where do and di are the object and image distances, respectively, for the objective lens as labeled in Figure 2. The object distance is given to be do=6.20 mm, but the image distance di is not known. Isolating di, we have

Polarization examples

Figure 5. Light rays from a specimen entering the objective. Paths for immersion medium of air (a), water (b) (n = 1.33), and oil (c) (n = 1.51) are shown. The water and oil immersions allow more rays to enter the objective, increasing the resolution.

To see how the microscope in Figure 2 forms an image, we consider its two lenses in succession. The object is slightly farther away from the objective lens than its focal length fo, producing a case 1 image that is larger than the object. This first image is the object for the second lens, or eyepiece. The eyepiece is intentionally located so it can further magnify the image. The eyepiece is placed so that the first image is closer to it than its focal length fe. Thus the eyepiece acts as a magnifying glass, and the final image is made even larger. The final image remains inverted, but it is farther from the observer, making it easy to view (the eye is most relaxed when viewing distant objects and normally cannot focus closer than 25 cm). Since each lens produces a magnification that multiplies the height of the image, it is apparent that the overall magnification m is the product of the individual magnifications: m = mome, where mo is the magnification of the objective and me is the magnification of the eyepiece. This equation can be generalized for any combination of thin lenses and mirrors that obey the thin lens equations.

A final distinctive and perhaps even unique feature of U.S. polarization is the powerful alignment of ethnicity, ideology, and religion on each side of the divide—what we call the “iron triangle” of U.S. polarization. In most other countries, just one or two of those three identity divisions is at the root of polarization; in the United States, all three are. As a result, America’s polarization is unusually encompassing and sharp.

compound microscope: a microscope constructed from two convex lenses, the first serving as the ocular lens(close to the eye) and the second serving as the objective lens

Both the objective and the eyepiece contribute to the overall magnification, which is large and negative, consistent with Figure 2, where the image is seen to be large and inverted. In this case, the image is virtual and inverted, which cannot happen for a single element (case 2 and case 3 images for single elements are virtual and upright). The final image is 367 mm (0.367 m) to the left of the eyepiece. Had the eyepiece been placed farther from the objective, it could have formed a case 1 image to the right. Such an image could be projected on a screen, but it would be behind the head of the person in the figure and not appropriate for direct viewing. The procedure used to solve this example is applicable in any multiple-element system. Each element is treated in turn, with each forming an image that becomes the object for the next element. The process is not more difficult than for single lenses or mirrors, only lengthier.

Linear polarization

The Democracy, Conflict, and Governance Program is a leading source of independent policy research, writing, and outreach on global democracy, conflict, and governance. It analyzes and seeks to improve international efforts to reduce democratic backsliding, mitigate conflict and violence, overcome political polarization, promote gender equality, and advance pro-democratic uses of new technologies.

Polarization then entrenches itself and becomes self-perpetuating. Polarizing actions and reactions feed on each other, dragging countries into a downward spiral of anger and division.

Now we must find the magnification of the eyepiece, which is given by [latex]m_{\text{e}}=-\frac{d_{\text{i}}\prime}{d_{\text{o}}\prime}\\[/latex], where di′ and do′ are the image and object distances for the eyepiece (see Figure 2). The object distance is the distance of the first image from the eyepiece. Since the first image is 186 mm to the right of the objective and the eyepiece is 230 mm to the right of the objective, the object distance is do′ = 230 mm − 186 mm = 44.0 mm. This places the first image closer to the eyepiece than its focal length, so that the eyepiece will form a case 2 image as shown in the figure. We still need to find the location of the final image di′ in order to find the magnification. This is done as before to obtain a value for [latex]\frac{1}{d_{\text{i}}\prime}\\[/latex]:

We wanted to know: Why has polarization come to a boil in so many places in recent years? Are there any telling similarities in the patterns of polarization across different countries? And perhaps most importantly, once societies have become deeply polarized, what can they do to start healing their divisions?

We normally associate microscopes with visible light but x ray and electron microscopes provide greater resolution. The focusing and basic physics is the same as that just described, even though the lenses require different technology. The electron microscope requires vacuum chambers so that the electrons can proceed unheeded. Magnifications of 50 million times provide the ability to determine positions of individual atoms within materials. An electron microscope is shown in Figure 7.

The term f/# in general is called the f-number and is used to denote the light per unit area reaching the image plane. In photography, an image of an object at infinity is formed at the focal point and the f-number is given by the ratio of the focal length f of the lens and the diameter D of the aperture controlling the light into the lens (see Figure 3b). If the acceptance angle is small the NA of the lens can also be used as given below.

When using a microscope, we rely on gathering light to form an image. Hence most specimens need to be illuminated, particularly at higher magnifications, when observing details that are so small that they reflect only small amounts of light. To make such objects easily visible, the intensity of light falling on them needs to be increased. Special illuminating systems called condensers are used for this purpose. The type of condenser that is suitable for an application depends on how the specimen is examined, whether by transmission, scattering or reflecting. See Figure 6 for an example of each. White light sources are common and lasers are often used. Laser light illumination tends to be quite intense and it is important to ensure that the light does not result in the degradation of the specimen.

Microscopes were first developed in the early 1600s by eyeglass makers in The Netherlands and Denmark. The simplest compound microscope is constructed from two convex lenses as shown schematically in Figure 2. The first lens is called the objective lens, and has typical magnification values from 5× to 100×. In standard microscopes, the objectives are mounted such that when you switch between objectives, the sample remains in focus. Objectives arranged in this way are described as parfocal. The second, the eyepiece, also referred to as the ocular, has several lenses which slide inside a cylindrical barrel. The focusing ability is provided by the movement of both the objective lens and the eyepiece. The purpose of a microscope is to magnify small objects, and both lenses contribute to the final magnification. Additionally, the final enlarged image is produced in a location far enough from the observer to be easily viewed, since the eye cannot focus on objects or images that are too close.

When we looked at the fierce polarization in many countries, we expected to find deep-seated differences between the opposing sides. So we were taken aback to discover that sometimes those differences seem slight.

5. (a) +18.3 cm (on the eyepiece side of the objective lens); (b) −60.0; (c) −11.3 cm (on the objective side of the eyepiece); (d) +6.67; (e) −400

Take the example of Bangladesh: acrimonious political competition there has led to violence, election fraud, and a complete breakdown of democracy. But polarization isn’t rooted in any fundamental ethnic, ideological, or religious division among voters. It is almost entirely the result of power struggles within a political elite that plays up and manufactures divisions.

Many other drivers of polarization struck us as surprising, even counterintuitive. You might expect, for instance, that a growing economy would ease polarization. Yet we found that in some places, such as India, it actually made things worse. Indeed, the growth of India’s middle class has led to rising support for polarizing Hindu nationalist narratives.

Look through a clear glass or plastic bottle and describe what you see. Now fill the bottle with water and describe what you see. Use the water bottle as a lens to produce the image of a bright object and estimate the focal length of the water bottle lens. How is the focal length a function of the depth of water in the bottle?

Amplifying the effect of these divisive figures is the technologically fueled disruption of the media industry, especially the rise of social media. Opposition leaders often fan the flames as well by responding with antidemocratic and confrontational tactics of their own. In Turkey, for instance, the head of the main opposition party stoked tensions by calling on the military to oppose Erdoğan’s potential bid for the presidency in 2007.

Figure 8. The image shows a sequence of events that takes place during meiosis. (credit: PatríciaR, Wikimedia Commons; National Center for Biotechnology Information)

We do not use our eyes to form images; rather images are recorded electronically and displayed on computers. In fact observing and saving images formed by optical microscopes on computers is now done routinely. Video recordings of what occurs in a microscope can be made for viewing by many people at later dates. Physics provides the science and tools needed to generate the sequence of time-lapse images of meiosis similar to the sequence sketched in Figure 8.

Polarization also reverberates throughout the society as whole, poisoning everyday interactions and relationships. Turkey is a particularly jarring example: almost eight out of ten people there would not want their daughter to marry someone who votes for the party they most dislike. Nearly three-quarters would not even want to do business with such a person.

Normal optical microscopes can magnify up to 1500× with a theoretical resolution of −0.2 μm. The lenses can be quite complicated and are composed of multiple elements to reduce aberrations. Microscope objective lenses are particularly important as they primarily gather light from the specimen. Three parameters describe microscope objectives: the numerical aperture (NA), the magnification (m), and the working distance. The NA is related to the light gathering ability of a lens and is obtained using the angle of acceptance θ formed by the maximum cone of rays focusing on the specimen (see Figure 3a) and is given by NA = n sin α, where n is the refractive index of the medium between the lens and the specimen and [latex]\alpha=\frac{\theta}{2}\\[/latex]. As the angle of acceptance given by θ increases, NA becomes larger and more light is gathered from a smaller focal region giving higher resolution. A 0.75 NA objective gives more detail than a 0.10 NA objective.

Our work identifies and analyzes eight different types of remedial actions, ranging from dialogue efforts and media reforms to international action. We’ll highlight just three examples here.

Figure 6. Illumination of a specimen in a microscope. (a) Transmitted light from a condenser lens. (b) Transmitted light from a mirror condenser. (c) Dark field illumination by scattering (the illuminating beam misses the objective lens). (d) High magnification illumination with reflected light – normally laser light.

Polarization tends to escalate at a dizzyingly fast pace, often in the span of just a few years. Just look at how rapidly the 2016 Brexit referendum has ripped the United Kingdom apart.

Yet despite these challenges, our research shows that a wide range of actors have tried inventive ways of addressing the problem—and sometimes achieved encouraging results.

Once a society becomes deeply divided, it is very difficult to heal. Before talking about remedial actions, it’s crucial to understand why this problem is so thorny and difficult to overcome.

Figure 2. A compound microscope composed of two lenses, an objective and an eyepiece. The objective forms a case 1 image that is larger than the object. This first image is the object for the eyepiece. The eyepiece forms a case 2 final image that is further magnified.

Polarization in Chemistry

Political leadership can also play a crucial role in de-escalating partisan divides. In Ecuador, President Lenín Moreno has rejected the polarizing tactics of his predecessor, even though the two come from the same political party. And in Turkey, opposition parties have achieved modest success by uniting to form a coalition: their candidate for mayor of Istanbul won a resounding victory in 2019 with a campaign that emphasized overcoming divisions.

The Democracy, Conflict, and Governance Program is a leading source of independent policy research, writing, and outreach on global democracy, conflict, and governance. It analyzes and seeks to improve international efforts to reduce democratic backsliding, mitigate conflict and violence, overcome political polarization, promote gender equality, and advance pro-democratic uses of new technologies.

Plane of polarization

While partisan warfare hasn’t eroded democracy in the United States to the same extent that it has in, say, Bangladesh or Turkey, it is testing our democratic guardrails in serious ways.

The degree of similarity we found across countries was startling. Even in democracies as different as Colombia, Kenya, and Poland, many of the roots, patterns, and drivers of polarization were the same.

Figure 3. (a) The numerical aperture of a microscope objective lens refers to the light-gathering ability of the lens and is calculated using half the angle of acceptance . (b) Here, is half the acceptance angle for light rays from a specimen entering a camera lens, and is the diameter of the aperture that controls the light entering the lens.

[latex]\displaystyle\frac{1}{d_{\text{i}}\prime}=\frac{1}{f_{\text{e}}}-\frac{1}{d_{\text{o}}\prime}=\frac{1}{50.0\text{ mm}}-\frac{1}{44.0\text{ mm}}=\frac{0.00273}{\text{mm}}\\[/latex]

From these, we extracted cross-cutting findings. And the sheer diversity of our cases—in terms of societal makeup, political institutions, and economic development—opened our eyes to discoveries that we might have missed if we had looked only at the United States and Europe.

And while the consequences of polarization are punishing, they don’t necessarily galvanize a government to respond, because the politicians who play the most significant role in exacerbating polarization mostly benefit from it and bear little of the cost.

Polarization definition psychology

Polarization is tearing at the seams of democracies around the world, from Brazil and India to Poland and Turkey. It isn’t just an American illness; it’s a global one.

In the first place, polarization in the United States isn’t primarily the result of polarizing politicians stoking divisions, as in most other countries. It has deep societal roots and is the outcome of a profound sociocultural struggle between contending conservative and progressive visions of the country. Consequently, U.S. polarization is not something that political leaders can easily reverse, even if they want to.

For one, several promising efforts to limit polarization have focused on institutional reforms, such as decentralizing political power or changing electoral rules. Kenya, for instance, adopted a new constitution in 2010 that sought to ease ferocious competition for national office by giving regional officials greater autonomy and control over state resources. But important reforms don’t always require changing a country’s constitution: in the United States, for example, Maine passed legislation in 2016 to enact ranked-choice voting, a system that favors centrist candidates and discourages negative campaigning.

We also found that patronage and corruption—two decidedly antidemocratic practices—can temporarily reduce polarization by helping politicians build very big tents. In the long term, however, the political rot that this causes frequently leaves voters disgusted with the traditional parties and fuels the rise of divisive populist figures, like Hugo Chávez in Venezuela and Jair Bolsonaro in Brazil.

numerical aperture: a number or measure that expresses the ability of a lens to resolve fine detail in an object being observed. Derived by mathematical formula NA = n sin α, where n is the refractive index of the medium between the lens and the specimen and [latex]\alpha=\frac{\theta}{2}\\[/latex]

Calculate the magnification of an object placed 6.20 mm from a compound microscope that has a 6.00 mm focal length objective and a 50.0 mm focal length eyepiece. The objective and eyepiece are separated by 23.0 cm.

Partisan conflict takes a heavy toll on civil society as well, often leading to the demonization of activists and human rights defenders. More seriously still, divisions can contribute to a spike in hate crimes and political violence: India, Poland, and the United States have all seen such increases in recent years.

Polarization is shaking societies across the world, from new democracies to long-established ones. Why are political divisions intensifying globally, and what can policymakers learn from other countries’ experiences?

We focused on nine diverse countries grappling with the problem: Bangladesh, Brazil, Colombia, India, Indonesia, Kenya, Poland, Turkey, and the United States. We assembled a group of scholars with deep local expertise on these countries, and they produced in-depth case studies.

Other efforts have involved legal or judicial action to limit polarization and majoritarianism—the idea that the feelings and rights of the minority should not constrain leaders with majority support. In India, for example, the Supreme Court has spoken out in defense of democratic institutions and demanded greater accountability for hate crimes and political violence.

Now is the moment to develop more sustainable security mechanisms in Palestine. The proven practice of unarmed civilian protection and accompaniment is a critical initiative toward such security.

Polarization of light

Can the NA be larger than 1.00? The answer is ‘yes’ if we use immersion lenses in which a medium such as oil, glycerine or water is placed between the objective and the microscope cover slip. This minimizes the mismatch in refractive indices as light rays go through different media, generally providing a greater light-gathering ability and an increase in resolution. Figure 5 shows light rays when using air and immersion lenses.

Particularly striking was just how decisive polarizing leaders often are. Figures like Narendra Modi in India, Jarosław Kaczyński in Poland, and Recep Tayyip Erdoğan in Turkey have relentlessly inflamed basic divisions and entrenched them throughout society (often with resounding electoral success). They’ve aggravated tensions not only by demonizing opponents and curtailing democratic processes but also by pushing for radical changes—like a total ban on abortion in Poland.

The more we looked at the experiences of other divided democracies, the more we realized that U.S. polarization stands out as unusual. It has several distinctive features, and unfortunately, all of them spell trouble for U.S. democracy.

It routinely undermines the independence of the judiciary, as politicians attack the courts as biased or pack them with loyalists. It reduces legislatures either to gridlock or to a rubberstamp function. In presidential systems, it frequently leads to the abuse of executive powers and promotes the toxic view that the president represents only his or her supporters, rather than the country as a whole.

These consequences generate a vicious cycle of rising polarization. Attacks on the judiciary, for example, only diminish its capacity to arbitrate conflict and heighten distrust between the opposing sides.

Polarization politics

Perhaps most fundamentally, polarization shatters informal but crucial norms of tolerance and moderation—like conceding peacefully after an electoral defeat—that keep political competition within bounds.

When using a microscope we do not see the entire extent of the sample. Depending on the eyepiece and objective lens we see a restricted region which we say is the field of view. The objective is then manipulated in two-dimensions above the sample to view other regions of the sample. Electronic scanning of either the objective or the sample is used in scanning microscopy. The image formed at each point during the scanning is combined using a computer to generate an image of a larger region of the sample at a selected magnification.

Still, these initiatives are small compared to the larger forces driving polarization. Democracies will need to rise to this challenge in new and determined ways if they are to swim successfully against the swelling global current of polarization.

While the numerical aperture can be used to compare resolutions of various objectives, it does not indicate how far the lens could be from the specimen. This is specified by the “working distance,” which is the distance (in mm usually) from the front lens element of the objective to the specimen, or cover glass. The higher the NA the closer the lens will be to the specimen and the more chances there are of breaking the cover slip and damaging both the specimen and the lens. The focal length of an objective lens is different than the working distance. This is because objective lenses are made of a combination of lenses and the focal length is measured from inside the barrel. The working distance is a parameter that microscopists can use more readily as it is measured from the outermost lens. The working distance decreases as the NA and magnification both increase.

Faced with Donald Trump’s return to the White House and his threat to transatlantic relations, the EU is woefully ill-equipped to act swiftly on foreign policy and security issues. An EU Security Council would go a long way in empowering it to respond more effectively.

Figure 7. An electron microscope has the capability to image individual atoms on a material. The microscope uses vacuum technology, sophisticated detectors and state of the art image processing software. (credit: Dave Pape)

Polarization in Physics

This situation is similar to that shown in Figure 2. To find the overall magnification, we must find the magnification of the objective, then the magnification of the eyepiece. This involves using the thin lens equation.

That finding gave us pause: it showed us that the potential for destructive divisions exists in almost all societies, even ones that seem relatively homogeneous. Our research underscores just how vulnerable democracies are to polarization—and how powerful the factors fueling divisions are.

A lot of research shows how populist and illiberal leaders are putting democracy in danger. But it rarely addresses what we feel is a more fundamental, underlying problem: severe political polarization.

As the f-number decreases, the camera is able to gather light from a larger angle, giving wide-angle photography. As usual there is a trade-off. A greater f/# means less light reaches the image plane. A setting of f/16 usually allows one to take pictures in bright sunlight as the aperture diameter is small. In optical fibers, light needs to be focused into the fiber. Figure 4 shows the angle used in calculating the NA of an optical fiber.

[latex]m_{\text{e}}=-\frac{d_{\text{i}}\prime}{d_{\text{o}}\prime}=-\frac{-367\text{ mm}}{44.0\text{ mm}}=8.33\\[/latex].

Figure 4. Light rays enter an optical fiber. The numerical aperture of the optical fiber can be determined by using the angle αmax.

Although the eye is marvelous in its ability to see objects large and small, it obviously has limitations to the smallest details it can detect. Human desire to see beyond what is possible with the naked eye led to the use of optical instruments. In this section we will examine microscopes, instruments for enlarging the detail that we cannot see with the unaided eye. The microscope is a multiple-element system having more than a single lens or mirror. (See Figure 1.) A microscope can be made from two convex lenses. The image formed by the first element becomes the object for the second element. The second element forms its own image, which is the object for the third element, and so on. Ray tracing helps to visualize the image formed. If the device is composed of thin lenses and mirrors that obey the thin lens equations, then it is not difficult to describe their behavior numerically.

Intense partisanship has gripped the United States for an unusually long time and thus become ingrained in social and political life. Today’s divisions date back at least to the 1960s and have been steadily intensifying for over fifty years. Most other current cases of polarization are more recent in origin.