring the shear displacement of a birefringent device in the second optical setup. Such a system, to implement the function of Fig. 5A, is made up of an illumination module 510 that delivers a focused light beam with mixed polarizations, the birefringent device 412, an imaging module 415, a polarizer 309, and a data acquisition module 218.

To illustrate the localization analysis of our method and without the loss of generality, an example that the optical shear of a birefringent device (either Fig. 1 A or B) is less or equal than

is given in Fig. 6. We represent three types of results collected by the data acquisition module 218: 1) Fig. 6A Intensity distribution 601 of I ₊ (x, y) +I _- (x, y) ; 2) Fig. 6B Intensity distribution 602 of I ₊ (x, y) ; 3) Fig. 6C Intensity distribution 604 of I _- (x, y) . It is obvious that the separation Δ cannot be resolved in Fig. 6A. By localization analysis, we determine the positions (x ₊, y ₊) and (x _-, y _-) of

centroids

603 and 605 of the

spots

602 and 604 corresponding to the intensity distributions I ₊ (x, y) and I _- (x, y) respectively. Then the separation between them can be calculated as

The corresponding shear angle is determined by Eq. (1) . The localization precision is estimated as

where σ is the standard deviation of the single spot intensity distribution, a is the pixel size of the data acquisition module, N is the number of photons collected, and b is the background noise. Because we do not have any limitation on the photon budget in principle and N is only limited by the camera sensor saturation. Accounting systematic perturbations from the environment, such as mechanical vibration and temperature fluctuation, a nanometer localization accuracy beyond diffraction limit is practically achievable.

For measuring the shear angle of the birefringent device of Fig. 2 and Fig. 3, we apply equation (1) to obtain the value of the shear angle

The shear angle measurement precision or resolution is given by

For a typical configuration with f=10 cm in Fig. 2 and 3, the nanometer localization precision leads to a shear angle measurement precision of 10 ^-8 rad.

For measuring the shear displacement of the birefringent device of Fig. 4 and Fig. 5, the shear displacement is determined by

Fig. 7 summarize the methods and procedures in measuring and determining the shear angle ε and shear displacement S. We start checking the type of birefringent device in step 701. If the task is to measure the shear angle ε, we go to step 702 and use the setup in Fig. 2 or Fig. 3. In step 703, we take frame 1 with I ₊ (x, y) and frame 2 with I _- (x, y) , and obtain their center positions (x ₊, y ₊) and (x _-, y _-) with localization analysis. In step 704, we obtain the separation distance between the two centers

In step 705, we obtain the value of the shear angle

where f is the local length of the lens system in Fig. 2 and Fig. 3. If the task is to measure the shear displacement S, from 701 we go to step 706 and use the setup in Fig. 4 or Fig. 5. In step 707, we take frame 1 with I ₊ (x, y) and frame 2 with I _- (x, y) , and obtain their center positions (x ₊, y ₊) and (x _-, y _-) with localization analysis. In step 708, we obtain the separation distance between the two centers

In step 709, we obtain the value of the shear displacement

Here M is the transverse image magnification of the imaging module in Fig. 4 and Fig. 5.

Claims

A method for measuring the shear angle of a birefringent device, comprising:

shining a light beam on the said birefringent device;

splitting two orthogonally polarized output beams (the first polarization beam and the second polarization beam) after the said birefringent device into two different directions with a shear angle;

using a lens system to focus the said two beams;

placing a light intensity distribution record device after the said lens system where the said beams are focused on;

recording the first frame in the said light intensity distribution record device with the said first polarization beam;

determining the first center position of the light intensity distribution in the first frame using localization analysis;

recording the second frame in the said record device with the said second polarization beam;

determining the second center position of the light intensity distribution in the second frame using localization analysis;

calculating the separation distance between the said first center position and the said second center position;

and calculating the said shear angle from the said separation distance and the focal length of the said lens system
The method according to claim 1, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent device.
The method according to claim 1, wherein the said lens system can be either a single lens or a combined multiple lenses system.
A first system for measuring the shear angle of a birefringent device, comprising:

a light illumination module outputting a light beam;

a polarization control unit;

the said birefringent device;

a lens module;

and a data acquisition module placed where the beams are focused.
The system according to claim 4, wherein the said polarization control unit is used to control the polarization of the input light beam so that the polarization of the output light beam from the said illumination module can be switched between the two orthogonal polarizations.
The polarization control unit according to claim 5, can be a polarizer if the said light beam is nonpolarized, or a half-wave plate, or a combination of a half-wave plate and a quarter-wave plate, or other polarization units.
The polarization control unit according to claim 5, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent prism.
The system according to claim 4, wherein the said lens system can either a single lens or a combined multiple lenses system.
The system according to claim 5, wherein the said data acquisition module can be a CCD camera or a CMOS camera.
A second system for measuring the shear angle of a birefringent device, comprising:

a light illumination module outputting a light beam;

the said birefringent device;

a lens module;

a switchable polarization filter;

and a data acquisition module placed where the beams are focused.
The system according to claim 10, wherein the said switchable polarization filter is used to select one of the two orthogonal polarization light components passing through the filter.
The system according to claim 10, wherein the said switchable polarization filter can be either a linear polarizer or a circular polarization filter, depending on the types of the two orthogonal polarizations.
The system according to claim 10, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent prism.
The system according to claim 10, wherein the said lens system can either a single lens or a combined multiple lenses system.
The system according to claim 10, wherein the said data acquisition module can be a CCD camera or a CMOS camera.
A method for measuring the shear displacement S of a birefringent device, comprising:

shining a light beam focused on the shear plane of the said birefringent device;

splitting two orthogonally polarized output beams (the first polarization beam and the second polarization beam) at the shear plane with a lateral shear displacement S;

using an imaging system to image the shear plane to a light intensity distribution record device with an image magnification M;

recording the first frame in the said record device with the said first polarization beam;

determining the first center position of the light intensity distribution in the first frame using localization analysis;

recording the second frame in the said record device with the said second polarization beam;

determining the second center position of the light intensity distribution in the second frame using localization analysis;

calculating the separation distance Δ between the said first center position and the said second center position;

and calculating the said shear displacement S from the measured separation distance d divided by the imaging magnification M. That is S=Δ/M.
The method according to claim 16, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent device.
A first system for measuring the shear displacement S of a birefringent device, comprising:

a light illumination module outputting a focused light beam;

a polarization control unit;

the said birefringent device with s shear plane when the light beam is focused on;

an imaging module;

and a data acquisition module placed where the image of the shear plane is formed.
The system according to claim 18, wherein the said polarization control unit is used to control the polarization of the input light beam so that the polarization of the input light beam can switched between the two orthogonal polarizations.
The polarization control unit according to claim 19, can be a polarizer if the said input light beam is nonpolarized, or a half-wave plate, or a combination of a half-wave plate and a quarter-wave plate, or other polarization units.
The polarization control unit according to claim 19, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent prism.
The system according to claim 18, wherein the said imaging module can be a single lens or a multiple lenses system.
The system according to claim 18, wherein the said data acquisition module can be a CCD camera or a CMOS camera.
A second system for measuring the shear displacement S of a birefringent device, comprising:

a light illumination module outputting a focused light beam;

the said birefringent device with s shear plane when the light beam is focused on;

a switchable polarization filter;

an imaging module;

and a data acquisition module placed where the image of the shear plane is formed.
The system according to claim 24, wherein the said switchable polarization filter is used to select one of the two orthogonal polarization light components passing through the filter.
The system according to claim 24, wherein the said switchable polarization filter can be either a linear polarizer or a circular polarization filter, depending on the types of the two orthogonal polarizations.
The system according to claim 24, wherein the said two orthogonal polarizations can be either two orthogonal linear polarizations or two orthogonal circular polarizations, depending on the type of the said birefringent device.
The system according to claim 24, wherein the said imaging module can be a single lens or a multiple lenses system.
The system according to claim 24, wherein the said data acquisition module can be a CCD camera or a CMOS camera.