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Review double intensity when phase is coherent (optics)

step inference rule input feed output step validity (as per SymPy)
1
  • 111981: 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
  • 111484: 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
  • 111253: 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
    \(Z^*=|Z| \exp( -i \theta )\)
  1. 4182362050
    \(Z=|Z| \exp( i \theta )\)
  1. 9191880568
    \(Z Z^*=|Z| |Z| \exp( -i \theta ) \exp( i \theta )\)
 invalid syntax (<string>, line 0)
4
  • 111457: 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
    \(Z Z^*=|Z|^2\)
 invalid syntax (<string>, line 0)
5
  • 111981: 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
  • 111886: 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. 4437214608
    \(Z\)
  1. 5623794884
    \(A + B\)
  1. 2236639474
    \((A + B)(A + B)^*=|A + B|^2\)
 invalid syntax (<string>, line 0)
7
  • 111556: 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
    \(I=| A + B |^2\)
  1. 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
  • 111474: 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
  • 111457: 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
  • 111886: 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. 9761485403
    \(Z\)
  1. 8710504862
    \(A\)
  1. 4075539836
    \(A A^*=|A|^2\)
 invalid syntax (<string>, line 0)
11
  • 111886: 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
    \(Z\)
  1. 6529120965
    \(B\)
  1. 7107090465
    \(B B^*=|B|^2\)
 invalid syntax (<string>, line 0)
12
  • 111556: 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. 4075539836
    \(A A^*=|A|^2\)
  1. 8065128065
    \(I=A A^* + B B^* + A B^* + B A^*\)
  1. 5125940051
    \(I=|A|^2 + B B^* + A B^* + B A^*\)
 valid
13
  • 111556: 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. 7107090465
    \(B B^*=|B|^2\)
  1. 5125940051
    \(I=|A|^2 + B B^* + A B^* + B A^*\)
  1. 1525861537
    \(I=|A|^2 + |B|^2 + A B^* + B A^*\)
 valid
14
  • 111984: 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. 1742775076
    \(Z\)
  1. 4583868070
    \(B\)
  1. 4192519596
    \(B=|B| \exp(i \phi)\)
 list index out of range
15
  • 111886: 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. 2064205392
    \(A\)
  1. 1894894315
    \(Z\)
  1. 1357848476
    \(A=|A| \exp(i \theta)\)
 LHS diff is pdg0003192 - pdg0004453 RHS diff is (Abs(pdg0003192) - Abs(pdg0004453))*exp(pdg0001575*pdg0004621)
16
  • 111484: 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
  • 111484: 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
  • 111797: 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. 4504256452
    \(B^*=|B| \exp(-i \phi)\)
  1. 1357848476
    \(A=|A| \exp(i \theta)\)
  1. 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
  • 111457: 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
  • 111886: 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
    \(\theta - \phi\)
  1. 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
  • 111182: 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
  • 111556: 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
    \(I=|A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))\)
  1. 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
  • 111981: 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
  • 111556: 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
    \(I=|A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
  1. 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
  • 111981: 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
  • 111556: 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. 2719691582
    \(|A|=|B|\)
  1. 8283354808
    \(I_{\rm coherent}=|A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )\)
  1. 8046208134
    \(I_{\rm coherent}=|A|^2 + |A|^2 + |A| |A| 2\)
 LHS diff is 0 RHS diff is -4*Abs(pdg0004453)**2 + 4*Abs(pdg0004698)**2
27
  • 111457: 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
  • 111981: 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
  • 111556: 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. 8602221482
    \(\langle \cos(\theta - \phi) \rangle=0\)
  1. 8497631728
    \(I=|A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )\)
  1. 6240206408
    \(I_{\rm incoherent}=|A|^2 + |B|^2\)
 LHS diff is -pdg0002435 + pdg0007882 RHS diff is 0
30
  • 111556: 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
    \(I_{\rm incoherent}=|A|^2 + |B|^2\)
  1. 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
  • 111457: 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
  • 111421: 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. 1172039918
    \(I_{\rm coherent}=4 |A|^2\)
  1. 3060393541
    \(I_{\rm incoherent}=2|A|^2\)
  1. 6556875579
    \(\frac{I_{\rm coherent}}{I_{\rm incoherent}}=2\)
 recognized infrule but not yet supported
33
  • 111341: 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


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