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Finesse and sensitivity gain in cavity-enhanced absorption spectroscopy of biomolecules in solution

Timothy McGarvey, André Conjusteau, and Hideo Mabuchi

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Abstract

We describe a ‘wet mirror’ apparatus for cw cavity-enhanced absorption measurements with Bacteriochlorophyll a (BChla) in solution and show that it achieves the full sensitivity gain (≈ 2.3 × 104) afforded by the finesse (3.4 × 104) and loss distribution of our optical resonator. This result provides an important proof-of-principle demonstration for solution-phase cavity-enhanced spectroscopy; straightforward extrapolation to a system with state-of-the-art low-loss mirrors and shot-noise-limited performance indicates that single molecule sensitivity in liquids is within reach of current technology. With the probe laser locked to the cavity resonance, our instrument achieves a sensitivity ≈ 3.4 × 10-8/√Hz (for a sample of length 1.75 mm) with 100 kHz bandwidth and can reliably detect sub-nM concentrations of BChla with 1 ms integration time.

©2006 Optical Society of America

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Figures (2)
Fig. 1.
Fig. 1. Schematic diagram of the experimental setup. PBS: polarizing beam-splitter; NBPS: non-polarizing beam-splitter; λ/4: quarter-wave plate; HV: high voltage. See text for other designations.

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Fig. 2.
Fig. 2. Cavity-enhanced estimate (αl)ce≈ versus nominal BChla concentration (see text).

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Equations (41)

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P sp − = P inc e − δ
P sp + = P inc e − δ e − αl
( αl ) sp = In [ P sp − ] − In [ P sp + ] .
Δ ( αl ) sp = { ∂ ( αl ) sp ∂ P sp − } Δ P sp − + { ∂ ( αl ) sp ∂ P sp + } Δ P sp +
= Δ P sp − P sp − − Δ P sp + P sp + .
P ce − = P inc ( r 1 2 − g rt − ) 2 r 1 2 ( 1 − g rt − ) 2 , g rt − = r 1 r 2 e − δ / 2
P ce + = P inc ( r 1 2 − g rt + ) 2 r 1 2 ( 1 − g rt + ) 2 , g rt + = r 1 r 2 e − δ / 2 e − αl = g rt − e − αl
( αl ) ce = In [ 1 − π 2 F − 1 + π 2 F − R − ] + In [ Q + π 2 F − R + Q − π 2 F − ] ,
R − ≡ P ce − P inc , R + ≡ P ce + P inc , Q ≡ 1 − R + 1 − R − .
( αl ) ce ≈ = π F − ( R + − R − 1 − R + ) = π F − ( P ce + − P ce − P inc − P ce + ) .
P inc → ε P inc , P ce → P rfl − − ( 1 − ε ) P inc , P ce + → P rfl + − ( 1 − ε ) P inc ,
R − → P rfl − − ( 1 − ε ) P inc ε P inc ≡ R − ε , R + → P rfl + − ( 1 − ε ) P inc ε P inc ≡ R + ε ,
( αl ) ce ≈ → π F − ( P rfl + − ( 1 − ε ) P inc − P rfl − − ( 1 − ε ) P inc ε P inc − P rfl + − ( 1 − ε ) P inc ) ,
∂ ( αl ) ce ≈ ∂ P ce + = π 2 F − P inc − P ce − P ce + ( P inc − P ce + ) 2 ,
∣ ∂ ( αl ) ce ≈ ∂ P ce + ∣ Δ P ce + = π 2 F − 1 − R − 1 − R + R + 1 − R + ( Δ P ce + P ce + ) .
∣ ∂ ( αl ) ce ≈ ∂ P ce + ∣ Δ P ce + = U ce ≈ − 1 ( Δ P ce + P ce + ) .
G op ≡ ∣ U ce ≈ U sp ∣ = U ce ≈ = ( π 2 F − 1 − R − 1 − R + R + 1 − R + ) − 1 .
1 − R − 1 − R + R + 1 − R + < 2 ,
∣ ∂ ( αl ) ce ≈ ∂ P ce + ∣ Δ P ce + < π F − ( Δ P ce + P ce + ) .
π 2 1 − R − 1 − R + R + 1 − R + ≈ π 2 R + 1 − R + < 1 ,
∣ ∂ ( αl ) ce ≈ ∂ F − ∣ Δ F − = π F − R − − R + 1 − R + ( Δ F − F − ) ,
∣ ∂ ( αl ) ce ≈ ∂ P inc ∣ Δ P inc = π 2 F − R − − R + 1 − R + 1 1 − R + ( Δ P inc P inc ) ,
∣ ∂ ( αl ) ce ≈ ∂ P ce − ∣ Δ P ce − = π 2 F − R − 1 − R + ( Δ P ce − P ce − ) ,
∣ ∂ ( αl ) cav ∂ P tr + ∣ Δ P tr + = π 2 F − 1 − R − ε 1 − R + ε R + ε 1 − R + ε ( Δ P tr + P tr + − ( 1 − ε ) P inc ) ,
Δ P ce + P ce + → ( S / q ) P ce + τ ( S / q ) P ce + τ = 1 ( S / q ) P ce + τ ,
⇒ ∣ ∂ ( αl ) ce ≈ ∂ P ce + ∣ Δ P ce + → U ce ≈ − 1 ( S / q ) P ce + τ .
Δ ( αl ) ce = ( ∂ ( αl ) ce ∂ P ce + ) Δ P ce + → ( ∂ ( αl ) ce ∂ P ce + ) η P ce + ,
Δ ( αl ) sp = ( ∂ ( αl ) sp ∂ P sp + ) Δ P sp + → ( ∂ ( αl ) sp ∂ P sp + ) η P sp + .
U th : ce = ( P ce + ∂ ( αl ) ce ∂ P ce + ) − 1 = 1 l P ce + ( ∂ P ce + ∂ α ) P ce + ,
U th : sp = ( P sp + ∂ ( αl ) sp ∂ P sp + ) − 1 = 1 l P sp + ( ∂ P sp + ∂ α ) P sp + .
P ce + = P inc ( r 1 2 − g rt + ) 2 r 1 2 ( 1 − g rt + ) 2 ,
∂ P ce + ∂ α = 2 l P inc g rt + ( r 1 2 − g rt + ) ( 1 − r 1 2 ) r 1 2 ( 1 − g rt+ ) 3 ,
( ∂ P ce + ∂ α ) P ce + = 2 l P ce + g rt + ( 1 − r 1 2 ) ( 1 − g rt + ) ( r 1 2 − g rt + )
⇒ U th : ce = 2 g rt + ( 1 − r 1 2 ) ( 1 − g rt + ) ( r 1 2 − g rt + ) .
G th ≡ U th : ce U th : sp = − 2 g rt + ( 1 − r 1 2 ) ( 1 − g rt + ) ( r 1 2 − g rt + ) ,
Δ ( αl ) th : ce ≥ U th : ce − 1 ( S / q ) P ce + τ
→ 2 ( 1 − g rt − ) ( r 1 2 − g rt − ) g rt − ( 1 − r 1 2 ) ( P ce − τ ) / ( h ̅ ω ) .
L − = 1 − g rt − 2 ≈ ( 1 − r 1 2 ) + ( 1 − r 2 2 ) + δ ,
L + = 1 − g rt + 2 ≈ ( 1 − r 1 2 ) + ( 1 − r 2 2 ) + δ + 2 αl .
2 π F + − 2 π F − = 2 π ℱ ( Δ f + − Δ f − ) = 2 αl ,
( αl ) Δ f = π ℱ ( Δ f + − Δ f − ) .
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