And the differences between right and left go beyond disagreements over politics, friends and neighbors. If they could choose anywhere to live, three-quarters of consistent conservatives prefer a community where “the houses are larger and farther apart, but schools, stores, and restaurants are several miles away.” The preferences of consistent liberals are almost the exact inverse, with 77% saying they’d chose to live where “the houses are smaller and closer to each other, but schools, stores, and restaurants are within walking distance.”

These are among the findings of the largest study of U.S. political attitudes ever undertaken by the Pew Research Center. Data are drawn from a national telephone survey of 10,013 adults, conducted from January through March of this year, and an ongoing series of follow-up surveys. This rich dataset, coupled with trends and insights from two decades of Pew Research Center polling, reveals a complex picture of partisan polarization and how it manifests itself in political behaviors, policy debates, election dynamics and everyday life.

ABOUT PEW RESEARCH CENTER Pew Research Center is a nonpartisan, nonadvocacy fact tank that informs the public about the issues, attitudes and trends shaping the world. It does not take policy positions. The Center conducts public opinion polling, demographic research, computational social science research and other data-driven research. Pew Research Center is a subsidiary of The Pew Charitable Trusts, its primary funder.

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If a collimated Gaussian beam with beam radius <$w_0$> hits a focusing lens with focal length <$f$>, the beam radius of the beam waist (focus) after the lens can be calculated with the equation

And while few Americans overall go so far as to voice disappointment with the prospect of a family member marrying a Democrat (8%) or a Republican (9%), that sentiment is not uncommon on the left or the right. Three-out-of-ten (30%) consistent conservatives say they would be unhappy if an immediate family member married a Democrat and about a quarter (23%) of across-the-board liberals say the same about the prospect of a Republican in-law.

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On measure after measure – whether primary voting, writing letters to officials, volunteering for or donating to a campaign – the most politically polarized are more actively involved in politics, amplifying the voices that are the least willing to see the parties meet each other halfway.

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Note that the locations of the left and right edges of the optical system (e.g. positions of outer lens surfaces, optical windows etc.) or its housing are not relevant for those definitions.

Ordinary lenses, working on the basis of refraction, have a focal length which is slightly wavelength-dependent due to the wavelength dependence of the refractive index (–> chromatic dispersion). This effect leads to chromatic aberrations of imaging systems and similar problems in other applications where an optical system is used for a wide range of optical wavelengths. Lens combinations (e.g., objectives for photographic cameras) can be designed such that chromatic aberrations are minimized. Most common is the use of achromatic doublets, i.e., lenses consisting of two different glass materials chosen such that the overall chromatic aberrations are largely canceled.

Among all Democrats, 27% say GOP policies are a threat to the well-being of the country;  among all Republicans, more than a third (36%) think Democratic policies threaten the nation.

With Barack Obama in the White House, partisan antipathy is more pronounced among Republicans, especially consistently conservative Republicans. Overall, more Republicans than Democrats see the opposing party’s policies as a threat and the differences are even greater when ideology is taken into account. Fully 66% of consistently conservative Republicans think the Democrats’ policies threaten the nation’s well-being. By comparison, half (50%) of consistently liberal Democrats say Republican policies jeopardize the nation’s well-being. Conservatives also exhibit more partisan behavior in their personal lives; they are the most likely to have friends and prefer communities of like-minded people.

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In 1994, hardly a time of amicable partisan relations, a majority of Republicans had unfavorable impressions of the Democratic Party, but just 17% had very unfavorable opinions. Similarly, while most Democrats viewed the GOP unfavorably, just 16% had very unfavorable views. Since then, highly negative views have more than doubled: 43% of Republicans and 38% of Democrats now view the opposite party in strongly negative terms.

Curved mirrors are often used for focusing or defocusing light. For example, within laser resonators curved laser mirrors with dielectric coatings are more commonly used than lenses, mainly because they introduce lower losses.

The focal distance should also not be confused with the working distance, which is the distance between a specimen and the lens housing. Note that a specimen is not necessarily placed in the focal plane, e.g. when the input light to an objective is not collimated.

If a divergent (rather than collimated) beam hits a focusing lens, the distance <$b$> from the lens to the focus becomes larger than <$f$> (Figure 2). The lens equation states that

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The rise of ideological uniformity has been much more pronounced among those who are the most politically active. Today, almost four-in-ten (38%) politically engaged Democrats are consistent liberals, up from just 8% in 1994. The change among Republicans since then appears less dramatic – 33% express consistently conservative views, up from 23% in the midst of the 1994 “Republican Revolution.” But a decade ago, just 10% of politically engaged Republicans had across-the-board conservative attitudes.

These sentiments are not shared by all – or even most – Americans. The majority do not have uniformly conservative or liberal views. Most do not see either party as a threat to the nation. And more believe their representatives in government should meet halfway to resolve contentious disputes rather than hold out for more of what they want.

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Partisan animosity has increased substantially over the same period. In each party, the share with a highly negative view of the opposing party has more than doubled since 1994. Most of these intense partisans believe the opposing party’s policies “are so misguided that they threaten the nation’s well-being.”

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The data in this report are based on two independent survey administrations with the same randomly selected, nationally representative group of respondents. The first is the center’s largest survey on domestic politics to date: the 2014 Political Polarization and Typology Survey, a national telephone survey of 10,013 adults, on landlines and cell phones, from January through March of this year. The second involved impaneling a subset of these respondents into the newly created American Trends Panel and following up with them via a survey conducted by web and telephone. The two surveys are described separately, in further detail, in the About the Surveys section of the report.

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The overall share of Americans who express consistently conservative or consistently liberal opinions has doubled over the past two decades from 10% to 21%. And ideological thinking is now much more closely aligned with partisanship than in the past. As a result, ideological overlap between the two parties has diminished: Today, 92% of Republicans are to the right of the median Democrat, and 94% of Democrats are to the left of the median Republican.

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Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

Curved laser mirrors usually have a curvature radius somewhere between 10 mm and 5 m. The fabrication of dielectric mirror coatings can be more difficult for very strongly curved mirror substrates, but with refined techniques it is possible to reach focal lengths of only a few millimeters, as required for some miniature lasers.

The second report, coming in a few weeks, is the new Pew Research Center Political Typology. The typology – the sixth such study since 1987 – looks beyond Red vs. Blue divisions to gain a clearer understanding of the dynamic nature of the “center” of the American electorate, and the internal divides on both the left and the right.

The formula ignores the constant part of the optical phase change as well as optical aberrations. Note that depending on the function of the lens – for example, focusing collimated input beams or refocusing divergent light –, higher-order terms in the phase profile may be required to avoid optical aberrations.

However, there is as much ideological uniformity on the left as the right. The share of Democrats holding consistently liberal views has grown steadily over the past 20 years, quadrupling from 5% in 1994 to 23% today. Social issues like homosexuality and immigration that once drove deep divides within the Democratic Party are now areas of relative consensus. And Democrats have become more uniformly critical of business and more supportive of government.

Later, the project will explore the various factors that contribute to political polarization, or stem from it. A September report will examine how political polarization is linked to people’s information environments: Their news sources, social media habits and interpersonal communication networks. Other reports will look at how political polarization relates to where people live, to their political environments, to how they view themselves and others around them, to their socioeconomic circumstances, to generational changes and to broader sociological and psychological personality traits.

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“Ideological silos” are now common on both the left and right. People with down-the-line ideological positions – especially conservatives – are more likely than others to say that most of their close friends share their political views. Liberals and conservatives disagree over where they want to live, the kind of people they want to live around and even whom they would welcome into their families.

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For a focusing lens, this means a reduced phase delay for increasing <$r$> coordinate. Note that there are different sign conventions in wave optics, where a phase delay can correspond to a positive or negative change of a phasor (complex amplitude).

And at a time of increasing gridlock on Capitol Hill, many on both the left and the right think the outcome of political negotiations between Obama and Republican leaders should be that their side gets more of what it wants.

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To chart the progression of ideological thinking, responses to 10 political values questions asked on multiple Pew Research surveys since 1994 have been combined to create a measure of ideological consistency. Over the past twenty years, the number of Americans in the “tails” of this ideological distribution has doubled from 10% to 21%. Meanwhile, the center has shrunk: 39% currently take a roughly equal number of liberal and conservative positions. That is down from about half (49%) of the public in surveys conducted in 1994 and 2004.

Different sign conventions for focal lengths are used in the literature. For example, one may have a negative front focal length if the front focal point lies before the front principal plane. Obviously, any equations involving focal lengths should be used with the assumed sign conventions.

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People on the right and left are more likely to say it is important to them to live in a place where most people share their political views

Beyond the rise in ideological consistency, another major element in polarization has been the growing contempt that many Republicans and Democrats have for the opposing party. To be sure, disliking the other party is nothing new in politics. But today, these sentiments are broader and deeper than in the recent past.

Many of those in the center remain on the edges of the political playing field … while the most ideologically oriented and politically rancorous Americans make their voices heard

Yet many of those in the center remain on the edges of the political playing field, relatively distant and disengaged, while the most ideologically oriented and politically rancorous Americans make their voices heard through greater participation in every stage of the political process.

Yet an equitable deal is in the eye of the beholder, as both liberals and conservatives define the optimal political outcome as one in which their side gets more of what it wants. A majority of consistent conservatives (57%) say the ideal agreement between President Obama and congressional Republicans is one in which GOP leaders hold out for more of their goals. Consistent liberals take the opposite view: Their preferred terms (favored by 62%) end up closer to Obama’s position than the GOP’s.

Considerable confusion arises from the fact that in the context of photo cameras the term effective focal length is also used with an entirely different meaning, as explained in the following.

Republicans and Democrats are more divided along ideological lines – and partisan antipathy is deeper and more extensive – than at any point in the last two decades. These trends manifest themselves in myriad ways, both in politics and in everyday life. And a new survey of 10,000 adults nationwide finds that these divisions are greatest among those who are the most engaged and active in the political process.

Whether this rule can also be applied to an extended optical system with focal length <$f$> depends on the applied definition of <$f$>. It is useful to specify an effective focal length which is valid for such relations.

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The Model 02-014-1 is an air spaced, computer designed multi-element lens that is diffraction-limited when used with fibers having core diameters as large as 1200 microns. Operational wavelength is 1064 nm. The 02-014 reimages the emitting surface of the fiber with a 0.67× demagnification. Focused spot sizes are significantly smaller than those achievable with single-element lenses used in a similar collimating/focusing configuration. All optical elements are fabricated from high laser-damage resistant glasses and are anti-reflection coated to reduce reflectance per surface to 0.13%. The working distance between the lens and the target is sufficiently large to allow use of a gas nozzle to enhance the cutting or welding process and to prevent debris from depositing on the lens surface.

where it is assumed that the beam radius at the focus is much smaller than the initial beam radius <$w_0$>. (This condition is violated for beams with a too small incident radius; the focus is then larger than according to the given equation.) Also, it is assumed that the beam radius is significantly larger than the wavelength <$\lambda$>, so that the paraxial approximation is valid.

To be sure, there are areas of consensus. Most Americans, regardless of their ideological preferences, value communities in which they would live close to extended family and high-quality schools. But far more liberals than conservatives think it is important that a community have racial and ethnic diversity (76% vs. 20%). At the same time, conservatives are more likely than liberals to attach importance to living in a place where many people share their religious faith (57% vs. 17% of liberals).

Various types of optical systems (e.g. microscope objectives and curved laser mirrors) can focus or defocus light, and the focal length is used for quantifying such effects. The simplest case is that of a thin focusing lens (Figure 1a). If a sufficiently large collimated beam of light is incident on the lens, the beam will be focused, and the focal length is the distance from the lens to that focus (assuming that the lens is surrounded by vacuum or air, not by some dense substance with a significant refractive index). For a defocusing lens (Figure 1b), the focal length is the distance from the lens to the virtual focus (indicated by the dashed lines), taken as a negative value. Some authors use different sign conventions, however, in particularly concerning the front and back focal length (see below).

The explained definition delivers a focal length which can also be used in equations for the size of the focus (see below), for example.

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Unfortunately, the terms are also used differently by other authors. For example, it happens that a focal distance is assumed to be the same as a focal length. Therefore, some product catalogs specify focal lengths, which should actually be called focal distances, and in addition the effective focal length.

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Lower apertures (small F-stop values) will result in less depth of field and sharpness but will allow you to capture more light. Higher apertures (high F-stop ...

The current report is divided into five parts: The first two focus on measuring the nature and scope of political polarization, emphasizing the difference between growing ideological consistency and rising partisan antipathy. The third looks closely at how polarization manifests itself in people’s personal lives. The fourth looks at the relationship between polarization and practical policymaking, and the fifth digs deeper into how political participation both amplifies and reflects polarization.

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For an optical system, which may consist of multiple lenses and other optical elements, the above definition of the focal length cannot be used, as it is not clear a priori for an extended system where to measure the distance to the focus: from the entrance into the optical system, from the exit, the middle, or some other position? In principle, an arbitrary definition of a reference point (e.g. the entrance or the middle) could be used, but that would in general mean that some common rules can not be applied, which e.g. hold for the radius of the beam waist at a focus behind some lens with a given focal length (see below), or the possible magnification of a telescope containing that optical system.

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Changes in ideological consistency on the right have followed a different course. In 1994, during the “Republican Revolution,” 13% of Republicans were consistent conservatives. That figure fell to 6% a decade later during George W. Bush’s presidency, before rebounding to 20% today. This increase has come despite more moderate views among Republicans on issues like homosexuality and immigration, as GOP thinking on issues related to government and the economy has veered sharply to the right.

A common (but not universally used) approach for the definition of focal lengths of extended systems is based on geometrical optics. For finding the front focal point, one calculates rays which are horizontal on the back side (see Figure 2), using the paraxial approximation. The optical system is considered as a “black box”, where one does not care about the actual ray paths; instead, one works with internal rays which are extrapolated from the outer rays. Based on those extrapolated rays, one can define the front principal plane (or first principal plane). The front focal length is then the distance between the front focal point (in the front focal plane) and the front principal plane (see Figure 2). Some authors define the focal length to be negative in the situation of Figure 2 because the focal point is located before the front principal plane; others take the absolute value.

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where <$a$> is the distance from the original focus to the lens. This shows that <$b \approx f$> if <$a \gg f$>, but <$b > f$> otherwise. That relation can be intuitively understood: a focusing power <$1 / a$> would be required to collimate the incident beam (i.e. to remove its beam divergence), so that only a focusing power <$1 / f - 1 / a$> is left for focusing.

And this shift represents both Democrats moving to the left and Republicans moving to the right, with less and less overlap between the parties. Today, 92% of Republicans are to the right of the median (middle) Democrat, compared with 64% twenty years ago. And 94% of Democrats are to the left of the median Republican, up from 70% in 1994.

Similarly, one can define the back focal plane (or second focal plane) and back principal plane (or second principal plane), where horizontal rays occur on the left side, while on the right side one has converging rays for a focusing system and diverging rays for a defocusing system. If the refractive index is the same on the input and output side (e.g. ≈1 for air), the front focal length and back focal length are identical (apart from possible sign differences used by some authors) and can thus simply called the focal length. The two principal planes, however, generally do not coincide for thick lenses, and they can even lie outside a lens.

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Liberals and conservatives share a passion for politics. They are far more likely than those with more mixed ideological views to discuss politics on a weekly or daily basis. But for many, particularly on the right, those conversations may not include much in the way of opposing opinions.

A lens with a given focal length <$f$> (taken as positive in the case of a focusing lens) creates a radially varying phase delay for a laser beam according to the following equation:

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One may eliminate chromatic aberrations altogether by using optical systems with mirrors only. A curved mirror with radius of curvature <$R$> has a focal length <$f = R / 2$> (for normal incidence), determined only by the geometry and thus independent of the wavelength. On the other hand, for non-normal incidence the focal length in the tangential direction is decreased by the cosine of the angle of incidence, and increased by the inverse cosine of that angle in the sagittal direction. Therefore, such mirrors can introduce astigmatism.

The dioptric power (also called focusing power) of a lens is defined as the inverse of the effective focal length (which is the same is the front and back focal length if the median on both sides of the optics is the same). This means that a strongly focusing lens has a small focal length, but a large dioptric power. For prescription glasses, it is common the specify the dioptric power, whereas the focal length is specified for standard lenses, microscope objectives, and photographic objectives.

Today 92% of Republicans are to the right of the median Democrat, and 94% of Democrats are to the left of the median Republican

Even these numbers tell only part of the story. Those who have a very unfavorable impression of each party were asked: “Would you say the party’s policies are so misguided that they threaten the nation’s well-being, or wouldn’t you go that far?” Most who were asked the question said yes, they would go that far. Among all Democrats, 27% say the GOP is a threat to the well-being of the country. That figure is even higher among Republicans, 36% of whom think Democratic policies threaten the nation.

The following equation allows one to calculate the dioptric power and thus the focal length of lens made of a material with refractive index <$n$> and with curvature radii <$R_1$> and <$R_2$> on the two surfaces:

When they look at a political system in which little seems to get done, most Americans in the center of the electorate think that Obama and Republican leaders should simply meet each other halfway in addressing the issues facing the nation.

For some applications, in particular for focusing of imaging systems, it is essential than the focal length of an optical system can be fine adjusted. The following physical principles can be used:

The equation shows that what determines the minimum possible beam radius is not the focal length <$f$> alone, but rather the ratio of <$f$> to the radius of the open aperture of the lens, which sets a maximum to the input beam radius <$w_0$>. For a focusing or collimation lens, that ratio is essentially the numerical aperture of the lens.

In contrast to focal lengths, focal distances are related not to the principal planes but rather to the vertex points of lenses (not caring about a housing, which may be further extended). The front focal distance is thus the distance between front focal point and the entrance surface of the optics, while the back focal distance is the distance between the back surface and the back focal point.

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Note that the lens equation applies for rays, assuming that the paraxial approximation is valid, i.e., all angles relative to the beam axis remain small.

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Nearly two-thirds (63%) of consistent conservatives and about half (49%) of consistent liberals say most of their close friends share their political views. Among those with mixed ideological values, just 25% say the same. People on the right and left also are more likely to say it is important to them to live in a place where most people share their political views, though again, that desire is more widespread on the right (50%) than on the left (35%).

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The angle of view of the camera is determined by the ratio of the image size on the film and the focal length. Film-based cameras have for a long time mostly used 35-mm film (also called 135 film according to ISO Standard 1007), where the image size on the film is typically 36 mm × 24 mm. (The width of the film spool is 35 mm, somewhat larger than 24 mm, as the picture does not extend to the edges of the spool.) A standard objective then has a focal length of 50 mm. However, modern digital cameras (particularly the more compact ones) often contain image sensors which are smaller than 36 mm × 24 mm, so that an objective lens with a correspondingly smaller focal length (e.g. 32 mm instead of 50 mm) is required for obtaining the same field of view. As many photographers are still used to the previously valid relation between focal length and angle of view, it has become common to specify the effective focal length of an objective of a digital camera as that focal length which would give the same angle of view in combination with ordinary 35-mm film. For example, an objective with a true focal length of 32 mm may then be said to have an effective focal length of 50 mm and thus function as a standard objective, rather than e.g. a macro or tele objective.

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Consistent liberals and conservatives define ideal political compromise as one in which their side gets more of what it wants

The curvature radii are taken as positive values for convex surfaces and negative for concave surfaces. Positive results are obtained for focusing lenses, negative results for defocusing lenses. The last term is relevant only for thick lenses with substantial curvature on both sides. The formula delivers the focal length within the paraxial approximation, not considering spherical aberrations, for example.

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For a defocusing system, the front focal plane can be located on the output side; it contains a virtual focal point. Again, the focal length is the distance between principal plane and focal plane.

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The signs of political polarization are evident on both ends of the political spectrum, though the trajectory, nature and extent differ from left to right.

This is the first report of a multi-part series based on a national survey of 10,013 adults nationwide, conducted January 23-March 16, 2014 by the Pew Research Center. The survey, funded in part through grants from the William and Flora Hewlett Foundation, the John D. and Catherine T. MacArthur Foundation and supported by the generosity of Don C. and Jeane M. Bertsch, is aimed at understanding the nature and scope of political polarization in the American public, and how it interrelates with government, society and people’s personal lives.

A mirror with a curvature radius <$R$> of the surface has a focal length <$f = R / 2$>, if the beam axis is normal to the mirror surface. (We take positive signs for concave curvatures and focusing mirrors.) If there is some non-zero angle <$\theta$> between the beam axis and the normal direction, the focal length is <$f_{\tan} = (R / 2) \cdot \cos \theta$> in the tangential direction (i.e., within the plane of incidence) and <$f_{sag} = (R / 2) / \cos \theta$> in the sagittal direction.