Optics is the branch of physics that studies the behaviour and properties of light , including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible , ultraviolet , and infrared light. Light is a type of electromagnetic radiation , and other forms of electromagnetic radiation such as X-rays , microwaves , and radio waves exhibit similar properties.
171-397: A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction . A simple lens consists of a single piece of transparent material , while a compound lens consists of several simple lenses ( elements ), usually arranged along a common axis . Lenses are made from materials such as glass or plastic and are ground , polished , or molded to
342-748: A general armistice ended the fighting. In the German Revolution (November 1918), Wilhelm II and the ruling princes abdicated their positions, and Germany was declared a federal republic . Germany's new leadership signed the Treaty of Versailles in 1919, accepting defeat by the Allies . Germans perceived the treaty as humiliating, which was seen by historians as influential in the rise of Adolf Hitler . Germany lost around 13% of its European territory and ceded all of its colonial possessions in Africa and
513-513: A sustainable energy supply, the debt brake for balanced budgets, measures to increase the fertility rate ( pronatalism ), and high-tech strategies for the transition of the German economy, summarised as Industry 4.0 . During the 2015 European migrant crisis , the country took in over a million refugees and migrants. Germany is the seventh-largest country in Europe . It borders Denmark to
684-567: A temperate climate, ranging from oceanic in the north and west to continental in the east and southeast. Winters range from the cold in the Southern Alps to cool and are generally overcast with limited precipitation, while summers can vary from hot and dry to cool and rainy. The northern regions have prevailing westerly winds that bring in moist air from the North Sea, moderating the temperature and increasing precipitation. Conversely,
855-408: A 1758 patent. Developments in transatlantic commerce were the impetus for the construction of modern lighthouses in the 18th century, which utilize a combination of elevated sightlines, lighting sources, and lenses to provide navigational aid overseas. With maximal distance of visibility needed in lighthouses, conventional convex lenses would need to be significantly sized which would negatively affect
1026-460: A NATO effort to provide security in that country after the ousting of the Taliban . In the 2005 elections , Angela Merkel became the first female chancellor. In 2009, the German government approved a €50 billion stimulus plan. Among the major German political projects of the early 21st century are the advancement of European integration , the energy transition ( Energiewende ) for
1197-405: A biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it
1368-430: A broad band, or extremely low reflectivity at a single wavelength. Constructive interference in thin films can create a strong reflection of light in a range of wavelengths, which can be narrow or broad depending on the design of the coating. These films are used to make dielectric mirrors , interference filters , heat reflectors , and filters for colour separation in colour television cameras. This interference effect
1539-658: A centralised totalitarian state , withdrew from the League of Nations , and dramatically increased the country's rearmament . A government-sponsored programme for economic renewal focused on public works, the most famous of which was the Autobahn . In 1935, the regime withdrew from the Treaty of Versailles and introduced the Nuremberg Laws which targeted Jews and other minorities. Germany also reacquired control of
1710-619: A changing index of refraction; this principle allows for lenses and the focusing of light. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n 1 and another medium with index of refraction n 2 . In such situations, Snell's Law describes the resulting deflection of the light ray: n 1 sin θ 1 = n 2 sin θ 2 {\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}} where θ 1 and θ 2 are
1881-399: A converging lens has positive focal length, while a diverging lens has negative focal length. Smaller focal length indicates that the lens has a stronger converging or diverging effect. The focal length of a simple lens in air is given by the lensmaker's equation . Ray tracing can be used to show how images are formed by a lens. For a thin lens in air, the location of the image is given by
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#17327872467082052-615: A framework laid out in the 1949 constitution known as the Grundgesetz (Basic Law). Amendments generally require a two-thirds majority of both the Bundestag and the Bundesrat ; the fundamental principles of the constitution, as expressed in the articles guaranteeing human dignity, the separation of powers, the federal structure, and the rule of law , are valid in perpetuity. The president , currently Frank-Walter Steinmeier ,
2223-543: A global power in industrial, scientific and technological sectors, it is both the world's third-largest exporter and importer . As a developed country , it offers social security , a universal health care system , and tuition-free university education . Germany is a member of the United Nations , Council of Europe , NATO and OECD , and a founding member of the European Union, G7 and G20 . It has
2394-680: A joint responsibility of the international community. It was the world's second-biggest aid donor in 2019 after the United States. Germany's military, the Bundeswehr (Federal Defence), is organised into the Heer (Army and special forces KSK ), Marine (Navy), Luftwaffe (Air Force) and Cyber- und Informationsraum (Cyber and Information Domain Service) branches. In absolute terms, German military spending in 2023
2565-536: A lens in air, f is then given by 1 f ≈ ( n − 1 ) [ 1 R 1 − 1 R 2 ] . {\displaystyle \ {\frac {1}{\ f\ }}\approx \left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\ \right]~.} The spherical thin lens equation in paraxial approximation
2736-572: A magnifying glass, or a burning glass. Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses". The oldest certain reference to the use of lenses is from Aristophanes ' play The Clouds (424 BCE) mentioning a burning-glass. Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period. Pliny also has the earliest known reference to
2907-547: A provisional capital, to emphasise its stance that the two-state solution was temporary. West Germany was established as a federal parliamentary republic with a " social market economy ". Starting in 1948 West Germany became a major recipient of reconstruction aid under the American Marshall Plan . Konrad Adenauer was elected the first federal chancellor of Germany in 1949. The country enjoyed prolonged economic growth ( Wirtschaftswunder ) beginning in
3078-477: A single scalar quantity to represent the electric field of the light wave, rather than using a vector model with orthogonal electric and magnetic vectors. The Huygens–Fresnel equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of superposition of waves. The Kirchhoff diffraction equation , which
3249-522: A single point on the image, while chromatic aberration occurs because the index of refraction of the lens varies with the wavelength of the light. In physical optics, light is considered to propagate as waves. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The speed of light waves in air is approximately 3.0×10 m/s (exactly 299,792,458 m/s in vacuum ). The wavelength of visible light waves varies between 400 and 700 nm, but
3420-437: A spectrum. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton. Some media have an index of refraction which varies gradually with position and, therefore, light rays in the medium are curved. This effect is responsible for mirages seen on hot days: a change in index of refraction air with height causes light rays to bend, creating the appearance of specular reflections in
3591-829: A spherical lens in air or vacuum for paraxial rays can be calculated from the lensmaker's equation : 1 f = ( n − 1 ) [ 1 R 1 − 1 R 2 + ( n − 1 ) d n R 1 R 2 ] , {\displaystyle {\frac {1}{\ f\ }}=\left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ \left(n-1\right)\ d~}{\ n\ R_{1}\ R_{2}\ }}\ \right]\ ,} where The focal length f {\textstyle \ f\ }
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#17327872467083762-638: A state, the East German authorities eased border restrictions, but this actually led to an acceleration of the Wende reform process culminating in the Two Plus Four Treaty under which Germany regained full sovereignty. This permitted German reunification on 3 October 1990, with the accession of the five re-established states of the former GDR. The fall of the Wall in 1989 became a symbol of
3933-712: A symbol of the Cold War . Tensions between East and West Germany were reduced in the late 1960s by Chancellor Willy Brandt 's Ostpolitik . In 1989, Hungary decided to dismantle the Iron Curtain and open its border with Austria , causing the emigration of thousands of East Germans to West Germany via Hungary and Austria. This had devastating effects on the GDR, where regular mass demonstrations received increasing support. In an effort to help retain East Germany as
4104-465: A thickness of one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550 nm. More complex designs using multiple layers can achieve low reflectivity over
4275-528: A total population of over 82 million in an area of 357,596 km (138,069 sq mi), making it the most populous member state of the European Union . It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France , Luxembourg , Belgium , and the Netherlands to the west. The nation's capital and most populous city
4446-476: A variety of technologies and everyday objects, including mirrors , lenses , telescopes , microscopes , lasers , and fibre optics . Optics began with the development of lenses by the ancient Egyptians and Mesopotamians . The earliest known lenses, made from polished crystal , often quartz , date from as early as 2000 BC from Crete (Archaeological Museum of Heraclion, Greece). Lenses from Rhodes date around 700 BC, as do Assyrian lenses such as
4617-525: A wide range of scientific topics, and discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light, basing it on the works of Aristotle and Platonism. Grosseteste's most famous disciple, Roger Bacon , wrote works citing a wide range of recently translated optical and philosophical works, including those of Alhazen, Aristotle, Avicenna , Averroes , Euclid, al-Kindi, Ptolemy, Tideus, and Constantine
4788-658: Is Berlin and its main financial centre is Frankfurt ; the largest urban area is the Ruhr . Settlement in the territory of modern Germany began in the Lower Paleolithic , with various tribes inhabiting it from the Neolithic onward, chiefly the Celts . Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity . A region named Germania was documented before AD 100. In 962,
4959-403: Is biconvex (or double convex , or just convex ) if both surfaces are convex . If both surfaces have the same radius of curvature, the lens is equiconvex . A lens with two concave surfaces is biconcave (or just concave ). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side
5130-407: Is convex-concave or meniscus . Convex-concave lenses are most commonly used in corrective lenses , since the shape minimizes some aberrations. For a biconvex or plano-convex lens in a lower-index medium, a collimated beam of light passing through the lens converges to a spot (a focus ) behind the lens. In this case, the lens is called a positive or converging lens. For a thin lens in air,
5301-445: Is h ), and v {\textstyle v} is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, u {\textstyle u} and v {\textstyle v} are also considered distances with respect to the vertex. Moving v {\textstyle v} toward
Lens - Misplaced Pages Continue
5472-427: Is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. In the absence of nonlinear effects,
5643-771: Is also what causes the colourful rainbow patterns seen in oil slicks. Germany – in Europe (light green & dark grey) – in the European Union (light green) Germany , officially the Federal Republic of Germany , is a country in Central Europe . It lies between the Baltic and North Sea to the north and the Alps to the south. Its sixteen constituent states have
5814-529: Is called the focal plane . For paraxial rays , if the distances from an object to a spherical thin lens (a lens of negligible thickness) and from the lens to the image are S 1 and S 2 respectively, the distances are related by the (Gaussian) thin lens formula : Optics Most optical phenomena can be accounted for by using the classical electromagnetic description of light, however complete electromagnetic descriptions of light are often difficult to apply in practice. Practical optics
5985-416: Is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most optical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, optical aberration is much worse than thin lenses, with the notable exception of chromatic aberration . For
6156-486: Is considered to travel in straight lines, while in physical optics, light is considered as an electromagnetic wave. Geometrical optics can be viewed as an approximation of physical optics that applies when the wavelength of the light used is much smaller than the size of the optical elements in the system being modelled. Geometrical optics , or ray optics , describes the propagation of light in terms of "rays" which travel in straight lines, and whose paths are governed by
6327-793: Is derived here with respect to the right figure. The 1st spherical lens surface (which meets the optical axis at V 1 {\textstyle \ V_{1}\ } as its vertex) images an on-axis object point O to the virtual image I , which can be described by the following equation, n 1 u + n 2 v ′ = n 2 − n 1 R 1 . {\displaystyle \ {\frac {\ n_{1}\ }{\ u\ }}+{\frac {\ n_{2}\ }{\ v'\ }}={\frac {\ n_{2}-n_{1}\ }{\ R_{1}\ }}~.} For
6498-491: Is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens–Fresnel principle can be found in the articles on diffraction and Fraunhofer diffraction . More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect
6669-648: Is devoted to agriculture, while 30% is forested and 14% is covered by settlements or infrastructure. Plants and animals include those generally common to Central Europe. According to the National Forest Inventory, beeches , oaks , and other deciduous trees constitute just over 40% of the forests; roughly 60% are conifers , particularly spruce and pine . There are many species of ferns , flowers , fungi , and mosses . Wild animals include roe deer , wild boar , mouflon (a subspecies of wild sheep), fox , badger , hare , and small numbers of
6840-460: Is further along in the direction of the ray travel (right, in the accompanying diagrams), while negative R means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, R 1 > 0 and R 2 < 0 indicate convex surfaces (used to converge light in a positive lens), while R 1 < 0 and R 2 > 0 indicate concave surfaces. The reciprocal of
7011-608: Is largely autonomous in regard to its internal organisation. As of 2017 , Germany is divided into 401 districts ( Kreise ) at a municipal level; these consist of 294 rural districts and 107 urban districts . Germany has a civil law system based on Roman law with some references to Germanic law . The Bundesverfassungsgericht (Federal Constitutional Court) is the German Supreme Court responsible for constitutional matters, with power of judicial review . Germany's specialized supreme court system includes
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7182-407: Is negative with respect to the focal length of a converging lens. The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it. Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of
7353-433: Is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, 1 f , {\textstyle \ {\tfrac {1}{\ f\ }}\ ,} is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (reciprocal metres). Lenses have the same focal length when light travels from
7524-653: Is the head of state and invested primarily with representative responsibilities and powers. He is elected by the Bundesversammlung (federal convention), an institution consisting of the members of the Bundestag and an equal number of state delegates. The second-highest official in the German order of precedence is the Bundestagspräsident ( President of the Bundestag ), who is elected by
7695-400: Is the radius of the spherical surface, n 2 is the refractive index of the material of the surface, n 1 is the refractive index of medium (the medium other than the spherical surface material), u {\textstyle u} is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height
7866-409: Is to the lens, the further the image is from the lens. With diverging lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at a spot one focal length in front of the lens. This is the lens's front focal point. Rays from an object at a finite distance are associated with a virtual image that is closer to the lens than the focal point, and on
8037-406: Is usually done using simplified models. The most common of these, geometric optics , treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically,
8208-488: Is vested in the parliament consisting of the Bundestag (Federal Diet) and Bundesrat (Federal Council), which together form the legislative body. The Bundestag is elected through direct elections using the mixed-member proportional representation system. The members of the Bundesrat represent and are appointed by the governments of the sixteen federated states. The German political system operates under
8379-1134: Is with respect to the principal planes of the lens, and the locations of the principal planes h 1 {\textstyle \ h_{1}\ } and h 2 {\textstyle \ h_{2}\ } with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex. h 1 = − ( n − 1 ) f d n R 2 {\displaystyle \ h_{1}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{2}\ }}\ } h 2 = − ( n − 1 ) f d n R 1 {\displaystyle \ h_{2}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{1}\ }}\ } The focal length f {\displaystyle \ f\ }
8550-463: The Gründerzeit period following the unification of Germany, Bismarck's foreign policy as chancellor of Germany secured Germany's position as a great nation by forging alliances and avoiding war. However, under Wilhelm II , Germany took an imperialistic course, leading to friction with neighbouring countries. A dual alliance was created with the multinational realm of Austria-Hungary ;
8721-491: The Bundestag and responsible for overseeing the daily sessions of the body. The third-highest official and the head of government is the chancellor, who is appointed by the Bundespräsident after being elected by the party or coalition with the most seats in the Bundestag . The chancellor, currently Olaf Scholz , is the head of government and exercises executive power through his Cabinet . Since 1949,
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#17327872467088892-480: The Book of Optics ( Kitab al-manazir ) in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment. He rejected the "emission theory" of Ptolemaic optics with its rays being emitted by the eye, and instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and then entered
9063-889: The Allies de jure abolished the German state and partitioned Berlin and Germany's remaining territory into four occupation zones. The western sectors, controlled by France, the United Kingdom , and the United States , were merged on 23 May 1949 to form the Federal Republic of Germany ( German : Bundesrepublik Deutschland ); on 7 October 1949, the Soviet Zone became the German Democratic Republic (GDR) ( German : Deutsche Demokratische Republik ; DDR). They were informally known as West Germany and East Germany. East Germany selected East Berlin as its capital, while West Germany chose Bonn as
9234-625: The Battle of Britain in the same year. In 1941, German troops invaded Yugoslavia , Greece and the Soviet Union . By 1942, Germany and its allies controlled most of continental Europe and North Africa , but following the Soviet victory at the Battle of Stalingrad , the Allied reconquest of North Africa and invasion of Italy in 1943, German forces suffered repeated military defeats. In 1944,
9405-798: The Battle of the Teutoburg Forest . The outcome of this battle dissuaded the Romans from their ambition of conquering Germania and is thus considered one of the most important events in European history . By 100 AD, when Tacitus wrote Germania , Germanic tribes had settled along the Rhine and the Danube (the Limes Germanicus ), occupying most of modern Germany. However, Baden-Württemberg , southern Bavaria , southern Hesse and
9576-597: The Berchtesgaden National Park . In addition, there are 17 Biosphere Reserves , and 105 nature parks . More than 400 zoos and animal parks operate in Germany. The Berlin Zoo , which opened in 1844, is the oldest in Germany, and claims the most comprehensive collection of species in the world. Germany is a federal , parliamentary , representative democratic republic. Federal legislative power
9747-706: The Black Death of 1348–1350. The Golden Bull issued in 1356 provided the constitutional structure of the Empire and codified the election of the emperor by seven prince-electors . Johannes Gutenberg introduced moveable-type printing to Europe, laying the basis for the democratization of knowledge . In 1517, Martin Luther incited the Protestant Reformation and his translation of the Bible began
9918-831: The Carolingian Empire in 800; it was divided in 843 . The eastern successor kingdom of East Francia stretched from the Rhine in the west to the Elbe river in the east and from the North Sea to the Alps. Subsequently, the Holy Roman Empire emerged from it. The Ottonian rulers (919–1024) consolidated several major duchies . In 996, Gregory V became the first German Pope, appointed by his cousin Otto III , whom he shortly after crowned Holy Roman Emperor. The Holy Roman Empire absorbed northern Italy and Burgundy under
10089-639: The Celtic , Iranian , Baltic , and Slavic tribes. Southern Germany was inhabited by Celtic-speaking peoples, who belonged to the wider La Tène culture . They were later assimilated by the Germanic conquerors. Under Augustus , the Roman Empire began to invade lands inhabited by the Germanic tribes, creating a short-lived Roman province of Germania between the Rhine and Elbe rivers. In 9 AD, three Roman legions were defeated by Arminius in
10260-1186: The Eurasian beaver . The blue cornflower was once a German national symbol . The 16 national parks in Germany include the Jasmund National Park , the Vorpommern Lagoon Area National Park , the Müritz National Park , the Wadden Sea National Parks , the Harz National Park , the Hainich National Park , the Black Forest National Park , the Saxon Switzerland National Park , the Bavarian Forest National Park and
10431-660: The Fall of Communism , the dissolution of the Soviet Union , German reunification and Die Wende ("the turning point"). United Germany was considered the enlarged continuation of West Germany so it retained its memberships in international organisations. Based on the Berlin/Bonn Act (1994), Berlin again became the capital of Germany, while Bonn obtained the unique status of a Bundesstadt (federal city) retaining some federal ministries. The relocation of
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#173278724670810602-513: The Imperial Estates . The legal system initiated by a series of Imperial Reforms (approximately 1495–1555) provided for considerable local autonomy and a stronger Imperial Diet . The House of Habsburg held the imperial crown from 1438 until the death of Charles VI in 1740. Following the War of the Austrian Succession and the Treaty of Aix-la-Chapelle , Charles VI's daughter Maria Theresa ruled as empress consort when her husband, Francis I , became emperor. From 1740, dualism between
10773-401: The Kingdom of Germany formed the bulk of the Holy Roman Empire . During the 16th century, northern German regions became the centre of the Protestant Reformation . Following the Napoleonic Wars and the dissolution of the Holy Roman Empire in 1806, the German Confederation was formed in 1815. Formal unification of Germany into the modern nation-state commenced on 18 August 1866 with
10944-467: The Molotov–Ribbentrop Pact that divided Eastern Europe into German and Soviet spheres of influence. On 1 September 1939, Germany invaded Poland , beginning World War II in Europe ; Britain and France declared war on Germany on 3 September. In the spring of 1940, Germany conquered Denmark and Norway , the Netherlands , Belgium , Luxembourg , and France , forcing the French government to sign an armistice . The British repelled German air attacks in
11115-506: The Nimrud lens . The ancient Romans and Greeks filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world . The word optics comes from the ancient Greek word ὀπτική , optikē ' appearance, look ' . Greek philosophy on optics broke down into two opposing theories on how vision worked,
11286-399: The North German Confederation Treaty establishing the Prussia -led North German Confederation , which became the German Empire in 1871. After World War I and the German Revolution of 1918–1919 , the Empire was replaced by the Weimar Republic . The Nazi rise to power in 1933 led to the establishment of a totalitarian dictatorship , World War II , and the Holocaust . In 1949, after
11457-430: The Saarland in 1935, remilitarised the Rhineland in 1936, annexed Austria in 1938, annexed the Sudetenland in 1938 with the Munich Agreement , and in violation of the agreement occupied Czechoslovakia in March 1939. Kristallnacht (Night of Broken Glass) saw the burning of synagogues, the destruction of Jewish businesses, and mass arrests of Jewish people. In August 1939, Hitler's government negotiated
11628-423: The Salian emperors (1024–1125), although the emperors lost power through the Investiture controversy . Under the Hohenstaufen emperors (1138–1254), German princes encouraged German settlement to the south and east ( Ostsiedlung ). Members of the Hanseatic League , mostly north German towns, prospered in the expansion of trade. The population declined starting with the Great Famine in 1315, followed by
11799-543: The Swabian Jura , including 42,000-year-old flutes which are the oldest musical instruments ever found, the 40,000-year-old Lion Man , and the 41,000-year-old Venus of Hohle Fels . The Nebra sky disk , created during the European Bronze Age , has been attributed to a German site. The Germanic peoples are thought to have emerged from the Jastorf culture during the Nordic Bronze Age or early Iron Age . From southern Scandinavia and northern Germany , they expanded south, east, and west, coming into contact with
11970-450: The Triple Alliance of 1882 included Italy. Britain, France and Russia also concluded alliances to protect against Habsburg interference with Russian interests in the Balkans or German interference against France. At the Berlin Conference in 1884, Germany claimed several colonies including German East Africa , German South West Africa , Togoland , and Kamerun . Later, Germany further expanded its colonial empire to include holdings in
12141-401: The emission theory , the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus . Some hundred years later, Euclid (4th–3rd century BC) wrote a treatise entitled Optics where he linked vision to geometry , creating geometrical optics . He based his work on Plato's emission theory wherein he described
12312-621: The inquisitorial Federal Court of Justice for civil and criminal cases, along with the Federal Labour Court , Federal Social Court , Federal Fiscal Court , and Federal Administrative Court for other matters. Criminal and private laws are codified on the national level in the Strafgesetzbuch and the Bürgerliches Gesetzbuch respectively. The German penal system seeks the rehabilitation of
12483-468: The intromission theory and the emission theory . The intromission approach saw vision as coming from objects casting off copies of themselves (called eidola) that were captured by the eye. With many propagators including Democritus , Epicurus , Aristotle and their followers, this theory seems to have some contact with modern theories of what vision really is, but it remained only speculation lacking any experimental foundation. Plato first articulated
12654-496: The revolutions of 1848 in the German states , raising the German question. King Frederick William IV of Prussia was offered the title of emperor, but with a loss of power; he rejected the crown and the proposed constitution, a temporary setback for the movement. King William I appointed Otto von Bismarck as the Minister President of Prussia in 1862. Bismarck successfully concluded the war with Denmark in 1864 ;
12825-448: The superposition principle , which is a wave-like property not predicted by Newton's corpuscle theory. This work led to a theory of diffraction for light and opened an entire area of study in physical optics. Wave optics was successfully unified with electromagnetic theory by James Clerk Maxwell in the 1860s. The next development in optical theory came in 1899 when Max Planck correctly modelled blackbody radiation by assuming that
12996-466: The surface normal , a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays and the normal lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection . For flat mirrors , the law of reflection implies that images of objects are upright and
13167-553: The third-highest number of UNESCO World Heritage Sites , 54 , of which 51 are cultural. The English word Germany derives from the Latin Germania , which came into use after Julius Caesar adopted it for the peoples east of the Rhine . The German term Deutschland , originally diutisciu land ('the German lands'), is derived from deutsch ( cf. Dutch ), descended from Old High German diutisc 'of
13338-479: The 11th and 13th century " reading stones " were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystal Visby lenses may or may not have been intended for use as burning glasses. Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of
13509-479: The 13th century. This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, and later in the spectacle-making centres in both the Netherlands and Germany . Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses (probably without
13680-593: The 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces. Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in
13851-514: The African . Bacon was able to use parts of glass spheres as magnifying glasses to demonstrate that light reflects from objects rather than being released from them. The first wearable eyeglasses were invented in Italy around 1286. This was the start of the optical industry of grinding and polishing lenses for these "spectacles", first in Venice and Florence in the thirteenth century, and later in
14022-924: The Austrian Habsburg monarchy and the Kingdom of Prussia dominated German history. In 1772, 1793, and 1795, Prussia and Austria, along with the Russian Empire , agreed to the Partitions of Poland . During the period of the French Revolutionary Wars , the Napoleonic era and the subsequent final meeting of the Imperial Diet , most of the Free Imperial Cities were annexed by dynastic territories;
14193-979: The Gaussian thin lens equation is 1 u + 1 v = 1 f . {\displaystyle \ {\frac {1}{\ u\ }}+{\frac {1}{\ v\ }}={\frac {1}{\ f\ }}~.} For the thin lens in air or vacuum where n 1 = 1 {\textstyle \ n_{1}=1\ } can be assumed, f {\textstyle \ f\ } becomes 1 f = ( n − 1 ) ( 1 R 1 − 1 R 2 ) {\displaystyle \ {\frac {1}{\ f\ }}=\left(n-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ } where
14364-504: The German Bundestag , though they have never been part of the federal government. In the 2017 German federal election , the right-wing populist Alternative for Germany gained enough votes to attain representation in the parliament for the first time. Germany is a federation and comprises sixteen constituent states which are collectively referred to as Länder . Each state ( Land ) has its own constitution, and
14535-622: The German Democratic Republic was a communist Eastern Bloc state and member of the Warsaw Pact . After the fall of the communist led-government in East Germany, German reunification saw the former East German states join the Federal Republic of Germany on 3 October 1990 . Germany has been described as a great power with a strong economy ; it has the largest economy in Europe by nominal GDP . As
14706-550: The Holocaust , the German government persecuted minorities , including interning them in concentration and death camps across Europe. The regime systematically murdered 6 million Jews, at least 130,000 Romani , 275,000 disabled , thousands of Jehovah's Witnesses , thousands of homosexuals , and hundreds of thousands of political and religious opponents . Nazi policies in German-occupied countries resulted in
14877-548: The Huygens–Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns. Interferometry is the science of measuring these patterns, usually as a means of making precise determinations of distances or angular resolutions . The Michelson interferometer
15048-482: The Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure . Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as
15219-471: The Latin translation of an incomplete and very poor Arabic translation. The book was, however, received by medieval scholars in the Islamic world, and commented upon by Ibn Sahl (10th century), who was in turn improved upon by Alhazen ( Book of Optics , 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century ( Eugenius of Palermo 1154). Between
15390-630: The Pacific and China. The colonial government in South West Africa (present-day Namibia ), from 1904 to 1907, carried out the annihilation of the local Herero and Namaqua peoples as punishment for an uprising; this was the 20th century's first genocide . The assassination of Austria's crown prince on 28 June 1914 provided the pretext for Austria-Hungary to attack Serbia and trigger World War I . After four years of warfare, in which approximately two million German soldiers were killed,
15561-684: The Pacific. On 11 August 1919, President Friedrich Ebert signed the democratic Weimar Constitution . In the subsequent struggle for power, communists seized power in Bavaria , but conservative elements elsewhere attempted to overthrow the Republic in the Kapp Putsch . Street fighting in the major industrial centres, the occupation of the Ruhr by Belgian and French troops, and a period of hyperinflation followed. A debt restructuring plan and
15732-599: The Reichstag after a special election in 1932 and Hindenburg appointed Hitler as chancellor of Germany on 30 January 1933. After the Reichstag fire , a decree abrogated basic civil rights and the first Nazi concentration camp opened. On 23 March 1933, the Enabling Act gave Hitler unrestricted legislative power, overriding the constitution, and marked the beginning of Nazi Germany. His government established
15903-672: The Soviets pushed into Eastern Europe ; the Western allies landed in France and entered Germany despite a final German counteroffensive . Following Hitler's suicide during the Battle of Berlin , Germany signed the surrender document on 8 May 1945, ending World War II in Europe and Nazi Germany. Following the end of the war, surviving Nazi officials were tried for war crimes at the Nuremberg trials . In what later became known as
16074-484: The amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in destructive interference and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect. Since
16245-552: The angle of incidence. Plutarch (1st–2nd century AD) described multiple reflections on spherical mirrors and discussed the creation of magnified and reduced images, both real and imaginary, including the case of chirality of the images. During the Middle Ages , Greek ideas about optics were resurrected and extended by writers in the Muslim world . One of the earliest of these was Al-Kindi ( c. 801 –873) who wrote on
16416-435: The angles between the normal (to the interface) and the incident and refracted waves, respectively. The index of refraction of a medium is related to the speed, v , of light in that medium by n = c / v , {\displaystyle n=c/v,} where c is the speed of light in vacuum . Snell's Law can be used to predict the deflection of light rays as they pass through linear media as long as
16587-404: The back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions. The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies, but in this article a positive R indicates a surface's center of curvature
16758-530: The communist-controlled Socialist Unity Party of Germany , supported by the Stasi , an immense secret service. While East German propaganda was based on the benefits of the GDR's social programmes and the alleged threat of a West German invasion, many of its citizens looked to the West for freedom and prosperity. The Berlin Wall , built in 1961, prevented East German citizens from escaping to West Germany, becoming
16929-469: The creation of a new currency in 1924 ushered in the Golden Twenties , an era of artistic innovation and liberal cultural life. The worldwide Great Depression hit Germany in 1929. Chancellor Heinrich Brüning 's government pursued a policy of fiscal austerity and deflation which caused unemployment of nearly 30% by 1932. The Nazi Party led by Adolf Hitler became the largest party in
17100-472: The creation of a more unified European political, economic and security apparatus. The governments of Germany and the United States are close political allies . Cultural ties and economic interests have crafted a bond between the two countries resulting in Atlanticism . After 1990, Germany and Russia worked together to establish a "strategic partnership" in which energy development became one of
17271-536: The criminal and the protection of the public. With the exceptions of petty crimes, tried by a single professional judge, and of serious political crimes, all charges are adjudicated by mixed tribunals where lay judges ( Schöffen ) and professional judges preside together. As of 2016, Germany's murder rate stood at a low of 1.18 murders per 100,000. In 2018, the overall crime rate fell to its lowest since 1992. Same-sex marriage has been legal in Germany since 2017, and LGBT rights are generally protected in
17442-461: The deaths of an estimated 2.7 million Poles , 1.3 million Ukrainians , 1 million Belarusians and 3.5 million Soviet prisoners of war . German military casualties have been estimated at 5.3 million, and around 900,000 German civilians died. Around 12 million ethnic Germans were expelled from across Eastern Europe, and Germany lost roughly one-quarter of its pre-war territory. After Nazi Germany surrendered,
17613-420: The development of lighthouses in terms of cost, design, and implementation. Fresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning. They were first fully implemented into a lighthouse in 1823. Most lenses are spherical lenses : their two surfaces are parts of the surfaces of spheres. Each surface can be convex (bulging outwards from
17784-449: The distance (as if on the surface of a pool of water). Optical materials with varying indexes of refraction are called gradient-index (GRIN) materials. Such materials are used to make gradient-index optics . For light rays travelling from a material with a high index of refraction to a material with a low index of refraction, Snell's law predicts that there is no θ 2 when θ 1 is large. In this case, no transmission occurs; all
17955-403: The distance from the lens to the spot is the focal length of the lens, which is commonly represented by f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature. Another extreme case of a thick convex lens is a ball lens , whose shape
18126-582: The early 1950s. West Germany joined NATO in 1955 and was a founding member of the European Economic Community . On 1 January 1957, the Saarland joined West Germany. East Germany was an Eastern Bloc state under political and military control by the Soviet Union via occupation forces and the Warsaw Pact . Although East Germany claimed to be a democracy, political power was exercised solely by leading members ( Politbüro ) of
18297-712: The ecclesiastical territories were secularised and annexed. In 1806 the Imperium was dissolved; France, Russia, Prussia, and the Habsburgs (Austria) competed for hegemony in the German states during the Napoleonic Wars . Following the fall of Napoleon , the Congress of Vienna founded the German Confederation, a loose league of 39 sovereign states . The appointment of the emperor of Austria as
18468-424: The effect of the lens' thickness. For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by n 1 u + n 2 v = n 2 − n 1 R {\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}} where R
18639-426: The exchange of energy between light and matter only occurred in discrete amounts he called quanta . In 1905, Albert Einstein published the theory of the photoelectric effect that firmly established the quantization of light itself. In 1913, Niels Bohr showed that atoms could only emit discrete amounts of energy, thus explaining the discrete lines seen in emission and absorption spectra . The understanding of
18810-533: The eye, although he was unable to correctly explain how the eye captured the rays. Alhazen's work was largely ignored in the Arabic world but it was anonymously translated into Latin around 1200 A.D. and further summarised and expanded on by the Polish monk Witelo making it a standard text on optics in Europe for the next 400 years. In the 13th century in medieval Europe, English bishop Robert Grosseteste wrote on
18981-535: The feud between the two lasted until Hooke's death. In 1704, Newton published Opticks and, at the time, partly because of his success in other areas of physics, he was generally considered to be the victor in the debate over the nature of light. Newtonian optics was generally accepted until the early 19th century when Thomas Young and Augustin-Jean Fresnel conducted experiments on the interference of light that firmly established light's wave nature. Young's famous double slit experiment showed that light followed
19152-474: The focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration . Curved mirrors can form images with a magnification greater than or less than one, and the magnification can be negative, indicating that the image is inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction occurs when light travels through an area of space that has
19323-411: The gloss of surfaces such as mirrors, which reflect light in a simple, predictable way. This allows for the production of reflected images that can be associated with an actual ( real ) or extrapolated ( virtual ) location in space. Diffuse reflection describes non-glossy materials, such as paper or rock. The reflections from these surfaces can only be described statistically, with the exact distribution of
19494-617: The government was completed in 1999, and modernisation of the East German economy was scheduled to last until 2019. Since reunification, Germany has taken a more active role in the European Union , signing the Maastricht Treaty in 1992 and the Lisbon Treaty in 2007, and co-founding the eurozone . Germany sent a peacekeeping force to secure stability in the Balkans and sent German troops to Afghanistan as part of
19665-1499: The imaging by second lens surface, by taking the above sign convention, u ′ = − v ′ + d {\textstyle \ u'=-v'+d\ } and n 2 − v ′ + d + n 1 v = n 1 − n 2 R 2 . {\displaystyle \ {\frac {n_{2}}{\ -v'+d\ }}+{\frac {\ n_{1}\ }{\ v\ }}={\frac {\ n_{1}-n_{2}\ }{\ R_{2}\ }}~.} Adding these two equations yields n 1 u + n 1 v = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) + n 2 d ( v ′ − d ) v ′ . {\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)+{\frac {\ n_{2}\ d\ }{\ \left(\ v'-d\ \right)\ v'\ }}~.} For
19836-416: The incident rays came. This is called retroreflection . Mirrors with curved surfaces can be modelled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces , parallel rays incident on the mirror produce reflected rays that converge at a common focus . Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing
20007-418: The indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. In most materials, the index of refraction varies with the frequency of the light, known as dispersion . Taking this into account, Snell's Law can be used to predict how a prism will disperse light into
20178-436: The interaction between light and matter that followed from these developments not only formed the basis of quantum optics but also was crucial for the development of quantum mechanics as a whole. The ultimate culmination, the theory of quantum electrodynamics , explains all optics and electromagnetic processes in general as the result of the exchange of real and virtual photons. Quantum optics gained practical importance with
20349-426: The interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light. Numerical modeling techniques such as the finite element method , the boundary element method and the transmission-line matrix method can be used to model
20520-483: The invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle making centres in the Netherlands. In the early 17th century, Johannes Kepler expanded on geometric optics in his writings, covering lenses, reflection by flat and curved mirrors, the principles of pinhole cameras , inverse-square law governing the intensity of light, and
20691-491: The inventions of the maser in 1953 and of the laser in 1960. Following the work of Paul Dirac in quantum field theory , George Sudarshan , Roy J. Glauber , and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light. Classical optics is divided into two main branches: geometrical (or ray) optics and physical (or wave) optics. In geometrical optics, light
20862-421: The knowledge of the rudimentary optical theory of the day). The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands . With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in
21033-504: The laws of reflection and refraction at interfaces between different media. These laws were discovered empirically as far back as 984 AD and have been used in the design of optical components and instruments from then until the present day. They can be summarised as follows: When a ray of light hits the boundary between two transparent materials, it is divided into a reflected and a refracted ray. The laws of reflection and refraction can be derived from Fermat's principle which states that
21204-417: The lens), concave (depressed into the lens), or planar (flat). The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through
21375-399: The lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens
21546-449: The light is modelled as a collection of particles called " photons ". Quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy , various engineering fields, photography , and medicine (particularly ophthalmology and optometry , in which it is called physiological optics). Practical applications of optics are found in
21717-422: The light is reflected. This phenomenon is called total internal reflection and allows for fibre optics technology. As light travels down an optical fibre, it undergoes total internal reflection allowing for essentially no light to be lost over the length of the cable. A device that produces converging or diverging light rays due to refraction is known as a lens . Lenses are characterized by their focal length :
21888-443: The mathematical rules of perspective and described the effects of refraction qualitatively, although he questioned that a beam of light from the eye could instantaneously light up the stars every time someone blinked. Euclid stated the principle of shortest trajectory of light, and considered multiple reflections on flat and spherical mirrors. Ptolemy , in his treatise Optics , held an extramission-intromission theory of vision:
22059-494: The merits of Aristotelian and Euclidean ideas of optics, favouring the emission theory since it could better quantify optical phenomena. In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for lenses and curved mirrors . In the early 11th century, Alhazen (Ibn al-Haytham) wrote
22230-477: The most important factors. As a result of the cooperation, Germany imported most of its natural gas and crude oil from Russia. Germany's development policy functions as a distinct sector within its foreign policy framework. It is formulated by the Federal Ministry for Economic Cooperation and Development and carried out by the implementing organisations. The German government sees development policy as
22401-660: The nation. Germany has a network of 227 diplomatic missions abroad and maintains relations with more than 190 countries. Germany is a member of NATO , the OECD , the G7 , the G20 , the World Bank and the IMF . It has played an influential role in the European Union since its inception and has maintained a strong alliance with France and all neighbouring countries since 1990. Germany promotes
22572-550: The north, Poland and Czechia to the east, Austria and Switzerland to the south, and France , Luxembourg , Belgium , and the Netherlands to the west. Germany is also bordered by the North Sea and, at the north-northeast, by the Baltic Sea. German territory covers 357,596 km (138,069 sq mi). Elevation ranges from the mountains of the Alps (highest point: the Zugspitze at 2,963 metres or 9,721 feet) in
22743-405: The object and image distances are positive if the object and image are on opposite sides of the lens. Incoming parallel rays are focused by a converging lens onto a spot one focal length from the lens, on the far side of the lens. This is called the rear focal point of the lens. Rays from an object at a finite distance are focused further from the lens than the focal distance; the closer the object
22914-401: The optical explanations of astronomical phenomena such as lunar and solar eclipses and astronomical parallax . He was also able to correctly deduce the role of the retina as the actual organ that recorded images, finally being able to scientifically quantify the effects of different types of lenses that spectacle makers had been observing over the previous 300 years. After the invention of
23085-613: The party system has been dominated by the Christian Democratic Union and the Social Democratic Party of Germany . So far every chancellor has been a member of one of these parties. However, the smaller liberal Free Democratic Party and the Alliance 90/The Greens have also been junior partners in coalition governments . Since 2007, the democratic socialist party The Left has been a staple in
23256-676: The path taken between two points by a ray of light is the path that can be traversed in the least time. Geometric optics is often simplified by making the paraxial approximation , or "small angle approximation". The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing , which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications . Reflections can be divided into two types: specular reflection and diffuse reflection . Specular reflection describes
23427-460: The people' (from diot or diota 'people'), originally used to distinguish the language of the common people from Latin and its Romance descendants . This in turn descends from Proto-Germanic * þiudiskaz 'of the people' (see also the Latinised form Theodiscus ), derived from * þeudō , descended from Proto-Indo-European * tewtéh₂- 'people', from which
23598-477: The permanent president reflected the Congress's rejection of Prussia 's rising influence. Disagreement within restoration politics partly led to the rise of liberal movements, followed by new measures of repression by Austrian statesman Klemens von Metternich . The Zollverein , a tariff union, furthered economic unity. In light of revolutionary movements in Europe , intellectuals and commoners started
23769-406: The physical centre of the lens. Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses that are used to correct astigmatism in someone's eye. Lenses are classified by the curvature of the two optical surfaces. A lens
23940-511: The propagation of light in systems which cannot be solved analytically. Such models are computationally demanding and are normally only used to solve small-scale problems that require accuracy beyond that which can be achieved with analytical solutions. All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing . Gaussian beam propagation
24111-414: The radius of curvature is called the curvature . A flat surface has zero curvature, and its radius of curvature is infinite . This convention seems to be mainly used for this article, although there is another convention such as Cartesian sign convention requiring different lens equation forms. If d is small compared to R 1 and R 2 then the thin lens approximation can be made. For
24282-416: The ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation. Some phenomena depend on light having both wave-like and particle-like properties . Explanation of these effects requires quantum mechanics . When considering light's particle-like properties,
24453-416: The rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarized much of Euclid and went on to describe a way to measure the angle of refraction , though he failed to notice the empirical relationship between it and
24624-423: The reflected light depending on the microscopic structure of the material. Many diffuse reflectors are described or can be approximated by Lambert's cosine law , which describes surfaces that have equal luminance when viewed from any angle. Glossy surfaces can give both specular and diffuse reflection. In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with
24795-574: The required shape. A lens can focus light to form an image , unlike a prism , which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as microwave lenses, electron lenses , acoustic lenses , or explosive lenses . Lenses are used in various imaging devices such as telescopes , binoculars , and cameras . They are also used as visual aids in glasses to correct defects of vision such as myopia and hypermetropia . The word lens comes from lēns ,
24966-774: The right infinity leads to the first or object focal length f 0 {\textstyle f_{0}} for the spherical surface. Similarly, u {\textstyle u} toward the left infinity leads to the second or image focal length f i {\displaystyle f_{i}} . f 0 = n 1 n 2 − n 1 R , f i = n 2 n 2 − n 1 R {\displaystyle {\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}} Applying this equation on
25137-415: The same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. The law also implies that mirror images are parity inverted, which we perceive as a left-right inversion. Images formed from reflection in two (or any even number of) mirrors are not parity inverted. Corner reflectors produce reflected rays that travel back in the direction from which
25308-407: The same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens. As with mirrors, upright images produced by a single lens are virtual, while inverted images are real. Lenses suffer from aberrations that distort images. Monochromatic aberrations occur because the geometry of the lens does not perfectly direct rays from each object point to
25479-410: The sign) would have zero optical power (as its focal length becomes infinity as shown in the lensmaker's equation ), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for
25650-405: The simple equation 1 S 1 + 1 S 2 = 1 f , {\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}},} where S 1 is the distance from the object to the lens, θ 2 is the distance from the lens to the image, and f is the focal length of the lens. In the sign convention used here,
25821-607: The south to the shores of the North Sea ( Nordsee ) in the northwest and the Baltic Sea ( Ostsee ) in the northeast. The forested uplands of central Germany and the lowlands of northern Germany (lowest point: in the municipality Neuendorf-Sachsenbande , Wilstermarsch at 3.54 metres or 11.6 feet below sea level ) are traversed by such major rivers as the Rhine, Danube and Elbe. Significant natural resources include iron ore, coal, potash , timber, lignite , uranium , copper, natural gas, salt, and nickel. Most of Germany has
25992-974: The southeast regions have more extreme temperatures. From February 2019 – 2020, average monthly temperatures in Germany ranged from a low of 3.3 °C (37.9 °F) in January 2020 to a high of 19.8 °C (67.6 °F) in June 2019. Average monthly precipitation ranged from 30 litres per square metre in February and April 2019 to 125 litres per square metre in February 2020. Average monthly hours of sunshine ranged from 45 in November 2019 to 300 in June 2019. The territory of Germany can be divided into five terrestrial ecoregions : Atlantic mixed forests , Baltic mixed forests , Central European mixed forests , Western European broadleaf forests , and Alps conifer and mixed forests . As of 2016 , 51% of Germany's land area
26163-464: The spectacle making centres in both the Netherlands and Germany. Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses rather than using the rudimentary optical theory of the day (theory which for the most part could not even adequately explain how spectacles worked). This practical development, mastery, and experimentation with lenses led directly to
26334-568: The standardization of the language; the 1555 Peace of Augsburg tolerated the "Evangelical" faith ( Lutheranism ), but also decreed that the faith of the prince was to be the faith of his subjects ( cuius regio, eius religio ). From the Cologne War through the Thirty Years' Wars (1618–1648), religious conflict devastated German lands and significantly reduced the population. The Peace of Westphalia ended religious warfare among
26505-408: The subscript of 2 in n 2 {\textstyle \ n_{2}\ } is dropped. As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the focal point ) at a distance f from the lens. Conversely, a point source of light placed at the focal point is converted into a collimated beam by
26676-762: The subsequent decisive Prussian victory in the Austro-Prussian War of 1866 enabled him to create the North German Confederation which excluded Austria . After the defeat of France in the Franco-Prussian War , the German princes proclaimed the founding of the German Empire in 1871. Prussia was the dominant constituent state of the new empire; the King of Prussia ruled as its Kaiser, and Berlin became its capital. In
26847-444: The superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances. This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are in phase , both the wave crests and wave troughs align. This results in constructive interference and an increase in
27018-472: The telescope, Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope , using two convex lenses to produce higher magnification. Optical theory progressed in the mid-17th century with treatises written by philosopher René Descartes , which explained a variety of optical phenomena including reflection and refraction by assuming that light
27189-440: The term "light" is also often applied to infrared (0.7–300 μm) and ultraviolet radiation (10–400 nm). The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated. The existence of electromagnetic waves
27360-862: The thin lens approximation where d → 0 , {\displaystyle \ d\rightarrow 0\ ,} the 2nd term of the RHS (Right Hand Side) is gone, so n 1 u + n 1 v = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) . {\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.} The focal length f {\displaystyle \ f\ } of
27531-1110: The thin lens is found by limiting u → − ∞ , {\displaystyle \ u\rightarrow -\infty \ ,} n 1 f = ( n 2 − n 1 ) ( 1 R 1 − 1 R 2 ) → 1 f = ( n 2 n 1 − 1 ) ( 1 R 1 − 1 R 2 ) . {\displaystyle \ {\frac {\ n_{1}\ }{\ f\ }}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\rightarrow {\frac {1}{\ f\ }}=\left({\frac {\ n_{2}\ }{\ n_{1}\ }}-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.} So,
27702-1258: The two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula . Applying Snell's law on the spherical surface, n 1 sin i = n 2 sin r . {\displaystyle n_{1}\sin i=n_{2}\sin r\,.} Also in the diagram, tan ( i − θ ) = h u tan ( θ − r ) = h v sin θ = h R {\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}} , and using small angle approximation (paraxial approximation) and eliminating i , r , and θ , n 2 v + n 1 u = n 2 − n 1 R . {\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.} The (effective) focal length f {\displaystyle f} of
27873-418: The two surfaces. A negative meniscus lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery. An ideal thin lens with two surfaces of equal curvature (also equal in
28044-466: The use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness , though the reference is vague). Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water. Ptolemy (2nd century) wrote a book on Optics , which however survives only in
28215-406: The war and a period of Allied occupation , Germany was organized into two separate polities with limited sovereignty: the Federal Republic of Germany, or West Germany , and the German Democratic Republic, or East Germany . Berlin continued its de jure Four Power status . The Federal Republic of Germany was a founding member of the European Economic Community and the European Union , while
28386-637: The western Rhineland had been incorporated into Roman provinces . Around 260, Germanic peoples broke into Roman-controlled lands. After the invasion of the Huns in 375, and with the decline of Rome from 395, Germanic tribes moved farther southwest: the Franks established the Frankish Kingdom and pushed east to subjugate Saxony and Bavaria . Areas of what is today eastern Germany were inhabited by Western Slavic tribes. Charlemagne founded
28557-615: The word Teutons also originates. Pre-human ancestors, the Danuvius guggenmosi , who were present in Germany over 11 million years ago, are theorized to be among the earliest ones to walk on two legs. Ancient humans were present in Germany at least 600,000 years ago. The first non-modern human fossil (the Neanderthal ) was discovered in the Neander Valley . Similarly dated evidence of modern humans has been found in
28728-400: Was a famous instrument which used interference effects to accurately measure the speed of light. The appearance of thin films and coatings is directly affected by interference effects. Antireflective coatings use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with
28899-540: Was emitted by objects which produced it. This differed substantively from the ancient Greek emission theory. In the late 1660s and early 1670s, Isaac Newton expanded Descartes's ideas into a corpuscle theory of light , famously determining that white light was a mix of colours that can be separated into its component parts with a prism . In 1690, Christiaan Huygens proposed a wave theory for light based on suggestions that had been made by Robert Hooke in 1664. Hooke himself publicly criticised Newton's theories of light and
29070-467: Was predicted in 1865 by Maxwell's equations . These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves. Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered. Many simplified approximations are available for analysing and designing optical systems. Most of these use
29241-495: Was the seventh-highest in the world . In response to the 2022 Russian invasion of Ukraine , Chancellor Olaf Scholz announced that German military expenditure would be increased past the NATO target of 2%, along with a one-time 2022 infusion of 100 billion euros, representing almost double the 53 billion euro military budget for 2021. In 2023, military spending according to NATO criteria amounted to $ 73.1 billion, or 1.64% of
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