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Review double intensity when phase is coherent (optics)

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 4182362050
    \(Z = |Z| \exp( i \theta )\)
 no validation is available for declarations
2
  • 0000111484: conjugate both sides
  • number of inputs: 1; feeds: 0; outputs: 1
  • Conjugate both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 4182362050
    \(Z = |Z| \exp( i \theta )\)
  1. 1928085940
    \(Z^* = |Z| \exp( -i \theta )\)
 recognized infrule but not yet supported
3
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 1928085940
    \(Z^* = |Z| \exp( -i \theta )\)
  2. 4182362050
    \(Z = |Z| \exp( i \theta )\)
  1. 9191880568
    \(Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )\)
 Not evaluated due to missing term in SymPy
4
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 9191880568
    \(Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )\)
  1. 3350830826
    \(Z Z^* = |Z|^2\)
 Not evaluated due to missing term in SymPy
5
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8396997949
    \(I = | A + B |^2\)
 no validation is available for declarations
6
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 3350830826
    \(Z Z^* = |Z|^2\)
  1. 5623794884
    \(A + B\)
  2. 4437214608
    \(Z\)
  1. 2236639474
    \((A + B)(A + B)^* = |A + B|^2\)
 Not evaluated due to missing term in SymPy
7
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8396997949
    \(I = | A + B |^2\)
  2. 2236639474
    \((A + B)(A + B)^* = |A + B|^2\)
  1. 1020854560
    \(I = (A + B)(A + B)^*\)
 LHS diff is -pdg0007882 + (pdg0004453 + pdg0004698)**2 RHS diff is -(pdg0004453 + pdg0004698)*(conjugate(pdg0004453) + conjugate(pdg0004698)) + Abs(pdg0004453 + pdg0004698)**2
8
  • 0000111474: distribute conjugate to factors
  • number of inputs: 1; feeds: 0; outputs: 1
  • Distribute conjugate to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1020854560
    \(I = (A + B)(A + B)^*\)
  1. 6306552185
    \(I = (A + B)(A^* + B^*)\)
 recognized infrule but not yet supported
9
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 6306552185
    \(I = (A + B)(A^* + B^*)\)
  1. 8065128065
    \(I = A A^* + B B^* + A B^* + B A^*\)
 valid
10
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 3350830826
    \(Z Z^* = |Z|^2\)
  1. 8710504862
    \(A\)
  2. 9761485403
    \(Z\)
  1. 4075539836
    \(A A^* = |A|^2\)
 Not evaluated due to missing term in SymPy
11
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 3350830826
    \(Z Z^* = |Z|^2\)
  1. 1511199318
    \(Z\)
  2. 6529120965
    \(B\)
  1. 7107090465
    \(B B^* = |B|^2\)
 Not evaluated due to missing term in SymPy
12
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8065128065
    \(I = A A^* + B B^* + A B^* + B A^*\)
  2. 4075539836
    \(A A^* = |A|^2\)
  1. 5125940051
    \(I = |A|^2 + B B^* + A B^* + B A^*\)
 LHS diff is pdg0004453*conjugate(pdg0004453) - pdg0007882 RHS diff is -pdg0004453*conjugate(pdg0004698) - pdg0004698*conjugate(pdg0004453) - pdg0004698*conjugate(pdg0004698)
13
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 5125940051
    \(I = |A|^2 + B B^* + A B^* + B A^*\)
  2. 7107090465
    \(B B^* = |B|^2\)
  1. 1525861537
    \(I = |A|^2 + |B|^2 + A B^* + B A^*\)
 LHS diff is pdg0004698*conjugate(pdg0004698) - pdg0007882 RHS diff is -pdg0004453*conjugate(pdg0004698) - pdg0004698*conjugate(pdg0004453) - Abs(pdg0004453)**2
14
  • 0000111984: change two variables in expression
  • number of inputs: 1; feeds: 4; outputs: 1
  • Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.
  1. 4583868070
    \(B\)
  2. 1742775076
    \(Z\)
  1. 4192519596
    \(B = |B| \exp(i \phi)\)
15
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 4182362050
    \(Z = |Z| \exp( i \theta )\)
  1. 1894894315
    \(Z\)
  2. 2064205392
    \(A\)
  1. 1357848476
    \(A = |A| \exp(i \theta)\)
 valid
16
  • 0000111484: conjugate both sides
  • number of inputs: 1; feeds: 0; outputs: 1
  • Conjugate both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1357848476
    \(A = |A| \exp(i \theta)\)
  1. 6555185548
    \(A^* = |A| \exp(-i \theta)\)
 recognized infrule but not yet supported
17
  • 0000111484: conjugate both sides
  • number of inputs: 1; feeds: 0; outputs: 1
  • Conjugate both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 4192519596
    \(B = |B| \exp(i \phi)\)
  1. 4504256452
    \(B^* = |B| \exp(-i \phi)\)
 recognized infrule but not yet supported
18
  • 0000111797: substitute LHS of four expressions into expression
  • number of inputs: 5; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} and LHS of Eq.~\ref{eq:#3} and LHS of Eq.~\ref{eq:#4} into Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.
  1. 1357848476
    \(A = |A| \exp(i \theta)\)
  2. 4504256452
    \(B^* = |B| \exp(-i \phi)\)
  3. 4192519596
    \(B = |B| \exp(i \phi)\)
  1. 7621705408
    \(I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)\)
 recognized infrule but not yet supported
19
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 7621705408
    \(I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)\)
  1. 3085575328
    \(I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))\)
 LHS diff is 0 RHS diff is -exp(-pdg0001575*pdg0004621 + pdg0004621*pdg0008586)*Abs(pdg0004453*pdg0004698*Abs(exp(pdg0001575*pdg0004621 - pdg0004621*pdg0008586)*Abs(pdg0004698) + Abs(pdg0004453))) + exp(pdg0001575*pdg0004621 - pdg0004621*pdg0008586 - re(pdg0001575*pdg0004621))*Abs(pdg0004453*pdg0004698*Abs(exp(pdg0001575*pdg0004621)*Abs(pdg0004453) + exp(pdg0004621*pdg0008586)*Abs(pdg0004698)))
20
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  1. 2293352649
    \(\theta - \phi\)
  2. 4935235303
    \(x\)
  1. 3660957533
    \(\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
 LHS diff is 0 RHS diff is exp(pdg0001464*pdg0004621)/2 - exp(-pdg0001575*pdg0004621 + pdg0004621*pdg0008586)/2 - exp(pdg0001575*pdg0004621 - pdg0004621*pdg0008586)/2 + exp(-pdg0001464*pdg0004621)/2
21
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 3660957533
    \(\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
  1. 3967985562
    \(2\)
  1. 2700934933
    \(2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
 valid
22
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 3085575328
    \(I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))\)
  2. 2700934933
    \(2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)\)
  1. 8497631728
    \(I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
 LHS diff is -pdg0007882 + 2*cos(pdg0001464) RHS diff is exp(-pdg0001575*pdg0004621 + pdg0004621*pdg0008586) + exp(pdg0001575*pdg0004621 - pdg0004621*pdg0008586) - 2*cos(pdg0001575 - pdg0008586)*Abs(pdg0004453*pdg0004698) - Abs(pdg0004453)**2 - Abs(pdg0004698)**2
23
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 6774684564
    \(\theta = \phi\)
 no validation is available for declarations
24
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8497631728
    \(I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
  2. 6774684564
    \(\theta = \phi\)
  1. 8283354808
    \(I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )\)
 LHS diff is pdg0001575 - pdg0008251 RHS diff is pdg0008586 - Abs(pdg0004453)**2 - Abs(pdg0004698)**2 - 2*Abs(pdg0004453*pdg0004698)
25
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2719691582
    \(|A| = |B|\)
 no validation is available for declarations
26
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8283354808
    \(I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )\)
  2. 2719691582
    \(|A| = |B|\)
  1. 8046208134
    \(I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2\)
 LHS diff is -pdg0008251 + Abs(pdg0004453) RHS diff is -4*Abs(pdg0004453)**2 + Abs(pdg0004698)
27
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8046208134
    \(I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2\)
  1. 1172039918
    \(I_{\rm coherent} = 4 |A|^2\)
 valid
28
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8602221482
    \(\langle \cos(\theta - \phi) \rangle = 0\)
 no validation is available for declarations
29
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8497631728
    \(I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
  2. 8602221482
    \(\langle \cos(\theta - \phi) \rangle = 0\)
  1. 6240206408
    \(I_{\rm incoherent} = |A|^2 + |B|^2\)
 LHS diff is -pdg0002435 + cos(pdg0001575 - pdg0008586) RHS diff is -Abs(pdg0004453)**2 - Abs(pdg0004698)**2
30
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 6240206408
    \(I_{\rm incoherent} = |A|^2 + |B|^2\)
  2. 2719691582
    \(|A| = |B|\)
  1. 6529793063
    \(I_{\rm incoherent} = |A|^2 + |A|^2\)
 LHS diff is -pdg0002435 + Abs(pdg0004453) RHS diff is -2*Abs(pdg0004453)**2 + Abs(pdg0004698)
31
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 6529793063
    \(I_{\rm incoherent} = |A|^2 + |A|^2\)
  1. 3060393541
    \(I_{\rm incoherent} = 2|A|^2\)
 valid
32
  • 0000111421: divide expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Divide Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 3060393541
    \(I_{\rm incoherent} = 2|A|^2\)
  2. 1172039918
    \(I_{\rm coherent} = 4 |A|^2\)
  1. 6556875579
    \(\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2\)
 recognized infrule but not yet supported
33
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 6556875579
    \(\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2\)
no validation is available for declarations

Symbols used in double intensity when phase is coherent (optics)

Steps and expressions for double intensity when phase is coherent (optics)

d3js visualization of steps and expressions in double intensity when phase is coherent (optics)


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