## review derivation: double intensity when phase is coherent (optics)

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.

Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:

Options
Alternate views of this derivation:
Edit this content:

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
24 substitute LHS of expr 1 into expr 2
1. 6774684564; locally 7781977:
$$\theta = \phi$$
$$pdg_{1575} = pdg_{8586}$$
2. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 8283354808; locally 2413866:
$$I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )$$
$$pdg_{8251} = \left|{pdg_{4453}}\right|^{2} + 2 \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is pdg7882 - pdg8251 RHS diff is 0 6774684564:
8497631728:
8283354808:
6774684564:
8497631728:
8283354808:
13 substitute LHS of expr 1 into expr 2
1. 7107090465; locally 2303305:
$$B B^* = |B|^2$$
$$pdg_{4698} \overline{pdg_{4698}} = \left|{pdg_{4698}}\right|^{2}$$
2. 5125940051; locally 4729665:
$$I = |A|^2 + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}} + \left|{pdg_{4453}}\right|^{2}$$
1. 1525861537; locally 8296872:
$$I = |A|^2 + |B|^2 + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
valid 7107090465:
5125940051:
1525861537:
7107090465:
5125940051:
1525861537:
29 substitute LHS of expr 1 into expr 2
1. 8602221482; locally 4842351:
$$\langle \cos(\theta - \phi) \rangle = 0$$
$$\cos{\left(pdg_{1575} - pdg_{8586} \right)} = 0$$
2. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 6240206408; locally 8093224:
$$I_{\rm incoherent} = |A|^2 + |B|^2$$
$$pdg_{2435} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is -pdg2435 + pdg7882 RHS diff is 0 8602221482:
8497631728:
6240206408:
8602221482:
8497631728:
6240206408:
18 substitute LHS of four expressions into expr
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
2. 4504256452; locally 1174231:
$$B^* = |B| \exp(-i \phi)$$
$$\overline{pdg_{4698}} = e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
3. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
1. 7621705408; locally 1405078:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)$$
$$pdg_{7882} = e^{pdg_{1575} pdg_{4621}} e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4453} pdg_{4698} \left|{\left|{pdg_{4453}}\right| + e^{- pdg_{1575} pdg_{4621}} e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|}\right|}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
no check performed 4192519596:
4504256452:
1357848476:
7621705408:
4192519596:
4504256452:
1357848476:
7621705408:
15 change variable X to Y
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
1. 2064205392:
$$A$$
$$pdg_{4453}$$
2. 1894894315:
$$Z$$
$$pdg_{3192}$$
1. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
LHS diff is pdg3192 - pdg4453 RHS diff is (Abs(pdg3192) - Abs(pdg4453))*exp(pdg1575*pdg4621) 4182362050:
1357848476:
4182362050:
1357848476:
25 declare initial expr
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
no validation is available for declarations 2719691582:
2719691582:
31 simplify
1. 6529793063; locally 5409843:
$$I_{\rm incoherent} = |A|^2 + |A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
1. 3060393541; locally 3246829:
$$I_{\rm incoherent} = 2|A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
valid 6529793063:
3060393541:
6529793063:
3060393541:
22 substitute LHS of expr 1 into expr 2
1. 2700934933; locally 8635275:
$$2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$2 \cos{\left(pdg_{1464} \right)} = e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}$$
2. 3085575328; locally 5595798:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))$$
$$pdg_{7882} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4453} pdg_{4698} \left|{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|}\right|}\right|$$
1. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is 0 RHS diff is (-2*exp(pdg4621*(pdg1575 - pdg8586))*cos(pdg1575 - pdg8586)*Abs(pdg4453*pdg4698) + Abs(pdg4453*pdg4698*Abs(exp(pdg4621*(pdg1575 - pdg8586))*Abs(pdg4698) + Abs(pdg4453))))*exp(-pdg4621*(pdg1575 - pdg8586)) 2700934933: error for dim with 2700934933
3085575328:
8497631728:
2700934933: N/A
3085575328:
8497631728:
17 conjugate both sides
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
1. 4504256452; locally 1174231:
$$B^* = |B| \exp(-i \phi)$$
$$\overline{pdg_{4698}} = e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
no check performed 4192519596:
4504256452:
4192519596:
4504256452:
33 declare final expr
1. 6556875579; locally 6088608:
$$\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2$$
$$\frac{pdg_{8251}}{pdg_{2435}} = 2$$
no validation is available for declarations 6556875579:
6556875579:
10 change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 9761485403:
$$Z$$
$$pdg_{3192}$$
2. 8710504862:
$$A$$
$$pdg_{4453}$$
1. 4075539836; locally 3404497:
$$A A^* = |A|^2$$
$$pdg_{4453} \overline{pdg_{4453}} = \left|{pdg_{4453}}\right|^{2}$$
Nothing to split 3350830826:
4075539836:
3350830826:
4075539836:
5 declare initial expr
1. 8396997949; locally 6461198:
$$I = | A + B |^2$$
$$pdg_{7882} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
no validation is available for declarations 8396997949:
8396997949:
19 simplify
1. 7621705408; locally 1405078:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)$$
$$pdg_{7882} = e^{pdg_{1575} pdg_{4621}} e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4453} pdg_{4698} \left|{\left|{pdg_{4453}}\right| + e^{- pdg_{1575} pdg_{4621}} e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|}\right|}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 3085575328; locally 5595798:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))$$
$$pdg_{7882} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4453} pdg_{4698} \left|{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|}\right|}\right|$$
LHS diff is 0 RHS diff is -exp(-pdg1575*pdg4621 + pdg4621*pdg8586)*Abs(pdg4453*pdg4698*Abs(exp(pdg1575*pdg4621 - pdg4621*pdg8586)*Abs(pdg4698) + Abs(pdg4453))) + exp(pdg1575*pdg4621 - pdg4621*pdg8586 - re(pdg1575*pdg4621))*Abs(pdg4453*pdg4698*Abs(exp(pdg1575*pdg4621)*Abs(pdg4453) + exp(pdg4621*pdg8586)*Abs(pdg4698))) 7621705408:
3085575328:
7621705408:
3085575328:
20 change variable X to Y
1. 4585932229; locally 7002927:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
1. 4935235303:
$$x$$
$$pdg_{4037}$$
2. 2293352649:
$$\theta - \phi$$
$$pdg_{1575} - pdg_{8586}$$
1. 3660957533; locally 9190817:
$$\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2} + \frac{e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2}$$
LHS diff is 0 RHS diff is exp(pdg1464*pdg4621)/2 - exp(-pdg1575*pdg4621 + pdg4621*pdg8586)/2 - exp(pdg1575*pdg4621 - pdg4621*pdg8586)/2 + exp(-pdg1464*pdg4621)/2 4585932229:
3660957533: error for dim with 3660957533
4585932229:
3660957533: N/A
6 change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 4437214608:
$$Z$$
$$pdg_{3192}$$
2. 5623794884:
$$A + B$$
$$pdg_{4453} + pdg_{4698}$$
1. 2236639474; locally 4137499:
$$(A + B)(A + B)^* = |A + B|^2$$
$$\left(pdg_{4453} + pdg_{4698}\right)^{2} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
Nothing to split 3350830826:
2236639474:
3350830826:
2236639474:
32 divide expr 1 by expr 2
1. 1172039918; locally 7442815:
$$I_{\rm coherent} = 4 |A|^2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
2. 3060393541; locally 3246829:
$$I_{\rm incoherent} = 2|A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
1. 6556875579; locally 6088608:
$$\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2$$
$$\frac{pdg_{8251}}{pdg_{2435}} = 2$$
no check performed 1172039918:
3060393541:
6556875579:
1172039918:
3060393541:
6556875579:
11 change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 6529120965:
$$B$$
$$pdg_{4698}$$
2. 1511199318:
$$Z$$
$$pdg_{3192}$$
1. 7107090465; locally 2303305:
$$B B^* = |B|^2$$
$$pdg_{4698} \overline{pdg_{4698}} = \left|{pdg_{4698}}\right|^{2}$$
Nothing to split 3350830826:
7107090465:
3350830826:
7107090465:
27 simplify
1. 8046208134; locally 2139818:
$$I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
1. 1172039918; locally 7442815:
$$I_{\rm coherent} = 4 |A|^2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
valid 8046208134:
1172039918:
8046208134:
1172039918:
3 multiply expr 1 by expr 2
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
2. 1928085940; locally 5663009:
$$Z^* = |Z| \exp( -i \theta )$$
$$pdg_{3192}$$
1. 9191880568; locally 4577339:
$$Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )$$
$$pdg_{3192}$$
Nothing to split 4182362050:
1928085940:
9191880568:
4182362050:
1928085940:
9191880568:
8 distribute conjugate to factors
1. 1020854560; locally 9192406:
$$I = (A + B)(A + B)^*$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
1. 6306552185; locally 2300056:
$$I = (A + B)(A^* + B^*)$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
no check performed 1020854560:
6306552185:
1020854560:
6306552185:
23 declare initial expr
1. 6774684564; locally 7781977:
$$\theta = \phi$$
$$pdg_{1575} = pdg_{8586}$$
no validation is available for declarations 6774684564:
6774684564:
1 declare initial expr
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
no validation is available for declarations 4182362050:
4182362050:
14 change two variables in expr
1. 7607271250; locally 5513927:
$$\theta$$
$$pdg_{1575}$$
1. 4182362050:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
2. 1742775076:
$$Z$$
$$pdg_{3192}$$
3. 4583868070:
$$B$$
$$pdg_{4698}$$
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
Nothing to split 7607271250:
4192519596:
7607271250:
4192519596:
12 substitute LHS of expr 1 into expr 2
1. 4075539836; locally 3404497:
$$A A^* = |A|^2$$
$$pdg_{4453} \overline{pdg_{4453}} = \left|{pdg_{4453}}\right|^{2}$$
2. 8065128065; locally 9934418:
$$I = A A^* + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4453}} + pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}}$$
1. 5125940051; locally 4729665:
$$I = |A|^2 + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}} + \left|{pdg_{4453}}\right|^{2}$$
valid 4075539836:
8065128065:
5125940051:
4075539836:
8065128065:
5125940051:
28 declare initial expr
1. 8602221482; locally 4842351:
$$\langle \cos(\theta - \phi) \rangle = 0$$
$$\cos{\left(pdg_{1575} - pdg_{8586} \right)} = 0$$
no validation is available for declarations 8602221482:
8602221482:
4 simplify
1. 9191880568; locally 4577339:
$$Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )$$
$$pdg_{3192}$$
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
Nothing to split 9191880568:
3350830826:
9191880568:
3350830826:
7 substitute LHS of expr 1 into expr 2
1. 2236639474; locally 4137499:
$$(A + B)(A + B)^* = |A + B|^2$$
$$\left(pdg_{4453} + pdg_{4698}\right)^{2} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
2. 8396997949; locally 6461198:
$$I = | A + B |^2$$
$$pdg_{7882} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
1. 1020854560; locally 9192406:
$$I = (A + B)(A + B)^*$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
LHS diff is 0 RHS diff is -(pdg4453 + pdg4698)*(conjugate(pdg4453) + conjugate(pdg4698)) + Abs(pdg4453 + pdg4698)**2 2236639474:
8396997949:
1020854560:
2236639474:
8396997949:
1020854560:
30 substitute LHS of expr 1 into expr 2
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
2. 6240206408; locally 8093224:
$$I_{\rm incoherent} = |A|^2 + |B|^2$$
$$pdg_{2435} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 6529793063; locally 5409843:
$$I_{\rm incoherent} = |A|^2 + |A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
LHS diff is 0 RHS diff is -2*Abs(pdg4453)**2 + 2*Abs(pdg4698)**2 2719691582:
6240206408:
6529793063:
2719691582:
6240206408:
6529793063:
26 substitute LHS of expr 1 into expr 2
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
2. 8283354808; locally 2413866:
$$I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )$$
$$pdg_{8251} = \left|{pdg_{4453}}\right|^{2} + 2 \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4698}}\right|^{2}$$
1. 8046208134; locally 2139818:
$$I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
LHS diff is 0 RHS diff is -4*Abs(pdg4453)**2 + 4*Abs(pdg4698)**2 2719691582:
8283354808:
8046208134:
2719691582:
8283354808:
8046208134:
9 simplify
1. 6306552185; locally 2300056:
$$I = (A + B)(A^* + B^*)$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
1. 8065128065; locally 9934418:
$$I = A A^* + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4453}} + pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}}$$
valid 6306552185:
8065128065:
6306552185:
8065128065:
2 conjugate both sides
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
1. 1928085940; locally 5663009:
$$Z^* = |Z| \exp( -i \theta )$$
$$pdg_{3192}$$
Nothing to split 4182362050:
1928085940:
4182362050:
1928085940:
16 conjugate both sides
1. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
1. 6555185548; locally 1584527:
$$A^* = |A| \exp(-i \theta)$$
$$\overline{pdg_{4453}} = e^{- pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
no check performed 1357848476:
6555185548:
1357848476:
6555185548:
21 multiply both sides by
1. 3660957533; locally 9190817:
$$\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2} + \frac{e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2}$$
1. 3967985562:
$$2$$
$$2$$
1. 2700934933; locally 8635275:
$$2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$2 \cos{\left(pdg_{1464} \right)} = e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}$$
valid 3660957533: error for dim with 3660957533
2700934933: error for dim with 2700934933
3660957533: N/A
2700934933: N/A
Physics Derivation Graph: Steps for double intensity when phase is coherent (optics)

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
7882 variable I
$$I$$
['real']
• mass: 1
• time: -3
intensity
9
1575 variable \theta
$$\theta$$
['real'] dimensionless angle
34
2435 variable I_{\rm incoherent}
$$I_{\rm incoherent}$$
['real']
• mass: 1
• time: -3
intensity of incoherent waves
4
4621 variable i
$$i$$
['imaginary'] dimensionless imaginary unit
74
4037 variable x
$$x$$
['real']
• length: 1
position
53
8586 variable \phi
$$\phi$$
['real'] dimensionless angle
10
4453 variable A
$$A$$
['complex'] dimensionless none
23
4698 variable B
$$B$$
['complex'] dimensionless none
19
1464 variable x
$$x$$
['real'] dimensionless 140
8251 variable I_{\rm coherent}
$$I_{\rm coherent}$$
['real']
• mass: 1
• time: -3
intensity of coherent waves
4
3192 variable Z
$$Z$$
['complex'] dimensionless none
9
MESSAGE:
• local variable 'all_df' referenced before assignment