31:
167:
366:
251:
187:
In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their
99:
256:
The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.
198:
85:
of the system, and lies close to the axis throughout the system. Generally, this allows three important approximations (for
162:{\displaystyle \sin \theta \approx \theta ,\quad \tan \theta \approx \theta \quad {\text{and}}\quad \cos \theta \approx 1.}
274:
Use of the small angle approximations replaces dimensionless trigonometric functions with angles in radians. When doing
404:
315:
17:
278:
on optics equations it is important to remember that radians are dimensionless and therefore can be ignored.
181:
420:
400:
335:
51:
34:
The error associated with the paraxial approximation. In this plot the cosine is approximated by
59:
275:
8:
331:
311:
376:
345:
43:
173:
55:
260:
414:
371:
189:
63:
30:
340:
268:
264:
82:
74:
192:
expansion is zero), while for cosine the second order approximation is
259:
For larger angles it is often necessary to distinguish between
90:
246:{\displaystyle \cos \theta \approx 1-{\theta ^{2} \over 2}\ .}
307:
201:
102:
245:
161:
412:
62:of light through an optical system (such as a
297:
295:
293:
291:
93:) for calculation of the ray's path, namely:
27:Small angle approximation in geometric optics
301:
184:is one method that uses the approximation.
288:
330:
29:
14:
413:
263:, which lie in a plane containing the
172:The paraxial approximation is used in
401:Paraxial Approximation and the Mirror
367:"Paraxial approximation error plot"
24:
405:The Wolfram Demonstrations Project
306:. SPIE Field Guides. Vol. 1.
25:
432:
394:
304:Field Guide to Geometrical Optics
143:
137:
121:
359:
324:
13:
1:
302:Greivenkamp, John E. (2004).
281:
182:Ray transfer matrix analysis
7:
10:
437:
77:that makes a small angle (
52:small-angle approximation
336:"Paraxial Approximation"
247:
163:
48:paraxial approximation
39:
248:
164:
33:
276:dimensional analysis
199:
100:
421:Geometrical optics
403:by David Schurig,
332:Weisstein, Eric W.
310:. pp. 19–20.
243:
159:
40:
239:
235:
141:
16:(Redirected from
428:
388:
387:
385:
383:
377:Wolfram Research
363:
357:
356:
354:
352:
346:Wolfram Research
328:
322:
321:
299:
271:, which do not.
252:
250:
249:
244:
237:
236:
231:
230:
221:
168:
166:
165:
160:
142:
139:
44:geometric optics
37:
21:
436:
435:
431:
430:
429:
427:
426:
425:
411:
410:
397:
392:
391:
381:
379:
365:
364:
360:
350:
348:
329:
325:
318:
300:
289:
284:
261:meridional rays
226:
222:
220:
200:
197:
196:
174:Gaussian optics
138:
101:
98:
97:
56:Gaussian optics
35:
28:
23:
22:
18:Paraxial optics
15:
12:
11:
5:
434:
424:
423:
409:
408:
396:
395:External links
393:
390:
389:
358:
323:
316:
286:
285:
283:
280:
254:
253:
242:
234:
229:
225:
219:
216:
213:
210:
207:
204:
170:
169:
158:
155:
152:
149:
146:
136:
133:
130:
127:
124:
120:
117:
114:
111:
108:
105:
26:
9:
6:
4:
3:
2:
433:
422:
419:
418:
416:
406:
402:
399:
398:
378:
374:
373:
372:Wolfram Alpha
368:
362:
347:
343:
342:
337:
333:
327:
319:
317:0-8194-5294-7
313:
309:
305:
298:
296:
294:
292:
287:
279:
277:
272:
270:
269:sagittal rays
266:
262:
257:
240:
232:
227:
223:
217:
214:
211:
208:
205:
202:
195:
194:
193:
191:
190:Taylor series
185:
183:
180:ray tracing.
179:
175:
156:
153:
150:
147:
144:
134:
131:
128:
125:
122:
118:
115:
112:
109:
106:
103:
96:
95:
94:
92:
88:
84:
80:
76:
72:
67:
65:
61:
57:
53:
49:
45:
32:
19:
380:. Retrieved
370:
361:
349:. Retrieved
341:ScienceWorld
339:
326:
303:
273:
265:optical axis
258:
255:
186:
177:
171:
86:
83:optical axis
78:
71:paraxial ray
70:
68:
47:
41:
178:first-order
60:ray tracing
351:15 January
282:References
382:26 August
224:θ
218:−
212:≈
209:θ
206:
154:≈
151:θ
148:
135:θ
132:≈
129:θ
126:
116:θ
113:≈
110:θ
107:
81:) to the
415:Category
334:(2007).
54:used in
91:radians
36:1 - θ/2
314:
267:, and
238:
46:, the
73:is a
50:is a
384:2014
353:2014
312:ISBN
308:SPIE
176:and
64:lens
58:and
203:cos
145:cos
140:and
123:tan
104:sin
89:in
75:ray
66:).
42:In
417::
375:.
369:.
344:.
338:.
290:^
157:1.
69:A
407:.
386:.
355:.
320:.
241:.
233:2
228:2
215:1
119:,
87:θ
79:θ
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.